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Koninklijke Akademie van Wetenschappen
te Amsterdam.

PROCEEDINGS

OF THE

Gd TON OF SCIENCES

>
06. 245( 4. Gd 2Ff

VOLUME VII.

(2nd PART)

AMSTERDAM,
JOHANNES MULLER.

July 1905.

(Translated from: Verslagen van de Gewone Vergaderingen der Wis- en Natuurkundige
Afdeeling van 24 December 1904 tot 22 April 1905. DI, XIII.)

Proceedin

»

»

os
ss

CRORN STEE NG ES.

Gb

of the Meeting of December 24

» January 28

» February 25

» March 25

» April 22

1904

»

»

487
. 537

595
. 635,

- p | ihk
(KEN MBO IL,
| NROTELK AKU KE

KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM,

PROCEEDINGS OF THE MEETING

of Saturday December 24, 1904.

IGC

(Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige
Afdeeling van Zaterdag 24 December 1904, Dl. XIII).

OENE eN ESE

D. J. KorreweG and D. pe Lance: “Multiple umbilies as singularities of the first order of
exception on point-general surfaces”, p. 386.

A. F. Horteman: “On the preparation of pure o-toluidine and a method for ascertaining its
purity”, p. 395.

Miss T. Tames: “On the influence of natrition on the fluctuating variability of some plants”.
(Communicated by Prof. J W. Mott), p. 398. (With one plate).

J. W. Morr: “On the nuclear division cf Fritillaria imperialis L”. Results from Dr. B.
SYPKENS’ thesis for the doctorate, p. 412.

J. M. Janse: “An investigation on polarity and organ-formation with Caulerpa prolifera”.
‘Communicated by Prof. HvGo pr Vries), p. 420.

P. ZEEMAN and J. Geest: “Double refraction near the components of absorption lines mag-
netieally split into several components”, p. 435. (With one plate).

H. A. Lorentz: “The motion of electrons in metallic bodies” I, p. 438.

S. Brok: “The connection betweea the primary triangulation of South Sumatra and that of
the West Coast of Sumatra”. (Communicated by Prof. J. A. C. OUuDEMANS), p. 453. (With one
plate).

Cu. M. van Deventer: “On the melting of floating ice”. (Communicated by Prof. J. D. van
DER WAALS), p. 459.

J. J. BrANKSMA : “On trinitroveratrol”. (Communicated by Prof. H. W. Baxuvis RoozEBoom),
p- 462.

S. Try stra Bz.: “On W. Marcxwatp’s asymmetrie synthesis of optically active valeric acid”.
(Communicated by Prof. H. W. Bakuvis Roozesoom), p. 465.

A. H. W. Aten: “On the system pyridine and methyl iodide”. (Communicated by Prof.
IL. W. Baxuvis Roozesoom), p. 468.

J. BorsekeN: “The reaction of FrieperL and Crarrs”. (Communicated by Prof. A. Fe
HorrEMAN), p. 470.

J. E. Verscuarrert: “The influence of admixtures on the critical phenomena of simple sub-
stances and the explanation of TEICHNER’s experiments”. (Communicated by Prof. H. Kamer-
LINGH ONNEs), p. 474. (With one plate).

J. A. C. OtpEMays: “Determinations of latitude and azimuth, made in 18%6—99 by Dr.
A. PANNEKOEK and Mr. B. Postnumus Meyses at Oirschot, Utrecht, Sambeek, Wolberg,
Harikerberg, Sleen, Schoorl, Zierikzee, Terschelling (the lighthouse Brandaris), Ameland,
Leeuwarden, Urk and Groningen”, p. 482.

Errata, p. 485.

The following papers were read:

i)
“I

Proceedings Royal Acad. Amsterdam. Vol. VII.

( 386 )

Mathematics. — “Multiple umbilics as singularities of the first
order of exception on point-general surfaces’. Communicated
by Prof. D. J. KorrpwrG and Mr. D. pr Lanes.

(Communicated in the meeling of November 26, 1904).

1. Let us suppose a point-general surface, i. e. general if considered
as a geometrical locus of points, in whose Cartesian equation parameters
appear; then for a continuous change of those parameters also the
surface will in general vary continuously in shape). If then we fix
our attention on any kind of singular points, plaitpoints, umbilies,
etc. appearing on a point-general algebraic surface in finite number,
it may happen during the deformation that two or more of those
singular points coincide. Such a point where this takes place may
be called a twofold or multiple singular point of that kind.

Now such a coincidence may generally occur, as the results tell
us, in more than one way. For some of these ways the coincidence
depends on a single relation between the coefficients of the Cartesian
equation being satisfied, whilst for others it depends on more suchlike
relations. The former cases belong to the singularities of the first order
of exception, the latter to those of a bigher order. It is only with
the former that we shall occupy ourselves in this paper’).

For plaitpoints the singularities of the first class, which must be
regarded as multiple plaitpoints, were investigated by the first
mentioned *). Two entirely different kinds of double plaitpoints were
found (the homogeneous kind and the heterogeneous one) ; furthermore
the points of osculation proved to be threefold plaitpoints, the nodes
of the surface twentyfourfold plaitpoints.

It seemed advisable to make an investigation also for other singular
points. This we have done for the umbilics. The results obtained
are communicated in this paper. For proofs and more elaborate
considerations see the dissertation by the second mentioned Mr. D.
DE LANGE issued recently.

a. The double umbilic at finite distance.

2. If we place the origin of a rectangular system of coordinates
at an umbilie and if we use the tangent plane in this point as zy-plane

1) See for more general considerations of the same kind as follow here: , Ueber
Singularitäten verschiedener Ausnahmeordnung und ihre Zerlegung”, Math. Ann.
41, p. 286—307 (1893).

2) See for the reason why these are asking in the first place our attention the
paper just quoted, on page 287.

3) D. J. Korrewre, ,Ueber Faltenpunkte’, Wien. Ber. 98, p. 1154~1191, (1889)
also Arch. Néerl. 24, p. 57—98, (1890).

( 387 )

the equation of the surface ean be written in the form:
zee Hy) de? + d,a*y + day? + dy? Heet +. (1)

By a slight deformation we arrive for the new surface at the
equation :

2—@- ia - B + em + Yao Hd (Cr HY) y° + d,2° + d,a*y . (2)
where the Greek letters represent small quantities, which can all be
regarded as of the same order, namely of the order of the small
variation which an arbitrary parameter appearing in the coefficients,
has had to undergo. Also the Latin letters must be regarded as
having been varied somewhat, which is however immaterial.

Let us now calculate by means of the wellknown conditions:

dz 22 072
Ou? Ow Oy Oy?
ee (3)

1 02\? Oz Oz ' 02\2
+(5 De "dy He

the position of the displaced umbilie ; then we shall find after neglecting
all terms which are small with respect to those which are retained,
the two linear equations :

y, +2d,a+ 2d,y=0; y, + (d, — 3d,)e + (8d,—d,)y=0 . (4)
from which in general we deduce without difficulty the sought for

displacement.
This however is different when the determinant

d, — 3d 3d, — d, |
KS Sata 8Gatad) ®)
L a

3 3

disappears. In that case no finite values satisfy the linear equations
(4). This proves, however, only that the displacement of the umbilic
has become of a lower order than the quantities indicated by the
Greek letters and that therefore the terms of the second order in &
and y must be included in the equations (4). If we do so we obtain
by comparing the two new equations and eliminating the linear
terms the new equation:

(d, —3d,)y,—24,y, +[12d,¢, +3(d,—3d,)e,—2d,e,—8c,2d,Ja? +
4+ [6d,e, + 4(d,—3d,)e, — 6d,e, —2c,°(d, —3d,) vy +[2d,e, +3(d, —3d,)e, —
— ie Calta Oh ise ate (ants) ah EE msi ge (0)
which must be combined with one of the equations (4).

This equation (6) is of order two in w and y, from which therefore
ensues: 1st that the displacement becomes of order } with respect
to that of the Greek letters used in (2), 24 that the umbilic originally
situated at the origin of the system of coordinates on the surface (1)

Ars

( 388 )

is broken up into two at the deformation of this surface, which two
umbilies diverge in general, real at a variation of the parameter in
one sense, and imaginary in the other. So we have to do with
a double umbilic, namely with such a one at whose effective *)
occurrence a transition takes place from the real to the imaginary.

3. Before considering the further properties of this double umbilie
we wish to observe that the condition A, =O was already known
as an important characteristic. It characterises namely the case
of transition between two of the three general kinds of umbilies
distinguished for the first time by Darsoux*) according to the manner
in which the lines of curvature bear themselves in their neighbourhood.

For the first kind, see fig. 1, lines of curvature are starting from
the umbilic in three different directions — namely in each direction a

Fig. 1. Fig. 2. Fig. 3.

single one, which we have represented by a right line because
its curvature depends on the terms of higher order of the equation
(1), to begin with those of the fourth. Those three directions have
the property that they cannot be represented in one quadrant, 1. e.
each of them lies inside the obtuse angle formed by the two others.
For this kind K, > 0°).

For the second kind, see fig. 2, also lines of curvature start from
the umbilie in three different directions; these directions are however
such that one of them falls inside the acute angle formed by the
two others, so that the three can now be contained in one quadrant.
Moreover an infinite number of lines of curvature — five of which, the
right line included, are indicated in fig. 2 — start in the firstmentioned
direction which might be called the :iddle one. For this kind A, < 0.

For the third kind, see fig. $, only one line of curvature starts
from the umbilic, the right line of that figure. The two other directions

1) See for the meaning of this term page 289 of the paper quoted in the first note.

2) G. Darroux. Legons sur la théorie générale des surfaces. Quatrième partie.
Gauthier-Villars, 1896, p. 448—465,

8) This characteristic K,>O means moreover as is proved in the dissertation
in a simple way, that the lines of curvature turn in the neighbourhood of O every-
where their convex side to the umbilic, but for K, <0 on the contrary their con- ~
cave side.

of departure have become imaginary. For this kind too A, <0. To
distinguish it analytically from the preceding one we can notice the
sign of the diseriminant of the cubic

de n° + (2d, — 3d) n° + (Bd, — 2d,)n—d,=0. . . (
which proves to serve for the determination of the three directions
of departure. If we call this discriminant A, chosen in such a way
that for A, > 0 the three roots are real, we have for the first kind
Kee ike OR for the second Ke <0, K, > 0; for. the third
ere he Ons Ay fourth kind Ke > 0; KC Or does mot: exist;
because as is demonstrated also algebraically A, > 0 includes A, > 0.
4. As is apparent from this explanation the double umbilic forms
the ease of transition between the first and the second kind, for which
case of transition A, must of necessity be equal to nought, and A, > 0.
The form of the lines of curvature now becomes very simple as long
as one confines oneself to the approximation which has led to the
figures 1, 2 and 3. Out of the differential equation

dy \°*
{d,«+d,y] |: — B |+ [(d, — 3d,)a + (8d, —d) 4]

C

ly x
=0 , (8)
dx

C

which serves to determine the lines of curvature, a factor separates
itself namely d‚v + d‚y, which made equal to zero represents a
right line, whilst the remaining furnishes two mutually perpendi-
cular pencils of parallel lines. In this manner, however, from each
point of the first mentioned right line three lines of curvature
would start, so that there would be an entire line of umbilics. This
is of course in general not the case, so that this representation of the
lines of curvature must undergo a considerable modification as soon
as the terms of higher order are taken into consideration. We shall
soon refer to this again.

5. We shall first mention the results of a closer investigation of
the deformation of the double umbilic. From this we were able to
prove, 1st. that for a variation of parameter in the sense in which the
two single umbilics diverge in a real manner, this diverging shall always
take place in the direction of the just discussed right line d,a—+-d,y—0,
which after that represents in first approximation for each of the
two separated umbilics one of the directions of departure of lines
of curvature, 2"¢. that these separated umbilics are always of a dif-
ferent kind, namely one of the first kind, the other of the second.
Moreover dr + d,y =O indicates for that of the second kind the
middle direction of departure, whilst also the remaining directions
of departure of the diverged umbilics nearly correspond to the direc-
tions of departure of the original double umbilic discussed in § 4.

( 390 )

All this being stated it is not diffieult to guess how in general the
form of the lines of curvature must be, shortly after the breaking
up of the double umbilic *).

Fig. 4.

Fig. 6.

That form is represented in fig. 4, where QO, indicates the umbilie
of the first kind, O, that of the second. At O, the angle of the two
other lines of curvature, starting from the umbilic, which contains
O, O, is a little larger than a right angle, at O, on the contrary it
is a little smaller.

If after that we allow the umbilies to coincide again, they meet
at about half the distance and the figure now formed where the lines
of curvature situated at some distance to the right and left of O,
and QV, must have retained in general the same direction, can hardly
be otherwise but such as has been indicated in fig. 5°), apart from
the symmetry which in general does not exist of course, no more
than in any of the other figures.

1) After the publication of the Dutch version of this paper we found that Mr.
A. Guuistranp already in 1900, in his memoir “Allgemeine Theorie der mono-
chromatischen Aberrationen und ihre nächsten Ergebnisse für die Ophtalmologie”
(see Nova Acta Regiae Societatis Scientiarum Upsaliensis, ser. 3, vol. 20, pp. 90
and 114) arrived also, starting from other considerations, at the investigation of
the double umbilie and its breaking up and that we obtained the same results.

2) However, a closer investigation of this subject by another method would not be
unwished for. It would have to be a systematic study of the lines, if possible in
their entire length, satisfying the differential equation :

dy»?
pets) jee ++ Be, + He, — 2c,*)ay Hey] +

de
di
+o [6d,y + 2(e,—Ge, + 4e,*)e? + 6(e,—e,)uy + 2(6e,—e,—40,)y?] =0.

For this is the form which the differential equation of the lines of curvature
assumes in the neighbourhood of a double umbilic at second approximation,
when we place the X-axis in the direction in which the two single umbilics diverge
by a slight deformation of the surface. We then have d;=0O and d; = 3d); the
Jatter on account of (5).

(391 )

If we then continue the deformation in the same manner so that
now the two umbilies diverge imaginarily, a figure seems to be
formed as is represented in fig. 6.

In no case there occurs a transition proper from the first kind
to the second on a point-general surface continuously deforming itself.
When the relation A,=O meets its fulfilment then we find that
two umbilies of different kinds approach each other to disappear from
the surface after the coincidence.

b. The nodes of a point-general surface as
twelvefold umbilics.

6. When there is a node, the equation of the surface in its neigh-
bourhood cannot be given in the form indicated in equation (1). After
a fit choice of the axes we can however start from:

ani +t byt dez 4+ A, fA, +....=20.... (9)
or after a slight deformation, from:

a+ Bx + By + Bz + aa? + by? Heet 4....=0. . (10)
It is soon evident that to determine at first approximation the
umbilies which appear in the neighbourhood of the place where
formerly the node existed, the terms of order two are sufficient.
So the surface may be treated there as a quadric, which immedia-
tely makes the behaviour of the umbilies clear. If namely we have
to do with an isolated point, made to appear after the gradual
disappearance of a sheet, then at the very instant four real *) umbilies
disappear, which were situated on that sheet, whilst eight others were
imaginary and become so again after the disappearance of the sheet.
If the node is a conical point then, when the two sheets are disunited,
four real *) umbilics make their appearance, becoming imaginary at the
union, whilst eight others again meet likewise for a moment in the
node, but are previously and afterwards imaginary. For an imaginary
node of course all the twelve umbilies coinciding there for a moment
remain imaginary.

The umbilics at injinity. General considerations.

7. The umbilies are distinguished from the plaitpoints and many
other singular points by the fact, that they cannot stand a projective

1) These are at first of the third kind. They can, however, gradually pass during
a continued deformation into those of the second kind without giving rise to the
appearance of a double umbilic.

2) Also for those holds good what was remarked in the preceding note,

( 392 )

transformation. The cause of this is that they are in a definite
relation to the plane at infinity and in particular to the spherical
points in that plane. This obliges us to give a separate consideration
of the cases of the first order of exception, where umbilies reach
infinity. It was a priori not improbable that this would be accom-
panied by the occurrence of multiplicity in all or in some of those
cases, as really it proved to be for some.

The method of investigation with respect to this was as follows :
first the umbilies were exchanged for a more general kind of
singular points which are capable of projective transformation. To
this end it is sufficient to observe that an umbilie can be defined

as such a point of a given surface which — when regarded as
a node of its section of the tangent plane — has the property

that both nodal tangents pass through the circular points of the
tangent plane.

After applying the general projective transformation the problem of
the umbilics of the original surface is in this way reduced to the
following :

Given a surface w, a plane a, and in that plane a conic ce; to define
on the surface w the points @ which have the property that the two
nodal tangents of the section of the tangent plane 9 im 2 pass
respectively through the points A, and A, where e is cut by g.

For this more general problem the plane at infinity has been
replaced by the plane a and we have but to study the points 2
which as singularities of the first order may appear in the section d of 2
and «@ which can be performed by choosing an appropriate system
of axes with such a point for origin, by calculating for this system
of axes the approximate equation of the surface, and by then applying
a slight deformation. The results obtained in this way can be imme-
diately applied to umbilies.

In this manner it became evident that umbilies can appear in four
different ways at infinity as singular points of the first order of
exception, which we shall successively describe in short.

e. The point of contact of a point-general surface with the
plane at infinity as a fourfold umbilic.

8. It is clear that whenever the surface touches the plane a,
such a point of contact must be regarded as an @-point; for its
tangents in the section of the tangent plane will certainly meet the
conic c in the plane «. By regarding the surface as a quadrie we
ean then by returning to the problem of the umbilics decide without
calculation that the point under observation is a fourfold @-point.

( 393 )

At the same time ensues from the behaviour of the quadries that
when there is a real contact with the plane at infinity, the point
of contact, if it appears in the section of the tangent plane as an
isolated point, breaks up at the deformation into two real and two
imaginary umbilics in whatever direction the deformation may take
place. In the opposite case we have to do with four imaginary
umbilies. So transition from real umbilies to imaginary ones never
takes place in this way.

d. The point of contact of a point-general surface with the curve
of the spherical points at infinity as a double umbilic.

9. It goes without saying that when w touches c the point of
contact must be an @-point, for the points A, and A, coincide with
this point of contact and so they are situated on the nodal tangents
in this same point.

By analysis it proves to be a double @-point. As the spherical
points at infinity are all imaginary, these umbilics and the single
ones into which they break up, are also always imaginary.

e. The points of infinity of the spinodal line as single wmbilics,
when the tangent of the spinode lies in the plane at infinity.

10. If we consider a point in which the spinodal line of w cuts
the plane a, it is easy to see that this point must be regarded as an
2-point as often as the cuspidal tangent of the section of the tangent
plane lies in plane «, which isa single condition. It appears, however,
that this point cannot be driven asunder by deformation, so it must
be regarded as a single @-point and the umbilic corresponding to it
likewise as a single umbilic. This umbilic can be real or imaginary.
The manner indicated here is the only one in which real umbilics
can reach infinity without passing into a multiple umbilic, i. e.
without meeting other umbilies there.

f. The points of intersection of the surface with the curve of
the spherical points at injinity as single umbilics, when
one of the nodal tangents in the section of the
tangent plane lies in the plane at infinity.
11. It is immediately evident that the corresponding points on w are
Q-points and after investigation they prove to be single ones. As
umbilies they are of course always imaginary.

Application to quadries.

12. The equation of a quadric can be brought with an appropriate

( 394 )

choice of axes when the origin is placed in one of its umbilies, into
the finite form:
z=, (# + y°) + hive + hye + ke? Ee (OU)
Bringing the value of z into the second member this furnishes
the development in series
ze, (#7? + y’) + khee? + hem y + hoeey* + hoe,y? + . (12)
Comparing this to (1) it is immediately evident that for the
umbilies on a quadrie we always find d, =d,, d, =d,, so K, < 0.
Furthermore the cubic (7) passes into (d‚n — d,) (n° + 1) == 0; so
K, <0. From this it is evident, as indeed is known, that on a
quadrie never other umbilics than those of the third kind can appear.
From this ensues again immediately that on a quadrie no common
double umbilics can appear. Indeed beside the nodes the only
possible multiple umbilies at finite distance on a quadric are the
vertices of a surface of revolution; but these are fourfold umbilies
whose occurrence on surfaces of higher order would demand more
than one relation between the coefficients of the equation. So it is
not astonishing that for such vertices the lines of curvature bear
themselves in an entirely deviating way.

13. Passing now to the umbilies of quadrics at infinity we observe
that the case given sub ec appears for paraboloids. If, however, we
regard more closely the section with the plane at infinity, then this
is evidently degenerated into two right lines. Each of these right
lines meets the curve of the spherical points in two points. If we
make tangent planes to appear in those points, then also there the
section of the tangent plane degenerates, namely, into one of the
recently considered right lines and into another. These two must at
the same time be regarded as the tangents of the section of the
tangent plane. One of these tangents therefore always happens to lie
in the plane at infinity and we are in case //.

To the fourfold umbilie at infinity four single umbilies are in this
way added for the paraboloid. For finite distances four such points
only are thus left, which furnishes here the proof to the sum.

Inversely case d requires as is easy to see, at least for quadrics
with real equation, that these should pass into surfaces of revolution.
There is then double contact of the surface and the curve of the
spherical points. Indeed in this case four umbilies pass into infinity;
the eight remaining ones coinciding four by four in both vertices.

The remaining case e cannot make its appearance for quadrics.
The case f has just been discussed. It can as is easy to see make
its appearance for quadries only in the manner indicated there.

( 395 )

Chemistry. — “On the preparation of pure o-toluidine and a
method for ascertaining its purity.” By Prof. A. F. HoLLeMan.
(Communicated in the meeting of November 26, 1904).

Whilst p-toluidine being a solid, well crystallised substance may
be very readily obtained in a perfectly pure state from the commer-
cial product by recrystallisation and distillation, this is by no means
the case with the liquid ortho-toluidine. The latter stands a good
chance of containing its para-isomer as it is prepared from o-nitro-
toluene, which is rather difficult to completely separate from the
p-nitrotoluene simultaneously formed in the nitration of toluene,
particularly because the ortho-nitrocompound is liquid. It is further
stated that o-toluidine sometimes contains aniline.

Of the various ways mentioned in the literature on the subject
for the purification of o-toluidine, the conversion into oxalate seemed
to me the most appropriate. According to Brmstrms Handbuch, the
solubility of ortho-toluidine oxalate amounts to 2.38 parts by weight
in 100 parts of water at 21°; that of the acid oxalate of p-toluidine
(the neutral compound does not exist) 0.87 parts in 100 parts of
water at 10°. If, therefore, the o-toluidine contains a few per cent
of para, the oxalate thereof must remain in the aqueous mother-
liquor when the mixture is submitted to recrystallisation, and the
use of ether, which is given as an accurate method of separating
the oxalates, becomes superfluous. Even any aniline which happens
to be present, may be removed in this manner.

In order to see whether a complete purification might indeed be
attained in this way, it was necessary to first obtain a characteristic
test for ascertaining the purity; for the processes found in the
literature for ascertaining the purity of o-toluidine, of HÄUssERMANN
(Fr. 26,750), Remuarr (Fr. 33,90) and Luner (Fr. 24,459) appeared
but little suitable for the detection of very small amounts of impurities.

For this purpose the determination of the solidifying point of the
acetyl compound proved serviceable. By determining a portion of the
solidifying point curve of o- and p-acetotoluidide the amount of the
impurity could then be ascertained quantitatively at the same time.

The following solidifying point figures were found :

Percentage Solidifying
of para. point.
0 109.715
1.12 108. 45
2.42 107. 75
9.58 103.°2
13.6 100.°8

( 396 )

That 109°.15 is the solidifying point of pure aceto-o-toluidide was
proved by reerystallising the oxalate prepared from a “chemically
pure” o-toluidine and then recovering the toluidine, which was then
treated once more in the same way.

After each crystallisation of the oxalate a small quantity of o-tolui-
dine was converted into the acetocompound; the observed solidifying
points were both the above figure, which moreover did not suffer
any change when the acetocompound was again recrystallised.

In order to ascertain how far small quantities of para-toluidine
and aniline may de detected by means of the solidifying point tigures,
the above purified o-toluidine was mixed with 2°/, of aniline and
another portion with 2°/, of p-toluidine and tested as follows:

25.2 grams of oxalic acid (*/; mol.) are dissolved in a litre of
boiling water and to this are slowly added 42.8 gram of toluidine
(?/, mol.). On cooling, the oxalate crystallises out; after placing the
flask in ice the liquid is thoroughly removed by suction and the
crystals washed once with a little water; the toluidine is then
recovered from the crystals as well as from the motherliquor by
adding alkali and distilling in a current of steam. In order to avoid
loss it is necessary to extract the water, which has also distilled
over, twice with ether. The toluidine so obtained is converted into
the acetocompound by adding per gram a mixture of 2 cc. of glacial
acetic acid and 1 ce. of acetic anhydride. The mass is now evaporated
on the waterbath and the dry residue once distilled in vacwo when
everything passes over leaving but a small black residue. The solidi-
fying point of both products is then determined. We found :

Added
2/, p-toluidine 2°/) aniline
Solidifying point of the acetotoluidide from the crystals : 109.715; 109.215
5 EN Hs 3 > » motherliquors: 103.°2 ; 103.°0

This shows that while the oxalate erystallised out, the added
impurities remained completely in the motherliquor and that the
acetocompound prepared from the latter shows the serious depression
of about 6°. If now we consider that the determination of the
solidifying point is accurate to 0.°2 and with practice even to 0.°1
it follows that we may detect in this way '/,, part of the impurities
now present, viz. */,, or 0.03 °/,.

Using this method I have examined two samples of o-toluidine
from different makers and both marked “chemisch rein” as to their
purity with the following result.

I. Converted into oxalate in exactly the same manner as described. Flask cooled
in ice water.

(397 )

From the erystals were obtained 31 grams, from the motherliquor 10.2 grams,
total 41.2 grams, 42.8 grams having been started with.

Solidifying point of the acetocompound from the crystals 109.°15. Therefore pure.

= a a 5 „ motherliquor 107. 15, corresponding

with 3.6%, of p-toluidine or 0.37 gram. The sample therefore contained

0.37 X 100

HQ = 0.9 0%, impurity.

Il. 42.8 grams of toluidine converted as before into oxalate. From the crystals
are taken 30.5 grams, from the motherliquor 11.2 grams, total 417 grams.

Solidifying point of the acetocompound from the crystals 108.°45 so this still
contained 1. °/, or 0.34 gram of byproduct. After having been converted once
more into oxalate, the newly prepared acetocompound now solidified at 109.°15.
Solidifying point of the acetocompound from the motherliquor 101.°9 corresponding
with 12.1%) or 1.36 gram. Total impurity present, therefore, 1.36 + 0.34 = 1.70
corresponding with 4.1 9/,.

Assuming the impurity to be either aniline or p-toluidine the
following plan was followed to ascertain which of these two was
present. Of a mixture of acetanilide (6 grams.) and acetoorthoto-
luidide (4 grams) the eutectic point was determined. For this was
found 64.°6 and 65.°1, mean 64.°8,. On adding to this mixture 0.1
gram of p-acetotoluidide, the said point was found to be 63.°1 and
63.°6, mean 63.°3,; the latter, therefore, seemed rather sensitive to
small additions of para.

5.64 grams of acetanilide were now mixed with 4.36 grams
of the acetocompound prepared from the motherliquor (1) which,
according the above examination, contain 4.20 gram of acetoorthoto-
luidide and 0.16 gram of an impurity, which might be p-acetotoluidide.

The point of initial solidification of this mixture was found to be
72.0 and 71°9, the point of complete solidification 62.°6 and 62.°8.
A mixtute prepared from 5.64 gram of acetanilide, 4.20 grams of
acetoorthotoluidide exhibited these same points at 72.°1 and 62.8, so
that the impurity seems to be indeed p-acetotoluide; acetanilide is
out of the question as then the point of complete solidification ought
to have coincided with the eutectic point of the pure mixture of
acetanilide and aceto-o-toluidide.

The above method will no doubt be found applicable in a number
of other cases as it is based on a general principle. By its means,
it is possible to ascertain the purity of organic preparations with a
greater degree of quantitative precision than has been the case up
to the present, particularly when dealing with liquid substances.

Mr. F. H. van per Jaan has ably assisted me in the experimental
part of this research.

Groningen, Chem. Lab. Univers. November 1904.

( 398 )

Botany. — “On the influence of nutrition on the fluctuating varia-
bility of some plants.” By Miss Tie Tammes. (Communicated
by. Prot. Jas Morr):

(Communicated in the meeting of October 29, 1904).

That nutrition has an influence on the development of plants has
long been known. Also that some parts are much more sensitive in
this respect than others and that, for example, the size of the stem
and leaf is much more affected by good or bad nutrition than the
number of stamens. As yet our knowledge on this point, especially
our quantitative knowledge, is very superficial. The introduction of
the statistical method, however, into botany has enabled us to for-
mulate more sharply the formerly vague and insufficiently defined
question of the influence of nutrition and also to interpret the results
obtained easily and accurately.

Although the number of statistical investigations on plant charac-
teristics, carried out in recent years, is fairly numerous, yet the
influence of nutrition on the value of these characteristics has not
often been studied.

Dre Vrins') carried out an extensive investigation in this direction
with Othonna crassifolia. He compared plants that had been grown
in a greenhouse in pots with very dry ground with garden-cultures
and found that with the plants from the greenhouse the median of
the length of the leaves was only about half that of the plants that
had grown in full ground, the average number of ray-flowers per
head being 12 with the former, 13 with the latter. In his work
“die Mutationstheorie’ pe Vrins*) describes experiments and obser-
vations, the chief object of which has been the comparisón of the
influence of nutrition with that of selection, but which at the same
time increase our knowledge about the influence of nutritive con-
ditions as such. He investigated the influence of these two factors
on the length of the fruit of Oenothera Lamarckiana and Oenothera
rubrinervis, on the number of umbel-rays of Anethum graveolens and
Coriandrum sativum.and on the number of ray-flowers of Chrysan-
themum segetum, Coreopsis tinctoria, Bidens grandiflora and Madia
elegans. From his observations pr Vrims coneludes that nutrition and
selection act in the same direction and that by stronger nutrition as
well as by positive selection the median value of a character is
increased. Moreover he generally observes that the variability of the

1) Hugo pe Vries, Othonna crassifolia, Bot. Jaarb. Dodonaea, 1900, p. 20.

2) Hueo pe Vries, Die Mutationstheorie. Bd. I, p. 368,

( 399 )

characters is increased when nutrition and selection act in opposite
directions, i.e. when, as in his experiments, strong nutrition goes
together with negative selection.

Also the experiments by Rermönr *) on the variability of the number
of stamens of Stellaria media show that with good nutrition the
median of this character possesses a higher value than with bad
nutrition. Besides Rreiönr finds that the index of variability, which
is a measure for the variability, becomes smaller under unfavourable
nutritive conditions.

Weissr *) investigated the influence of nutrition on various charac-
ters of Helianthus annuus and found that the arithmetical mean for
all the characters studied is smaller with plants cultivated on a sandy
soil than with well-fed plants. His numbers, (for each culture about
forty) are too small, however, to allow us to calculate the constants
for median and variability from them and to draw conclusions from
these.

Mac Lrop*) made experiments in order to determine the influence
of nutrition on the number of ray- and disk-flowers of Centaurea
Cyanus and found that this number is the smaller the more the
nutritive conditions are unfavourable. Besides he investigated the
influence of good and bad nutrition on the number of stigmatic-rays
of Papaver Rhoeas coccineum aureum. He arrived at the result that
with the badly-fed plants the median is considerably smaller, but
that the variability of the character is increased by the bad nutrition.

From this short summary it will appear that in very few cases
only the quantitative change, caused in the median by varying nutri-
tion, has been determined. It is desirable to extend the number of
observations on this point, but it is especially important to learn the
influence of nutrition on the variability for several characters and
plants. Two questions here arise, in the first place whether this
influence is different for different parts of the same plant, in agree-
ment with VerscHarrent’s *) result that the variability itself of diffe-

1) Frrepricn Reréut, Die Variation im Andröceum der Stellaria media Cyr. Bot.
Zeit. 1903, p. 159.

2) AntHur Wersse, Die Zahl der Randbliithen an Compositenköpfchen in ihrer
Beziehung zur Blattstellung und Ernährung. Jahrb. f. wiss. Bot. Bd. 30, 1897, p. 453.

5) J. Mac Leop, On the variability of the disk- and ray-flowers in the cornflower
(Centaurea Cyanus). Hand. v. h. 3de Vlaamsch Nat. en Geneesk. Congres, Sept.
1899, p. 61 (in Dutch) and On the variability of the number of stigmatic-rays in
Papaver. Hand. v. h. 4de Vlaamsch Nat. en Geneesk, Congres, Sept. 1900, p. 11
(in Dutch).

1) Ep. Verscuarrett, Ueber graduelle Variabilität von pflanzlichen Wigenschaften.
Ber. d. d. bot. Gesellsch. Bd. XII, 1894, p. 350.

( 400 )

rent parts differs considerably, and secondly whether bad nutrition
causes either an increase or a decrease of the variability for all
characters, or an inerease for some and a decrease for others.

With the object of answering these questions, I made some culture
experiments in the botanical garden at Groningen in the summer of
1903. The description and results of these experiments will be found
in what follows.

For the cultures four beds of 2 metres breadth and 6 metres length
were prepared in April. Two of them were manured with hornmeal,
about half a kilogram per square metre. The other two beds were
dug out to a depth of about half a metre and filled with a very
meagre loamy sandsoil, originating from Harendermolen, a sandy
region in the neighbourhood of Groningen. In the middle of April
on one of the manured beds and on one of sandy soil equal quan-
tities of seed were sown of J/beris amara Linn., obtained from
Haace and Scumpr at Erfurt, Ranunculus arvensis Linn., obtained
from various botanical gardens and mixed, and of Malva vulgaris Fr.
(Malva rotundifolia Linn.), obtained from the botanical garden at
Leiden. The seeds of three other species, which were sown at the same
time on the remaining two beds, did not germinate in sufficient
numbers, so that about the middle of June we resolved to weed
them all out and to sow afresh. This time Anethum graveolens
Linn., from the trade, Scandix Pecten- Veneris Linn. and .Cardamine
hirsuta Linn., both obtained from various botanical gardens were
chosen, three species of which it might be expected that, although
sown so late in the summer, they might still fully develop. This
seed was sown in germinating dishes, each species partly in meagre
and partly in fertile earth taken from the beds in the garden. In the
course of the following days part of the germplants were placed
into small pots with meagre as well as with manured earth, special:
care being taken that no selection from the germplants should be
made. At the middle of July the young plants were placed in the
beds at such distances from each other that each could freely develop.

Already at the beginning a considerable difference between the two
cultures could be observed in all three species sown in the garden.
The seed in the bed that had been manured with hornmeal came
up sooner and the plantlets developed much more vigorously. With
Malva vulgaris the difference between the plants’ of the two beds was
at first very great. Those on the fertile soil showed already abundant
leaves and flowers when the plants on the sandy soil had only
formed few and small leaves. This difference remained till the begin-
ning of July, when suddenly also the plants on the meagre soil

( 401 )

began to develop vigorously, so that in the autumn scarcely any
difference could be observed. The reason of this late, very rapid
development appeared when the plants were dug out. It turned out,
namely, that some of the strongest roots had reached the underlying
earth through the layer of sand. As long as the plants only obtained
their food from the sand, they remained tiny and backward, but
when the roots had penetrated into the fertile earth they still deve-
loped vigorously and with great rapidity. Also with Zberis amara
the roots appeared to have reached the earth underneath but in a
much less degree. It was difficult here to trace the fine terminals
of the principal roots as far as the underlying earth, whereas the
roots of Malea vulgaris, where they passed from the sand into the
earth below, were strong and penetrated at least a few decimetres.
Of Ranunculus arvensis only few roots had reached the underground
with their tips, the same being the case with Scandix Pecten- Veneris
and Anethum graveolens; the roots of Cardamine hirsuta were restricted
to the sand, as far as I could see.

Although with most of the species studied the nutrient material
was not entirely derived from the sandy soil, yet all these plants
were in less favourable nutritive conditions than the plants on the
manured soil. So the experiments will show us the consequences
of the difference in nutrition.

For the investigation I chose some characters that are easily
expressed quantitatively and numerically and took care that the
determination was made at the same time for both cultures and that
the same parts of both were always taken.

In this way I determined in the first place the length of the leaf
of Lheris amara. In July the length of the five oldest leaves, which
were already adult then, was measured. Besides in the autumn, after
the plants had been dug out, the length of the plant was determined
from the base to the top of the inflorescence of the principal stem ;
at the same time were counted the number of branches of the second
order, the number of branches of the third order and the number
of fruits on the inflorescence of the principal stem.

Of Malvea vulgaris the number of akenes of the schizocarp, the
length of the leaf-blade and the length of the leaf-stalk were deter-
mined. These countings and measurements were made in the beginning
of July, when a very distinet difference in the development between
the two cultures was visible, hence probably before the roots of the
plants on the meagre soil had penetrated the layer of sand, and in
any case before a better nutrition had any perceptible effect.

In the case of Anethum graveolens and Scandia Pecten- Veneris the

28

Proceedings Royal Acad. Amsterdam. Vol. VII.

( 402 )

number of lobes of the first leaf was counted in the plants that
had survived in germinating dishes. Besides I determined in adult
plants of Scandi Pecten-Veneris the number of umbel-rays and
with Anethum graveolens also the number of umbel-rays and at the
same time the number of flowers of the umbellet. For the determi-
nation of this latter character only the umbellets of the oldest umbel
of each plant were taken. Of Panunculus arvensis the number of
fruits per flower was determined and of Cardamine hirsuta the length
of the silique, of each plant the siliques of the principal stem being
measured.

For each of the characters mentioned | took of each of the cultures
on fertile soil and on sandy soil 300 measurements or countings, a
number which, according to the calculations of Prof. KarrryN, gives
in investigations of this kind a sufficient guarantee of accuracy. For
certain characters I had to be contented with a smailer number
since the material in these cases was deficient. For those cases in
which the variability concerns the number, the numbers were noted
increasing by unity; for those characters that vary in leneth, the
length was determined in fractions of a millimetre, in millimetres
or in centimetres, depending on the absolute size of the parts. By
means of the numbers obtained, curves were plotted in order to have
a general survey of the observations and to facilitate a comparison
of the observations of the culture on fertile soil with that on sandy
soil. In most cases the observations were combined into groups, so
that from seven to seventeen intervals were obtained. In this way
curves are obtained that admit of easy inspection and in which the
smaller irregularities have disappeared. Only for the number of
branches of the third order of /beris amara, tig. V, the observations
of the plants on the fertile soil had to be combined to 28 groups,
since only then a comparison with the plants from the sandy soil
was possible.

The curves for the various characters are reproduced on the
accompanying plate. Since for all cases the frequencies have been
calculated, all the curves have the same area and can be mutually
compared. For each character the curve of the well-fed plants has
been drawn as a continuous line and that of the badly-fed plants as
a dotted one, both having the same absciss. Of both the observations
have been combined to groups with the same interval. In all the
figures the size or the number of the part in question increases from
left to right.

These curves now show us the way in which the studied characters
vary and the limits of this variation.

( 403 )

Looking at the various figures we notice that the studied charac-
ters generally give fairly symmetrical curves, disregarding smaller
irregularities. Only in a few cases, as with Anethum graveolens for
the number of umbel-rays of plants on the sandy soil, fig. VI, for
the number of lobes of the leaves of the well-fed plants, fig. VIII,
and besides for the number of lobes of the leaves of Scandiv Pecten-
Veneris of the fertile soil, fig. IX, the curve is markedly oblique.
Only for the number of branches of the third order of /beris amara
from the sandy soil, fig. V, a semi-curve has been obtained.

Examining in the various figures the position of the two curves
with respect to each other, it appears that they partly coincide.
This means that in the two corresponding cultures plants are found
in which the organ under consideration is as large or occurs in
equal number in the well-fed and in the badly-fed plants. But at
the same time they show that in one culture individuals occur, in
which a definite part is so strongly or feebly developed, as are not
to be found in the other cultures. The figures further show that in
all cases except of the number of akenes of JZa/va vulgaris, fig. NI,
the curve of the plants on sandy soil has been shifted to the left
with respect to that of the well-fed plants.

The observations now enable us to determine how great the
influence of the nutritive conditions is in the various cases and
whether this difference in development between the two cultures is
the same for various parts of the same plant.

Examining the figs. I—V, relating to the characters of /beris
amara; figs. VI—VII of Anethum graveolens and XI—XIII of
Malra vulgaris it appears that, whereas with the two former plants
the shifting of the curve is very different in the various cases, it is
about the same for the three characters of Malea vulgaris and for
all three of them relatively small. So the curves enable us to form
an approximate idea of the influence of various nutritive conditions,
but a clear insight is only obtained when the curves are defined by
definite constants and these are mutually compared. In this way it
is possible to determine what influence feeding has not only on the
median value of the character, but also on its variability. In order
to obtain these values, the median value J/ and the quartile Q were
deduced from the observations. From these the coefficient of varia-
bility Ww which is a measure of the variability and enables us to

mutually compare the variability of different characters, was calcu-
lated by the method introduced by VerscrarreLT *). Also for the

1) Ep. VerscHarreLt, |. ¢.

28%

( 404 )

somewhat skew curves these values have been determined, since
these curves do not considerably deviate from the symmetrical ones
and besides, in all cases the average of both quartiles has been taken.
Only from the semi-curve for the branches of the third order of
[beris amara, fig. V, no constants were calculated. This curve will
be dealt with later on.

I give here the values found for the various characters in the plants
studied in the same order as that of the curves of the plate. In the
table, G means the constants of the well-fed, B those of the badly-
fed plants. For each character are given: the median value, the
quartile, the variability-coefficient and the minimum and maximum
value. Besides the differences of these values in the well-fed and the
badlv-fed plants have been calculated as well for the median as for
the variability-coefficient. This difference, divided by the value for
the well-fed plants and consequently expressed as a fraction of this
value, I will call the sensibility-coesfcient of the median or the
variability. This coefficient is given in. the table under the two values.
A + sign for the sensibility-coefficient means that the value is
greatest with the well-fed plants, a — sign that with these the value
is smallest.

It appears from this table as well as from the curves that in
general the median value of the characters of the badly-fed plants
is smaller than of the well-fed ones. Only with Mafra vulgaris the
median value of the number of akenes of the plants from the sandy
soil is slightly larger, the difference being very small, however. The
sensibility-coefficient is only — 0.015. With the remaining characters
the sensibility-coefficient of the median is positive and differs very
much; on the whole it varies between 0.015 and + 0.54.

Let us now see from the table whether nutrition has the same
influence on the median value of the different characters of the same
species. We shall leave Malva vulgaris out of account here since,
as was mentioned above, its roots had in the bed of unfertile earth
penetrated into the fertile underground and possibly on this account
the differences were very slight for all the characters considered.
Comparing the sensibility-coefficients of the median of the various
characters of one species, we find that they diverge largely.

While the sensibility-coefficient of the median of the number of
branches of the second order of /beris anura is — 0,54, it is
+ 0,15 for the number of silicles of the principal stem ; the sensi-
bility-coefficients of J/ for the length of the plant and the length of
the leaf lie between these values and amount to + 0,24 and + 0,28.
With Anethum graveolens the sensibility-coefficient of the median of

¢ ae :
M Q = Minimum. | Maximum.
M
Iberis amara.
\ G il 1 cM. | 4.65 cM. | «0.114 | 26 cM. | 56 cM.
I. Length of the plant |
|B SAIS enen 10-4030) ME > 754.8" on
sensibility
coefficient. .…… + 0.24 + 6.09
| { G 7.9 eM. | 4.085 cM.) 0.137 4.5 cM.) 14.2 cM,
II. Length of the leaf
(2B dS. hs 01895- oi) 0-160) 273 »°| Bees
sensibility
coefficient... + 0.28 — 0.17
G Gld MANE BE 0.13 29 91
II. Number of silicles | |
B 47 6.8 0.14 11 MS
sensibility
coefficient...) + 0.15 =~ 0.08)
; 99 a qe 5 pn ae
LV. Number of bran- a 22.4 3.35 Dodo 9 | 35
ches of the 2d order} B 10.3 3.78 0-36 0 99
sensibility
coefficient. … 4- 0.54 — 1.40
Anethum graveolens.
Ee) . NT fi
VI. Number of umbel- \ G 32.8 6:40 GEA 15 9
pee eee | (B 184 6.45 | 0.33 i Al
sensibility
| coefficient... + 0.44 OA
|
| |
|
| 4 EE 6 Iers G ; 5
VII. Number of flowers) | G 33.3 6.90 0.19 Ì 67
in the umbellet.| | B 26.8 5.7 0.21 fi 45
| |
| sensibility | 5
| coefficient..., + 0.20 — 0.105
| | DE AE
VIII. Numb. of lobes | \ G 18 } 3.25 | OCAS 3 40
of the first leaf. | B 16.8 We Ol 0127 7 28
sensibility 7
coefficient... + 0.08 | + 0.29) |
| | |
| | |

Scandix Pecten-
Veneris.
1X. Number of lobes
of the first leaf. .

‘X. Number of umbel-
EAYSE eiser.

Malva vulgaris.

XI. Lengthof the blade

XII. Length of the
leaf-stalk...

XIII. Number of

alkenes... vn.
|
|
|

Ranunculus arvensis.

XIV. Number of

AKENES Beene |

Cardamine hirsuta. |

XV. Length of the
SITE rte sisters

sensibility
coefficient. .

sensibility

coefficient. ..

G

B

sensibility
coetlicient. .

G
)
,

(NEE

sensibility
coeflicient. .

| B

sensibility
coetlicient...

G

|
Ls

sensibility

coellicient. . .

sensibility

coefficient

M Q a Minimum. Maximum.
97.2 3.85 0.14 | 416 56
54 | 26 0.105) 4 | ag
+ 0.08 | 0.26 |
6.05 0.7 0.101 3 10
5.03 0.55 0.100 4 7
+ 0.17 + 0.01!
| | |
| 53.8 mM.) 3.85 mM. 0.071 40 mM. 65 mM.
(51.9 » |3.95 » | 0.0751 30 » | 70 »
| | |
0503 — 0.055,
| | |
172.4 mM. 15.4 mM.) 0.089 | 128 mM.) 289 mM.
167 » |43.65 » 0.081) 145 » | 44 »
Je 150703 + 0.09 |
| |
| |
13.38 0.7 0.05 Om 17
13.6 0.6 0.044 11 ee ly/
— 0.015 + 0.12
8.5 0.75 0 09 5 12
6.9 0.775 0.11 4 11
+ 0419 | — 0.22

17.5 mM.) 2.75 mM.| 0.15 | 4 mM.

|
lezen gen Ie Os DEE

24.1 mM.

PRY

( 407 ) :

the number of umbel-rays is + 0,44, that of the number of lobes
of the first leaf only + 0,08. To some extent this may be explained
by the circumstance that the influence of nutrition on the first leaf
is not so great as on characters which appear later, since the food,
stored in the seed, is the same for both cultures and possibly has
not been entirely used when the first leaf develops. In agreement
with this the sensibility-coefficient of the median of the number of
lobes of the first leaf of Scandiv Pecten- Veneris is + 0,08, whereas
it is + 0,17 for the number of umbel-rays of the same plant.
From what precedes it will be seen that the influence of nutrition
of the median value of different characters of the same plant varies
greatly, some organs being very sensitive for differences in nutrition,
others experiencing little difference in their development on this account.
Concerning the value of the quartile the table shows that we do
not obtain in all the cases studied, a variation in the same sense
by bad nutrition, as was the case with the median value. In some
cases (J is greater in the plants from the fertile soil, in other cases
it is smaller, as great or nearly as great as with the plants from
» compare the variability of the

meagre soil. In order to be able t
characters in both cultures, however, and to draw conclusions from
this comparison about the influence of nutrition on the degree of
variability, we must not take the quartile but the variability-coefficient Ik

If, to begin with, we consider the value of this variability-coefficient
in the various cases, we see from the table that it varies between
wide limits 0,044 and 0,36. Also VerrscnarreLrt ') found equally

divergent values of a for the characters of different plants studied

by him. The smallest variability is found with the different cha-
racters of Malea vulgaris, as well in the well-fed as in the
badly-fed plants. Hence this plant appears to be little variable.
Comparing the variability of the different characters of the same
species with each other, we see that they diverge relatively little
with the well-fed plants, as well with /beris amara, as with Anethum
graveolens and Malva vulgaris. For the different characters of Zberis
amara 5 is respectively 0,114, 0,137, 0,18, 0,15; for Anethum

ál

graveolens 0,19, 0,19 and 0,18 and for Malva vulgaris 0,071, 0,089
and 0,05.
It will be seen that for the same species these values are nearly

1) VerscHarreELT, |. c. p. 353.

( 408 )

the same, while they differ considerably among the three species.
Doing the same with the badly-fed plants we find a much greater
difference between the variability-coefficients of the various characters
Q

of the same plant. For this culture = varies between 0,10 and
0,36, for the characters ot J/beris amara and between 0,127 and
0,35 for those of Anethum graveolens. Hence it follows that the
influence of nutrition on the variability of the different properties of
a plant is not the same; how much this influence varies will be
seen from what follows.

Comparing for each character separately the variability of the
well-fed with that of the badly-fed plants, we find that the difference
between the variability-coefficients for the two cultures varies greatly
in different cases; for some characters it is very considerable, for
others small. In order to compare these differences, they were divided

Q
by the value of = of the well-fed plants, as stated. The resulting

number is the sensibility-coefficient of the variability. This sensibility-
Q
coefficient of yy Sppears to vary between — 0,140 and + 0,29. In

a comparison of various characters of the same species the fact that
the roots of the bad culture had more or less penetrated into the
subsoil, obviously is of no consequence, so that the results obtained
with Malva vulgaris are also available here.

()
Or == ¢

The sensibility-coefficient f Theris amara is for the four

characters respectively — 1,40, — 0,17, — 0,08 and + 0,09; for
the characters of Anethum graveolens — 0,74, — 0,105 and + 0,29;
and for those of Malra vulgaris — 0,055, — 0,09 and + 0,12. Especi-
ally with the first two plants these sensibility-coefficients diverge
considerably, which proves how very different the influence of
nutrition is on the variability of the different characters of a plant.
By the same change in nutrition the variability of one character is
hardly modified at all and that of another character of the same
plant very considerably increased or diminished.

It is very important to know in what direction the nutrition reacts
on the variability, whether under unfavourable nutritive conditions
the variability is either always greater, or generally smaller or whether
the two cases are equally frequent. In this respect the table shows
Q_

5

us that for 6 out of 14 characters the sensibility-coeflicient of 7

( 409 )

positive and the variability-coefficient of the well-fed plants greater
than of the badly-fed ones, whereas in the other characters the
sensibility-coefficient is negative and the variability-coefficient greatest
in the badly-fed plants.

Even with the same species one character shows a greater, another
a smaller variability when the cultures grown under favourable and
unfavourable nutritive conditions are compared. With /berzs amara
the length of the plants from the fertile earth is more variable than
that of the plants from the sandy soil, other characters, on the other
hand, show greater variability in the badly-fed culture. In the same
way in Anethum graveolens the variability is greatest with the num-
ber of lobes of the well-fed plants and with the number of flowers
and umbel-rays of the badly-fed ones, while with Malra vulgaris
the length of the leaf-stalk and the number of akenes of the well-fed
plants, but, on the other hand, the length of the blade of the plants
from the sand, show the greatest variability.

Summarising the results obtained, we see that nutrition influences
the median value and the variability of the characters. Besides it
appears that the sensibility-coefficient of the median is very different:

1. for different species compared among each other.

2. for different characters of the same species.

And about the variability we saw:

Q

1. that with good nutrition the variability-coefficient is fairly

constant for different characters of the same species, but very diver-
gent for the different species.

2. that with bad nutrition two of the species studied show great
differences between the variability-coefficients of the different charac-
ters of the same species, while with one species the variability-
coefficients of the various characters diverge relatively little.

Q
F En BEI malas ee 5 . Ayo
3. that the sensibility-coefficient of — diverges greatly for different
species and characters and varies between —1,40 and —+ 0,29.
: ese ve ORL,
4. that for some characters the sensibility-coeffieient of us

positive and good nutrition results in an increase of the variability:
while for other characters, even of the same species, this coefficient
is negative.

In what precedes, there has only been question of those charac-
ters which show symmetrical or sensibly symmetrical curves and
Which, when expressed in constants, yielded the results mentioned.

From these the curve of the number of branches of the third

( 410 )

order of Jheris amara, grown on the sand, deviates entirely, being
a semi-curve. For the culture on fertile earth, however, this same
character gives a symmetrical curve. In fig. V this latter is very
flat and extended in length, as the observations were divided over
a great number of intervals in order to allow a comparison of the
two curves. If, however, the observations are arranged to a number
of groups equal to that of the other figures, the curve thus obtained
is not different from those of the other characters. For this culture

the median is 58, the quartile 17.25 and the variability-coefficient

— 0.32, the minimum number of side-branches being 1, the maxi-
mum 162.

With this character now, bad nutrition does not result in a simple
shifting of the curve to the left, accompanied by greater or smaller

ê QO
changes in the values of J/, Q and ar as in the other cases, but

here the symmetrical curve changes into a semi-curve of which the
apex lies at zero.

We can explain the origin of this semi-curve in the following way.
The lower limit for the number of branches of the third order
of Jberis amara is 0. Since the plant also blooms on the principal
stem and on the branches of the second order, it may exist without
branches of the third order. Under favourable nutritive conditions
the development of the plant is so vigorous that in all individuals
branches of the third order are formed, but in greatly diverging
numbers, as is shown by the curve of fig. V for this culture. With
unfavourable nutrition, however, also individuals arise in which no
branches of the third order are originated and as nutrition becomes
worse the number of these individuals will become greater. Hence
we see that with the very bad nutrition of the sandy soil, a great
number of plants has no branches of the third order and so has
reached the lower limit, the other specimens bearing a greater or
smaller number of these side-branches, as is shown by fig. V for
this culture. This leads us to the conviction that the semi-curve for
this character is a necessary consequence of the fact that by the
unfavourable nutritive conditions the variation-curve is shifted in
such a way that it strikes against the lower limit of the whole
range of variation of this character, a great many of the individuals
showing this lower minimum value.

Also with Anethum graveolens a great difference is noticed in the
shape of the curves of the number of umbel-rays in the two cultures,
fig. VI. The curve of the well-fed plants is nearly symmetrical, while

T. TAMMES “On the influence of nutrition on the fluctuating variability of
some plants.”

ms il

OMG, ig 24 30 J6 42 4g 54 60 66 2 18 04 90 96 102 108 114 120 126 132 138 144 150 156 162 168

ie tf B

(OS Ch MD UP EG I the A

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26 28 30 32 34 36 38 40

CT a OP

IX =

es es

30 33 36 39 42 15 ET 5/ 54 gy ce 63 66 69 72

FE

105 720 135 150 165 180 195 210 225 240 255 20 285 300

Tí sr \ if
| bee | LOTUS 12 IS) TE on hommes

G4 AG F BY WW Ht G2 FO 45 60 SF 90 105 120 195 130 165 180 195 SW 225 240 255

Proceedings Royal Acad. Amsterdam. Vol. VII.

( 411 )

that of the plants from the sandy soil is asymmetrical in such a
way that the top of the seurve lies nearer the minimum. It can not
be stated with certainty whether in this case we have the same
phenomenon as with /beris amara, ie. whether the lack of symmetry
of the curve indicates that it has been shifted to the proximity of
the lower limit. But the fact that the minimum now obtained, viz. 7,
is already very small compared with the maximum 41 and that this
lower limit cannot be zero, renders this view probable. Yet we must
bear in mind in cases like the present, that the appearance of an
asymmetrical curve need not in general be a proof that the curve
is located near one of the limits of the range of variation, but that
the asymmetry of the curve may also be the consequence of entirely
different causes.

Botanical laboratory at Groningen. July 30, 1904.

EXPLANATION OF THE FIGURES.

The figures are all reproduced at about half size. In the original figures the
distances of the intervals, placed along the absciss, are 1 em, each mm. of the
ordinates having a value of 1°/). So we can find from the length of the ordinates
the percentage number for each interval. In most figures the ordinates are drawn
between the two numbers indicating the interval, only in figs. X, XUL and XIV,
where the observations are not arranged in groups, the ordinates stand above the
number. The curves of the well-fed plants are drawn in continuous lines, those
of the badly-fed plants are dotted.

Fig. I. Iberis amara. Length of the plant from the base of the principal
stem to the top of the inflorescence of this latter, in cm.

yall Iberis amara. Length of the leaf, in cm.

» HL Zberis amara. Number of silicles of the inflorescence of the principal stem.

» IV. Iberis amara. Number of branches of the second order.
ores Iberis amara. Number of branches of the third order.

» VL Anethwm graveolens. Number of umbel-rays.

» VIL Anethum graveolens. Number of flowers in the umbellet.
„ VIL. Anethwm graveolens. Number of lobes of the first leaf.
» IX. Scandix Pecten-Veneris. Number of lobes of the first leaf.
ATA Xe Scandix Pecten-Veneris. Number of umbel-rays.

» AL Malva vulgaris. Length of the leaf-blade, in mm.

» All. Malva vulgaris. Length of the leaf-stalk, in mm.

» XII. Malva vulgaris. Number of akenes of the schizocarp.

» XIV. Ranunculus arvensis. Number of fruits per flower.

» XV. Cardamine hirsuta. Length of the silique, in mm.

( 412 )

Botany. — Prof. J. W. Morr presents the thesis for the doctorate
of Mr. B. Sypxens: “On the nuclear division of Fritillaria

imperialis 1 and gives a summary of the results.
(Communicated in the meeting of October 29, 1904.)

The subject of this investigation is especially the nuclear division
in the embryo-sac of /y7ti//aria, formerly a favourite material for inves-
tigations on the subject of nuclear division.

Mr. Sypkens studied the free nuclear divisions in the parietal layer
of protoplasm as well as the nuclear divisions in the first layer of
endosperm-cells which are directly followed by tangential cellular
divisions. Besides some observations were made on the nuclei in the
ovules of Tulipa and in the growing-point of the root of Vicia
Faba.

All the material was fixed by means of the strong chromo-aceto-
osmie acid of FreMMiNG. It was for the greater part imbedded in
paraffin in various ways and was examined in series of sections
of 2 to 4 wu thickness, stained with gentian violet. Some observations
were also made by means of the method introduced by var WISSELINGH,
in which the nuclei are dissolved in chromic acid of about 50°/,. These
two methods supplement each other; the chromic acid method is to
be preferred for observations about the chromatic parts, sections
give more information about the nuclear spindle. But in this investi-
gation the excellence of both methods was again proved as compared
with the observation of the nucleus as a whole, which in many
cases renders it impossible to form an accurate idea about its
internal structure.

I will briefly mention the chief results obtained by Mr. SYPKENs
for the various stages of nuclear division.

The resting nucleus was studied by means of sections and of
chromic acid and the results so obtained were in the main a complete
confirmation of the results published by van WIsseLINGH and by
GreGorrE and his co-workers Wyeanrts and Brerens. The framework
of the resting nucleus consists of numerous larger and smaller lamps
of chromatin, connected by fine threads so that an anastomosing
network is formed. There is no reason for assuming in this network
the existence of two constituents, chromatin and linin; the chromic
acid method as well as coloured nuclear sections show the contrary,
if only partial washing out of the stain is prevented, as Mr. SypKENS
did. Those who wish to maintain the assertion about the existence
of linin-connections will have to bring forth new and valid proofs,

( 413 )

Also for the nuclei of the integuments and nueellus of Fritillaria and
of the ovules of Tudipa the same results were obtained.

Concerning the individuality of the chromosomes van WisseLINGH
has shown that it exists in the spirema, since at that stage a
continuous thread is never found. But his further observations as well
as those of GRÉGOIRE and WyGarrts indicate that probably, even in the
resting stage, this individuality never entirely disappears. Mr. SypKuns
was led to the same conviction by his observations about the
formation of the spireme and of daughter-nuclei from the daughter-
spiremes. He speaks of a “centralisation and decentralisation of a
number of chromatine masses, which in certain stages form as
many chromosomes.”

About the behaviour of the chromosomes during the process of
division little that was new could be found in this investigation for
the reason mentioned. The number of chromosomes was fixed at
about 60, but in certain nuclei it decidedly is much smaller. Neither
is the shape of the chromosomes constant; in the same nucleus U-
shaped, as well as W- and J-shaped ones could be found.

The study of the nuclear spindle on the other hand gave important
results, not so much about the formation of the spindle as about its
further history and the part played by it in cellular division.

The formation of the spindle could be followed in details. Round
the free nuclei in the parietal layer of protoplasm of the embryo-sac
granular protoplasm occurs with many very small adventitious vacuoles ;
round the nuclei of the first endosperm-cells also protoplasm with
several small vacuoles. Now, when the nuclei begin to divide and the
nuclear membranes are dissolved, the surrounding protoplasm pene-
trates into the nuclear space, at first without many vacuoles, and forms
at the interior the spindle-threads, which at first consist of coherent
granules and later become smoother. They gradually assume parallel
directions and are connected to a bundle without strongly converging
towards its poles. The nuclei are then in the spireme-stage. Later, in
the aster-stage, besides the threads already mentioned, others are
formed in exactly the same way, which grow thicker and only
proceed from the poles to the equator, where they are attached to
the chromosomes, which have been formed in the mean time. They
are found not only at the circumference of the spindle, but also in
the interior part of all the longitudinal sections of a nucleus. Srras-
BURGER has called the former sort of threads, running from pole to
pole, “Stiitzfasern”, the shorter and thicker ones “Zugtasern”.

Now metakinesis follows and in the dyaster-stage a separation
of the two sorts of spindle-threads has taken place. The shorter

( 414 )

and thicker ones have much contracted and form at both poles,
adjacent to the daughter-nuclei, two small caps which soon disappear
in the protoplasm. The long threads on the other hand remain between
ihe daughter-nuclei, extending from one to the other and hence are
often called connecting threads. They occur in numbers from 200 to
300 and cross-sections show that they form a massive bundle lying
free in the surrounding protoplasm, which can freely penetrate
between them.

Hence Mr. SYPKENS arrives at the conclusion that the nuclear
spindle is entirely formed from the cytoplasm within the nuclear
space and so agrees with what has been found by most other inves-
tigators and on main points also with the results obtained by STRASBURGER
and Heeser for the nuclei of Pritdllaria.

Now with regard to the part played by the nuclear spindle in
cell-division zoologists and botanists have divergent views. Con-
cerning animal cells the general opinion is that the nuclear spindle
is dissolved in the cytoplasm after the nuclear division has been
completed and takes no active part in cell-division, the cell subse-
quently dividing by constriction. Botanists on the other hand, attach
great importance to the spindle in cell-division and especially in
the formation of the wall. Their generally accepted representation
is that the above-mentioned connecting threads of the spindle grow
thicker in the equatorial plane and form so-called dermatosomes.
By fusion of the dermatosomes the so-called cell-plate is then formed,
which subsequently participates in some way or other in the for-
mation of the new cell-wall between the nuclei. STRASBURGER is one
of the chief representatives of this much spread conception.

It is a consequence of the fact that the study of this phenomenon
has for the greater part taken place with nuclei that were seen
from the outside. By means of his sections Mr. SYPKENs was enabled
to prove that, for the objects studied by him, the opinion now
prevailing in botany is incorrect and that, at any rate as far as the
behaviour of the nuclear spindle is concerned, the phenomena have
great resemblance with those of animal cells.

In describing the later phases of the nuclear spindle it is desirable
to distinguish three different cases of nuclear division. In the first place
we have the free nuclear divisions in the parietal layer of protoplasm of
the embryo-sac of Fritdlaria, whieh will be followed by still other
nuclear divisions before there is question of cell-divisions. Here in the
beginning a system of connecting threads between the daughter-nuclei
appears, as in all other cases, but this soon becomes narrower at the

equator and so assumes the shape of an hour-glass and is then

( 415 )

absorbed in the protoplasm and disappears. So this case needs no
further consideration.

The second case regards the parietal Jayer of protoplasm of the
embryo-sac, which has already become partly divided into cells. Now
when here also free nuclear divisions take place, the nuclear spindle,
consisting of connecting threads, behaves at first in exactly the same
manner as in tissue-cells in which the cell-division follows immediately :
the system of connecting threads swells laterally and forms a so-called
nuclear barrel. After this, however, the spindle here is also lost
in the protoplasm and not until later one sees successive divisions
take place between these nuclei, progressing regularly from that part
of the parietal layer of protoplasm that is already divided into cells, so
that finally a complete pavement of endosperm-cells is formed from
the protoplasm. This description renders the existence of a connec-
tion between the nuclear spindle and cell-division not very probable.

The most important case is the third, in which the just-mentioned
endosperm-layer divides into two layers of cells by tangentially
directed walls. Here the nuclear divisions are immediately followed
by cell-divisions, in the same way as in the formation of various
sorts of tissues.

Hence this case, as was proved by comparative observations, must
be considered as completely analogous with what happens in the
cells of the growing-point of the roots of Wicia Faba.

From Mr. Sypkens’ sections it appears that in the two latter cases
the connecting threads soon cease to deserve that name, as their
extremities are not attached to the daughter-nuclei but end freely in
the protoplasm. In Vici faba moreover, the equatorial parts are
soon dissolved so that the system of connecting threads falls asunder
into two halves.

Meanwhile the protoplasm round the nuclei of the parietal layer of
protoplasm penetrates with its small adventitious vacuoles into the
space between the daughter-nuclei where the massive complex of
connecting threads is found. These threads are consequently forced
asunder towards the circumference and thereby united to spindle-
shaped bundles, which lie free in the protoplasm; they form what is
usually called the nuclear barrel. The result is that the two daughter-
nuclei are at last separated from each other by the same granular
protoplasm, which also surrounds them and in which also the remains
of the connecting threads are found. The spindle-shaped complexes,
formed from these, are united to a barrel-shaped, equatorially swol-
len, cylindrical mantle, which, if the nuclei are only observed from
the outside, still seems to join them, although in reality this is no

( 416 )

longer the case by any means. On the contrary, the remains of the
connecting threads gradually disappear as if they were dissolved in
the protoplasm and this process has long been completed when the
cell-walls successively appear between these nuclei also.

Also in the divisions of the endosperm-cells ef /riti/laria and in
the root-tip of Vieta Kaba mainly the same occurrences take place
although there are some points of difference to which I shall refer
presently, and although the formation of the cell-wall follows sooner here.

How this wall-formation takes place has for the present not been
investigated by Mr. SrPKeNs, but that it stands in no relation to the
nuclear spindle or to a cell-plate formed by it, is pretty clear from
what precedes. A cell-plate in the sense of botanical authors does not
even occur.

Although the opinion, so generally spread in botany, that in many
cases the formation of cell-walls is dependent on nuclear spindles,
may have a certain probability when we only think of the eross-
divisions of the cells of growing-points and suchlike, it lacks,
generally speaking, every foundation. For any one knows that the
formation of cell-walls can in many cases have nothing to do with
a nuclear spindle. Not to mention all possible cases of thickening of
the cell-wall whieh do not correspond to the formation of a primary
membrane, [ will only mention zoospores which, after having come to
rest, form a wall; plasmolysed protoplasts of Sperogyra and other
Algae which cover themselves with a new cell-wall; Caulerpa and
other Coeloblasts, the protoplasm of which after a lesion produces a
new wall-piece.

But also in other cases, which resemble more the cell-divisions in
erowing-points, it is often easy to show how newly-formed cell-walls
cannot possibly have been formed in the nuclear spindle. I mention
the antipodal cells, which so frequently are formed projecting inwardly
in the embryo-sac connected only for a small part of their surface
with the cell-wall of the embryo-sac; in any case no more than a
small part of the free wall-surface can have been formed here in a
nuclear spindle. A corresponding case is that of the U-shaped walls
in the epidermal cells of the leaves of ferns, by which the mother-
cells of stomata are formed. More clearly still one sees the same
thing in the formation of the stomata of Anemia fravinifolia: the
stomata lie in the middle of an epidermal cell of the leaf and the
nucleus of this cell is still pressed against the stoma. A nuclear divi-
sion has taken place here before the stoma-mother-cell was formed
in the epidermal cell, and between the two cells so formed there
certainly was a spindle at first. But in the subsequent cell-division a

( 417)

eylindrical wall was formed at a certain distance round onc of the
nuclei, which consequently could for a small part only have been
formed in the spindle. Finally we have the formation of the first
pavement of endosperm-cells from the parietal layer of protoplasm of
the embryo-sac as well in Fivtillaria as in many other plants. When
the number of nuclei of this laver of protoplasm has very greatly
increased, separations between the nuclei arise, so that a layer of flat,
pentagonal or hexagonal cells is formed, which at last are separated
from each other by cell-walls. These cell-walls are formed at a period
when of the originally present nuclear spindles no trace is left.

In relation with these facts the result of Mr. Sypkens about the
negative part played by the nuclear spindle in cell-division cannot
surprise us and it even gains in inner probability by them. This
result also shows the way to a more profound study of the pheno-
mena of cell-division and wall-formation in the vegetable kingdom.
The cell-divisions in growing-points, in the above-mentioned epider-
mal cells of ferns, also in the parietal layer of protoplasm of the
embryo-sac, must now be more closely investigated, preferably by
the method applied by Mr. Sypkens, and important results may be
expected of this investigation. Also the study of living, dividing cells,
in the same sense as was formerly done by Tree *) deserves again
our attention in this respect.

It is by no means impossible that by such investigations the con-
ception of cell-division in plants will come still nearer to that of
the same phenomenon in animals than is the case at present.

From all that precedes it appears that the nuclear spindle is formed
entirely from the ¢ytoplasm and returns to it. Besides, all investi-
gators agree that in nuclear division the nuclear membrane and the
nucleoli are dissolved and later are formed anew in the daughter-
nuclei. An uninterrupted individual position with regard to the cyto-
plasm is consequently, among all the parts of the nucleus, occupied
by the chromosomes alone, only here there is question of a hereditary
organisation.

The opinion of some authors that the nucleus during the whole
process of division would form an isolated whole with respect to
the cytoplasm and that at first there would be a sort of vesicle,
joining the daughter-cells and separating the spindle from the cyto-
plasm, must consequently be abandoned.

In relation with this IT may briefly point out the complete agree-
1) M. Treus, Quelques recherches sur le rôle du noyau dans la division des cellules
végétales. Publié par Académie Roy. Néerl. des Sciences. 1878.

29

Proceedings Royal Acad. Amsterdam, Vol. VIL.

( 418 )

ment between the results of Mr. Sypkexs and the theory of pr Vries
and Went, which looks upon the vacuoles as hereditary organs of
the protoplast. If the nucleus were, during division, an isolated whole,
the question about the origin of the vacuoles, present inside the
spindle, would perhaps give some difficulty. But we saw, how the
observations of Mr. SypkmNs prove that we have here ordinary
vacuoles, already present in the granular protoplasm and which are
shoved in between the spindle-threads from the outside with the
protoplasm.

Yet it will be desirable to give some nearer information about
this process, since two somewhat divergent cases occur and here
again a distinction must be made between the nuclear divisions in
the parietal layer of protoplasm of the embryo-sac and those in the
first endosperm-layer or in the meristem of the roots of Viera.

In the latter cases, in which ordinary division of tissue-cells takes
place, Mr. SypKens observed what follows. In these cells there are
a number of vacuoles, which are about equivalent and lie round
the nucleus in the granular protoplasm. After nuclear division this
protoplasm with its relatively large vacuoles, penetrates into the spindle
between the connecting-threads, as we saw above. This penetration
here occurs as well in the equator as more in the neighbourhood of
the daughter-nuclei. Hence it is the ordinary vacuoles of the mother-
cell, which shove in between the daughter-nuclei with the protoplasm
in which they he. Later, when the connecting-threads have been
dissolved and cell-division takes place, these vacuoles, as well as
those which did not penetrate into the spindle, are divided equally
between the two daughter-cells. So the question is here very simple
and in complete accordance with what vaN WIssELINGH found in
Spirogyra. Ouly in this latter case the mother-cell has not several
equivalent vacuoles but a single large one which penetrates laterally
into the nuclear spindle.

Somewhat different are the circumstances in the divisions of the
parietal layer of protoplasm of the embryo-sac. This cell not only
contains many nuclei but has also a somewhat different structure
with regard to its vacuoles. If has namely one single large vacuole,
filling the middle part of the cell, but besides in the parietal layer
of protoplasm a great number of very small adventitious vacuoles,
which were very conspicuous in the preparations of Mr. Sypkens,
stained without washine out of the stain. Now, after nuclear
division, the granular protoplasm with its many adventitious vacuoles
penetrates between the daughter-nuelei and the free extremities of
the connecting threads. From there it penetrates further towards

(419)

the equator between the connecting threads. Hence the daughter-
nuclei are finally separated from each other by granular protoplasm
with adventitious vacuoles of the embryo-sac. Now, when later the
parietal layer of protoplasm divides into cells, the large embryo-sac
vacuole does not partake in this process, but each newly formed
endosperm-cell is provided with a certain number of adventitious
vacuoles.

So there is a certain antithesis here with what happens in ordinary
cell-divisions in young cells, but with the vacuole theory of bn Vrins
and Went this process also is in complete harmony, for Want has
shown that small adventitious vacuoles can occur in large numbers
in all sorts of ordinary cells and can in all respects be compared
with the large vacuole, from which they can also be produced
by division. I should not he surprised if further investigation
showed that their occurrence is much more general still than is now
supposed.

The case met with in the embryo-sac of Mritillaria and many
other plants stands by no means isolated, and is also met with in
the division of other multinuclear cells. Werr mentions some cases
of this kind in his investigation about the vacuoles of Algae.
(Chaetomorpha aerea, Acetabularia mediterranea, Codium tomentosum).

I had an opportunity personally to observe a similar case of
division in the formation of asexual zoospores in the cells of Hydro-
dictyon utriculatum. While the zoospores, which had been formed by
division of the parietal layer of protoplasm, were partly in motion and
partly had already arranged themselves to a network, all this inside
the wall of the large mother-cell, L saw the middle part of this
cell occupied by three great tonoplast vesicles, having their origin
in the great central vacuole of the cell and which, upon being
heated under the microscope, first shrank and then burst. Hence
here, no more than in the embryo-sac of Fritti/laria, the great
central vacuole took part in the formation of new cells. That the
zoospores were provided with very small vacuoles, present in the
granular protoplasm, cannot be doubted according to the above-
mentioned investigations of Went. I also observed them very distinctly
in the cells of the young nets very soon after their formation.

Finally it requires to be mentioned that the doctoral dissertation
of Mr. Sypkens will soon appear in a German translation in the
second Part of Volume I of the Recueil des travaux botaniques
Neerlandais.

29%

( 420 )

Botany. — “An investigation on polarity and organ-formation with
Caulerpa prolifera.’ By Prof. J. M. Janse. (Communicated by

Prof. Hugo pr Vriss).

(Communicated in the meeting of October 29, 1904).

Polarity is a property of very many of the lowest organisms as
well as of a great part of the cells in the body of the higher plants
and animals.

The regular exterior shape and internal structure of organs must
be partly attributed to the agency of polar influences during their
development, while the definite vital phenomena of organs must
also, among other causes, be ascribed to polar actions of the con-
stituent cells.

The cause of this polarity, i.e. the property of acting or reacting
in a certain direction otherwise than in the opposite direction, is
unknown, and the great difficulty of finding suitable material for
investigation is perhaps the principal cause of this.

Former observations made with Cuulerpa prolifera had convinced
me *) that this unicellular, relatively gigantic and morphologically
highly differentiated alga must be suitable for this purpose.

Having had the opportunity during last summer, of submitting this
plant to a renewed investigation at the Zoological Station at Naples,
I wish to relate briefly the principal results obtained.

For a description of the structure of Caulerpa prolifera, as well
as of its protoplast and the very intense currents that take place
in it, I refer to my quoted paper.

For the new experiments the “leaves” were exclusively used, namely
the outgrowths of the “rhizome” measuring in extreme as much as 22
centimetres in length and 20 millimetres or a little more in breadth.
Their little thickness allows us to examine them also microscopically
in a living condition, while their considerable length and breadth
make them particularly fit for experiments. Moreover cut leaves or
parts of leaves can form new rhizomes and rootlets and so can
regenerate to complete plants by neo-formation.

Formerly already L used these leaves for experiments concerning the
course of the protoplasm-currents, in which it was often required to
make large incisions in the leaves.

These plants, to be true, often sustain serious lesions which heal

1) Die Bewegungen des Protoplasma von Caulerpa prolifera, Pringsheim’s
Jahrb. f. wiss. Bot. 1889, Bd. XXI, pag. 163—284, with 3 plates.

( 424 )

In one day, but this is always accompanied by a great loss of proto-
plasm by which the cell is much enfeebled.

This time I succeeded in finding a new method in order to get at the
same result, based on the observation that every laceration of a
part of the numberless protoplasm-threads, which run through the
whole plant as an extremely fine network, is immediately followed
by the loeal secretion of a white, tough, wiry substance, which very
soon becomes stiffer, assumes a bright yellow colour and then forms
a perfect partition. If at the same time the cell-wall had been injured,
the external wound is closed in this way. But the same laceration
of the plasm-threads can be brought about by pressure and without
external lesion; the partition is then restricted to the place where
sufficient pressure was exerted. In this manner one can at any
arbitrary point of the leaf produce, as it were, a eross-wall, to which
any desired direction and length can be given. If one proceeds with
care this partition is no broader than */, millimetre.

In this way one can also physiologically, namely without external
lesion, divide a leaf into two or more parts.

This treatment, which in all respects has the same consequences
as are observed with a wound, is not accompanied by weakening of
the cell, since no protoplasm is lost, and besides the plant is already
after a minute fit for further manipulations or for examination.

Caulerpa prolifera derives its specifie name from the circumstance
that the “leaves” which spring forth from the “rhizome” very often
produce new leaves, prolifications. Especially by this circumstance
I succeeded formerly in showing that, in accordance with pm Vrins’
views, there exists also in this plant a direct relation between the
intensity of the motion of the protoplasm in various places and that
of the nutrition in these places. The bundles of protoplasm bands,
coloured dark green by chlorophyl grains and very often visible to
the naked eye, which pass from the stalk of the prolification into
the primary leaf and then tend to the leaf-stalk of this latter were
a very important aid in this investigation.

These bundles are lacking where very young prolifications are
found and only gradually develop in the leaf, and in doing so always
begin at the stalk of the prolification and extend towards the base
of the leaf. These stream-bundles are never seen developing in the
opposite direction, i.e. beginning at the leaf-stalk and extending
towards the stalk of the prolification, neither do they proceed from
the prolification to the top of the leaf. So they originate from above
and tend downward.

Moreover if an existing prolification is cut off, one sees the bundles

(422)

gradually disappear: this disappearance also proceeds from above
down ward.

Both phenomena point to the existence of a polarity in the regulation
of the protoplasmic currents, of which the impulse proceeds in the
direction from the organic top to the base.

If this stream-bundle is interrupted by a large cross-wound the
communication is restored round the end of the wound. Now, however,
the currents above and below the wound behave quite differently :
the bands which proceed from the stalk of the prolification remain
on the whole unchanged until they have arrived near the wound; they
then partly deviate transversely and bend round the end of the wound,
after which they go in a straight line to the leaf-stalk. Another part
often turns back with a bend, namely if the currents are strong. So
above the wound there occurs as it were a thrust and often a
reflection which are entirely absent below the wound.

Also this difference in the course of the currents above and below
the wound points to a polarity in the regulation of the protoplasmic
currents, the impulse evidently here also proceeding from the top
and being directed to the organic base.

We must add here in the first place that the currents, running
in a non-proliferous leaf, which assemble like a fan from the top
and the edge of the leaf and all pass into the leat-stalk, behave in
exactly the same manner, when interrupted, as the stream-bundle
which proceeds from a prolification downward; only this latter is
generally more powerful and so more suitable for experimenting.

Secondly we must remember that everywhere in the leaf there
exists a very complicated network of currents, stretching between
the numerous (+ 800 per sq.mm.) eross-beams which join the two
sides of the leaf; so there exists an almost straight, but little intensive
connection between any arbitrary pair of points on the leaf; so when
we speak here for simplicity’s sake of the generation of currents, we
mean the strengthening or thickening of the currents in such a way
that they become visible with the naked eye or with the eye-glass.

Thirdly the protoplasm in all currents moves continually or alter-
nately in both directions and this applies also to those which develop
from above and to those which disappear from above.

Thus far my previous investigations had led me.

The renewed investigation was begun again with these experiments ;
they gave entirely concordant results.

As the experiments with cross-wounded leaves had shown that
it is possible to deviate the large nutritive currents from their way
and to cause them to assume a lateral direction, the question was

( 423 )

whether it would be possible to go farther still and
to lead the current in an opposite direction.

Formerly already [ had made similar experiments,
which had given a favourable result, but there was
reason to repeat them now on a more extensive scale.

The arrangement of the experiments was such that
two internal partitions were produced forming two
hooks, embracing each other, and the short arms of
which extended as far as the edge of the leaf (fig. 1).

Hence the connection between top and base lay
through the whole middle piece between the two
longitudinal partitions and in this piece the develop-
ment of the current would have to take place in a
direction opposite to that in the normal leaf.

Now the experiments proved that indeed such a
development, and thus so to say the ‘reversion’, in this
middle piece is possible, and that the typical direction
of the currents is then as indicated in the figure by

the continuous line. The experiments proved besides:

Bickel 1. that it takes a long time before by this route
a powerful connection between top and base is formed, some weeks
being required ;

2. that the attempt is successful only when the distance between
the cross-wounds is not too considerable, 25—50 mm. being the
extreme limit;

3, that for success it is desirable that the impulse from above
be powerful, which is the case, for example, if above the highest
cross-wound one or more vigorous prolifications occur ;

4. that the leaf always strongly opposes the reversion.

Concerning this latter point | must add what follows:

When a leaf of Caulerpa is cut off, either at the leafstalk or at
a higher level, rootlets are formed at the cut piece and this nearly
always exactly at the sectional plane i.e. at the organic lower side;
a middle piece from the leaf does exactly the same.

The plant thus makes an attempt at beginning an independent life
by neo-formation. (In nature this is the most powerful, if not the only
means of propagation for Cau/erpa, since it seems to have no sexual
organs).

Now one sees the same happen with the double-hooked wounds:
along the whole breadth above the lower cross-wound rootlets often
grow, proving that communication has become so much impeded,
that the first piece of the leaf (Ll, which is in communication with

( 424 )

the top) and the middle piece, II, evidently meet their want of con-
nection sooner and perhaps better by means of the neo-formation of
rootlets, than by strengthening the existing but feeble communica-
tion with the old ones. *)

The third part of the leaf, UL, (which consequently is in direct
communication with the base) never shows any inclination to the
formation of rootlets, obviously because the communication has
remained unimpaired here.

If we must assume that the stream-bundle in the uninjured leat
is regulated by a basally directed impulse, then, when the “reversion”
has succeeded, the newly formed current in the middle piece must be
directed by an opposite impulse, or, to speak more correctly, by the
same impulse, after it has, so to say, been reflected by the eross-wound.

That this current in fact behaves in this manner, follows at once
from the fact that the new current is first visible below in the middle
piece and is gradually prolonged upward.

A still more convincing proof of this can be given by a further
experimental operation : if namely these new currents are interrupted
in I and II by a small cross-wound (as in fig. 1) one sees the thrust
in I occur above, in II on the other hand below the wound, and
the currents take their way as is indicated by the dotted line in the
figure. This is a proof that these two adjacent pieces behave oppositely.

Though we finally often succeeded in bringing about the “reversion”
in the middle piece, vet this reversion is very incomplete, as I infer
from the following observation. In one of the leaves with a double-
hooked wound a prolification had been formed above in the middle piece,
while the complete reversion was being brought about; the new
leaflet lay a little sideways of the current. Proceeding from this
leaflet a little bundle of three currents had developed. One of them
proceeded along the lower side of the upper eross-wound into the
third part of the leaf, after having joined the main current coming
from below. The other two, however, took their way straight down-
ward as if the connection with the base of the leaf were still exactly
as before the lesion. Hence one of the currents, when coming forth
from the leallet, obeyed the action of the reflected impulse, whereas
the other two experienced no influence. In that place of the middle
piece the old basipetal impulse must consequently have been preserved.

A similar case, occurring in another experiment, will be men-
tioned later.

1) Above the upper cross wound also rootlets are sometimes formed, although
only when the top-part is large and so powerful enough, or when prolifications
occur on it.

( 425 )

Also one of the leaves, mentioned above, in which a small cross-
wound was made in the middle piece and at the same time in the
first piece (as in fig. 1), showed a phenomenon which Lean only
explain in this manner. It was pointed out already that above wounds
new rootlets are regularly formed; we shall hereafter describe the
phenomena preceding the formation of the rootlets, phenomena which
always make themselves felt in a basipetal direction. These preli-
minary phenomena now appeared in that leaf in the first part (1)
above the small wound, as usual, and in the middle piece also
above the small wound. If the polarity of this whole piece had been
reversed, these changes should have appeared there below the small
wound. Now this indicates, in my opinion, that the reflected impulse
was localised and had no influence on the lateral part of the middle
piece, after it had been withdrawn from its direct action, and
this piece, having retained the old impulse, reacted therefore as
normally.

After I had succeeded in “reversing” a current, it was probable
that it would also be possible to cause a whole plant to develop
inversely.

When, however, this experiment was made in such a way thata
whole plant, with rhizome and rootlets, was reversed and the leaf-
tops were buried in mud, it gave no result; for seven weeks such
a plant remained absolutely unaltered; only the top of the leaf
became white on account of tbe loss of chlorophyl-grains, caused
by the darkness, while the rhizome grew a little and made some
new rootlets.

Cut leaves, freely suspended upside down or planted with their
top in mud, gave quite different results, however.

Nearly all the leaves, and especially and most quickly the youngest,
first formed new rootlets, which also in this position of the leaf always
arose at the end of the stalk ; very many appeared already after two days.

After that several prolifications appeared generally in various
places, and then a first consequence of the reversion could be observed
in the course taken by the stream-bundles coming out of the proli-
fications and continuing their way through the old leaf.

In a cut but erectly planted leaf these go always, without exception,
to the base of the leaf; here in nearly all prolifications the greater
part of these currents went to the base also, but some of them took
their course towards the top of the leaf, without reaching it however.
Gravity, acting in the opposite direction during their formation and
development, had evidently deviated them.

Stull more clearly the existence of an antagonism between gravity

( 426 )

and the basipetal impulse was visible in some of these leaves from
the fact that stream-loops were formed. From the young prolifications,
namely, some currents were seen going to the top of the leaf, which
however later suddenly returned with a very sharp bend and then
went back straight to the leaf-stalk. So gravity had first deviated them,
but the continuous counteracting influence of the basipetal impulse,
which was evidently. felt in every point of the leaf, had at last
overcome gravity and got the upper hand. These loops had in large
leaves a leneth of 5 to 10 mm.

All these changes took place before at the apical side rootlets
developed. This occurred finally with very many leaves; the earliest
appeared after 9 days, the majority came later, but after almost four
weeks they had not yet developed in all of them. That most of them
had formed rootlets at the extremity of the leaf-stalk much earlier,
proves that they possessed to the fuli the power of forming them.
That the presence of rootlets at the leaf-stalk was no impediment
for the development of new rootlets elsewhere, appeared also from
the fact that, with respect to these latter, no difference could be noticed
between leaves with and without rootlets at the stalk.

If now these leaves were planted in mud with their apical rootlets
(which, however, were hardly ever placed exactly at the top, but
at a smaller or greater distance from it) the prolifications grew on
or, if they had not been present beforehand, they always appeared
after this. A connection was generally formed between the stream-
bundle issuing from them and the rootlets and so a plant was obtained
in which, under the impulse proceeding from the prolification, the
nutritive current had developed in a direction opposed to the impulse
existing in the leaf.

Here also it could be proved in the same way,
A as before by means of a cross-wound (as in fig. 2,
in which the basal half of a leaf was planted
upside down), that this bundle in the leaf obeyed
indeed the impulse of the prolification A, since

© from this side the thrust occurred.

Yet here also the reversion appeared to be
only local. A small prolifieation 4 had namely been
formed below the cross-wound, after this had
been made (consequently at the side of the apical
rootlets). This new leaflet in its turn formed a

Fig. 2. small stream-bundle of which some thinner currents
went in the direction of these rootlets; one thicker current however
took his course athwart alongside the wound, turned at the end with

( 427)

a sharp bend and went to the old base of the leaf. In doing so this
current crossed?) the bundle going from the other prolification to the
apical rootlets, but even this did not cause it to change its direction.

So here also a reversion was obtained, this time by the influence
of gravity, but it also was proved to have a very local character.

The inverted leaves gave me material for still another experiment.

These leaves, as has been remarked, had at last for the greater
part formed prolifications and rootlets towards the apical side. Now
in some of them a prolification and a rootlet were found at about
the same height, but the one on the left side of the leaf, the other
on the right.

What would happen now if this piece of the leaf, isolated from
the other basal rootlets and prolifieations by a eross-wound, were
planted separately ? Since a manifold direct, but feeble communication
actually existed between the two organs by means of the numerous
fine protoplasmic currents, it was possible that the direet communication
would be strengthened and so a cross-current would arise, in the
same way as above a large cross-wound. But it was also possible
that the basipetal impulse of the prolification and of the piece ot
the leaf would not admit a communication or not one in that
direction.

The three experiments for which suitable material was obtained,
were not entirely at an end at my departure. Yet it then appeared
already with perfect distinctness that nowhere a strengthening of
the cross-communication had taken place. On the contrary, the currents
everywhere went from the prolifications straight to the basal wound ;
currents Communicating with the rootlet showed the same. Even an
indirect Communication between the two, via the basel wound, was
not established.

In one of the leaves a young rhizome was formed beside the
prolification at a distance of 1'’, millimetre from it; a communication
between the two was established, but by a very roundabout way,
mm.

In another leaf a rhizome was developed near the rootlet and
another a little above the prolification. Im both cases the communi-

viz. via the basal wound, which lay at a distance of 6°/

2

cation between each of the two groups was again established via
the basal wound, but none between prolification and rootlet. The
complete physiological separation between prolification and rootlet

1) The current seemed to intersect the bundle, but as the currents proceed
from beam to beam and often two currents are attached to one beam at different
heights, these currents must have crossed each other. The same often happens with
stream-loops.

( 428 )

finally appeared from the fact that in a small spot at the wound,
exactly in the place where the currents from the prolification reached
if, a number of small rootlets were formed.

Hence the basipetal impulse was so strong that it entirely prevented
a cross-communication, as a consequence of which each of the parts
of the leaf formed two individuals, cohering in a morphological
sense but scarcely in a physiological sense.

We spoke above of currents that were reflected by the wound;
this expression was chosen because the direction of the wound evidently
influences the direction which the current assumes afterwards and
this in a similar way in which a solid wall affects an impinging
wave-front.

This influence is most clearly seen when of three leaves the top
part is cut off, (this latter being taken as large as possible) following
in the first leaf a transverse line, in the second a V-shaped one, the
point of the V bemg downward, and in the last leaf an inverted
V-shaped line. After a few days the currents are seen to bend near
the wound in such a way that the lines bisecting these current arches,
are in the first case parallel to the longitudinal axis of the leaf, in
the second converge and in the third diverge. These currents are
often so strong that one can follow them over long distances with
the naked eye.

However, only those parts of the currents that lie near the wound
must be taken into account, firstly because the reflection is not
sudden but gradual, so that the currents assume a more or less sharp
bend with a radius of '/, to 2 mm, secondly because the leaves are
rather narrow and so the reflected currents cannot, for a long
distance, freely continue their new course.

That in the formation of wound-cork in higher plants the new
cross-walls in the phellogen are always parallel to the direction of
the wound in the nearest place, suggests a similar influence of the
wound in these plants.

The basipetal impulse, indicated by the experiments mentioned,
shows itself no less distinetly in the formation of new organs in eut
leaves. ')

1) | never saw rootlets or rhizomes arise on intact leaves, attached to the rhi-
zome; cut rootlets die off at once, while loose rhizomes, when they are strong
enough, form new organs, but always in an entirely normal way.

( 429 )

Wakker *) already pointed out that in these the young rhizomes
and rootlets always arise above the basal wound.

Investigation has shown that immediately after the lesion the
formation of these organs is prepared, namely by a division in the
protoplasm. This I could only observe in leaves which were in very
good condition of life; in these, however, the changes were well
visible with the naked eve or else with the hand-magnifier.

Above the basal wound a clear white spot is gradually seen to
arise, often several millimetres in size. In these places only the
rootlets are later formed, while in the immediate vicinity of them
the rhizomes appear.

Where it was mentioned above (pag. 425) that the wounded leaves
showed an inclination for forming rootlets, the arising of such a
white spot was meant.

The first question now was: what causes this white spot?

In vigorous cut leaves one sees often already one day after the
cutting whitish streaks occur, of which no trace can be detected in
the intact plant. As far as they are rendered visible by a strong hand-
magnifier they begin at some distance from the top as well as from
the edges of the leaf, but become soon thicker and proceed in a
feeble curve (which is concave towards the edge) towards the middle
and there assemble and proceed together to the leafstalk ; here and
there they are connected among each other. So their mode of proceeding
is exactly the same as that of the green currents.

But also in other respects they behave like these latter: if the
cut leaf bears a prolification, from the stalk of this latter a bundle
of these currents passes into the leaf; when the currents meet a
cross-wound they proceed as far as this, move sideways and when
they have arrived at the corner of the wound, continue their way
straight to the leaf-stalk.

These currents, which sometimes appear light greenish because they
are seen through the peripheral layer of chlorophyl, consist of a
very fine-granulated and therefore milky white protoplasm, very dif-
ferent from the much clearer protoplasm of the green currents, in
which the chlorophyl-grains are moved along. The white currents
partly originate from the green ones: these latter are namely seen
to become feebler when the latter arise, while at the disappearance
of the white currents the green ones gradually become more distinct
again.

1) Die Neubildungen an abgeschrittenen Blättern von Caulerpa prolifera: Versl.
en Meded. der Kon. Akad. van Wetenschappen te Amsterdam, 1886, 3d series,
part 2, p. 252.

( 430 )

From all points of the wounded leaf consequently fine-granulated
protoplasm flows together; it gathers immediately above the wound,
replaces the chlorophyl-containing peripheral layer and currents and
so causes the leaf locally to assume a white colour.

When the white currents are observed microscopically, also in them
a distinet streaming is observed, mostly in the two opposite directions
at the same time or otherwise alternately, while a number of unco-
loured granules are dragged along. But chlorophyll grains are entirely
lacking. Yet after the lesion the quantity of plasm above the wound,
‘white’ as well as “green”, increases, while in the top it diminishes,
occasionally to such an extent even that the top becomes empty
and dies. From this follows that the mass of plasm, conveyed down-
ward by the currents is greater than the mass which is taken back
upward, so that the resultant of the two motions is equivalent to a
current going to the organic base.

So the white currents behave exactly like the green ones; yet
there is a difference between them, although only a quantitative one:
while both groups of currents obey the same basipetal impulse, this
latter appears to exert a somewhat greater influence on the white
currents than on the green, for the green protoplasm is always
observed to be pushed aside by the white.

Now, when it had appeared that the white currents move towards
the basal wound, the question arose whether they only strove to
reach this wound or, perhaps not contented with this, would also
try to occupy the very lowest (most basal) place near this wound.

öxperiments showed this latter to be indeed the case.

If a wound be made in a slanting direction with regard to the
diameter of the leaf, the white plasm flows down along the wound
and assembles in the sharp point; if the wound be V-shaped all
gathers in the middle, while with a 4-shaped impediment the white
currents flow off to the two points near the edges of the leaf. With
these lesions the green currents behave exactly as the white ones,
but again their terminal point is left a few millimetres behind that
of the uncoloured currents. From this it appears more clearly still
that these latter feel the basipetal impulse more strongly.

Especially the current of uncoloured protoplasm which flows off
along the wound is seen to follow a wavy course, since it consists
of very short pieces of current, which go longitudinally downward,
are then retlected by the wound and soon afterwards bend down
again. Not unfrequently two currents run close to each other and
in doing so cross each other repeatedly. The height of these waves

is small, no more than '/, to */, mm.

The basal part of the so wounded leaves underwent no change
when it remained in connection with the rhizome. If, however,
shortly before, it had been separated from it, the lower piece behaved
like a cut entire leaf. White currents here also appear; they however
only begin at some distance below the lesion as fine lines and,
growing thicker, pass all into the leaf-stalk. At the lower end of
this latter the accumulation of white protoplasm then takes place.

The very sharp antithesis between the phenomena below and
above the wound, again furnishes a striking proof of the existence
of the basipetal impulse and of its influence on the white plasm.

With regard to the origin of this plasm it must be remembered
that all organs during their growth always contain a large quantity
of such plasm at their top. Behind it, when growth has been com-
pleted, it is clear and contains in leaves and rhizomes a very great
number of chlorophylgrains. When an organ stops growing, the
white top soon disappears.

For this reason and on account of its appearance, | compared
already formerly ') this fine-granulated, turbid protoplasm to that which
fills the meristem-cells in higher plants.

Hence Caulerpa possesses, notwithstanding its being unicellular, a
“meristem-plasm” which, however, is only to be found during growth
and in the growing tops. After the growth has ceased it disappears,
which disappeareance must be regarded as a mixing up with the
remaining protoplasm *).

The experiments now showed that after serious lesion this meristem-
plasm is secreted again (which can take place evidently in all points
of the leaf), after which it unites to currents of increasing thickness
and flows together at the organic base.

On the thus formed white spots the rootlets are produced, while
the rhizomes take their origin in the immediate vicinity of them,
mostly on the transition of the white spot to the dark green part,
but still within reach of the white currents, So both arise in conse-
quence of the resulting descending current after the lesion.

Hence the rootlets and rhizomes derive their meristem-plasm from
the same confluent, turbid protoplasm and therefore this latter may
in itself be regarded as meristem-plasm.

Although the source of the meristem-plasm of rootlets and rhizomes

Ae cp. 203:

*) Such a secretion of meristem-plasm from the protoplasm of the cell and its
resolution in it, has recently been described by Nout for a closely related plant
(Bryopsis); cf.: Beobachtungen und Betrachtungen über embryonale Substanz;
Biologisches Centralblatt, 1903, No 8.

( 432 )

is the same, yet there exists a sort of antagonism between the two.

So, for example, it is not unfrequently seen that when somewhere
on a leaf a rootlet has been formed, immediately behind it a rhizome
arises, or the reverse. ;

The most striking case in this respect I observed with a leaf whieh
had formed two rhizomes laterally of the leafstalk (which is a rare
occurrence) one close above the other: at the other side of the leafstalk,
exactly behind each of the rhizomes, a well-developed rootlet was
found.

Properly speaking this antagonism is already observed when a
rootlet is formed on a rhizome in the ordinary way; it namely does
not arise at some distance from the top, but quite close to it, so that
sometimes the impression is given as if the top, of the rhizome would
divide dichotomically, i.e. into two equivalent branches, whereas
later one point develops into a rhizome, the other into a rootlet.

So the two meristem-plasms are very nearly equal, until at a
certain moment a division takes place. The principal cause of this
division is, in my opinion, light.

The rootlets can very well form and develop in light, but, if
possible, they seek the shaded side or turn away from the light.
Rhizomes, on the other hand, as well as prolifications, generally
are formed at the bright side.

Now taking into account that Norr *) has shown that a rhizome of
Caulerpa forms rootlets at the upper side if this is shaded above
and only illuminated from below, I think we have every reason to
look upon the difference in the intensity of the light on both sides
of the rhizome as the principal cause of the separation, which takes
place in the at first homogeneous meristem-plasm, and hence also of
the antagonism between rhizome and rootlet.

That also internal causes play a part here, follows already from
the fact that the rootlets as well as the leaves, are formed on the
rhizome at distances which for each of them are pretty regular.

A rhizome-top is even occasionally seen to dissolve entirely into
rootlets, which proves that there can be no considerable difference
of origin between them.

So we are naturally led to the question: how do the leaves arise?

In this respect I must restrict myself to a few hints, since the
investigation of this point has not been completed vet.

In the intact plant they arise either on the rhizome or as prolifi-

1) Kinfluss der Lage, u. s. w. : Arbeiten aus dem botanischen Institut in Wiirzburg,
Bd. III, 1888, S. 470.

( 433 )

cations on the leaves. On the rhizome they arise on the upper side
but, in opposition with the rootlets, always at a great distance (a few
centimetres) from the top and consequently quite out of reach of
the meristem-plasm there. I presume that their formation on the upper
side is also determined by light, although this has not been proved yet.

In unwounded leaves, and hence in the normal life of the plant,
they are formed on full-grown leaves and then generally near the
top, in cut leaves they very rarely occur near the top ; in these they
as a rule arise on the lower two thirds of the leaf, preferably even
on the lowest third part, but hardly ever immediately above the
wound. So here also they arise out of reach of the meristem-plasm.

The formation of the leaf begins with the appearance ofa very small
white spot on the dark green organ. This rapidly grows out into
a cylindrical, soon broadening appendix, which often remains entirely
white until it has reached a length of one centimetre, after which
it becomes green from below during further growth. The top remains
white as long as the leaf increases in length, but turns green when
growth is arrested, either by the leaf having reached maturity or by
unfavourable external circumstances.

In no case the formation of a leaf was preceded by the appearance
of a large white spot with affluent streams of meristem-plasm. This
leads to the conclusion that the young leaf derives its meristem-plasm
evidently from the protoplasm of the whole neighbourhood; so a
preferred direction of motion, as a consequence of a basipetal or
acropetal impulse, cannot be detected. As a consequence of this each
of these currents is so feeble, that it could not be observed with the
hand-magnifier. So the formation of leaves appears to be independent
of the descending current of meristem-plasm.

In one case only I have seen white currents in connection with
a young leaf; in a cut leaf I observed that a strong white bundle
had differentiated itself, running close to the base of a young leaflet
(prolification) that had arisen after the cutting. This was a little over
a centimetre long and still as white as ivory. From its stalk six
white streams passed into the leaf; they all ran in a basal direction
and soon became absorbed in the white principal bundle.

So these also obeyed the basipetal impulse; they had gradually
formed during the development of the leaflet and so had not appeared
as preliminaries to the formation of it, as is the case with the rootlets.
Also the large white spot was absent here.

Since the white currents in this cut leaf also flowed together
at the base in order to prepare there the formation of a rhizome
and of rootlets, we may infer from this that there is no essen-

30

Proceedings Royal Acad. Amsterdam. Vol. VII.

( 434 )

tial difference between the meristem-plasm of leaves on one hand
and that of rhizomes and rootlets on the other.

Hence there must be causes which in the cut leaf or in the
rhizome bring about a division in the plasm for the formation of new
prolifications and leaves, and since in cultures it is regularly observed:
that new prolifications arise on the lighted side of the leaf, light
certainly plays a part here. Undoubtedly, however, there are still
other, internal, at present unknown, factors which cooperate in deter-
mining the origin and place of origin of the prolifications.

From the here briefly described observations we may infer that
in the leaves of Cuulerpa a basipetal impulse is active, proceeding
from every point of the leaf and revealing itself in two ways:

1. in leaves, connected with rhizomes and rootlets, in the course
of the “green” currents of protoplasm in the unwounded leaf as
well as in the severely wounded leaves.

2. in cut, vigorous leaves in the occurrence and course of the
“white” currents of meristem-plasm which partly assemble at ‘the
most basally situated place. It is this descending current which
prepares the formation in that place of new rhizomes and rootlets.

None of the observations, on the other hand, gave reason for
assuming also the existence of a contrary, acropetal impulse, thus
even not with the formation of leaves and prolifications.

It proved possible, by certain lesions and by planting invertediy,
to cause the formation of currents which developed contrary to the
currents in the normal leaf. It could be proved however that this
was not an inversion of the impulse itself, but that gravity or reflection
by a wound caused a change in the direction of the current, whereas
the basipetal impulse underwent no or hardly any change, and even
then in any case only a very local one.

The chief phenomena observed with Caulerpa remind us of obser-
vations which have been known for a long time in higher plants.

The consequences of annular wounds or of large transverse wounds
in the bark-tissue of trees, the formation of much plasm-containing
tissue (callus) above the wound, the frequent mortification of the
tissue below the wound, the formation or sprouting of adventitious
roots exclusively above the wound and in the very lowest place of
the living bark-tissue, are, mutatis mutandis, evidently analogous.
The same may be said of the formation of adventitious roots exclu-
sively at the base of cut leaves of very many plants, of the regene-
rative phenomena of the fruit-stalks of Marchantia, ete.

The former phenomena were explained by the older physiologists

(435 )

by means of the hypothetical descending “sap-current’”’, by the newer
physiologists they were, as a rule, not explained at all. Now that the
existence of a resulting deseending current could be proved with
Caulerpa, which shows so many analogous phenomena, it seems to
me to be probable that on closer investigation it will also be found
with higher plants, although perhaps in an entirely different form
than was originally thought.

Physics. — “Double refraction near the components of absorption
lines magnetically split into several components’’, according to
experiments made by Mr. J. Gerst. By Prof. P. Zeman.

It has already appeared from experiments which | had the honour
to communicate to the Academy on a former occasion that the
magneto-optie theory of Vorer'), who established a simple and
rational connexion between the magnetic splitting up of the spectral
lines and dispersion, accounts extremely well for all the phenomena
observed in the region of the absorption lines.

If light traverses parallel to the lines of force very attenuated
sodium vapour placed in the magnetic field, the plane of polarization
is rotated im the positive direction for all periods lying outside the
components of the doublet, but in the negative direction, and very
strongly *), for periods intermediate between those of the components.

If light traverses the vapour normally to the field, there is double
refraction as predicted by Vorer from theory. When placed in a
magnetic field, all isotropic bodies should show double refraction,
but to a measurable degree only in the neighbourhood of the absorption
lines. Vorer in collaboration with Wiscnert experimentally verified
this result, using a small grating and a flame with relatively much
sodium vapour.

I have extended these results*) by working with sodium vapour
so dilute that, in a strong magnetic field, there were seen the four
absorption lines corresponding to the components of the quartet into
which the line D, is split by the magnetic field. The mode of depen-
dency of double refraction on the period could, in this special case
with some reserve, be predicted from Voiert’s theory. Observations,
in which Mr. Grrsr took part, confirmed the theoretical result. Mr.

1) Voter, Wrepemann’s Annalen. Bd 67, p. 359, 1899.

2) Zeeman, Proc. Acad. Amsterdam, May 1902, see also Harro, Thesis for the
doctorate, Amsterdam, 1902.

5) Zeeman en Geest, Proc. Acad. Amsterdam, May 1903.
TOF

( 436 )

Grest has now extended these observations and will give a more
detailed exposition of his results elsewhere’); I intend to give here
a short explanation of them.

The arrangement of the apparatus was for the most part the
same as in our former experiments. Plane polarized light, under
azimuth 45° to the vertical, falls on a Basiner’s Compensator with
horizontal edges. The light then traverses a second Nicol with its
plane of polarization perpendicular to that of the first. An image of
the system of parallel imterference bands in the compensator, is
thrown on the slit of the spectroscope. The light is then analysed
by means of a large RowraNp grating mounted for parallel light.
The greater part of the experiments were made with a compensator
of which the prisms had angles of about 50’, but for the study of
some details compensators were used with angles of 10’ or of 3°. In
the spectroscope a few dark horizontal interference bands are observed
as long as the magnetic field is off. The fine absorption lines of the
vapour are then coincident with the reversed sodium lines due to
the are light. As soon as the field is on, the bands become distorted.
Their vertical displacements are, with the method used, proportional
to the difference of phase between vibrations respectively parallel
and normal to the field.

For the simplest case of a line split by the field into a triplet,
Vorer deduced a formula giving the difference of phase as a function
of the wave length *). The sodium lines YD, and D, being split,
however, by the magnetic field into a quartet and a sextet, it was,
in order to compare theory with observation, necessary to deduce
the formulae for these cases. Mr. Guest has made these calculations
according to the method already indicated by Vorer *) on another
occasion. According to his calculation, the difference of phase between
vibrations normal and parallel to the field, the light having traversed
a layer / of the absorbing vapour, is given by:

woel 4d —d' po? 1
Vv: | (4d?—d'? 90)? ARO Agde

In this formula V indicates the velocity of light in the aether,
fh the strength of the field, ¢,d,d' and ¢ being constants characte-
ristie of the medium. Moreover 2273, =— tr, is the period of vibration
and d=0%. The formula given applies to the case of the sextet; for
the quartet, d' =o and for the triplet, moreover d= o0. Figs. 1—3

1) Gerst, Thesis for the doctorate, Amsterdam, 1904.

2) Vorer |. c.

5) Vorer Wied. Ann. 68 p. 352. 1899,

P. ZEEMAN and J. GEEST, Double

refraction near the components of absorption lines magnetically split
into several components,

II
|] |
) || IN
| \
\
iN
EN uf
Trip L. = == = JEE ———— >
AN
\| |
ae
= 1 IL —
Wig. 1. Wig. 4. Fig. 5.
(theoretical curve) (observed) (observed)
= = =
|
|
|
|
|
|
N=
Quartet. — 4 Jl Z ———
| [ |
In
| |
||
|
IE
— Dan, 1 1 at _ = B
Fig. 2. Fig. 6. Fig. 7.
(theoretical curve) (observed) (observed)
ial ES
|
|
| | |
1 |
| \ ||
| \
| \
NS
Sextet. ——— an 1H —— a
= \} Wf |
\ 7
| |
We = b
‘ (observed)
Fig. 3.

(theoretical curve)

Proceedings Royal Acad. Amsterdam. Vol. VIL

( 437 )

give the graphical representation of 4 as a function of d for each
of these three cases.

The result of the observations is represented in figs. 4—8.
These drawings are made with the aid of photographic negatives.
We have not yet succeeded in getting negatives that showed all
details simultaneously and equally well. Hence ocular observations
had to supply the imperfections of the photographic records.

Figs. 1, 4, 5 refer to the triplet (type line D, in feeble fields) ;
figs. 2, 6, 7 to the quartet (type line D,); figs. 3, 8 to the sextet
‘type line JD,).

When comparing the results of observation with theory, it should
be taken into account that the theoretical curve indicates the distortion
which one single interference band would undergo. With the method
of observation used, the central part of the field of view contained
also parts originating from bands lying higher and lower than the
one considered. The theoretical figure must therefore be completed
with parts of theoretical curves lying above and below the one
represented.

We will first of all consider the quartet. We indicate the bands
by a, 6, c, a being the superior one, and by 1, 2, 3, 4, we indicate
the positions in the spectrum which would be occupied by the com-
ponents. The double curved line between 2 and 3 shows entirely
the same character in both figures. This sinuous line (figs. 6 and 7)
thickens out at the extremities into more intense parts (where the
double refraction is at a maximum or at a minimum) turning their
concave side towards band 6. These intense parts correspond to the
loop of the theoretical curve, the loop between 1 and 2 belonging
to band ce, and the one between 3 and + to band a. It was not to
be expected that the two branches which asymptotically approach
the components, would be seen separated from the loops. The distance
is too small by far to allow that. The vertical central line in the
figure is the reversed sodium line due to the are. With increased
vapour density the loops increase their distance from their band.
Fig. 7 relates to this case, which is also in accordance with theory.
As the vapour deusity increases, fewer details become visible, but
we will not go further into this point now.

The observations concerning the sextet are very difficult on account
of the extremely small distance of the components. It is already
difficult to observe the inverse sextet, and hence so much the more to
observe phenomena occurring between its components. Only under
very favourable circumstances could the phenomenon be observed
as it is represented in fig. 8. The other phenomena observed with

( 438 )

D, are most readily interpreted by considering them as originating
from a triplet and not from-a sextet.

It seems rather superfluous to give any further explanation of
figs. 8, 4, 5; in the case relating to fig. 5, the vapour density is
again greater than in fig. 4. All the phenomena we have considered
are qualitatively in excellent accordance with Vorer’s theory.

The phenomena described for D, and PD, again demonstrate the
existence of very characteristic differences between different spectral
lines, differences no less striking here than in the case of the related
phenomena of the magnetic separation of the spectral lines and of
the rotation of the plane of polarization in the interior‘) of, and
close to, the absorption line. It is certainly very interesting that the
theory explains the entirely different behaviour of D, and D, in the
case now considered by differences between the velocities of propa-
gation of vibrations normal and parallel to the field, assuming, of
course, the magnetic division of the lines.

Physics. — “The motion of electrons in metallic bodies’. 1. By
Prof. H. A. Lorenrz.

It has been shown by Rigcke*), Drupr*) and J. J. THomson ‘*)
that the conductivity of metals for electricity and heat, the thermo-
electric currents, the THomson-effect, the Harr-effeet and phenomena
connected with these may be explained on the hypothesis that a
metal contains a very large number of free electrons and that these
particles, taking part in the heat-motion of the body, move to and
fro with a speed depending on the temperature. In this paper the
problems to which we are led in theories on these subjects will be
treated in a way somewhat different from the methods that have
been used by the above physicists. .

§ 1. I shall begin by assuming that the metal contains but one

1) Zeeman, Proc. Acad. Amsterdam May 1902, see also the description of another
phenomenon in Vorer, Göttinger Nachrichten, Heft 5, 1902.

2) E. Riecxe, Zur Theorie des Galvanismus und der Wärme, Ann. Phys. Chem.
66 (1898), p. 353, 545, 1199; Ueber das Verhältnis der Leitfiihigkeiten der Metalle
für Wärme und für Elektrizität, Ann. Phys. 2 (1900), p. 835.

3) P. Drupe, Zur Elektronentheorie der Metalle, Ann. Phys. 1 (1900), p. 566;
3 (1900), p. 369.

4) J. J. Tuomson, Indications relatives à la constitution de la matière fournies
par les recherches récentes sur le passage de |’électricilé à travers les gaz, Rapports
du Congrès de physique de 1900, Paris, 3, p. 138

( 439 )

kind of free electrons, having all the same charge e and the same
mass 1; the number of these particles per unit volume will be
represented by MN, and I shall suppose their heat-motion to have
such velocities that, at a definite temperature, the mean kinetic
energy of an electron is equal to that of a molecule of a gas. Deno-
ting by 7 the absolute temperature, I shall write for this mean
kinetic energy a7’, where a is a constant.

We shall further consider a cylindrical bar, unequally heated in
its different parts, so that, if « is reckoned along its length, 7’ is
a function of this coordinate. We shall also suppose each electron to
be acted on, in the direction of OY, by a force mX, whose intensity
is a function of x. Such a force may be due either to an electric
field or, in the case of a non-homogeneous metal, to a molecular
attraction exerted by the atoms of the metal. Our first purpose will
be to caleulate the number of electrons » and the amount of energy
W crossing an element of surface perpendicular to the axis of w in
the positive direction, or rather the difference between the numbers
of particles in one case and the quantities of energy in the other
that travel towards the positive and towards the negative side. Both
quantities » and JI’ will be referred to unit area and unit time.

This problem is very similar to those which occur in the kinetic
theory of gases and, just like these, can only be solved in a rigourous
way by the statistical method of Maxwerr and BorTZMANN.

In forming our fundamental equation, we shall not confine ourselves
to the cylindric bar, but take a somewhat wider view of the subject.
At the same time, we shall introduce a simplification, by which it
becomes possible to go further in this theory of a swarm of electrons
than in that of a system of molecules. It relates to the encounters
experienced by the particles and limiting the lengths of their free
paths. Of course, in the theory of gases we have to do with the
mutual encounters between the molecules. In the present case, on
the contrary, we shall suppose the collisions with the metallic atoms
to preponderate; the number of these encounters will be taken so
far to exceed that of the collisions between electrons mutually, that
these latter may be altogether neglected. Moreover, in calculating the
effect of an impact, we shall treat both the atoms and the electrons
as perfectly rigid elastic spheres, and we shall suppose the atoms
to be immovable. Of course, these assumptions depart more or less
from reality; I believe however that we may safely assume the
general character of the phenomena not to be affected by them.

§ 2. Let dS be an element of volume at the point (w, y, 2). At

( 440 )

the time f, this element will contain a certain number (in fact, a
very large number) of electrons moving in different ways.

Now, we can always imagine a piece of metal of finite dimensions,
say of unit volume, in which the ‘concentration’, as we may
call it, of the electrons and the distribution of the different velocities
among them are exactly the same as in the element /S. In studying
the said distribution for the N electrons, with which we are then
concerned, we shall find a diagram representing their velocities to be
very useful. This is got by drawing, from a fixed point O, N vectors,
agreeing in direction and magnitude with the velocities of the electrons.
The ends of these vectors may be called the velocity-points of the
electrons and if, through the point O of the diagram, we draw axes
parallel to those used in the metal itself, the coordinates of a velocity-
point will be equal to the components &, 7, § of the velocity of the
corresponding electron.

Writing now

HS n, 5) da
for the number of velocity-points within the element dé at the
point (§, 9,6), we make the exact solution of all problems relating
to the system of electrons depend on the determination of the
function f(S, 1, 5).
We may also say that
PSs NUSA rn A)

is the number of electrons in the element dS, whose velocity-points
lie in dà; in particular
FE n, 8) dSd and. 3 2 ae}

is the number of electrons for which the values of the components
of velocity are included between § and §+ds5, 4 and 4 + dy,
5 and 6+ do. The expression (2) is got from (1) by a proper choice
of the element dà.

If the function in (1) were known, we could deduce from it the
total number of electrons and the quantities » and WW mentioned
in § 1. Integrating over the full extent of the diagram of velocities,
we have

N= f G5) a Bs. fs) te ote (5)

v= f EF Enda, | ERE

and if, in treating of the flux of energy, we confine ourselves to
the kinetic energy of the particles,

—_—— = ay oe

—)

_

Winfried re)

In the latter formula, 7 denotes the magnitude of the velocity.

It ought to be observed that, in general, the state of the metal
will change from point to point and from one instant to another. If
such be the case, the function /(§, 7,6) will depend on «x, y, 2 and
t, so that the symbol may be replaced by /(6, 7,5, 7, y,2,t). We
shall, however, often abbreviate it to //.

As to the integrations in (3), (4) and (5), in performing these, we
must treat w, 7,2 and f as constants.

$ 3. We shall now seek an equation proper for the determina-
tion of the function 7. For this purpose we fix our attention on the
electrons present, at the time 4, in the element dS at the point (, y, z),
and having their velocity-points within the element (2; we shall
follow these particles, the number of which is

FG: 1,5, 4,y,2,)dSda.. ... . . (6)
in their course during the infinitely short time df. At the end of
this interval those particles of the group which have escaped a
collision with an atom will be found in an element dS’, which
we may get by shifting JS in the directions of the axes over
the distances &dt, dt, Sdt. At the same time, if there are external
forces, the velocities will have changed. I shall suppose each elec-
tron to be acted on by the same force (mX,mYV,m/). Then, for
each of them, the components of the velocity will have increased
by Adt, Ydt, Zdt and, at the end of the interval dt, the velocity-
points will be found in the element d2/, which may be considered
as the original element d2, displaced over those distances.

We must further keep in mind that, while travelling from dS to
dS’, the group (6) loses a certain number of electrons and gains
others. Indeed, all particles of the group that strike against an atom
have their velocities changed, so that they do not any longer belong
to the group, and, on the other hand, there are a certain number of
encounters by which electrons having initially different velocities,
are made to move in such a way, that their velocity-points lie
within dà. Writing

adSdidt
for the number of electrons leaving the group and
bdSdidt
for the number entering it, we may say :
If, to the number (6), we add (h— a) d Sd àdt, we shall get

( 442 )

the number of electrons which, at the time {+ dé, satisfy the condi-
tions that they themselves shall be found in the element dS’ at
the point (e+ Sdt.y+ydt,2+6d2) and their velocity-points in
the element d2’ at the point (§-+ Xdt,y-+ Ydi,é+Zdt). Hence,
since dS’ = dS and dd’ = da,
f(&n § 2 Ys 2 t) + (6 — a) dt=
=f (§ + Xdt, 4 + Vdt,$ + Zdt, x + §dt, y + dt, 2 + Sdt, t + dt),

or

òf òf

6b—-az=—X+

Of i Often Oe ONE
5 On

st SF 3 2 Ze
En aes

This is the equation we wanted to establish *).

(7)

Ger

It is easily seen that, in calculating the numbers of collisions
adSdadt and 6dSdidt, we need not trouble ourselves about
the state of the metal varying from one point to another; we may
therefore understand by adàdt the decrease, and by bdadt the
increase which the group of electrons characterized by dà would
undergo, if we had to do with a piece of metal occupying a unit
of volume and being, in all its parts, in the state that exists in the
element dS.

§ 4. We are now prepared to calculate the values of a and 5.
Let Rk be the sum of the radii of an atom and an electron, 7 the
number of atoms in unit space, and let us in the first place con-
fine ourselves to encounters of a definite kind. I shall suppose that
in these the line joining the centres falls within a cone of the infinitely
small solid angle dw.

Taking as axis of this cone one of the straight lines that may be
drawn in it, and denoting by & the acute angle between the axis
and the direction of motion of the group (6), I find for the number
of electrons in this group undergoing an encounter of the kind chosen,

DL (Sy ACO ONC CLD 5 ny 5 (sh)
per unit time, a result which leads to the value
DNA IAN EP oo oe co. 0 os (OO)

if we take into account a// encounters, whatever be the direction of
the line joining the centres.

Now, if we ascribe to a metallic atom so large a mass, that it
is not sensibly put in motion by an electron flying against it, the
velocity of the latter after the encounter is given by a very simple
rule. We have only to decompose the initial velocity into one

1) See Lorentz, Les équations du mouvement des gaz et la propagation du
son suivant la théorie cinétique des gaz, Arch. néerl. 16, p. 9.

( 443 )

component along the line of the centres and another perpendicular
to it; the latter of these components will remain unchanged and the
former will have its direction reversed.

In applying this to the encounters of the particular kind specified
at the beginning of this §, we may take for all of them the line of
centres to coincide with the axis of the cone dw. Our conclusion
may therefore be expressed as follows: Let V be a plane through
the origin in the diagram of velocities, perpendicular to the axis of
the cone. Then, the veloeity-point of the electron after impact will
be the geometrical image of the original point with respect to this
plane. It is thus seen that all electrons whose velocity-points before
the encounters are found in the element J2 will afterwards have
their representative points in d2,, the image of dà with respect to
the plane V.

By this it becomes also clear, in what way the number 5 ean
be calculated; indeed, in encounters taking place under the circum-
stances considered, velocity-points may as well jump from di, to
dk as from dà to dj,. The number of cases in which the first takes
place is found from (8), if in this expression we replace §, 4, § by
the coordinates §', 7,6 of the image of the point (&, 7, $) with respect
to the plane V. It is to be remarked that the factor 7 cos 9 d 2 may
be left unchanged, because the lines drawn from the origin of the
diagram to the points (6, 7,5) and (§', 7), 6’) have equal lengths and
are equally inclined to the axis of the cone. Also d’,=d4. The
increase per unit volume of the number of electrons in the group
(6), insofar as it is due to encounters in which the line of centres
lies within the cone da, is thus found to be

n I? fF (S', nj, 8) r cos Hdd
and, in order to find 6, it remains only to divide this by da and to
integrate with respect to all cones that have to be taken into account.

Using the formula (8) we may as well calculate directly the
difference /—a. By this the equation (7) becomes

n R? „fire. n,8) — 7 (En, $)} cos Bd w =

Oe Oene wf òf

SE ya : eae Ee PRT ;
We must now express §',7/,6' in §, 7,6. Let Jig, be the angles
between the axes of coordinates and the axis of the cone dw, this
last line being taken in such a direction that it makes the acute
angle 9 with the velocity (§, 7, 5). Then

'

(10)

=§ — 2r cos dos f, n= — 2rcos Heosg, § =§ — Arcos Heos h, (11)

( 444 )

These formulae show that, as we know already, the magnitude of
the velocity (S, 1,5), which I shall call 7’, is equal to the magnitude
r of the velocity (6, 7, 5).

As to the integration in (10), it may be understood to extend to
the half of a sphere. Indeed, if in the diagram of velocities, we
describe a sphere with centre O and radius 1, and if P and Q are
the points of this surface, corresponding to the directions (&, 4, 6) and
(f,g,h), we must give to the point Q all positions in which its
spherical distance from P is less than $a. For dw we may take a
surface-element situated at the point Q.

§ 5. At the time ¢ and the point (a, y,2) the metal will have a
certain temperature 7’ and the number N, the concentration of the
swarm of electrons, a definite value.

Now the assumption naturally presents itself, that, if 7 and NV
had these values continually and in all points, the different velocities
would be distributed according to Maxwerr’s law

PE n= Ae * EEE)

Here, the constants A and / are related to the number NV and the

mean square of velocity 7»? in the following way

he
A=N —, . EEn
a

Since 4m7?= aT, the latter relation may also be put in the form
3m
h en =
4aT
It appears from this that the way in which the phenomena depend
on the temperature will be known as soon as we have learned in
what way they depend on the value of /.

ee (i)

§ 6. The function / takes a less simple form if the state of the
metal changes from point to point, so that A and / are functions of
x,y,z. In this case we shall put

FE n= Ar En
where p is a function that has yet to be determined by means
of the equation (10). We shall take for granted, and it will be eon-
firmed by our result, that the value of p(&, 1,5) is very small in
comparison with that of Ae#”. In virtue of this, we may neglect
the terms depending on ¢ (§, 7,6) in the second member of (10), this

( 445 )

having already a value different from 0, if we put #—= Ae#*. For
a stationary state and for the case of the bar mentioned in § 1, the
member in question becomes
dA dh
(— Qh AX 4 "A ) gee Eee, whee STE)

da dx

As to the left hand side of the equation (10), it would become 0,
if we were to substitute f — dek“. Here, we must therefore use
the complete value (15), the deviation from Maxwerr's law being
precisely the means by which this member may be made to become
equal to (16).

The occurrence of the factor § in this last expression makes it
probable that the same factor will also appear in the function p. We
shall therefore try to satisfy our equation by putting

PSN Gan) ce ee ee sy (LZ)

This leads to

F (Ss M$) = Ae” + $x (r)
and
Ff. 4, $) =A e-tr? +8 y (r)),
consequently, since 7’ =r, if we use (11),
£63 1',5) —f 6, 1, 5) = — 2 r cos B cos f x (r),
so that the first member of (10) becomes

a

—2nk ry foo GROND a a va UE)

Denoting by w the angle between the velocity (&,n, 5), i.e. the
line OP, and the axis of z, and by wp the angle between the planes
QOP and X OP, I find for (18)

„

Qa :
— 2nR*r* y (7) foe D (cos B cos u + sin D sin u cos Wy) sin Hd Hd =
9

©

== rt (7) cos waren RIE r (5):
If this is equated to (16), the factor § disappears, so that y (7) may
really be determined as a function of 7. Finally, putting
1
an Rt
we draw from (15) and (17)
Ennn Aten eee i( 2 AX — = + 9 1) 2 e—hr? | (20)

da dre) r

LEE Ve ast ive eneen (0)

T must add that, as is easily deduced from (9), the quantity /
defined by (19) may be called the mean length of the free paths of
dA

the electrons, and that, in the equation (20), the terms in
Lv

and

( 446 )

dh
— are very small in comparison with A e—*, provided only the state
av

of the metal differ very little in two points whose mutual distance
is /. This is seen by remarking that the ratios of the terms in
question to Ae—”* are of the order of magnitude
dA
l
da =e dh

atl fp
4 at

= 1
and that, in the second of these expressions, 7? is of the same order as re:

If the term in (20) which contains X, is likewise divided by
Ae-*”, we get
2hl X.

Now, 2/ X is the square of the velocity an electron would acquire
if, without having an initial motion, it were acted on by the external
foree m X over a distance /. If this veiocity is very small as com-
pared with that of the heat-motion, the term in Y in our equation
may also be taken to be much smaller than the term A e#.

It appears in this way that there are many cases in which, as
we have done, the function p (S, 1,5) may be neglected in the second
member of the equation (7).

The above reasoning would not hold however, if, in the case of

two metals im contact with one another, there were a real discon-
tinuity at the surface of separation. In order to avoid this difficulty,
I shall suppose the bodies to be separated by a layer in which the
properties gradually change. I shall further assume that the thick-
ness of this layer is many times larger than the length /, and that
the forces existing in the layer can give to an electron that is initially
at rest, a velocity comparable with that of the heat-motion, only
if they act over a distance of the same order of magnitude as the
thickness. Then, the last terms in (20) are again very small in com-
parison with the first.

As yet, a theory of the kind here developed cannot show that
the values we shall find for certain quantities relating to the contact
of two metals (difference of potential and Prxtimre-effect) would still
hold in the limit, if the thickness of the layer of transition were
indefinitely diminished. This may, however, be inferred from thermo-
dynamical considerations.

§ 7. Having found in (20) the law of distribution of the veloci-

( 447 )

ties"), we are in a position to calculate the quantities vand W (§ 1)
with which we are principally concerned. If the value (20) is sub-
stituted in (4) and (5), the term A e#” leads to an integral containing
the factor §; this integral vanishes, if taken over the full extent of

o2

the diagram of velocities. In the remaining integrals the factor 8

7 Ee ]
occurs ; these are easily found, if we replace §* by at the element

dà by 4277 dr, and if then we integrate from r=0O to r =o.
Taking 7? = s as a new variable, we are led to the integrals

ie}

aoe ao)
|: ehsds, fe eAsds and fs ehs ds,
0

0 0

whose values are

Finally, the “stream of electrons’ and the flux of heat are given by

ee Mee lie (2 ae =) ae ae |
3 h° da hs dx

Aen E (2 hAX a) mie =| ey)
3 1 Ge) Se hide

These are the equations that will be used in all that follows.
For the sake of generality, I shall suppose (though, of course, this
is not strictly true) that, if only a proper value be assigned to /,
the formulae may still be applied even if we make other assumptions

concerning the metallic atoms and their action on the electrons. From
this point of view, we may also admit the possibility of different
kinds of electrons, if such there are, having unequal mean lengths
of free paths, and of, for each kind, / varying with the temperature.

‘Provisionally, we shall have to do with only one kind of electrons,
reserving the discussion of the more general case for a future com-
munication.

§ 8 From the equation (21) we may in the first place deduce a
formula for the electric conductivity 6 of the metal.

Let a homogeneous bar, which is kept in all its parts at the same
temperature, be acted on by an electric force # in the direction of
its length. Then, the force on each electron being e 4, we have to put

1) It may be observed that, as must be the case, the value (20) gives N for

: 33 B
the number of electrons per unit volume and B) for the mean square of velocity.
aly

( 448 )

. @ly
yee
m
Also,
dA dh
== (and) ——— (0)
Av (a
so that (21) becomes
4 cl. le
p= Sy.
hm

Multiplying this by e, we find an expression for the electric current
per unit area, and in order to find the coefficient of conductivity,
we must finally divide by /. The result is

Oe ee te ien Hee 23
dhm a:
or, taking into account the relations (18) and (14) and denoting by

€

u a velocity whose square is the mean square Oh of the velocity of
L

2 Neu
6 == —_ . —— o A ° . ° e (24)
Bh gd

Drupr gives the value

heat-motion,

§ 9. The determination of the coefficient of conductivity for heat,
which we shall call & (expressing quantities of heat in mechanical
units) is rather more difficult. This is due to the circumstance that,
if initially N—O, the equation (21) implies the existence of an
electric current in a bar whose parts are unequally heated. This
current will produce a certain distribution of electric charges and
will ultimately cease if the metal is surrounded on all sides by non-
conductors. The final state will be reached when the difference of
potential and the electric force arising from the charges have increased
to such a degree that everywhere » = 0.

Since it is this final state, with which one has to do in experiments
on the conduction of heat, we shall calculate the flux of heat in the
assumption that it has been established.

In the first place we have then by (21), putting » = 0,

A A dh
2h AX = 2

da Ch de

and next, substituting this in (22) and again using the formula (14),

( 449 )

Shida:
Consequently, the coefficient of conductivity has the value
8alAa

— EA Lea Geena MoS (5)
Qh? (28)

8 EN
rei Finan, add)

Drupe’s result for this case is

or

1
k==I Nau.
3

The ratio of the two conductivities is by my formulae

k 8 z

w=5(*)r- vir SEREN
0 9 \e

k >(<)

—==| =| 7.

0 3 \e

Here again, the difference between the two formulae consists
merely in the numerical coefficients.

and by those of Drupe

k
Just like Drepe we may therefore conclude that the value of —
o

does not depend on the nature of the metal and that it varies pro-
portionately to the absolute temperature, consequences that have been
verified with a certain approximation in the case of many metals.

It need hardly be observed that these conclusions could only be
arrived at because we have neglected the mutual encounters between
electrons *). Im fact, these would tend to diminish the conductivity
for heat, but not that for electricity, since they cannot have an
influence in a phenomenon in which all electrons move in the same
way. It is clear that, under these circumstances, a value of “ inde-
pendent of the nature of the metal could hardly be expected.

Let us next consider the absolute values.

ia al

The value of that can be deduced from those of / and o and

é

for which, using (28), I find

yp 9k,
eee De shat ie BEES)
e 86

1) See Tuomson, l.c., p. 146.

Proceedings Royal Acad. Amsterdam. Vol. VII.

( 450 )

may be compared, as has been observed by Drupr and REINGANUM '),
to a value of the same expression that is obtained from other data.
I shall suppose that the charge e of an electron is equal to that of
an ion of hydrogen in an electrolytic solution and I shall represent
by p the pressure that would be exerted, at the temperature 7, by
gaseous hydrogen, if a unit of volume contained one electrochemical
equivalent. Then
al

— =p.

The proof of this formula is as follows. We may write for the

: ; 1
number of atoms in unit volume of the gas considered — for the
e

1
number of molecules =, and, since the mean kinetic energy of a
ze

rp

. Asis well

molecule amounts to «7, for the total kinetic energy

ae

known, the numerical value of the pressure per unit area is two
thirds of this.

Using the C.G.5. system and electromagnetic units, we have for
the electrochemical equivalent of hydrogen 0,000104 and, putting,
T = 273° + 18°,

op == 38.

On the other hand, the measurements of JanGeR and DrrsseLHORST

have given for silver at 18° C.
ke
Den Gyiskr << TOE,

whence, by (29),
al
— = 47.
e
The agreement between the results of the two calculations, for
which the data have been furnished by widely different phenomena,
though not quite satisfactory, is close enough to make us feel con-
fident that Drupr’s theory rests on a sound basis”).

§ 10. We might now return to the formula (25) and, denoting
by @ the electric potential, so that

1) Remeanum, Theoretische Bestimmung des Verhältnisses von Wärme- und
Elektrizitätsleitung der Metalle aus der Drupe’schen Elektronentheorie, Ann. Phys.,
2 (1900), p. 398.

2) A better agreement is found if, instead of (28), we use Drupe’'s formula.

we might deduce from it expressions for the fall of potential in each
point and for the difference of potential between the ends of the bar.

It is more interesting, however, to make a calculation of this kind
for a more general case. Before doing so, we may observe that the
equations (21) and (22) may be applied to a thin curved wire or
bar and that we may as well suppose the normal section slowly to
change from one point to another. The line passing through the
centres of gravity of the normal sections may be called the axis of
the conductor and we shall understand by « the distance from a
fixed point, measured along this axis. We shall also assume that in
all points of one and the same normal section the properties of the
bar and the temperature are the same, but that, generally speaking,
both depend on z, changing from one section to the next. By making
different assumptions in this respect, we come to consider circuits
of different kinds, composed of one or more metals and with any
distribution of temperature we like.

For the sake of generality we shall introduce the notion of
“molecular” forces of one kind or another exerted by the atoms of
the metal on the electrons and producing for each electron a resulting
force along the circuit in all points where the metal is not homo-
geneous. Actions of this nature have been imagined long ago by
HermnorLtz for the purpose of explaining the phenomena of contact-
electricity. We may judge of their effect in the simplest way by
introducing the corresponding potential energy WV of an electron
relatively to the metallic atoms. This quantity, variable with »
wherever the metal is not homogeneous, will be a constant in any
homogeneous part of the circuit; we shall suppose this even to be
so in case such a part is not uniformly heated. If, as before, we
write p for the electric potential, the force X divides into two parts

X = Xn + Xe,
1 dV en

We shall now consider an open circuit, calling the ends Pand Q,
and reckoning wv from the former end towards the latter. Putting in
(21) vr =O and attending to (30), we obtain for the stationary state

dp 1dV md ( -) m dlog A

ite eda eo dc Nh 2eh dx

be Bue et)

whence by integration
31*

( 452)

1 Vv Vv te m 1 1
Pa ap = ye Pe Q e \ Ap ha

Q

ee lee
| a de EE

Ze h dea

P
a formula which may now be applied to some particular cases.

a. Let all parts of the circuit be kept at the same temperature.
Then, h is a constant, and

1 m
va =~ (Ye— Va) tale Ap — Ay) . (33)

The potential-difference will now have a positive or negative value,
if the ends of the circuit are made of different metals. It appears
in this way that the differences that have been observed in this case
may be attributed either to an inequality of Vp and Vg, i.e. to
“molecular” forces acting at the places of junction (HELMHOLTZ), or
to an inequality of Ap and Ag, i.e. to a difference in the “con-
centrations” proper to the metals (Drupr).

It need hardly be added that (83) becomes 0 whenever the ends
are made of the same metal and that the law expressed in Vouta’s

tension-series is implied by the equation.

b. Let the metal be the same everywhere. Then A is a function
of h and (32) will always be 0, if the ends P and Q are kept at
the same temperature, whatever be the distribution of temperature

in the intermediate parts.

c. Let us next examine the potential-difference between the ends
of an open thermo-electric circuit, a difference that may be regarded
as the measure for the electromotive force F existing in it. Starting
from P and proceeding towards Q, the state of things I shall
consider is as follows: 1s" Between P and a section F’', the metal /
maintained at a temperature varying from 7'p to 7" in R'. 2"d Between
R' and S', a gradual transition ($ 6) from the metal / to the metal //,
at the uniform temperature 7”. 3rd From S' to S”, the metal ZZ with
temperatures varying from 7” to 7”. 4 Between S" and R", a
gradual transition from the metal // to the metal /, the temperature
being 7” in every point of this part of the circuit. 5 Finally, between
R" and Q, the metal / with a temperature changing from 7” to
Tq= Tp. It being here implied that the ends of the circuit consist
of the same metal and have the same temperature, the equation (32)
reduces to the last term, and we find, after integration by parts,

e

(433 )

Q
FN gta 34
rd wal) ED
JP

This integral may be divided into five parts, corresponding to the
above parts of the circuit.

Distinguishing by appropriate indices the different values of hand A
that have to be considered and keeping in mind that / is a constant
both in the second and the fourth part, we have

IS, SEL
d fl iS d fl
loge dai; log A—| — Jd = 0,
“ da\h Se aa Na
feo Se
Re Q kt

log A DE 1, log A gl te ] 1 A afs dh
Og £ As 7a at + Og £ EE nl ab == 0g Ith A ills
P R” h”

h and h" being the values corresponding to 7 and 7”, the tempe-
ratures in #' and FR". Similarly

"

fees az te = fi an 5(; at.
Ss’

If we combine these results, re formula (34) for the electromotive
force becomes

h"
m Aged
Sa phe ae
2e “ Aqzh
h'
, if we use (13) and (14),
7 Add
2a ( Nu
Te lo CL sams. oe leden (35
los (35)
Geodesy. — “The connection between the primary triangulation of

South-Sumatra and that of the West Coast of Sumatra.” By
Mr. S. Brox. (Communicated by Prof. J. A. C. OupEMmans).

I. Short description of the triangulations of South-Sumatra
and the West Coast of Sumatra’).

Towards the end of 1896 the measurements for the primary
triangulation, which will serve as a basis for the topographical sur-

1) For a more detailed description I refer to the papers of Dr. J. J. A. Mutter,
occurring in the proceedings of the International Geodetic Association of 1892,
1896 and 1903.

( 454 )

vey of South-Sumatra, were begun at the station Langeiland P68.
These measurements were carried from the West Coast of Java over
the Strait of Sunda and are lately completed at the station B' Gadang
P39, situated in the Government of the West Coast of Sumatra.

The triangulation consists of one continuous chain of triangles,
which, beginning at the side Langeiland ? 68 — Gs Radja Basa P 67,
is connected with the side Gs Talang P38 — Bt Gadang P 39 of
the triangulation of the West Coast of Sumatra.

It is true that this side does not exceed the length of 17120 meters,
but a connection with the longer side Bt Poenggoeng Parang P45 —
Gs Talang P38 had to be abandoned after it was found that the pillar,
erected at Bt Poenggoeng Parang during the triangulation of the West
Coast, was so damaged that it no longer could be used for this purpose.

The experience made during the measurements of the base at
Padang for the triangulation of the West Coast of Sumatra, executed
by means of a 20 meters steel tape, did not tempt us to measure
also the base line for South Sumatra with this comparatively unreli-
able apparatus; and as an instrument admitting of a high degree of
accuracy was not available, no special base was measured, but the
length of the first side of the chain was based upon the two sides
Batoe Hideung P 15 — Gs Karang P35 and Gs Karang P35 —
x3 Gede P36 of the Java triangulation. For the Java triangulation
3 base lines had been measured with an apparatus of RersoLp, which
had been sent back to Europe in 1882.

For the orientation of the South-Sumatra chain, determinations of
latitude and azimuth were made at the station Gf Dempoe P 71 in
the Lampong Districts in 1897. The geographical longitudes were
reckoned from the meridian of 3°15’ West of Batavia. This meridian,
which nearly passes over the middle of South-Sumatra, is deter-
mined by the geographical longitude of the Java station Gs Karang
P35, as given in Abtheilung V der Triangulation von Java, p. 207.

To obtain a zero mark for the determinations of altitude, tidal obser-
vations were made during a year at Telok Betong in 1897 and 1898.
From these the mean height of the sea level in Lampong Bay, the
Lampong-zero, was derived. This was transferred to the pillar 71559
at Telok Betong by levelling, and thence by reciprocal but not
simultaneous zenith distances measurements to the primary point
Gs Betoeng P 70°).

1) In 1902 and 1903 tidal observations were also made at Benkoelen and from
them the Benkoelen-zero (the mean height of the sea level at that place) was
derived, which wili be used afterwards, when the secondary measurements will
be so far advanced.

( 455 )

With respect to the triangulation of the West coast, I have remar-
ked above that the steel tape, with which a base line near Padang
of 4860 M. was measured in 1883, did not admit of a high degree
of accuracy. The length of the steel tape was determined before
and after the operation by measuring with it under the necessary
precautions a line of 200 M., of which the true length was accurately
known from measurements with the base apparatus of RePsoLD.

Determinations of latitude and azimuth for the orientation of the
chain were made at the West end of the base in 1885 *).

The geographical longitudes were reckoned from the meridian of
Padang, which passes through the West end of the base, for which
meridian 6° 26’ 42’ West of Batavia has been preliminarily accepted,
a difference in longitude formerly determined by chronometers.

As zero mark for the altitudes was taken the Padang-zero, the
mean sea level at Padang, formerly determined by observations
during some months of 1874 *).

For the astronomical determinations, the measurements of the
horizontal angles and those of the altitudes, the 10-inch Universal
instruments of Piston and Martins and of WeGENER were used in
both triangulations.

The telescopes of these instruments are placed excentrically ; each
circle is read with two micrometer microscopes.

With the exception of the Padang base-net, where directions
were measured, the triangulation was made according to SCHREIBER’s
method; the measurements of all combinations of angles were repeated
so often that the weight of a direction adjusted at the station was
about 24, the weight of one observation of a direction being adopted
as unit.

For the trigonometrical determinations of altitude, reciprocal but
not simultaneous measurements were made; at each station, whenever
possible, 6 zenith distances were measured for each point, under
conditions as favourable as possible. With the exception of the first
measurements on the West Coast, where signals were employed,
all observations were taken on heliotropes.

As to the adjustments and computations I remark that, for the
South-Sumatra chain, exclusive of the connecting pentagon with Java,
which was adjusted according to the method of least squares, the

1) In 1896 determinations of azimuth and latitude were also made at the station
Tor Batoe na Goelang, P 62.
2) In 1889 the mean sea level at Siboga, about 350 kilometers off Padang,

was determined by tidal observations; the connection of the two marks showed a
difference of 0,85 M.

( 456 )

adjustment was effected by equally distributing the error of closure of
each triangle over the 3 angles. The computation was made in a
plane by transference by means of a Mercator’s projection according
to the method of ScHors.

The adjustments of the triangulation of the West Coast of Sumatra
were made in portions; only for the base-net and for the Northern
part the method of least squares was applied; in most cases an
approximation method was used. The computations were made on
the ellipsoid.

The following remarks may be useful for a judgment of the
accuracy which may be expected in the different connections.

(1). The distance between the base of Simplak, on which the
triangulation of South-Sumatra rests, and that of Padang is about
850 kilometers; the least number of triangles, necessary for the
transference of the length of the side Poetri-Dago of the Simplak base-
net to the first side G3 Gadoet P1 — Poelau Satoe P2 of the
triangulation of the West Coast of Sumatra, is 49.

(2). The distance between the stations Gs Dempoe and the West
extremity of the base at Padang, used for the orientation of the net,
is about 700 kilometers; the least number of triangles, by means of
which the azimuth of the line GS Dempoe — G+ Tenggamoes can
be transferred to the first side of the triangulation of theWest Coast, is 40.

(3). The distance between Telok Betong and Padang, where the
tidal observations were made, is over 700 kilometers; the least number
of steps, necessary for the transference of the altitude of the pillar
at Telok Betong to the zero mark of Padang, is 24.

The difference between the two values of the logarithm of the
length of the connecting side is expressed in units of the 7
decimal and corresponds to about a of the length of the side
or to 43.2 mms. per kilometer.

The differences found are comparatively so small that their origin
may be easily explained by the accumulation of errors of observation
and by the irregularities of refraction. The difference between the
values found for the latitudes does not indicate a local deviation
at GS Dempoe with respect to Padang.

For the length of the connecting side a better result might have
been expected, if for the base measurement at Padang a more
suitable apparatus had been available.

For the rest, the differences are such that for the purpose of the
triangulation, namely, to afford a basis for the topographical work,
they do not come into consideration.

( 457 )

IJ. Mean Errors.

7 : South- | WEE]
Nature of the errors. : __ | Coast of Remarks.
| Sumatra |
| Sumatra |
In the determination of the | | |
geographical latitude | 1) 0.21 0".35 | (1) For the orientation of
In the determination of the{ “| ; | | the nets.
azimuth of a night signal | OEZ OL
(2) Determination derived
In the azimuth of the 1st side, in from 37 closing errors
so far as they arise from errors | 4 (see appendix).
of the base-net. | = 0.85

| (3) Determination derived
In the angles adjusted at the from 73 closing errors.
station (weight 12) ;

(4) If we consider only the
a from the results of the ad- | 10 triangles’), which
justments at the station; 0.34 0.52 in the shortest way
| (4) connect the first side
b from the errors of closure of | with the connecting side,
the triangles according to the we find: @0."59, 4 0".86,
formula : | |
= | | i (3)
m= vier) | 07.64 | 0.96
ve (2) | (4)

Ill. Differences found in the adjustments.

South West Coast | Diffe-

Nature of the determination. meant
rhea REWER TRE JD Sumatra | of Sumatra rence,
Logarithm of the connecting 4,2335135.7 | 42334918 0 87.7
side _ |
Length » > > 17120.39 M | \7119.65 M. |0.74M.
| |
Azimuth » » » | 24799618 '.07 24799613"A3 |4!'. 94

Geogr. latitude Gs. Talang 9°6'9". 312 S | 281.699 S. |0'.613

|
» » Bt. Gadang
2°9'43" 1655 | 29949555 S. |0” .610
Altitude above the sea level | 1375.5 M 1376.7 M. 1.2 M.
Gg. Talang
) ) » Bt. Gadang 281.8 M. | 2841 M. 2.3 M.

| |

From the triangulation of South Sumatra we derive:
the geographical longitude of Gz Talang 5° 32’ 48/',525 | West of
cs Hi se ,, Bt Gadang 5° 41’ 20’7,236 \ Batavia
and from that of the West Coast of Sumatra:

*) These triangles occur under the numbers 1, 2, 16, 17, 35, 36, 43, 50, 51

and 52 on pp. 603 and 604 of Comptes Rendus des séances de la dixiéme conférence
générale de l'Association Géodésique Internationale.

( 458 )

Appendix.

Errors of closure of the triangles in the South Sumatra chain.

2 4 =f Closing ee:
> ED = 28| for each
Hig Nineteen gE tale) a
Zi En
1 | Pistor and Martins, 27 cm. AD WU | 0.44) — | 0.1936
2 | ) 4" 12 (24.25, 41.82 | — | 3.314
3 | > 4" 12 |94.95|) 0.44) — | 0.1936
4 | » 4! (2 |94.95) — | 0.63 | 0.3969
5 » WENDE SN OR Oat
6 | Wegener, Pistor and Martins, 27 cm.) 2”.4"| 2 | 24.25) — | 0.65 | 0.4225
il » » Av) 2 |24.25) 0.58 | — | 0.3364
8 » » QA") 2 24,5) — | 0.93 | 0.8649
9 » » | 2.4") 2 | 24.95) 0.21 | — | 0.0444
10 » » UNK, 24 — | 1.74 | 3.0276
14 » » ee le 24 — | 0.29 | 0.0841
42 » » QA 2 24 | 0.04!) — | 0.0016
13 ) ) Qn AN) 2 24 | 41.20} — | 1.4400
14 | Pistor and Martins, 27 cm. URS | 24 — | 0.73 | 0.5329
15 | Pistor and Martins, Wegener, 27 cm.) 1”.2") 2 24 — | 41.33 | 4.7689
46 » » A 2, 24 14.93) — | 3.7249
47 > N | avon) 2) OR — AAN RL
18 > » | 4".9r] 2 | 4 | — | 4.76 | 3.0976
19 | Pistor and Martins, 27 em. AOE Tiley MU | 2.36} — | 5.5696
20 » dir 2 24. — | 1.70 | 2.8900
21 » Wegener, 27 cm. 4" 2") 2 Dh — | 1.06 | 4.4236
22 > > 4.9" 2 | 2% | 0.388) — | 0.4444
23 | Wegener, 27 cm. AD 2 24 — | 0.03 | 0.0009
24 » Ye alle) 24 — | 4.44 | 4.9881
25 » Ue 2 24 — | 1.36 | 1.8496
26 » U | 2 24 — | 0.66 | 0.4356
27 » Oe 24 — | 0.55 | 0.3025
98 » U | 2 24.25 1.83 | — | 3.3489
29 » Qu | 2) 24°25) — |) 405) Aeros
30 » 2 | 2 124.95) — | 0.42 | 0.0144
31 » Qin | 2 124.95) 0.75 | — || OFS605
32 » en 2 24 — | 0.56 | 0.3136
33 » ar | 2 24 {4.42} — | 4.2544
34 » u | 2 24 | 0.81 | — | 0.6564
35 » Ee) 24 |1.56 | — | 2.4336
36 » De 02, 24 | 0.40) — | 0.1600
37 » DA EIER) 24 | 0.43 | — | 0.1849

S. BLOK: The connection bety

ele
Sed Satay

1/25 2h
WE Deddo AS tal

2g
BS Lang sai
Connection with
the first side of
the WestCoast

of Sumatra

1a Gjarater

$5 Rady dodou

EN y
Wes

S. BLOK: The connection between the primary triangulation of South-Sumatra and that of the West Coast of Sumatra.

Primary

Triangulation
of

South Sumatra.

Scale 1:1250,000.

a
Shuliing Sadaet
4

VEEN

0 fuebvel Sontang

Connection with

the first side at

the WestCoast
of Sumatra

Connection with the
Javanet.

epee — Satara h
Pre qoereg

Proceedings Royal Acad. Amsterdam. Vol. VII.

( 459 )

the geographical longitude of G= Talang 0° 53’ 37,833 ) East of
ie 4 2 , Bt Gadang 0°45’ 6’’,151 \ Padang,
whence for the longitude of the West end of the base of Padang
respectively : 6° 26’ 26/7,858 and 6°26’ 26'’,387, of which the mean
value is 6° 26’ 26’’,373 West of Batavia, corresponding to
100° 22’ 10//,68 East of Greenwich.

Hence follows that the difference in longitude between Padang and
Batavia, as determined by geodetic measures, is less by 16’’ or over
1s than that found by chronometers.

Probably the difference is due for a small part only to the accu-
mulation of errors of observation in the triangulation and almost
exclusively to the method of determination by means of chronometers.

Physics. — “On the melting of floating ice’, by Dr. Cu. M. van
Deventer. (Communicated by Prof. van DER WAALS).

In what follows I shall discuss a physical fact, which though one
of the simplest and most important of phenomena, seems to have
escaped the attention of physicists up to now. The author asked at
least some twenty men versed in physics after it, and not one of
them had heard about it: many of them and specially the most
experienced in this branch of science were not a little astonished
at it. It is therefore not devoid of interest to discuss the faet in
question, though the explanation can be followed even by beginners
in physics.

§ 1. In order to show how surprising the fact is, we put the
following

Problem.

Given a tray of a certain dimension, in which water is up to a
certain Jevel, and in which floats a piece of ice of a certain weight,
everything at O°. Required to find: in what way will the level of
the water be changed, when the ice melts?

Solution : the level of the water does not change.

$ 2. This answer may be derived as a simple application of
the law of ARCHIMEDEs. ‘)

If the piece of ice weighs A kg., the upward pressure is also
A kg., and so the weight of the displaced water also A kg. As now
the melted ice weighs also A kg., the melted mass will occupy

1) The weight of the air is neglected in this discussion.

( 460 )

exactly the place of the immerged part, and accordingly leave the
position of the surrounding water intact.

In short: the ice when melting contracts into the volume of the
immerged piece.

§ 3. A more elaborate, but perhaps more graphical demonstration
is the following.

Let the piece of ice have the volume of A liter. Then the volume
of the free part is 83 A cm. and of the immerged volume 917
A cm. The immerged part gives, when melting, 841 A em. of water,
the free piece 76 A cm. of water. The water of the free piece supplies
therefore what the immerged piece had lost in volume when melting,
and there is no reason for change in the level of the surrounding
water.

In this the specific gravity of ice is put at 0,917.

§ 4. It is obvious, that the same reasoning applies to other sub-
stances, so that the following general rule may be drawn up : when
a substance, floating in its own melting-liquid, melts, the level of
the liquid will not change.

$ 5. An application of everyday interest is this: if a glass is
filled to the brim with water, in which ice floats, the water will
not flow over when the ice melts.

We should, however, take care, when making the experiment with
a full glass, not to mistake water that is condensed on the cold
outside wall, and runs down, for water flowing over. A better proof
is furnished by a glass which is not quite full of water, and on
which the first level is indicated: after melting we must find the
same level.

§ 6. Attention may be called to the fact that not only after, but
also during the melting the level is the same as before.

For if one gramme of ice (or a given part of it) melts and gives
one cm. (or an equally large part of it) of water, the weight of the
floating piece and so also the upward pressure will be diminished
by one gramme (or an equally large part of it), and consequently
the volume of the immerged piece will be decreased by one em. (or
an equally large part of it). For the additional water room has been
made by the decrease in displacement.

§ 7. The law of the permanent level holds also when the floating
ice has empty cavities.

( 461 )

This is obvious for cavities which are in the piece rising above the
surface, as these cavities have no influence on the upward pressure.

If the inrmerged piece has a cavity of A em, the upward pressure
is equally large as for a solid piece of ice of the same weight, but
there are A grammes of ice more above the water. When melting,
these A grammes of ice form A grammes of water, just sufficient
for filling up this cavity.

The law of the permanent level holds also when the ice contains
air bubbles, at least by the same degree of approximation, with
which we may neglect the weight of the air.

§ 8. When fresh water ice floats and melts in salt water, the
level does rise, though slightly, the immerged part now being smaller
than before, and so the melted ice cannot be contained in the volume
of the immerged part.

Here and in what follows the change of volume, caused by the
mixing of salt water and fresh water, is neglected, which is certainly
permissible when the proportion of the salt is slight.

For A liters of ice, which weigh 917 A grammes and float in
salt water of 1,03 specific gravity, the volume of the immerged piece
is 890 A cm.; the available space can therefore hold 890 A em. of
the melted water, but the remaining 27 A em. raise the level.

This remaining part is about one fourth of the piece which rises
above the surface of the water (110 A em).

§ 9. If in salt water a piece of one liter of ice floats, which has
a cavity under water of B em, then there are (1000-B) em. of solid
ice of a weight of 0,917(1000-4) grammes.

The upward pressure is therefore 0,917 (1000-8) gramme, and
with a specific gravity of the salt water of 1.03, the immerged
volume is 0,89(1000-B) em. When melting, we get 0,917(1000-B) em.
and so there is a surplus of 0,027(1000-B) em. of water to the
volume yielded by the immerged piece, which raises the level.

The piece of ice rising above the surface was 1000-0,89(1000-B) em.
or (110 + 0,894) cm, and the ratio of the remaining piece mentioned

27—0,027 B
The smaller B is, the more this relation approaches to about
a fourth.

B
to this volume is as one to (41 + ure

§ 10. History. A remark made two years ago by a pupil of the
third year of the “Gymnasium Willem III” at Batavia to the writer

( 462 )

suggested this paper. This pupil, called van Erprcum, said that he
had observed that a glass filled to the brim with water and floating
ice, does not flow over, when the ice melts.

This fact leading easily to the law of the permanent level and
this law — as the writer is bound to believe — having up to
now escaped the attention of physicists, physical science owes the
discovery of a remarkable fact and the addition of a paragraph to
this pupil.

Amsterdam, Dec. 1904.

Chemistry. — “On _ trinitroveratrol’. By Dr. J. J. BLANKSMA.
(Communicated by Prof. H. W. BaKnuis RoozrBoom).

It has been previously stated *) that the dimethylether of trinitro-
pyrocatechin is formed by the nitration of the dimethylether of 3.5
dinitropyrocatechin. As the nitro-group might have been introduced
either in the position 4 or 6, it was still necessary to ascertain the
constitution of this compound. The substance which melts at 146°—
147° is identical with trinitroveratrol, which has already been des-
eribed by Tremann and Marsmoro ®) and is obtained by nitration of
veratrol (the dimethylether of pyrocatechin) or of veratrie acid.
Tiemann and Matsmoro have shown that veratrie acid on nitration
yields nitroveratrol and nitroveratric acid. Afterwards, ZINCKm and
Francke *) have proved that nitroveratrie acid formed by nitration
of veratric acid has the following constitution:

OCH,
Zo NOCH
xsd

ONS
COOH.

Now, on further nitration with fuming nitric acid this nitroveratric
acid yields trinitroveratrol so that the constitution of trinitrovera-
trol is

OCH,

7 OCH

teu
NO,\ ANO,
NO,.

1) Recueil 28, 114.
2) Ber. 9, 937.
5) Ann. der. Chem. 293, 175.

( 463 )

Dinitroveratrol prepared by nitration of veratrol ') and of meta-

hemipinie acid?) and which is consequently formed as follows:

OCH, OCH,

NOCH, WS OCH,
TU wo /

COOH NG?
also gives on subsequent nitration the same trinitroveratrol, again
showing that the constitution of that substance may be expressed
by C, H(OCH,), (NO,), 1, 2, 3,4, 5.

Now, trinitroveratrol obtained from veratrol is identical with
that from the dimethylether of 3.5 dinitropyrocatechin ; the melting
points of both substances are the same; a mixture of the two sub-
stances shows no lowering of the melting point, whilst the same
reaction products are obtained from both substances by the action
of alcoholic ammonia or methyl-aleoholie sodium methoxide. We
therefore see that in the nitration of 3.5 dinitroveratrol, the nitro-
group is introduced between the two existing nitro-groups.

OCH, OCH,
HAN OCH, as a
NON Ao, NO NO:
NO,

TipMANN and Marsmoro*) have already demonstrated that trini-
troveratrol reacts readily with alcoholic ammonia. As they thought
that the two OCH, groups were replaced by NH,, they have not
been able to identify the product formed in this reaction.

On repeating the experiment, I noticed that ammonium nitrite is
formed so that also one of the NO, groups is replaced by NH,. The
substance formed melts at 247° and is identical with the compound
afterwards obtained by Nimrzkr and KurreNBACHer *) which is formed
by the action of alcoholic ammonia on trinitrohydroquinonedimethy|-
ether.

OCH, OCH, OCH,
7 \OCH, / \NH, JNNO,
aged = Wesel 5 He

NO,\/NO, NO,\ NO, NO,\ NO,

NO, NH, OCH,

1) BRÜGGEMANN, Journ. f. prakt. Chem. (2). 53, 252.

*) Rossin, Monatsh. f. Chem. 12, 491. Heiniscu, ibid. 15, 229,
SPBermo moa ent sin

4) Ber. 25, 282.

( 464 )

This also shows that the NO, groups in trinitroveratrol are situated
in the positions 3, 4 and 5

If this dinitrodiamidoanisol is treated with KOH we obtain the
monomethylether of dinitrotrioxybenzene, a substance already obtained
by Ninrzkr and KurrENBACHER from the said reaction-product of
trinitrohydroquinonedimethylether and ammonia.

In quite an analogous manner the same result was obtained for
the oxyethyl compound :

OCH, OCH, OC,H, OCH,
UN NO! AN C,H J NOCHE
| — 40") —_ | 78° per [122°|
o\ no, NON NO, NON /N0. NO;N/NO,
NO,
J
OC,H OC,H, OC,H,
aS ICH NOCH, ANNES
= DDS = |245°|
7 NO. oe: NO,\ ANO?
NO, NH,

This latter substance has been formerly obtained by NrietzkKr *) by
treating trinitrohydroquinonediethylether with alcoholic ammonia.

Although now the constitution of trinitroveratrol and of trinitro-
pyrocatechindiethylether seemed to be satisfactorily determined, I
have tried to furnish additional evidence by treating these substances
with sodium ethoxide or methoxide ; then it was to be expected that
the following changes might occur:

OCH, OCH, OCH,
DANSEN) JN OCH, /\ OCH,
Ley = | = Mie

NO,\ NO, NO,\ ANO, NO,\ NO,

OCH, OCH, NO,

If now trinitrohydroquinonedimethylether (1) is treated with a
solution of sodium methoxide in absolute methylaleohol the addition
of each drop causes a brownish coloration which nearly instantly
disappears. After a partial evaporation of the solvent, crystals are
formed which melt at 92°; according to an analysis this is the
trimethylether of dinitro-oxyhydroquinone.

OCH,
Qe
|

NON NO:
OCH,

1) Ann. der Chem. 215, 153.

( 465 )

When we treat this substance with alcoholic ammonia two OCH,’s
are readily replaced by NH, and we obtain the same dinitro-
diamidoanisol as that obtained from trinitrohydroquinonedimethy ether.

Trinitroveratrol (LI) however behaves quite differently from Na OCH,
If to the methyl-aleoholie solution is added sodium methoxide a
purple-red coloration is obtained, which only disappears after heating for
a few minutes on the waterbath, after which the liquid turns yellow.
On cooling, fine yellow crystals are deposited (m.p. 152°) which are
not affected by alcoholic ammonia or by potassium hydroxide.

The motherliquor contains besides Na NO, a small quantity of a
substance which is perhaps identical with that from trinitrohydro-
quinonedimethylether.

Fine crystalline compounds are also obtained by the action of
potassium cyanide on trinitroveratrol in alcoholic or methyl-alcoholic
solutions; in either case two different substances are produced.

It is probable that trinitroveratrol (in common with other nitro-
compounds) first forms an additive product with Na OCH, or KCN’),
which then suffers decomposition and causes the formation of the
said products.

The fact that the course of the reaction is a somewhat unusual
one is most likely to be attributed to the presence of three adjacent
nitro-groups in the benzenecore. | hope a further study will throw
some more light on the subject.

AMSTERDAM, Dec. 1904.

Chemistry. — “On W. MaroKwarp’s asymmetric synthesis of optically
active valerie acid.” By Dr. S. TisMstra Bz. (Communicated
by Prof. BAkHuis RoozeBoom).

Some time ago, MarckwaLp’) prepared active valeric acid in a
manner which according to his opinion entitled him to look upon
this synthesis as the first purely asymmetric one. Shortly afterwards
this opinion was challenged in an article from Messrs. Conen and
Parrerson *), who denied that the synthesis could be an asymmetric
one as being opposed to the theory of electrolytic dissociation. After-

1) Lorine Jackson. Amer. Chem. Journ. 29, 89, (1903).

_Losry pe Bruyn. Rec. 23. 47.

2) Ber. 37, 349.

5) Ber. 37, 1012.

Proceedings Royal Acad. Amsterdam. Vol. VII.

( 466 )

wards, Marckwaup *) defended his standpoint in such a manner that
no further controversyhas taken place.

Although the theoretical aspect of the question might be considered
as solved, it still occurred to me that from an experimental point
of view, the synthesis might be capable of some improvement.
MareKkwarp starts from methylethylmalonic acid; of this the acid
brucine salt is made in which now occurs an asymmetric carbon atom in

CH,

“the residue of the methylethylmalonic acid: CO,H .C———CO,H . Br.

Non,

Owine to the influence of the active brucine, the two possible
forms will not be produced in equal quantities and as a transfor-
mation between the two forms is possibly owing to the ionisation,
the solution, on evaporation, will only deposit one salt, as during the
crystallisation the equilibrium between the two forms is constantly
being restored. The active brucine salt is now heated at 170° when
earbon dioxide is eliminated and the brucine salt of methylethyl-
acetic acid is formed. As this elimination of carbon dioxide will
take place exclusively, or nearly so, at the free carboxyl group, the
result will be a brucine salt of active methylethyl acetic or in other
words /-valerie acid. By acidification with dilute sulphuric acid, distil-
lation in steam and rectification, Marckwa.p obtained a product which
in a 10 e.m. tube showed a rotation of [elp = —1°.7 which corre-
sponds with not quite 10°/, of /-valeric acid. Marckwarp attributes
this low yield of active material to the high temperature employed
(170°), which may have caused atomic shiftings.

The problem appeared to me of too great importance not to try
and obtain a better yield of active valeric acid by altering the
modus operandi. The idea struck me that it ought to be possible to
considerably lower the temperature at which carbon dioxide is
eliminated and thus remove one cause of atomic shifting.

In my preliminary experiments 1 used the methylethyimalonic acid
itself which melts at 118° and of which it is stated in the literature
that it rapidly loses its carbon dioxide at 180°. As a rule the acids,
which possess two carboxyl groups attached to one carbon atom,
lose carbon dioxide when heated above their melting point; we rarely
find, however, in the literature cases where this temperature is correctly
indieated and very often, at least in the case of substances melting at
low temperature, the uniform temperature of 170°—180 is accepted.

The methylethylmalonic acid was now heated in vacuo at 130° in

1) Ber. 37, 1368.

( 467 )

a tube connected with a mercury barometer and also with a mer-
cury airpump, which caused a fairly rapid decomposition ; the pressure
rose beyond 1 atmosphere. Even at 100°, decomposition takes place
if we only take care to continually evacuate; the mass first becomes
partially liquid owing to the valerie acid formed and now we can
plainly see the evolution of carbon dioxide from the solid particles
of methylethylmalonie acid still suspended in the liquid. We cannot,
therefore, speak of a definite decomposition point of acids with two
carboxyl groups attached to one carbon atom. The statement that
these substances lose carbon dioxide by heating above their melting
point is consequently not only very vague but to some extent also
incorrect as methylaethylmalonie acid already loses CO, when still
in the solid condition.

Whilst, however, it takes days before the methylethylmalonic acid
is decomposed at 100° at the ordinary pressure, this process is
finished in a few hours if we continually evacuate. This would he
most readily explained by assuming that this decomposition is a
dissociation phenomenon. At each temperature, there would then
exist a definite dissociation tension and if now by a continual
evacuation care is taken that one of the decomposition products is
always being removed, it is obvious that finally all must be decom-
posed. The only difference between this phenomenon and the classical
example of Ca CQO, is this that one has never succeeded in obtaining
an acid with two carboxyl groups by heating an acid, containing
one carboxyl group, in carbon dioxide. This may be explained either
by assuming false equilibria, or by supposing that the velocity of
reunion of the decomposition products is exceedingly small. I intend
to further investigate this point.

As it had now been proved that the temperature of decomposition
of acids with two carboxylgroups to one carbon atom could be greatly
decreased by diminution of pressure, it was obvious that the synthesis
of |-valerie acid might also be improved by allowing the CO,-elimination
to take place in vacuo at least if Marekwarp’s idea was correct
that the bad yield of active material was due to atomic shiftings').

I have now heated the acid brucine salt of methylethylmalonic
acid with continual evacuation at 120°, therefore far below its melting

1) It is easy to understand that a decrease of the temperature at which carbon
dioxide is expelled is in itself not capable of improving the synthesis. The velocity
of the atomic shiftings would no doubt have much diminished but then also the
velocity of the carbon dioxide elimination, and the complete decomposition of the
substance would take a much longer time. The evacuation, therefore, merely serves
to accelerate the decomposition process.

( 468 )

point (155°), and after the whole mass had turned to a thick liquid
and no more carbon dioxide was evolved the product was dissolved
in boiling water. The solution was acidified with sulphurie acid and
distilled in a current steam. The distillate was shaken with ether,
the etheral solution was dried and after the ether was evaporated, the
residual valerie acid was rectified and its boiling point found to be
174°—176°. The rotation of this was determined at [¢|p = —4’.3
which corresponds with 25,8°/, -valeric acid. It made no difference
whether the first or last fraction of the distillate was taken.

The synthesis of /-valeric acid has, therefore, been much improved
and it is possible to still further increase the yield of active acid by
operating at still lower temperature as ] have observed that the acid
salt of methylethylmalonic acid possesses even at 100° a fairly large
decomposition tension.

Amsterdam, Org. Chem. Lab.

Chemistry. — “On the system pyridine and methyl vodide”” By
Dr. A. H. W. Aten. (Communicated by Prof. Bakaurs RoozEBoom).

Among the binary systems which have been studied up to the
present in the gaseous, liquid and solid condition there are many in
which occur chemical compounds formed from the two components.
In most of those cases, those compounds possessed but little stability
so that the conditions of formation and decomposition were situated
within an easily attainable range of temperatures.

In the case of the more stable chemical compounds, however,
those conditions of gradual formation and decomposition are less
easy to attain. Still, their study promises a clearer insight into the
changes which a system undergoes when a chemical compound is
formed therein, and in the systems which form very stable com-
pounds; such a comparison can be made all the more readily at a
lower temperature because the reaction velocities are then generally
so reduced that the system can be studied at will in the presence
or absence of the compound so that these two cases may be compared.

A first example in which this could be at least partially attained
is given by the system pyridine and methyl iodide. These two sub-
stances are capable of forming a quaternary ammonium compound
C,H,N.CH,I which possesses a fairly great stability. At 60° and
higher temperatures this compound is rapidly formed in the mixtures
of the two liquids; at the ordinary temperature this formation takes
place rather slowly and exceedingly slowly on cooling. On cooling

( 469 )-

rapidly, we should therefore undoubtedly get from the liquid mixtures
solid pyridine (m.p. —50 ) and solid methyl iodide (m.p. below —80°).
Moreover, all those liquid mixtures in which no compound has
formed as yet are homogeneous.

If, however, the liquid mixtures are kept for some time, the
compound is formed with a considerable evolution of heat and it
separates at the ordinary temperature in the solid condition, the
amount depending on the temperature and the proportion of the
mixture. At higher temperatures, however, it may cause the formation
of two liquid layers. The peculiar behaviour shown is elucidated in
the annexed figure in which the composition of the mixtures is
expressed in molecule-percents of pyridine.

Let us first glance at the

7/20 5 : D
Ee right side of the figure. In this
‚ 1 is the melting- or solubility-
4 4 L /00

line of the compound, commen-
cing with the melting point of
the compound (117°) and ex-
tending to a _ eutectic point
very close to the melting point
of pyridine, because at lower
temperatures the compound is
but little soluble in pyridine.
ar. AY 61° from dS Dm ol

oe, eee L go°
go"
zo"
bo

JY of pyridine, the line 1 is however
a interrupted as no homogeneous
-10°

liquid can exist between the
-20° two concentrations. The line 2
ineloses with its two branches,

-49 which meet in M, an immisci-
-§°° bility-region which becomes

|S! —t 20" ‘ 10° » 1: HAR
0 0 20 go 40 0 60 ~ 90, 92,100 enlarged at higher temperatures.

EC The fused compound is therefore
miscible with pyridine toa limited extent only. The point M, however,
ean only be reached when there is no separation of solid compound
which may be easily prevented for some time.

At the left side of the figure we meet with nearly the same
series of phenomena: 3 is here the solubility line, 4 the two branches
of the immiscibility line. The immiscibility region is here very great,
at 88° from about 0.5 to 41 mol.°/, of pyridine, whilst no change
could be observed at higher temperatures and consequently no
critical mixing point is known.

( 470 )

The line 5 is the solubility line of a metastable form of the solid
compound; this line, however, can only be partially determined in the
presence of an excess of pyridine. With a large excess the stable
form was formed too readily. If the line could have been continued,
it would have been continuous, in distinction from 1, as it is situated
entirely below the mixing point M.

It could not be determined at the side of the mixtures which are
richer in CH,I as these erystallise very slowly and then we always
obtain the stable form.

The most noteworthy result of this research is, however, that two
liquids which are miscible in all proportions, may yield two sets of
coexisting liquids owing to the formation of a chemical compound.
In the formation of less stable compounds such has never as yet
been observed and the better known stable compounds have not as
yet been studied from this point of view chiefly because the com-
parison of combined and uncombined liquids is so often rendered
difficult by the great differences in the melting points of the components.

The sharp intersection of the melting point lines 1 and 3 at 117°
and the strong elevation of the boiling point after the combination
(pyridine 116°, CH,I 42°, combined liquid 270°) show that even in
the liquid state, the compound is certainly for the greater part undis-
sociated. .

Probably the partial miscibility of this combined liquid with its
components is connected with the fact that the chemical nature of
the compound differs so greatly from those of the components. On
this point also we possess but very little knowledge at present.

Chemistry, — “The reaction of Frepen and Crarts’. By Dr. J.
BörseKEN. (Communicated by Prof. A. F. HorLEMAN).

As is well known, the reaction of FRisppr and Crarts does not
always proceed uniformly. Sometimes traces only of the catalyzer
seem to suffice for the preparation of large quantities of the desired
product; in other cases equimolecular quantities of the products to

to 1’,, mol. of the reagent. Inva

be condensed require from !/, 5

20
great many condensations it has been shown that at least 1 mol.
of AICI], is required in order to obtain the highest possible yield.
The reaction is also dependent on a number of circumstances
which are either connected with a secondary action of aluminium
chloride (Ree. XXII p. 302) on one of the substances present during

the reaction, or else depend on the nature of these substances
themselves.

It strikes me that the number of different syntheses made since
the discovery of the catalytic action of aluminium chloride is. large
enough to enable us to explain the cause of this different conduct
by a somewhat systematic consideration.

It must be well remembered that aluminium chloride can only
then exert its power when it is capable of rendering the chloride
(or anhydride) active; that is to say it must be present in the mass
either in a free or loosely-combined state.

This mass contains besides the catalyzer (and eventually some
diluent, such as CS,): A the chloride (or anhydride), B the benzene
derivative, C the formed product. If now we disregard the above-
mentioned secondary decomposition phenomena the following cases
may occur.:

I. The aluminium chloride combines with zone of these substances
or the compounds are completely dissociated at the reaction-tempe-
rature.

We are then dealing with the catalytic action in the truest form.
A trace of aluminium chloride will suffice to convert unlimited quan-
tities of A and B into C. This is for instance the case in the chlo-
rination (bromination) of benzene at the ordinary temperature. If the
substances used have been carefully dried more than 1 kilo of chloro-
(bromo) benzene may be prepared with the aid of 0.5 gram of
aluminium chloride without a visible diminution of the quantity of
the catalyzer. When preparing diphenylmethane from benzyl chloride
and benzene we can also work with very small quantities of the
catalyzer if the strongly diluted benzyl chloride is poured into a
large excess of benzene and the reaction-mass is from time to time
replaced by new benzene; yet the decomposition of the benzyl
chloride by the catalyzer cannot be entirely prevented (Recueil
XXIII p. 98).

II. The aluminium chloride combines with the chloride (A) to a more
or less strong additive product. When these compounds are very
stable, the reaction may not take place at all: the phosphorus oxy-
chloride combines with strong evolution of heat with a mol. of
aluminium chloride (Cassetmann, Ann. 98 p. 220), and this product
is not attacked by benzene or toluene. In the other cases the reaction
proceeds, however, very favourably. As aluminium chloride also
combines with the formed product (C) only one mol. of the catalyzer
is required for equimolecular quantities of the components.

Here we must still distinguish between the following categories;

( 472.)

a. the catalyzer is situated closely to the place where the conden-
sation takes place, which is the case in all syntheses of ketones,
sulphones ete from the corresponding acid chlorides, where it is
linked to the carbonyl {sulphuryl) group, for instance:

CH,COCIAICI, + C,H, = CH,COC,H, AlCl, + HCl
(Recueil XIX p. 20).

Presumably, this ought to include the syntheses of GATTERMANN,
B 1897 p. 1622, where the aromatic aldehydes are constructed from
CO and HCl and the amides of the aromatic carboxylic acids are
‘obtained from carbamine chloride (Cl CO NH,) B. 1899 p. 1117.

4. the aluminium chloride is combined to the chloride but not
near the place where the condensation occurs, for instance :

AICI,p.NO,C,H,CH,Cl + C,H, = AlCl,p.NO,C,H,CH,C,H, + HCI

(Recueil XXIII p. 103),
the catalyzer is here combined to the nitro-group.

UL The aluminium chloride combines with the benzene derivative
(B) and not or with great difficulty, with the chloride (A).

In this case, the benzene-group which has combined with the
catalyzer may increase (a) or diminish (/) the activity of the other
H-atoms.

a. In the first case although the reaction may take place it will
be much retarded.

Anisole, for instance, which yields a well-defined additive product
with aluminium chloride hardly reacts at the ordinary temperature
with carbon tetrachloride; the chlorine atoms of this chloride do
not, apparently, get under the influence of the combined catalyzer.

Benzyl chloride, which acts very violently on benzene, attacks
anisole so slowly that the velocity of the reaction could be measured
at the ordinary temperature. (H. Goxpscumipr, Central-Blatt 1903
II p. 820).

h. In the second case, the reaction does not take place. Nitro-
benzene, aceto- and benzophenone, sulphobenzide ete. do not suffer
condensation with carbon tetrachloride, chloroform, methylene chlo-
ride, sulphur chloride ete. by means of aluminium chloride.

IV. The aluminium chloride unites both with the chloride (A)
and the benzene derivative (B). In this case it will depend chiefly
on the influence of the groups present in the benzene whether the
condensation takes place or not.

Whilst nitrobenzene cannot be acetylised or benzoylised, the nitro-
anisoles may be converted into the corresponding acetyl compounds.

One does not succeed in introducing a second acetyl group into
acetophenone, but on the other hand m-xylene, mesytilene, sym.

(473 )

triethylbenzene, and sym. durene appear to be diacetylised ; from the
experiments it appears that at least two mols. of the catalyzer are
wanted (V. Meyer, B. 1895 p. 3212; B. 1896 p. 846; B. 1896 p.
2564; H. Wem, B 1897 p. 1285).

VY. The catalyzer combines but little or not at all with the chloride
(A) or the benzene derivative (B) combines only with the formed product.
When this is a molecular compound (as in the cases known up to the
present) at least one mol. will be required for one mol. of the chloride.

I have found that one mol. of carbon tetrachloride exactly requires
the molecular quantity of aluminium chloride for the formation of
triphenylmethane chloride

CCl, + 3C,H, + AICI, = 3HCI + (C,H),CCI. AICI,.
S,Cl, and SCI, also require one mol. of the catalyzer when being
condensing with benzene.

The behaviour of sulphur itself towards benzene is very interesting
in this respect; from an investigation, the details of which will be
published elsewhere, it appears that this condensation must be repre-
sented by the following scheme :

S; + 6C,H, + 3AlCl, = 2(C,H,),S. AICI, + (C,H,)S,. AICI, + 4H,S
diphenylsulphide thianthrene

For one mol. of sulphur, three mols. of the catalyzer are absorbed;
the element itself does not combine with aluminium chloride.

As stated above we have only mentioned the cases where no
secondary actions occur or where these may be greatly prevented.
In a number of syntheses this is very difficult to realise particularly
where we start from chlorides where the carbon atom which carries
the chlorine atom is also combined with hydrogen atoms (Recueil
XXII p. 306), or where hydrogen and chlorine oceur near adjacent
carbon atoms. ((Moungyrat, Bull. Soe. chim [8] 17 p. 797; [3] 19
p. 179, p. 407 and p. 554).

To this belong all the syntheses of the homologues of benzene
where we also have the circumstance that the more alkyl groups
enter into the benzene, the more readily it will be decomposed by
aluminium chloride ; the quantities of aluminium chloride required there-
fore become larger and vary in each individual case. In order to get
a better insight in the actual catalytic action of aluminium chloride
these last reactions will furnish in my opinion, a less suitable ma-
terial than the first five categories which I have mentioned. These
will have to be submitted to a systematic and, if possible, also
quantitative research.

I have been engaged for some time in experiments in this
direction, which will be published from time to time.

Assen, Dee. 1904. Chem. Lab. H. B. S,

Physics. — “The influence of admixtures on the critical phenomena
of simple substances and the explanation of Tricarer’s expe-
riments.” By J. B. VrrscuarreLt. Supplement N°. 10 to the
Communications from the Physical Laboratory at Leiden by
Prof. KAMERLINGH ONNEs.

§ 1. Introduction. That small proportions of any admixture cannot but
have a great influence on the critical phenomena of a simple substance
has repeatedly been demonstrated by KAMERLINGH Onnus and his pupils.
This conviction led them to look for an explanation of the abnormal
phenomena at the critical point — on which some observers base their
doubts of the validity of the theories of ANprEws and VAN DER
Waars — by preference in small quantities of admixture, and gave rise
in the Leiden laboratory to several researches in which the greatest
care was bestowed upon the cleaning of the substances investigated.

As early as Oct. °98, in Comm. N°. 8, p. 15, Kuenen has demon-
strated the importance of phenomena of retardation, due to the
irregular distribution of admixtures. In Comm. N°. 11 (Proc. May ’94)
he proved experimentally that, when pure substances were used, the
deviations found by Gatirzins were not observed. The subject of
Comm. N°. 68, p. 4 (Proc. April ’01, p. 629) was a difference in
opinion between pr Hern and KAMeERLINGH ONNEs about the significance
of the former’s well-known experiments, of which the results were
ascribed by the latter to admixtures. I have taken part in some
preliminary experiments undertaken in consequence of this difference
of opinion. They gave us the conviction that pr HwreN'’s observations
required systematic corrections and that, if these were applied, the —
observations would agree with the theories of ANDREWs and VAN DER
Waals’).

Indeed, according to KAMERLINGH ONNkES’ opinion, maintained by
him in Comm. N°. 68, p. 18 (Proceedings, April ‘01, p. 637), the
deviations found should be ascribed for a good deal to impurities, and
should be explained by means of vaN DER W aars’ theory of mixtures ’*),
Le. p. 6 (Proc. p. 681). Moreover, if attention were paid to the
variation of the molecular pressure the deviations to be expected in
consequence of ,admixtures would show exactly the same nature as
those observed by pe Heen, while the variation of the molecular pressure
owing to impurities, however small it may be for a small quantity of
admixture, would yet cause considerable differences of density owing

1) A more careful repetition of those preliminary researches is begun at Leiden
soon after the controversy wilh pe Heen.

2)Cf. also Harrman, Suppl. N°.3 to the Comms. from the Phys. Lab. at Leiden, p. 47.

to the high degree of compressibility of the substance in the critical
state; l.c. p. 13 (Proc. p. 637).

At the time it was not possible to form a true judgment about
the influence of admixtures and the required corrections. While other
corrections, which had probably to be applied and which might
have the same influence, were fully discussed, about the correction
for admixtures, nothing could be said but that (Le. p. 6, Proe. p.
631) measurements were being made at the laboratory, which would
spread the desired light on the influence of small admixtures.

Since that time have been published those measurements by myself
on mixtures of carbon dioxide and hydrogen (Comms. N°. 45, Jan.
"99 and N°. 47, Febr. ’99) and those by Kersom on carbon dioxide
and oxygen (Comm. N°. 88, Jan. ’04). In the series of “Contributions
to the knowledge of var DER WAA1S’ y-surface” occur several calculations
of Krgsom (Comms. N°. 75, Dec. OL and N°. 79, April 02) and of
myself (Comm. N°. 81, Oct. 02 and Suppl. N°. 6, May and June ’03).
These calculations in which the law of corresponcing states has been
applied according to KAMERLINGH OnNES for substances with admix-
tures, reduce all the deviations from the properties of the pure

ve 1 (dT x
substances to the knowledge of the two quantities a = ~ *)
zl,

1 dps . Zn i 7
En Hs and of the empirical equation of state.
Pk an en

I have availed myself of the obtained results to investigate what
differences of density will be observed in a tube of CAGNIARD DE LA
Tour, containing carbon dioxide mixed with a small molecular
composition of oxygen, if in the manner indicated by Trtcuner *)
floats are placed into it to determine the density. I found it confirmed
that the nature of the deviations which would be observed in TrrcunEr’s
experiments in consequence of small admixtures (if pressure and
temperature are in equilibrium), corresponds entirely to that of the
deviations observed.

It seems to me an important result that, on the strength of the
knowledge of the behaviour of the mixtures of carbon dioxide and
oxygen, we can calculate that even very small quantities of oxygen
in carbon dioxide (a few 0.001 mol.) are sufficient to produce the

1) Drupe’s Ann., 13, 595, 1904. The explanation of Trrcuner’s experiments
covers that of Gauirzine’s experiments, where the density was determined at dif-
ferent heights by an optical method. In tubes filled with carbon dioxide, Gouy
(C. R. 116 p. 1289, June 1893) has observed a slow displacement of the meniscus
a little below the critical temperature, and has ascribed this phenomenon, and
rightly I hold, to impurities.

( 476 )

differences of density which pr Heen observed in carbon dioxide.
Small admixtures of the same kind as those by which pr HreN'’s
experiments can be explained, may, until we have a proof to the
contrary, also be assumed in the carbon tetrachloride with which
TricHNer experimented. I therefore hold that Tricuyer’s researches,
which from an experimental point of view leave less to be desired
than those of pe HeEN's, must be explained in the same way.

They are now being repeated at the Leiden laboratory with carbon
dioxide of the greatest possible purity, while in order to omit all
- doubts of temperature equilibrium 9, thermoelements are sealed in
the tube.

§ 2. Difference in density between two phases with slightly differing
proportions of admixture, when equilibrium of pressure and of tem-
perature exists. We imagine that in a tube, at a temperature Te
which differs only little from the critical 7) of the pure substance,
there are two layers of which the one contains per gramme molecule
x, mol. of the admixture, the other z, mol; the pressure is supposed
to be the same’), i.e. equal to p, and also to differ little from the
critical pressure p, of the pure substance. In order to determine the
density of a mixture with an (infinitely small) composition z, we
may proceed as follows. The quantities a, 3, and y = «—@ determine
the critical elements Tir, per, Von Of the point which for the mixture
corresponds to the critical point of the pure substance, in first
approximation (Comm. N°. 81 equation (14)) by the equations :

Ta= T, (+62), par = pe + 82), viz =m +72):

Hence to the temperature of observation 7’, i.e. the temperature

of the mixture, a temperature 7” of the pure substance corresponds

El 7

: k : :
in such a way that = =F. and we may therefore write in first
xk

approximation: 7” = T'(A — aw). In the same way the pressure
p =p(1— ge) of the pure substance corresponds to the observed
pressure p (pressure of the mixture). Suppose that at the temperature
T’ and the pressure p’ the pure substance occupies the molecular
volume v’, a volume which may be derived from the empirical equation
of state or which may be read on a diagram of isothermals, then,
under the circumstances observed (7, p), we have for the molecular
volume of the mixture considered v =v’ (1+ y2).

1) Cf. Vizarp, C. R. 118 and Comm. N°. 68 (April ’01).
2) Doing so, we neglect the influence of gravitation, which is much smaller than
at of the admixtures, and moreover increases the differences of density.

Determining the value of v, either by calculation or by means of

3
. . . t “Te ~
a diagram, we find that, if the proportion — differs much from
a

Ti. (07 0 : es
Es On = (5) = 4.8!) (ef. Suppl. N°. 6, p. 14; Proc. June ’03,
pe \OT Jer ON

p. 121) owing to the particular shape of the isothermals near the
critical point, the difference v’—v, is much larger (of lower order, viz.

Co

if ; : a
3) than the correction term © ye. For that reason and also because

of the uncertainty about the volumes which belong to a definite
pressure, again owing to the shape of the isothermals, we need not
distinguish between ¢ and v’, in other words, we may neglect
the correction term v ya’). As, however, we intend to determine
the density of the mixture, we must bear in mind that v is the
volume occupied by JZ, (1—.«a) + M, x gr, JZ, and M, representing
the molecular weights of the pure substance and the admixture. Thence
M, (1—2z) + M‚ z

follows the density ———— ——, for which, for the same reason
5

i

4 . . . .

as above, we may put —, i.e. the density of the pure substance itselt
5

at the temperature 7” and the pressure p’.

On the strength of this consideration we may conclude that the
densities of the two mixtures x, and z,, at a temperature which is
about the critical temperature of the pure substance 7, may be read
approximately on a p, @ diagram of the isothermals of the pure
substance; on the isothermals of the temperatures 7’ = 7; (1—e.r,)
and 7,’ = Trax) we seek two points for which the pressures
are p, = pr (ABe) and p,’ = pe (1— 2.) respectively.

Besides these two layers, however, the tube really contains still
several others of different composition, because the composition varies
gradually *). If for different compositions we determine the densities

1) Further on we shall see that, in the cases known thus far, this condition is.
satisfied.

2) The circumstance that we must determine the difference between the v’s for
two mixtures, does not alter this conclusion in the least. For also the difference

between v,’ and v,’ is found to be of a lower order i than the first.

3) What has been said here about accidental impurities, holds also for the
experiments of Carrerer and Coranpeau (C, R. 108, 1280, 1889) where jodine,
which had been dissolved in liquid carbon dioxide, was not diffused equally through
the tube at the temperature at which the meniscus disappeared; it also holds for
similar experiments of Hagenpacu (Drude’s Ann., 5, 276, 1901), who dissolved

( 478 )

at the same temperature and pressure, we obtain points which
all lie in one curve, this curve therefore represents the variation of
the density in the tube; from the shape of this curve, which very
much resembles that of an isothermal in the neighbourhood of the
critical point, it is evident that the substance in the tube cannot but
show considerable differences of density.

I assume that between the two ends of the tube there is a ceftain
difference in composition; then the greatest difference in density
depends not only on the @ and the 6 of the substances considered,
and on the difference in composition, but also on the temperature, on
the mean density and on the mean composition. For those mixtures
of which the « and the @ are known, I now shall give the difference
in density which corresponds to za, = 0.001, if the temperature
is about the critical temperature of the pure substance and the mean
density is also the critical density :

CO, with 0.001 mol. CHsCl ,#= 0.878 , B= 0.088 , n=34"/, of the
CHC,» » CO ,2=—0221,@=- 0281 AS
CO. vene He Stam Bh 1669 ae
6 a ER 6 ag y_ \density.
GO NRO a= 066~ B 1.09) = BONA

The following differences in density would be observed in carbon
dioxide with small quantities of oxygen, with different temperatures
and differences of concentration, the mean density being still the
eritical one :

i a 2, = 01008 c.— 2, ON aa (OO

d 2 2
31°.0 A =36°/, A = 30°/, A =17/,
31°.5 24 17 6
32° 17 10 2.5
33° 12 5 + 1/,
34° 6 3 =e

How the difference in density depends on the mean density of
the substance may be seen from the following table, which relate to
carbon dioxide with oxygen at a temperature of about 31° C. and
for #,—v, — 0.001.

Mean density 1.39, [AS
1.2 8
: Areal 24
1.0 36
0.9 24
0.8 6
0.707 1.5°/,

salts in liquid sulphur dioxide. These experiments, therefore, where an admixture
had intentionally been added, have been erroneously adduced as arguments against
the theory of Axprews and van per Waars; for the rest Hagensacu himself has
understood the cause of the deviation he had found.

Dr. J. E VERSCHAFFELT, “The influence of admixtures on the critical phenomena

of simple substances and the explanation of Teichner’s experiments.”

282.0 22 2.9 33° 39 42 46 50 58 66 Sd 60 42 2.9 deler EA Peo)

Q0005m Oz
oe)
0.00005moPO,

20005 mol 0,

000025 mol 0,

Proceedings Royal Acad. Amsterdam. Vol. VII.

The next table shows how for carbon dioxide with oxygen, at a tem-
perature of about 31° C., the mean density being the critical density
and zz, — 0.001, the difference in density depends on the mean

2
composition.
4 (v, + 4) = 0.0005 A= 36"),
0.005 WY,
0.01 12
0.015 6

All these numbers relate to carbon dioxide with oxygen as admixture ;
it is probable that these results will also be more or less applicable
to carbon dioxide with nitrogen, hence also with air, and as in
carbon dioxide, which had been purified with great care, KrEsom
detected about 0.00025 mol. of air, the possibility is not excluded of
explaining the anomalies observed with carbon dioxide, by impurities
of air.

The variation of the difference in density with the mean density
reminds of a diagram concerning DE Hren’s experiments, formerly
made by me (ef. Comm. N°. 68, Appendix p. 26; Proc. April 1901,
p. 695); in Comm. N°. 68, Appendix p. 22 (Proc. April 1901 p. 691)
KAMpRLINGH Onnes has derived the same diagram for the course of
the differences in density that would result from differences of tempe-
rature; therefore part of the deviations observed by pr HEEN are
perhaps due to differences of temperature.

$ 8. Survey of the experiments of Tuicuner. In the influence of
impurities we have a complete qualitative explanation of TricHNnEr’s
observations. The results of his second series of observations, of
which I have used only those above the critical temperature, are
represented in fig. 1. The positions of the floats are indicated on
vertical lines and the points occupied by the same bulb at different
temperatures are combined by lines. In this manner curves of equal
densities are obtained; for each curve I have given the corresponding
density. In this series of experiments TricHNer has first made obser-
vations at gradually increasing, and then at decreasing temperatures;
after each variation of temperature the observer waited till the
temperature had become the same throughout. As abscissae I have
not taken the temperatures themselves, but I have placed the different
observations at equal distances, that is to say, I have taken time
as abscissa, thus assuming that between two observations there is
always the same interval of time, which will not probably be far
wrong. The temperature 282°.0 C. (uncorrected). is that at which

( 480 )

the meniscus with increase of temperature was seen last and reap-
peared when the temperature was lowered; hence very nearly the
critical temperature. It will be seen that most of the curves of
equal density, when the temperature is raised, leave the point where
the meniscus was seen last, bend away from that point more and more
rapidly, turn round at about the highest temperature observed and
return to the same point, which only few, however, reach when the
temperature is fallen to the eritical temperature.

From this last circumstance we conclude that the course of
the curves of equal density is not only governed by the variation
of the temperature but also by diffusion. Both through increase of tem-
perature and also through diffusion, the distribution of the substance
becomes more regular, and hence the curves of equal density ascend
and would finally project beyond the drawing, if not the decrease
of temperature in the second part of the experiment caused the
withdrawing curves, partly at least, to return. But the very fact that
the curves of equal density in the second part le higher than those
at equal temperatures in the first, is a proof that the progressing
diffusion opposes the influence of the temperature; the following
numbers may show which is about the course of the greatest difference
of density in the tube throughout the series of experiment :

t — 282° 283° 284° 285° 286° 288° 286° 284° 283° 282°
A —= 50 DA AQ) ie 30 he 25 mie 20 He 15 Os HED 20 SA 2D 30 bi

It will be seen that the difference in density first decreases, then
inereases, but the values at equal temperatures are lower in the
second part than in the first and the deviation increases; from this
appears the influence of diffusion.

The value of 4 is not even smallest at the highest temperatures ;
the smallest value is not reached until the temperature is falling, in
harmony with which is the fact that the bulbs 0.555 and 0.578 have
reached their highest position not at 281°.1 C. but at 286°.0 C.,
hence during the period of decreasing temperature. This proves that,
at least at the beginning of the decrease of temperature, the diffusion
has a preponderating influence. That the heaviest bulbs did not show
this peculiarity must probably be ascribed to the circumstance that
in the lower part of the tube, where the substance is much denser,
the diffusion takes place much more slowly; in those lower curves of
density, however, we can clearly distinguish a point of inflection,
which also, though less striking, points at the progressing diffusion.

That these circumstances can actually be explained by the diffusion
of impurities I have tried to demonstrate by calculating and by

(481 )

representing graphically in the same way as in fig. 1 how the
density of a substance is distributed in a tube which is filled with
carbon dioxide, mixed with a small proportion of oxygen, if that
admixture increases in concentration from the bottom upwards. I
also suppose that the temperature first rises from the critical tempe-
rature of 31° C. to 38° C., and then falls again to 31° C. Further
I assume that the concentration of the oxygen which at first decreased
regularly from the top downwards, so that the greatest difference
of concentration was 0.001 mol., at last, owing to a more rapid
diffusion in the upper space, varies there less rapidly with the height
than in the lower space’). Fig. 2 thus obtained, may really be
looked upon as a diagrammatical reproduction of fig. 1; in the
falling period the density curves, as in fig. 1, show a point of in-
flection; in the upper half no maximum has yet been reached by
the curve 0.450, but by adopting a more rapid diffusion in that
space I might have brought about also this circumstance.

§ 4. Conclusion. On the strength of what precedes we can there-
fore firmly deny that Tricunnr’s observations’), at least with respect
to the nature of the phenomena, are incompatible with the theory of
ANDREWS and vaN DER Waars. Down to details these phenomena can
be explained by the presence of admixtures, which are slowly diffu-
sing through the substance; and calculations based on existent data
have shown that in order to reach a quantitative agreement, we
must assume a proportion of the admixture of the same order as that
which actually was present in other experiments with so-called pure
substances. Whether in the carbon tetrachloride, used by Turcunmr, the
required proportion of any admixture, of which neither the nature
nor the @ and 3 are known with certainty, has occurred, is a ques-
tion that cannot be answered. It does not seem impossible, however,
because carbon tetrachloride is a substance which, owing to the manner
in which it is prepared, might contain many foreign components,
and the constancy of the boiling point (to within 0°.1 CP) is not
deemed by us a guarantee for sufficient chemical purity. We are
even inclined to consider the existence of the deviations as a proof
to the contrary, and the non-existence of the deviations (other

1) Starting from a given condition, [ might evidently have worked out this problem
in perfect harmony with reality; it appeared to me, however, that this would have
been useless trouble, and that the scheme, I have given of it, does at any rate
represent the phenomena qualitatively.

2) The same conclusion holds for similar observations (pe HeeN, Gaxirzine, etc.)
about the so-called abnormal phenomena near the critical point.

30

Proceedings Royal Acad. Amsterdam. Vol. VII,

( 482 )

causes taken into account) as the only certain physical criterium
of purity.

As long as it has not been proved that existing impurities cannot
account for the phenomena quantitatively, 1 see no reason to aban-
don the thesis that each substance shows a critical point at which
the two coexisting phases become identical, so that one single critical
density belongs to the critical temperature and the critical pressure.

Geodesy. — “Determinations of latitude and azimuth, made m
1896—99 by Dr. A. PANNEKOEK and Mr. R. Posrnumus Mrysrs
at Oirschot, Utrecht, Sambeek, Wolberg, Harikerberg, Sleen,
Schoorl, Zierikzee, Terschelling (the lighthouse Brandaris),
Ameland, Leeuwarden, Urk and Groningen.” Short account
of the report published under this title by Prof. J. A. C,
OUDEMANS.

Besides the stations mentioned in the title, the programme, as
drawn by the Dutch Geodetic Committee, contained also the stations
Leyden and Ubagsberg, where the observations were made under
superintendence of Prof. H. G. VAN DE SANDE BAKHUYZEN, who himself
will publish them.

The observations of Messrs. PANNEKOEK and Postaumus MryJzs at the
above named thirteen stations, have been made under my super-
intendence, and in an introduction I have given an account and a
criticism of them. Here the following details may suffice :

The mean latitude of the four northernmost stations, Terschelling,
Ameland, Leeuwarden and Groningen is 53°18'39", that of Schoorl,
Urk and Sleen 52°42'45", that of Leyden, Utrecht, Wolberg and Hari-
kerberg 52°10'40", that of Zierikzee, Oirschot and Sambeek 51°35'51",
while the latitude of the southernmost station Ubagsberg is 50°50'53".

The entire are of meridian, of which the length will be computed
as soon as the results of the entire triangulation will be known,
amounts therefore to 2°27'46" and may be considered to consist of
four parts of 35/54", 325", 3449" and 44/58" respectively. Thus
it will appear afterwards whether the curvature of the meridian, as
found here, agrees with the form adopted.

The Universal instruments used for the observations were of
Rersorp; they were provided with a horizontal circle of 315 mms.,
and a vertical circle of 245 mms. in diameter, and belonged to the obser-
vatories of Leyden and Utrecht respectively. The circles were gradu-
ated to 4, whereas the microscopes of the Utrecht instrument are
read directly to 2", those of the Leyden instrument to single seconds.

( 483 )

The micrometer screws, the levels and the differences in diameter
of the pivots were accurately investigated and all irregularities were
accounted for. For the illumination, electric lamps were always used,
for which the current was supplied by accumulators.

The latitudes were determined by zenith distances of northern and
of southern stars. For the northern stars only the two pole stars,
a and d Ursae Minoris were used; the southern stars were chosen
so that they had a northern declination from 6 to 14°, and conse-
quently culminated at zenith distances almost equal to that of the
pole, i.e. equal to the co-latitude.

As a rule, for each determination 16 zenith distances of the pole
stars were observed, without regard to the point on the parallel
they occupied; of the southern stars, four in number, 8 zenith dis-
tances were observed, four before and four after culmination ; so
that each complete determination of latitude rests on 32 zenith
distances north and 32 south of the zenith.

At each station four such determinations were made in four
positions of the circle which differed by 45 degrees. If we bear
in mind that the reading was always made by two opposite
microscopes, the zenith distance of each star may be said to be
determined by eight different ares of the circle, hence the periodic
error of the graduation may be considered as almost entirely elimi-
nated.

The declinations of the stars used were taken from the Berliner
Jahrbuch, while due account was taken of the latest corrections,
published by Auwers in nos. 3927 -29 of the Astronomische Nach-
richten. Finally the latitudes found were corrected for the polar
motion, according to the latest data furnished by ALBruEcHT.

For the azimuth determinations only the Polar star was used at

different points of its parallel. The horizontal distance between the Polar
star and the object was measured four times in 12 positions of the
circle, differing 15 degrees; this was done aceording to the follow-
ing scheme :
Object, Polar star, Polar star, Object, reverse the instrument 180°;
Object, Polar star, Polar star, Object, while for each pointing at
the Polar star the level was read in two positions. Accordingly
each determination of azimuth consisted generally of 12 series of 8
observations i.e. 2 complete determinations each ; hence of 24 complete
determinations.

As object was used either a lamp, or a heliotrope, in most cases
a heliotrope. Its position with relation to the adopted centrum of
the station was determined by the Triangulation Service.

( 484 )

The following may be remarked about the accuracy attained :

For the mean error of one result from two zenith distances += 0"4551)
was found as mean value; the mean error of each final result,
derived from say 128 double observations, was then caleulated in
different manners to be + O"065.

For the determinations of azimuth the mean error

of a single determination was found to be + 122,
hence that of the mean of 12 determinations + 0,355.

The amount of all these mean errors can very well stand a com-
parison with the determinations of other observers.

To this criticism of the determinations executed for geodetic pur-
poses two appendices are added, namely :

I. “A comparison between the latitude, determined at the station
Utrecht, Cathedral tower (Domtoren), by Mr. Posrnumus Mryses, and
the determinations made at the Observatory.”

The final result of this investigation was the following: Latitude
of the Universal instrument at the Observatory :

derived from observations of cireummeridian

zenith distances. ss. te ke) Se) ek RODE

derived from the observations in the prime vertical 52 5 10,29,
= » >» result of Mr. Postoumus Mrysns, reduced

from the “Domtoren” to the Observatory . . . 52 5 9,84.

This agreement is quite satisfactory, especially if we consider that
the observations of the cireummeridian zenith distances at the Obser-
vatory, which had been made for exercise, were executed in only one
position of the vertical circle, which was also a motive for neglecting
the polar motion.

II. “A comparison between the azimuth of Amersfoort, determined
by the author in 1879 and ’80, and the same azimuth determined
by Mr. Postnumus Mrysrs in 1896.”

The final result of this comparison, after due regard was paid to
all reductions, was: Azimuth Utrecht (Centre) — Amersfoort (Centre):
Determination of 1879,80: 68° 22’ 44"71 + 0'31,

5 a) SoG 15159 = 0:29.

Between these two determinations there is a difference of 0’’88
+ 0’’42 (mean error), which partly may be explained by the acci-
dental errors of the observation and the graduation, and partly by
the uncertainty in the different reductions which occur in this com-
parison. We should also bear in mind that in the results of Mr. Posruumus

1) For Mr. Pannekoek + 0'49, for Mr. P. Meyses + O42, two numbers that
are nearly reciprocal to the magnifying powers of the telescopes of the two
instruments (60 and 68 times).

( 485 )

Meyses three out of twelve differences from the arithmetic mean exceed
the negative quantity —0’’88, whereas in the author’s results five
out of nineteen differences exceed the positive quantity —+ 0".88.
Accordingly the difference between the two results may be considered
as purely accidental.

(The last sentence does not occur in the original. It should be
remarked that in the publication of 1880, the last difference from the
arithmetic mean for 1879, must be + O",74 instead of + 1,74).

ERRATA.

Page 238, line 5 from bottom, for “increases” read “decreases.”
op HAO) ey ee gg) me = ze IN rgemol INU
edt ep SOE rp » I" read T" (twice).
» 241, in the formula for Xip7, X,, Xipr read zipr, 2,, XypT-

(January 25, 1905).

KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM,

PROCEEDINGS OF THE MEETING
of Saturday January 28, 1905.

IC

(Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige
Afdeeling van Zaterdag 28 Januari 1905, DI. XIII).

GSOEN KEEN ESS

P. H. Scuoute: “The formulae of GuLpIN in polydimensional space”, p. 487.

W. Karreyn: “On a series of Bessel functions”, p. 494.

Hi. G. Jonker: “Contributions to the knowledge of the sedimentary boulders in the Nether—
lands. I. The Hondsrug in the province of Groningen. 2. Upper Silurian boulders. — First
communication: Boulders of the age of the Eastern Baltic zone G”. (Communicated by Prof.
K. Marri), p. 500.

J. J. van Laar: “On some phenomena, which can occur in the case of partial miscibility of two
liquids, one of them being anomalous, specially water”. (Communicated by Prof. H. A. Lorentz),
p. 517. (With one plate).

J. CarpryaaLt: “The equations by which the locus of the principal axes of a pencil of
quadratic surfaces is determined”, p. 532.

The following papers were read:

Mathematics. — “The formulae of GurpiN in polydimensional
space.” By Prof. P. H. Scnoure.

(Communicated in the meeting of December 24, 1904).

9 el : é 5 B (a)
We suppose in space S, with 7 dimensions an axial space S,

and in a space S,4: through this SP a limited part with p + 1

. C D (a B a 5 5
dimensions rotating round SS)’. Then an arbitrary point P of this
5 Pp d

limited space, which may be called a polytope independent of the
shape of its limitation and may be represented by the symbol (P0),+1,
describes a spherical space of n—p dimensions lying in the space
Sj through P perpendicular to Sy) having the projection Q of P
od
Proceedings Royal Acad. Amsterdam, Vol, VII.

( 488 )

on Si as centre, PQ as radius; so it can be represented by the symbol
Spr—p (Q, PQ). 4
The question with which we shall occupy ourselves is as follows:
“How do we determine volume and surface of
the figure of revolution generated by (Po)Aq ro-

re a
» if we assume that opp and Sy
though lying in the same space S,4; have no points

; (a
tating round §

in common?”

This theorem is solved with the aid of a simple extension of the
well known formulae of GurpiN, which serve in our space to deter-
mine the volume an the surface of a figure of revolution. To prove
these generalized formulae we have but to know that the surface of
the above-mentioned spherical space Sp,—p (@, PQ) is found by multi-
plying PQr-P—! by a coefficient s,—, only dependent on n—p; for
its application however it is desirable to know not only this coeffi-
cient of surface s,—, but also the coefficient of volume wv, by which
PQ» must be multiplied to arrive at the volume of the same

spherical space. To this end we mention beforehand — as is learned
by direct integration — that between these coefficients the recurrent
relations
2a 27
Un == Vn—2 A = 5 Sn? eae you oe (IL)
1 Med

exist, whilst the well known relation between volume and surface
leads in a simpler way still to the equation

1
RB oes Oph cao oro ((24))

VW

Un =

In this way we find as far as and inclusive of m= 12 out of the
well known values of »,, v7, and s,, s,

] | | d | k
| |
dE 3 4 NG 7 8 9 10 | 11 |) 42
| | | | | | |
| 4 Nel Sr ae ee ee eral eee se CN
el KE: Di eae |) GO) OAS Oo O0 [0395 | 720 ~
EJ y >i g
Sn ars Ax 97x? © me 7 ze ie : a ie ic” ee rsa J xf
| 3 15 3 105 | 42 945 | 69

L Determination of volume. If « indicates the length
of the radius PQ and the differential dr the p + 1-dimensional
volume-element, lying immediately round ?, of the rotating polytope
(Po), then the demanded volume is

( 489 )

*

ws fer dv,

if the integral is extended to all the elements of volume of (20),41. If
now Vj is the volume of (70),41, we can imagine a quantity ,
satisfying the equation

fare i mrt fas =a" p—l Vo

and we can insert this quantity in the above formula. By this it passes into
== VA » Sn—p min.

If we call x the “radius of inertia of order n—p—1” of the
volume Vr, of the rotating figure (Po), with relation to the

( L . . . Y kl .
axial space Se lying in its space S,41, we find this theorem :
We find the volume of the figure of revolution
generated by the polytope (Popp rotating round

\
i}

. ct . . nies
an axialspace S,’ not cutting this polytope of its
space Sp, if we multiply the volume Vo of (Po)
by the surface of a spherical space SD having

the radius of inertia of order n—p—1 of Von

E 9 ola) c 5
with relation to S,’ as EO Orne

2. Determination of surface. If in the above we
substitute the p-dimensional element of surface for the p + 1-dimen-
sional element of volume and in accordance with this for the volume
Vp41 and its radius of inertia the surface Suy and its radius of
inertia, we arrive in similar way at the theorem:

We find the surface of the figure of revolution gener-
ated as above if we multiply the surface Su, of (Po),41
by the surface of aspherical space Sp,_,, having for
radius the radius of inertia of order n—p—1 of Su

with relation to Sa

3. The segment of revolution. The opinions will differ
greatly about the use of the n-dimensional extension of the Gurpin
formulae proved above. Those regarding only their generality and
their short enunciation may rate them too high, though reasonably
they cannot go so far as to believe that these formulae allow the
volume and the surface of a figure of revolution to be found when
the common principles of the calculus leave us in the lurch, as
the quadratures can be indicated but not effected in finite form.

34%

( 490 )

Others, whose attention is drawn to the fact that these formulae
displace the difficulties of the quadratures but apparently — in this
case displace them from definition of volume and of surface to the
definition of radii of inertia — will on the other hand perhaps fall
into another extreme and will deny any practical use to the formulae
in question. Here of course the truth lies in the mean. Though it
remains true that the GerpiN formulae help us but apparently out
of the difficulty in the case where the direct integration falls short,
yet by the use of those formulae many an integration is avoided
because the radii of inertia appearing in those formulae of volume
and surface of the figure of revolution are known from another
source, which latter circumstance appears in the first place when
p=n—2, thus each point P of the rotating figure describes the
circumference of a circle and the radu of inertia relate therefore
to the centre of gravity of volume and surface of that figure, whilst
3 the knowledge of the common radius of inertia of

for p=n
mechanics gives rise to simplification.

As simplest example of the case p=n—2 we think that a
segment Sp (7,9) of a spherical space Sp,—1 with 7 and @ as
radii of spherical and base boundary generates a segment of revo-

lution Sp (7,0, @, by rotation round a diametral space SN situated
in its space S,—, having no point in common with it and forming
an angle « with the space S,—» of the base boundary. For this we
find the following theorems:

“We find the volume of the segment of revolution
Sp(r‚e,a)n by multiplying the volume of a spherical
space Spn with @ for radius by eos a.”

“We find the surface of the segment of revolution
Sp(v,e,@, Which is described by the spherical boundary
of Sp,-i(r,9) when rotating, by multiplying the circum-
ference of a circle with r for radius by the volume of
the projection of the base boundary of Sp,—i(r,e) onthe

axial space Gree

These theorems are simple polydimensional extensions of well
known theorems of stereometry. They can be found by direct inte-
eration where the case «=O is considerably simpler than that of an
arbitrary angle « And now the formulae of GuLpry teach us exactly
io avoid the integration in the general case, showing us immediately
that the theorems are true for the case of an arbitrary angle «, as
soon as they are proved for « — 0. If namely w, and «ws are the
distances from the centres of gravity of volume |, and surface

( 491 )

’ . 1 — 41) al . . . N
Stu of Spr—i (7, 9) to Ss » where Sj now again indicates.exclusive-
ly the spherical boundary, then the formulae of Gerpix furnish us with

Vs = 27 2, cos a . Vay Sie = Er Lg COS . Sn |

Vi den En Su, = 20 Es 5 yaaa

and from this ensues immediately

Ve — Ve cosa B Sit, == Sito cos a

and therefore what was assumed above, so that only for « = 0 the
proofs have vet to be given. We commence with the volume. If w

. . a ya - ’
is the distance from Ss to a parallel space Ss cutting Sp,—1(”, @)

in a spherical space Spo. with 7 =V 7? — 2’ for radius, then the
demanded volume is

rr
V = 2 voe y"—2 ada
=a

and this passes, as #° + y? =7" and «dx + ydy = 0, into

f

2
V = Ar vn—o f yr | dy = — O2 0" = Un OP,
n

o
with which the special case of the theorem for the volume has
been proved.

In the special case of the theorem for the surface we regard the
superficial element generated by the rotation of the surface Su, (r, ©)

. y(t) rahe dx 7 2
situated between the parallel spaces S,> and SEE) If ds is the
apothema of this frustum the demanded surface is

mr

Su = 22 a ads,

niel Ors

With the help of the relations yds =rdu and ade + ydy=0 this
passes into
hd .
Su = 2ar maf dy = 5 PS. = Zr. Ue "2,

==
o

Le. the desired result.
Of course we can represent to ourselves the more general segment
of revolution Sp(7, 0, jr of order & generated by the rotation of a

( 492 )

spherical segment Spn—1 (1.9) round a diametral space core of its
space S,—z; of the various possible cases
NVE GEen » n—2

the first is the one treated above extensively. As any point generates
at the rotation the surface of a spherical space Spy, we find —
if along the indicated way by means of the formulae of GuLpin the
general case of an arbitrary angle « is reduced to the special
case a=0 — for volume Vz and the surface Sur of Sp(7, 0, Onze
_ the formulae

r=r

Vik = Onkel Sk 1C08k ef ak da

SN bee ET
r= VP
c=—r

Sine Sen INS costa wba wk da

he PAR Ee
t=VP=-Z |

and from this ensues the general relation

Sun, = 290 7 cos? a Vn—» ks
by which all cases of determination of surface except Stn,n—2 and
Stn 3 ave deduced to simpler cases of the determination of volume.

When determining the volume the integral gives a rational result,
an irrational one or a transcendental one according to # being odd,
n odd and # even, or n even and # even. And this is evidently
likewise the case for the determination of surface.

4. The torusgroup. By rotation of aspherical space Sp,—z(r)
around a space So, -; of its space S, 7 at a distance a >>r from
the centre a ring is generated in S,, the ring or “torus” 7'(7,a)n2-
For volume Vr, a), and surface Su(r, dop of this figure of revo-

lution of order # we find

a
z el
VAO ne — SANO kel Vr—2 (a+ex)kde
=a
54)
a
8 ln
Su (1, ane = 7 SH ni f Vr—2? (atea)Fde
—a

from which ensues again the formula of reduction
Sui 27 Vion Oehoe OPENEN Dr (4)
For the case 4=1 and k=2 the results are calculated more
easily by means of the formulae of Gerpin, if one makes use of

( 493 )

the centre of gravity and of the oscillation centre of the rotating
spherical space.
Case k=1. The centre of gravity of volume and surface of
the spherical space Sp,—1(”) lying in the centre, we find
VES cra, Goa ite en Sh Er TN

Case k}=2. The radii of inertia of volume and surface of a

5)

5 y 5 il

spherical space Spp—s (r) with respect to the centre are 7 pe
1

Ge

and 7, those with respect to a diametral space S,—3 are thus 7 ==

nr

1
and 7 4 . So we find
n—2

1 1
V= Ar (« dn open? , Suda? + - a ) «Sg 78,
n n—2

If instead of a whole spherical space Prik (7) we allow only

. vs (a) 2 > 8
half of it to rotate around a space Sj in its space S,—; parallel
to its base at a distance a, then the limits (—7,7) of the two
integrals (1) change into (0,7) or (—r,0) according to the half
spherical space Spy (7) turning its base or its spherical boundary
. ‚(a) 7
to the axial space Sj. We shall occupy ourselves another moment
with the former of these cases, namely for k—=1 and #=—=2.
Case (0,7), k=1. We find immediately

2 Un. 9 . Zie OE

2 ; ea yeas , n—2 5

nf adr }. Op? ‚SUR Ar. Sn 0-2,
Un) n—2 Sn—1

Case (0,7), k=2. We determine the moments of inertia of

Y=

volume and surface first with respect to the base Sa and then
: : : (2
suecessively with respect to the parallel space Shae through the centre

: ; : : Sl .
of gravity and with respect to the axial space S,“’5. Thus we
finally find the formulae

r 2 Un—2 3 2 Un—2 3
Vl See — er Hf — +: ra » U_—9 1—2,
n (0, Basil 1D Brea
„3 a 2 29 N 2
7 2 Sn—2 a Spn—2
Su = 27 — —— r}] Hf —— or a 8 n—8
= . =D
n—2 NA Snel N—2 Sn sy wi :
or
4 Un—2 r?
VS 27 | a? = art — » Un—2 72,
n Un—1 n
4 89 je
Su= 2 f a a ar Fats) yn—3,
n—2 Bp) | n—2

Which pass for « =O appropriately into volume and surface of the
spherical space Sp, (7).

( 494 )

Mathematics. — “On a series of Bessel functions.” By Prof. W.
KAPTEYN.

(Communicated in the meeting of December 24, 1904).
In the following we shall try to determine the sum of the series
T, (@) I, (©) +3 1, (@) A, (@) +57,@1,(@)+-..= = n In (@) In («).
To this end we begin to determine the sum of tie simpler series
Ss = = Lj (w) cos ng.

If we introduce, being an odd number, for the Bessel function
the form

then
x Le
ka) iz)
S= de ST (t cos p Ht cos 3p -+ ..-),
and
t (1—+t?’) cos p
tcosgttcs8y+...= TE (mod t <1),
hence
x 1
Gas) ea
g ij e (1—#’) cos p
faa fe Ae Ea yas EE
If we put
ae cos p En
1—2 t? cos 2p-+t*
then
x 1
mins
S= — bal an — t*) R,
or
: A
= 1, cosng = — En e (i — t°) R.
1.3
Differentiating this equation, we get
x 1
aa IR

wD » r
= nl,(x) sin ng= re) e
1.3 0

( 495 )
If now we multiply this equation by

we integrate between the limits 0 and a we find

1

pn 1 p li nva en
= n I, (2) 1, (0) = = Cy 2 (1 — t°) ie sin (a sin p) dep
1.3 JT dp
0
a, 1 ic
od ap al cay
na co
0
Putting for the further reduction

ny
u =x cos (a sin p) cos p dp
0

cos” p cos (ct sin p) :
== Ct
1—2 cos 2p”
0

we arrive at

* cos? psin(asing) ,
rend = | ———_—————_ sing dg,
dea 1—2 #? cos 2g Ht!

“cos? pcos(asing) . . 1

en a = — sin P ap

da’ 1—2 ¢? cos 2g tt /
0

and because

EN rl et cos 2p tt’ (1 - 2)?
se? (QS eS

. At At
we find
du à iL aigere ;
re =mu— raf p cos (a sin p) dp,
0
1—??
where m=
2t
1 + cos 2p
If we replace cos’ p by = ga we. can easily reduce this
differential equation to
au A n 7 7
ei NEET a a
_ sa Ho (a) + 1, (0)

xz I, (a)
JE

sin (a@ sin p) dp and if
7

(1 — #?) | RR cos (a sin ~) cos p dp.

( 496 )

Let us now determine the integral of this equation satisfying the
conditions that for « — 0

5 cos? pd wy ld
Ui =
1—2# cos? pt (lt)
0

and
du

da

We then find

a

7 mz — mz J Fk ne ne
= aes ae +e fa = a ‘| (9) dp |e oe) aes e Kf)
4 (1l—t®) 8 t° m B

0

and by this

Er t ! ra 7 ]
hee =e ea ie
1.3 4 0

sk 1 ) dm 1

a er (5) 1 2 1 9 ‘--) :
+ +f 3 dp Sie Ë —e

0

Remembering now that

ag)

ae =D HtLD HELD H..
il 1
mn 0/0
we see that the residues are easily determined. We have

Sn In () In (0) = TUe Le +o] +

« (2,8)
+ $f SUG et ete. 0
0

From this result another important relation may be deduced. To
show this, we shall again develop
Le —ea) + L(e +e)
into a series.
From

1 7
Ie) = { sin p sin (x sin p — asin p) dy

wT

0
and

( 497 )

ger
L(e + ea) = = sin @ sin (w sin g + asin p) dy
Xt
0
follows
Sade
I,(@—a +L (we +e) = =) sin p sin (a sin p) cos (a sin p) dp.

x
0

If we write
sin (a sin gp) = 2 I, sng + 21, sin Sp +.

we obtain

4
L(«—-a+hwt+a=—-Jf of so p cos (a sin p) dp
I
0
4 Peon |
+ — I, (x) | sing sin 3p cos (a sin p) dp
x

sin p sin 5@ cos (a sin yp) dip

+
En
n

+.
or as
2 sin p sin (2n + 1) p cos (a sing) dp =
0
= fes 2np — cos (2n + 2) y] cos (a sin 7) dip
= a [Lon (@) — Tan 42 (@)]
gh: d Tan 41 (@)
da
we get

I, (@—a)+1,(@+a)= aE ‘+ 1, () ELD |
Substituting here 7

dI,,
= =n TI, (a) — a Ini (a)

we arrive finally at

4 ®
I, (e—a)4+ 1, (e +e)= = Sn I, (a) ZL (a) — 4 > Ln +1 (a) Zo (2).

( 498 )

With this equation the result (1) may be written

<< 1 lij (3) - Ns
== Ini (e) I, (@) = fi | = 5 WE a+ Srl (ta - B) (2)
1.3 Ee ‘

0

If here we develop
LE, (ew—ea+8) = ZL, (x) J, (e—P)+27, (2) 7, (e—B)+27, (x) L, (a) ...
Inese PB) Lo (2) 1, (a SB) (@) 2, (¢ 2) (LEN nn
we find

a

Sy (AS ‘ ay £1 (B)
Ent (a) Ine) = = T, (x) NC 8)
1.3 1.3

7 dp
B

and consequently by comparing the coefficients of J, (x)

naor nt 1 8) dB el ES)

By means of this formula we ean give equation (1) another form.
For, according to (9),

I, (@—a)= ie (v — a — B) a dp
= je (« — a + 9) a
ata

Leta= [Le fae Pas

0
hence the second member of (1) takes the form

dt Lt

a t I, (8) fg a ne: 1, (8)
t|-fue-ate 5 zalen a ‘a
A torie (7 +a— 8) Pas]
0

or

pn utr
Ry a if (5) A: Ih (8)
an Temata Pua [he ba Ao a8

4—wDZ

( 499 )

If we now put in the first integral 8 = a@ — y and in the se-
cond one B=a + y this becomes

x

a ik JAG) a L (ay)

Se ie OY + fz, Bj |}

4 Omi a+y el
0 0

with which equation (1) assumes the final form

wc

2 : Ge L (ay) . I, (a+y)
= n I, (a) I, (x) = A fr (& — 7) | : Ii — t dy. .. (4)
. oH

d a—y a+y

A closer investigation of formula (3) teaches us, that it holds
good for even values of too, also that many analogous relations
exist. So we find inter alia, # being any integer,

eile —p Ly
{2108s
a—p n
0
* Lijn 3 Ihe C
fn («—8) apa

0

a

fA (e—8) Te (3) ‘ 3 = Zaft (a)
a—p n

0

el aml 3 1 Te Js 5
iS ze TN (8) dp: =S (a) 2 (a) .
(a—Bp)’ 2n li n— 1 ntl
v0

4

| I, (a— 8) I, (8) d8 = sin a.

0
We shall not dwell upon this at present; we only remark, that
when a very great positive value is assigned in (1) to 2, so that

ee 2n+1
NDSS ye COS (« == Sar ),
HU 4

I, («—a)+ 7, (#—a)=2 yok cos (« — =) cos Ct,
Ey di 4
2 mr
I, (@ —a-+ Bs) —TJ, (e + ae — ~)=2 sin (« — =) sin (a — B).
nt

This changes (1) into

we find

24
ra nr a OD (len (B)
Sn Re ds pe TENG ;
= n Ly, (a) sin Dg sat =a | ar sim (a — B) dp
es < a =. t

0

(500 )

or, noticing that
e, , mat a
Xn Ly (a) sin Ses I, (a);
1.3 ra pe

we have

7
I, (a) = cosa +f a sin (a — B) dp.
t
0

If we differentiate this equation, we find

a

I, (a) = sina (8 cos (a— 3) dp

0
from which we conclude that
TB), pe
F sin BdB = 1 — cosal, (a) — sin a 1, (a),
0
UP
eo cos 8 dp = sin a I, (a) — cosa I, (a).
J |
Geology. — ‘Contributions to the knowledge of the sedimentary

boulders in the Netherlands. 1. The Hondsrug in the province
of Groningen. 2. Upper Silurian bouiders. — First commu-
nication: Boulders of the age of the Eastern Baltic zone G.”

By Dr. H. G. Jonker. (Communicated by Prof. K. Marry).

This communication introduces the description of the Upper Silurian
boulders of Groningen and its surroundings, in which my contribution
that treats of the Cambrian and Lower Silurian erraties and appeared
in 1904, is continued (36). The circumstance that in the summer
of last year I had an opportunity of getting more intimately acquainted
with the Seandinavian-Baltie strata by investigations of my own
has aided me considerably in the study of these younger rocks.
Owing to nearly a month’s stay in Gothland I managed to collect
a great number of different species of rocks together with fossils
characteristic to them in order to compare them with erratics that
are found here. Much I owe to the kindness and assistance of
Drs. O. W. WeNNERSTEN, who accompanied me on some excursions
and whom I had very often reason to admire for his extensive
knowledge of his native country, the classical ground for the study
of the Upper Silurian formation, But T have as yet not been able

( 501 )
to pay a visit to Scania and Oesel; the material for comparison
from those regions (present in the Min.-Geol. Institution in this
town, for the greater part collected by Mr. J. H. Bonnema), however,
will make up for it to a large extent, though not all questions can
be solved.

Some days’ stay at Upsala enabled me, thanks to the kindly
assistance of Dr. C. Wiman, to examine the collections present in
the Geological Institution from the different Lower Silurian regions
of Sweden. This examination, which of course had to be made in
haste, obliges me to introduce some alterations into my former
description which however are not very important. By this time the
material has been increased by new finds, and as more recent
publications always make some alterations or completions necessary,
I have made up my mind not to introduce them now but to collect
all these corrigenda and addenda in an appendix at the end of the
treatise of the Groningen erratics.

The real description of the Upper Silurian species of boulders of
which two have been dealt with in this communication, is preceded
by some pages which, from an_ historical point of view, are not
unimportant. After the appearance of my first contribution Dr. L.
Hormsrröm at Akarp was so kind as to draw my attention to some
parts of his lately published biography of Orro Torerr. From this
I learned that, in 1866, the latter had written a prize-essay on a
subject suggested by the Dutch Society of Sciences at Harlem, and
treating of the origin of the stones and fossils of the Groningen
Hondsrug. His essay was rewarded, but was never published and
not given up to the Dutch Society till after the author's death.
Thanks to the kindness of its secretary, Prof. dr. J. Bosscna, I have
been able to study TorerL's essay, and now comprehend his relation
to the Groningen boulders which formerly really puzzled me. His
ideas about our subject are a necessary completion of the historical
outline.

Finally, it pleases me to state that this year as well as last year
the support of the Groningen University Fund fell to my share,
while the expenses of my investigations in Sweden have for the
greater part been defrayed by a subsidy granted to me by the
“Central Bureau for the promotion of the knowledge of the province
of Groningen” after receiving the approval of “the Board of the
Physical Society at Groningen.” This obvious interest taken in the
subject of my study has been a source of much delight to me.

( 502 )
Supplement to the Historical Outline.

O. Torell.

The prize-subject of the Dutch Society of Sciences at Harlem
(1865) ran as follows:

“On sait, surtout par le travail de M. Roemer a Breslau, que
plusieurs des fossiles, que Von trouve pres de Groningue appartien-
nent aux memes espèces que ceux que on trouve dans les terrains
siluriens de Tile de Gothland. Ce fait a conduit M. Rormer a la con-
clusion, que le diluvium de Groningue a été transporté de cette ile
de Gothland ; mais cette origine parait peu conciliable avec la direc-
tion dans laquelle ce diluvium est déposé, direction qui indiquerait
plutôt un transport de la partie méridionale de la Norvège. La
Société désire voir décidée cette question par une comparaison exacte
des fossiles de Groningue avec les minéraux et les fossiles des terrains
siluriens et autres de cette partie de la Norvège, en avant égard
aussi aux modifications que le transport d'un pays éloigné et ses
suites ont fait subir à ces minéraux et a ces fossiles.”

ToRELL’S answer to this question consists of two parts. The first
part deals with the essential question and is entitled: “Essai sur la
question proposée de la Société Hollandaise des Sciences à Harlem.”
Here the author enumerates the Groningen fossils known to him with
their geological occurrence and the literature on this subject. Hardly
any new fossils are mentioned, so that this description is little
more than a development of Rormer’s treatise of the Groningen
fossils. Nor is this wonderful, because he, too, had received the
greater part of this material from ConeN, whom he had paid a visit
in 1865. No doubt there were among the collection sent to him by
the museum of Natural History at Groningen, about which I have
spoken in my first essay (36, p. XXXII), various fossils unknown
to Rormer, but Torri seems not to have paid much attention to
the determination of new fossils. From his enumeration he arrives
at the conclusion that the sedimentary boulders might originate in
Norway, but that there is not the least proof for it and that most
likely the origin from Oesel-Gothland is much more probable.

In the second part of this first essay, however, he deals with the
rocks themselves. By a comparison with limestones from Norway
and Gothland he is led to exclude the first region altogether and
this result is further on supported by what the crystalline boulders
teach, which are described next. The dispersion of the different
erraties being examined, his conelusion with regard to the question
which had been put runs;

( 503 )

“Le résultat de ces recherches tend ainsi àconfirmer
l'opinion déjà émise par M. Ferp. Rormer, que les
bloes siluriensde Groningue proviennent de l’Esthonie
et de l’ile de Gothland, mais nullement de la Norvège”.

This first essay was inserted by him in 1866; the next year.
followed a second, entitled :

“Recherches sur les phénomenes glaciaires de PEurope du Nord”,
which for more than one reason is a remarkable treatise. As, how-
ever, its contents do not refer directly to the question we are discussing
it be only said that this treatise is, in short, a pleading in all its
details for the glacial theory, which is here for the first time con-
sistently adopted and asserted, under a motto borrowed from L. von
Bven (Ueber die Ursachen der Verbreitung grosser Alpengeschiebe,
1811, Abhandl. d. Berlin. Akad., p. 185, 186), too interesting not
to be cited here:

“Wer sich etwas mit den Blöeken beschaftigt hat, welche in so
zahlloser Menge die Ebenen des nördliehen Europa bedeeken, wird
nicht einen Augenblick zweifeln, dasz nicht auch in dieser Zerstreu-
ung dasselbe Phänomen wiederholt ist, was in der Schweiz so auf-
fallend wird. Ware die Granitzone des Wallisausbruchs nicht von
den Jurabergen zuriickgehalten worden, so würde sie an den
Ufern des Doux und der Saone eben so zerstreut über die Flachen
gelagert sein, eben so dicht wie in soviel Gegenden der Mark
Brandenburg, von Pommern, Meklenburg, Holstein... ... Das
nordische Phänomen ist daher wohl bei weitem grösser als das
schweizerische, allein von derselben Natur; und wahrscheinlich liegt
thm deswegen auch eine ühntiche Ursache zum Grunde’’.

Toren, gives here a compendium of his opinions, founded on
insights acquired by many travels about the origin of diluvial
deposits and the grounds which in his opinion argue a glacial
covering. The older theories are amply criticised, and after describing
the formation of ice in Greenland, where the inland-ice covers an
extent of country of °/, of the North-European erratic zone, he says:

“Serait-il done absurde de supposer, qu'une couverture glaciaire
semblable, mais plus grande des */

/,, a existé aussi dans l'Europe
du Nord pendant une époque, où la faune marine du Spitzberg
vivait entre les 50° et 60° de Lat., où le Betula nana croissait dans
le Devonshire et ot le renne avait son domicile dans la France
méridionale |”

This quotation sufficiently illustrates the importance of this essay. It
has however never been printed. Torerr did claim back his work
from the Dutch Society to revise it for the press and various emen-

Bo

Proceedings Royal Acad, Amsterdam. Vol. VII.

( 504 )

dations and marginal notes have been introduced, but he did not get
farther than that. This is much to be regretted as it was now not
until 1875 that his insights and opinions found adherents among
the German geologists; it was the year when TorerL on the me-
.morable day of Nov. 3 by his lecture for the German geological
society in connection with the glacial scratches once more discovered
by him on the Muschelkalk of Rüdersdorf convinced different colleagues
of the correctness of his theory. For, if the above-mentioned essay
had been printed as early as 1867 it would have contributed in a
high measure to propagate the novel ideas more rapidly.

The bulky manuscript written in French and provided with French,
Swedish and Dutch annotations (the Dutch annotations are by STARING,
who was a members of the jury, as well as Bosqver and vAN BREDA)
is at present again in possession of the Dutch Society of Sciences.
The maps (2) and plate mentioned in the text are not wanting.
For further details about the contents the reader is referred to
ToreLL’s biography by Hotmstrém (35, p. 18—25).

UPPER SILURIAN BOULDERS.

In the description of the Upper Silurian boulders various difficulties
present themselves, which all may be reduced to the fact that the
exact succession of strata in the Seandinavian-baltic zone is not known
for certain. Especially with regard to the eastern balticum the struc-
ture has long ago been made out by Scumipr and never refuted by
anybody that I know of. His division of the strata in Gothland, on
the contrary, corresponding with MurcHison’s conceptions has found
but few adherents, and is especially called in question by LINDSTRÖM,
who has a quite different opinion. This discrepancy as to the strue-
ture of Gothland, which has already existed many years, has not
yet been satisfactorily removed. If must be said, however, that well-
nigh all other investigators who have pronounced their opinion about
this question, have taken Linpstrém’s side; a.o. Dames, who has
made a division which differs but a little from Laxpsrröm’s; then
Srouuey, Wimax, Barunr, Kayser and others. | myself, owing to
my short stay in Gothland, am not so fortunate as to be able to
pronounce a decided opinion, though it does seem to me that, on
the whole, Scumipt’s arguments are stronger than LINDSTRÖM's, so
that it appears scarcely possible to me that new investigations will
confirm the opinions of the latter in every respect. In collecting fos-
sils im Gothland, I frequently doubted of the correctness of Linpstr6m’s

division, and in some cases noticed certain contradictions. Anyhow
for the present it is impossible to parallel the Upper Silurian strata
of Gothland with those of Oesel, a question, indeed, which for a
determination of boulders of that age can hardly be dispensed with.
We may sincerely hope that the researches by Horm, who has been
engaged in this question, may not be long in coming, and that this
solution may finally settle the question!

(The chief literature about this controverted question follows here :
Aes 215 225 27; 28, p- 416; ete).

Nevertheless in enumerating the species of boulders we must
adopt a certain succession of strata to arrive at the determination
of their age. I select for this purpose Scumipt’s division of the
Eastern Baltic Upper Silurian (8, p. 41—54), corresponding to the
method hitherto followed in the museum:

G.1. Jérden Beds.

2. Borealis bank.

3s. Ratkull Beds.
Pentamerus-esthonus zone.
Lower Oesel zone.
Upper Oesel zone:

Lf sNorthern yellow zone:

2, Southern grey zone.

Py

Dames (22), Srorrer (30), SrecertT (32), and others have founded the
determination of their erratics on the division of Gothland by Linp-
STROM; as it seems to me, however, that our boulders approach the
Eastern baltic rocks much more, I did not follow this example,
the more so, as I have said before, the above-mentioned opinion,
which is quoted below with the alteration introduced by Damns,
does not appear to me to be the right one in every respect.

a. Oldest red shale beds with Arachnophyllum.

b. Stricklandinia-shale.

ec. Shale beds and sandstone.

d. Bands of limestone and shale, in some parts oolite.

e. Pterygotus-beds.

Crinoid- and Coral-limestones with intermediate Stromatopora-
riffs, Gastropoda- and Ascoceras-limestones, together with Mega-
lomus-banks.

g. Upper Cephalopodan strata.

The material may best be subdivided into four groups: I Boulders
of the age GG; Il... H; Wl... 2; IV... K. The last division
will appear to be by far the most important. Besides there are some

35*

(506 )

characteristic boulders, which cannot be placed in the Eastern baltic
scheme; these, together with others whose age lies between limits
too far apart to reduce them to one of these divisions, will be
described at the end.

After these introductory remarks we may proceed to the
description of the boulders of the first-mentioned group.

G,—G,.

The boulders belonging to the oldest zone G',, those of the Jorden
beds in Esthonia with Leptocoelia Duboisii pe VerN., which are
occasionally mentioned by German geologists, are not found near
Groningen. The two younger zones G, and G,, however, have
been met with.

29. Borealis-limestone.

These well-known and characteristic boulders consist of limestone
or dolomite, and usually contain in large quantities remains of

Pentamerus borealis Ercuw.,

while other fossils are absolutely wanting. As regards the kind of
rock my material from Groningen may be divided into two varieties:

a. Limestone, as a rule distinctly crystalline but somewhat marly,
as may be easily observed on its weathered surface. The slightly
variegated colour of the ground-mass shades from gray to brownish-
yellow at the fresh fracture; if weathered, however, it has mostly
a sallow-vellowish-gray tinge. In this ground-mass the valves of the
above-mentioned species of Pentamerus always occur in great numbers;
they are invariably changed into crystalline calcite and this is very
often of a bright white colour, so that the always very thick shells
sometimes stand out very distinctly against the surface of the boulders,
which is sometimes polished. Besides the ground-mass weathers more
readily than this calcite, so that the fossils appear in relief. The
number of these petrifactions has influenced the exterior appearance
of the boulder. Though always numerous, the ground-mass may yet
occur in sufficient quantities to give a compact character to the stone,
These limestones which are rather hard when not partially weathered
make up the majority of the stones found. The dimensions of some
of them amount to about 17 ¢.M. In other pieces the ground-mass
recedes much to the background and the stone consists almost

exclusively of fragments of the valves of this species of brachiopoda

( 507 )

and thus forms a real shell-breccia. The ground-mass then is commonly
weathered to a more or less earthy yellow mass, which also covers
the surface of the shells, by which the whole assumes a yellow
colour. In other cases, however, the weathered ground-mass is almost
white, sometimes also brown-ochre-yellow. Though they differ so
much in exterior appearance, all specimens have in common that
this Pentamerus occurs almost always only in single valves which
themselves are, for the greater part broken into more fragments.
I have never been able to produce a wholly preserved specimen,
though some fragments actually show that parts of both valves occur
in natural position. So this confirms in the main the results of
Rormer’s examination (13, p. 74), though I doubt of the truth of
his opinion, according to which these boulders should contain only
ventral valves of this species. This conception was supported by
KIcHWALD’s communication that also in the parent rocks both valves
were never seen in connection. Nevertheless Ererwarp did know the
smaller dorsal valve and describes it as having half the length of
the larger one, being much broader and much less vaulted. This
can hardly be right, for afterwards Scumipr found complete specimens
at Weissenfeld in the neighbourhood of Hapsal in Esthonia. Among
my material for comparison there are three such specimens from
the above-mentioned place, collected by Boxnema. These, however,
show a dorsal valve, but little smaller than the ventral one, but
much flatter and so comparatively wider. This causes the great
difference between the two shells to disappear, and so there is no
reason left for the inexplicable fact that in boulders only the ventral
valve should occur. Meanwhile the interior structure of the small
shell has to be examined still to confirm this. I have not been
occupied with this work.

6. Dolomite, very fine-grained, sometimes even impalpable, of a
light-gray or light-brownish-yellow colour. This dolomitie ground-
mass also contains great numbers of nuclei of Pentamerus borealis
Ercuw., which are covered all over with little, graceful, dolomite-
rhombohedra, which, however, are easily perceptible by the naked eye.

Of these boulders, which in literature is usually called “Penta-
merenkalk” are found here :

Limestone : Boteringesingel, Groningen 2
Behind the “Sterrebosch’, _ ,, A
Helpman 1
the “Huis de Wolf”, near Haren 1
“Old Collection” 9
Dolomite : Boteringesingel, Groningen 2

( 508 )

So in all 16 pieces. From this list appears that at an early time
already these boulders have attracted the attention. Quite in corre-
spondence with this is the fact that as early as 1878 Martin men-
tioned 11 pieces from Groningen (6, p. 21, a and c), and even
earlier still Rozmrr observed such boulders from here (1, p. 387,
n°. 16; 3, p. 269, n°. 27). Afterwards van CALKER also pointed out
their occurrence in the Hondsrug (19, p. 357; 25, p. 363).

As regards the further spreading of this species of boulders, I refer
to Rormer’s excellent treatise about everything known at the time
about this subject (18, p. 75), and only wish to state here, that in
Germany these boulders are found in a great many places, but nowhere
in large quantities. So everywhere in East- and West-Prussia (20,
p. 58), in Posen, Silesia and Brandenburg, near Sorau in the district
of Frankfurt on the Oder, in South-Holstein in various localities
(18, p. 45). Further north they seem not to occur, more westward,
on the other hand, Lüneburg in Hannover and Jever in Oldenburg
are still to be called as places where they are found. Afterwards
WaAHNscHAFFE has made mention of a specimen found near Havelberg
(14), and various observations attached to it as to the value of these
boulders for the determination of the direction of the ice-flow and the age
of the diluvial deposits, in which they are found. [ hope afterwards
to recur to this question. In Pomerania the Borealis-limestone is not
(vet) known (31, p. 83), no more, it seems, in Mecklenburg. SToLLEY
afterwards states that he has found it again in Sleswick-Holstein,
but differs in this respect from all other notations known to me that
he has come upon greater numbers of dolomites than of limestones
(30, p. 98). Lastly, these boulders are neither rare in the regions
south of the Russian Baltie provinces.

While, as we see, an enormous tract is taken up by the erratics,
the Borealis-limestone occupies but a very small part as solid rock.
In the eastern baltie (8, p. 48) it forms Scumipr’s zone G,, the
Borealisbank, which stretches in E.-W.-direction throughout Esthonia,
in the shape of a zone narrowing to the west, which also appears
in the island of Dagö. The rock consists of limestone or dolomite,
just like the boulders, and for a long time only single valves of
Pentamerus borealis Ercnw. have been found in it. Afterwards ScumiptT
has discovered also complete specimens of this species, in a marly
variety of the rock from the neighbourhood of Hapsal, as already
stated (27, p. 130).

Of this eastern-baltie occurrence I possess limestones for comparison
from Risti in the extreme west of this zone on the mainland and
dolomite from Pantifer in East-Esthonia, Our boulders correspond

( 509 )

very well with those limestones as regards the principal features,
though they are not interchangeable with the latter. More perfect
still is the correspondence of our dolomites with the sample from
Pantifer. From this it appears sufficiently that we have to look for
the origin of our boulders in the eastern-baltic zone. Besides all
authors agree about this question. Of course we do not mean to say
that these Groningen erratics must of necessity originate in the zone
now known. As the Borealis-bank is also found in Dagö, it may be
surmised that it stretches, or formerly stretched, still farther westward
under the sea, and the very uniform petrographical character of the
rock throughout Esthonia leads us to adopt the opinion that this sub-
marine continuation may also be considered as the possible place of
origin of our boulders. Of course it is impossible to indicate a definite
point in this zone,
30. Elegans-limestone.

With this term, referring to one of the most important fossils of
this species of boulders, I denote a erystalline-limestone, generally
fine-grained, sometimes almost impalpable, but still oftener rather
coarse-grained. Calcite, bright as water often occurs rather regularly
spread through the stone, but not in large quantities. The limestone
is not perceptibly dolomitic nor marly, as in the solution in nitric
acid only a small part is left and this solution produces no or hardly
any reaction with magnesia. Its colour is bright-gray, sometimes
rather yellowish-gray, rarely blewish-gray ; when weathered, however,
the stone shades from white to yellowish white. Its surface is very
often marked by distinetly visible glacial scratches. Layers are but
seldom perceptible and moreover not very distinct. Fossils are by
no means rare, but belong to a relatively small number of species,
which are mentioned here :

Phacops elegans Sars and Borek sp.
Leperditia Hisingeri Scar.
Strophomena pecten L.
Vincularia nodulosa Erenw.
Vineularia megastoma Ercnw.
Enerinurus punctatus WAHLB.
Calymene sp.

Orthoceras sp.

Proetus sp.

Ptilodictya sp.

Beyrichia sp.

Murehisonia sp.

( 510.)

The first five species almost occur in every piece. Head-shields of
the said Phacops-species are very common, pygidia occur as well,
and an almost complete thorax (which has not been figured by ScHMipT)
has also been found. I have named these boulders after this charac-
teristic species. Equally important is further the presence of the
Leperditia-species, whose valves, both right and left, are occasionally
present in large numbers in a single stone; in the unweathered rock
they are bright brown, weathered nearly white. The mentioned
Strophomena-species is very plentiful, while especially the two bryozoa-
species, mentioned next, sometimes give the stone a peculiar appea-
rance. Though occurring in each of these boulders they are hardly
perceptible in the unweathered rock; they are together in great
numbers at the fractured surface, split along the foliaceous “Mittel-
schicht”, like graceful little white feathers. But the structure of this
fossil may be more distinctly perceived at the weathered surface of
the boulders. The other fossils mentioned are found but rarely and
do not contribute in a great measure to the diagnosis of the rock.

Besides these fossils, however, remains of brachiopoda are very
frequently met with, which no doubt are characteristic, though I
have failed to determine them satisfactorily. Some Phynchonella- and
Orthis-species are undeniably present among them. One piece also
contains white, globular and angular erinoid-stems. Also the presence
of graptolite-remains is most interesting ; these, however, have been

y
preserved too incompletely to be specifically determined.
Of this species of boulders, thus petrographically and palaeontologi-
cally characterized there are among my material 33 pieces from the

following places :
o

“Noorderbegraafplaats”’, Groningen 2
“Boteringesingel’, 7 9
“Noorderbinnensingel ”, EE 1
“Nieuwe Boteringestraat”’, 5 il
Between “Parklaan” and ‘“Heerebrug”, ,, 1
“Nieuwe Veelading”, 5 3
Jehind “het Sterrebosch’’, 5 1
“Schietbaan”, A 1
Café “de Passage”, Helpman 2
“Hilghestede”, ‚r 4
55 1

Between Helpman and Haren 2
Villa “Edzes” near a alt
3 1

Groningen 1

“Old Collection” 2

(511)

To determine the age of these boulders, which, as the above list
shows, are by no means rare near Groningen, all that is known
about the oceurrence of the characteristic fossils is communicated
below as completely as possible.

Phacops elegans Sars and Borck sp.

Scominm 8, -peo02, 0. Tf 4; TeX ef, 10—12: TE XL, £17,

is said by Scumpr to oceur in the Raiküll strata and the Estonus-
zone in Esthonia. It was first found in the oldest of the two zones
G,, near Wahhoküll in the centre of East-Esthonia, together with
Strophomena pecten L. and Diplograpsus estonus Schot ; its locality
in the H-zone is almost straight to the south of it near Törwe in
the neighbourhood of Talkhof, on the border of Livonia. Complete
specimens, however, have not been found ; the thorax found here is
there unknown.

Most probably P. quadrilineata ANG. Laxpsrröm, 12, p. 43; 17, p. 2;
is identical with this species; it has been described by the latter
from the oldest strata of the Upper Silurian formation « and h, near
Wisby. Moreover Scumipr mentions Farö and Lau there, places which
according to him belong to his middle and youngest zone in Gothland
(8, p. 74); this notation borrowed from Lixpsrröm seems to me to
want confirmation. In Sweden this fossil is also found in Dalarne
(17, p. 27) and if P. elliptifrons Esmarck must be identified with
this species (which I cannot state with perfect certainty), in Jemtland
(29, p. 269) as well. The stage there argues a conclusion in the
affirmative. This fossil is not known from Scania. On the other hand
it is found together with Leperditia Hisingeri Scar in Malmö
in the bay of Christiania, it seems in a corresponding stratum
(8, p. 74).

This species is not known in the literature of German boulders,
though Wicanp makes mention of Phacops Stokes’ Mitxu Epwarps,
the English fossil, which is most like our species (16, p. 40). The
illustrations of this fossil found near Rostock in Phacites-sandstone
prove, however, that this species certainly does not correspond in
all respects with our specimens. Lhacops prussica Pomprcks, may
also be taken into consideration but neither the latter is completely
corresponding with those from Groningen; the rock in which this
species occurs in East-Prussia, “krystalliner, gelblich-grauer ober-
silurischer Kalk” would not argue against it (23, p. 19). Roruer
does not mention our species.

( 512 )
Leperditia Hisingeri Scumwpr.
ScuMIDT, 10, p. 14—16, T. I, f. 5—7.

identical with Leperditia Schmidtii Kormopix, has already been known
for a long time from the neighbourhood of Wisby, where it frequently
occurs in the Stricklandinia-shale; esp. near Snäckgärdet I found
beautiful loose specimens. But it also occurs south of Wisby in
LinpstrOMs stage c,; according to KormopiN moreover also in the
shale of Westergarn (c‚) and Capellshamn (7, p. 138). In Esthonia
this fossil belongs to the zones G, and G,, and is found there in
many places, also in Dagö. Our specimens are on an average much
smaller than those of Gothland, but for the rest correspond very
well in their relative dimensions with the description of the true
form. As already stated, this species is also found near Christiania.
Linpstrém states moreover, that it is found in Seania (17, p. 25);
I failed to find out on what grounds this notation is based, and
have reasons to doubt of the truth of it.

Kirsow writes that he has found it in German boulders from
Spengawsken in West-Prussia and in a limestone (not corresponding
with ours) which curious enough also contains Leperditia baltica
His. (11, p. 274). Cnumietewskt on the other hand has not come
upon the true species in East-Prussia and Kowno (34). KRaAusE,
again, has found it in Neubrandenburg (24, p. 7) and Sroruey in
a bright yellow, erypto-öolitie limestone from Sleswick-Holstein (30,
p. 109).

Strophomena pecten L.

is a fossil generally occurring in the Jörden and Raiküll beds in
Esthonia; in Gothland it is frequently found near Wisby and our
specimens correspond most with this occurrence. Linpsrröm, however,
mentions it from e-h; hence I should not be surprised if different
varieties of this species were to be distinguished. Wiman also states
to have found it in Jemtland in the quartzite with Phacops ellipti-
frons Eso. (29, p. 270).

Gace has deseribed it from boulders of Beyrichia-limestone
(20, p. +7) from East- and West-Prussia ; various authors moreover
mention it in boulders of different age, which strengthens my opinion
to draw no important conclusions from this species.

Vincularia nodulosa Eicuw, and V. megastoma Etcuw.
Eicuwatp, 5, T. XXIV, f 8 and 9,

are very characteristic of the Raikill stratum in Esthonia and are

(513)

found there everywhere, though they oecur in the Estomus-zone as

well (8, p. 43).

Encrinurus punctatus Wane.
is only present in a single piece and is a fossil found in all Upper
Silurian regions throughout all zones so that this species is of no

value for the determination of the age.

If we take these results together we get:

Esthonia. Gothland.
Phacops elegans Sars and Borek. GC, — H a—b.
Leperditia Hisingert Scumipr. Gas Gs be.
Strophomena pecten L. Gs e—h.
Vincularia nodulosa Eicuw. G,—H
Vineularia megastoma Ercuw. G,—H

It appears from this distinctly, that these boulders are remains of
an equivalent of the Raiküll zone G, in Esthonia. As to Gothland,
the comparison with Lixpstrém’s zone 6, if a comparison is desired,
is the most probable one.

Moreover this result is especially interesting, because boulders of
this age are not known in literature that I know of. In the Groningen
collection on the other hand some pieces have been brought to this
zone long since. But once NorrLinG mentions a stone belonging to
this stratum which, however, contained no determinable fossils and
was only under reservation by reason of the great correspondence
to a piece of limestone from Raiküll, counted as a representative of
this zone (9, p. 291). Roemer doubts of this (13, p. 77).

As regards the origin of these boulders, it may first be stated,
that none of the regions where only one or a few of the fossils
characteristic of this occur, viz. Norway, (Scania), Dalarne and Jemt-
land, can be taken into account. Besides the petrographic nature of
these deposits precludes this supposition altogether. In Gothland on
the contrary these fossils, with the exception of the two bryozoa-
species, are all found. But the rock occurring there (almost always
shales) does not show petrographically the least correspondence with
our limestone. In fact these boulders must not be considered to origi-
nate in Gothland.

Lastly as regards Esthonia: The Raiküll zone, G,, (8, p. 43)
extends from Laisholm in Livonia and Wahhoküll in East-Esthonia
westward as far as Dagö; in the eastern part the zone is wider and

(514)

narrows westward. It almost always consists of two systems, now
limestone, then dolomite. In the above-mentioned passage SCHMIDT
gives no further petrographical description of the rock; but afterwards
he speaks once more (33, p. 308) of a “dichten, festen, etwas kiesel-
haltigen hellgelben Kalkstein, der demjenigen unsrer Raiküll’schen
Schicht am meisten gleicht”. For want of material for comparison: 1
dare not conclude from this a great correspondence with our lime-
stone. Further it is striking that Scumipr says that petrifactions are
comparatively rare in the Raiküll stratum, except corals. Now our
boulders contain a comparatively great number of fossils, whereas
corals are altogether wanting. Just the reverse argues the fact that
graptolites occur in both, which though shortly described as Deplo-
graptus estonus Scum. (2, p. 226), are not vet figured. Perhaps the
same species may be found in our pieces.

By reason of the differences adduced above, I deem it little probable,
that the Raikiill stratum in Esthonia itself can be considered as the place
of origin. It is not impossible that the submarine continuation of this
zone consists of a rock more corresponding with our boulders. For
the present this question cannot be solved more completely though
material for comparison esp. from G, in Dagö, may render valuable
services.

LITERATURE.

1. Roemer, F. — » Ueber holländische Diluvialgeschiebe’.
Neues Jahrbuch fiir Mineralogie, etc, 1857, p. 385—592.

2. Scmuprt, F. — » Untersuchungen über die Silur-Formation von Ehstland,
Nord-Livland und Oesel’’.

Sep.-Abdruck a. d. Archiv f. d. Naturkunde Liv-, Ehst-und Kurlands, le Ser.,
Bd. IL, Lief. 1, p. 1—248. Dorpat, 1858.

3. Rormer, F. — »Versteinerungen der silurischen Diluvialgeschiebe von

Groningen in Holland”.
Neues Jahrbuch, etc, 1858, p. 257—272.

4. Scrmpr, F. — »Beitrag zur Geologie der Insel Gotland, nebst einigen
Bemerkungen ueber die untersilurische Formation des Fest-
landes von Schweden und die Heimath der norddeutschen.
silurischen Geschiebe’’.

Archiv f. d. Naturk, Liv-, Ehst- und Kurlands, Ie Ser. Bd. II, Lief, 2, no. 6,
p. 403—464; 1859.
5. Ercuwatp, E. p’ — »Lethaea rossica ou Palaeontologie de la Russie”.
Atlas. Ancienne Période.
Stuttgart, 1859,

( 515 )

6. Martix, K. — »Miederländische und nordwestdeutsche Sedimentärgeschiebe,
ihre Uebereinstimmung, gemeinschaftliche Herkunft und
Petrefacten”.
Leiden, 1878.
7. Kormopin, L. — »Ostracoda Silurica Gotlandiae’’.
Ofvers. af Kong]. Svensk. Vet.-Akad. Förhandl., 1879, no. 9, p. 133—139;
Stockbolm, 1880.
8. Scuampr, F. — »Revision der ostbaltischen silurischen Trilobiten, nebst
geognostischer Uebersicht des ostbaltischen Silurgebiets’’.
Abtheilung I.
Mém. de Acad. Imp. d. Se. de St. Pétersbourg, 7e Sér., T. XXX, no. 1; 1881.
9. Noeriinc, F. — »Die Cambrischen und Silurischen Geschiebe der Pro-
vinzen Ost-und Westpreussen’’.
Jahrbuch d. k. pr. geol. Landesanstalt ete, fiir 1882, p. 261—324 ; Berlin, 1883.

10. Scumipt, F. — »Miscellanea Silurica LIL.

1. Nachtrag zur Monographie der russischen silurischen
Leperditien.
2. Die Crustaceenfauna der Eurypterenschichten von
Rootzikiill auf Oesel”’.
Mém. de l’Ac. Imp. d, Se. de St. Pétersbourg, 7e Sér., T. XXXI, no. 5; 1883.

11. Kirsow, J. — »Ueber silurische und devonische Geschiebe Westpreussens”.

Schriften d, naturf. Ges. in Danzig, N. F., VI, 1, p. 205—300; 1884.

12. Lixpsrrém, G. — »Frteckning pa Gotlands Siluriska Crustacéer’’.

Ofvers. af Kongl. Vet.-Akad. Férhandl., 1885, no. 6. p. 37—100.

13. Roemer, F. — »Lethuea erratica oder Aufzählung und Beschreibung der
in der norddeutschen Ebene vorkommenden Diluvialgeschiebe
nordischer Sedimentirgesteine’’.

Palaeont. Abhandl., herausg. v. W. Dames und E. Kayser, IL, 5, 1885.

14. Wannscnarre, F. — »Bemerkungen zu dem Funde eines Geschiebes mit
Pentamerus borealis bei Havelberg’’.

Jahrbuch d. k. pr. geol. Landesanstalt etc. f. 1887, p. 140—149; Berlin, 1888.

15. Linpsrrém, G. — »Ueber die Schichtenfolge des Silur auf der Insel Got-

land”.
Neues Jahrbuch, 1888, I, p. 147—164.

16. Wicanp, G. — »Ueber die Trilobiten der silurischen Geschiebe in Mecklen-
burg”. I.

Inaug.-Dissert , Rostock; Berlin, 1888
17. Lanpsrröm, G. — » List of the fossil faunas of Sweden. II. Upper Silurian’.
Stockholm, 1888.

18. Zerse, O. — »Beitrag zur Kenntniss der Ausbreitung, sowie besonders der
Bewegungsrichtungen des nordeuropdischen Tnlandeises in
diluvialer Zeit”.

Inaug.-Dissert., Königsberg, 1889.
19. Van Carker, F. J. P. — »Die zerquetschten Geschiebe und die nihere

Bestimmung der Groninger Mordnen-Abla-
gerung”.
Zeitschr. d. deutsch. geol. Ges., XLI, p. 343—358, 1889.

( 516 )

20. Gacen, ©. — »Die Brachiopoden der cambrischen und silurischen Ge-

schiebe im Diluvium der Provinzen Ost- und Westpreussen”’.
Beitr. z. Naturk. Preussens, herausg. v. d. phys.-oekon. Ges, zu Königsberg,
6; Königsberg, 1890.

21. Scumipr, F. — »Bemerkungen über die Schichtenfolge des Silur auf
Gotland”.

Neues Jahrbuch, 1890, IL, p. 249—266.

22. Dawes, W. — »Ueber die Schichtenfolge der Silurbildungen Gotlands und
ihre Beziehungen zu obersilurischen Geschieben Nord-
deutschlands”’.

Sitz-Ber. d. k. pr. Akad. d. Wiss. zu Berlin, 30 Oct. 1890, Bd. XLII,
p. L11—1129.
23. Pourncks, J. F. — »Die Trilobitenfuuna der ost- und westpreussischen
Diluvialgeschiebe”.
Beitr. zur Naturk. Preussens, herausg. v. d. phys.-oekon. Ges. zu Königsberg,
7; Königsberg, 1890.

24. Krause, A. — »Die Ostrakoden der silurischen Dituvialgeschiebe”’.

Wiss. Beilage z. Programm der Luisenstädtischen Oberrealschule zu Berlin;
Ostern, 1891.

25. Van Carker, F. J. P. — »De studie der erratica’.

Hand. v. h, 3e Natuur- en Geneesk. Congres te Utrecht, p. 360—370; 1891.

26. SreusLorr, A. — »Sedimentärgeschiebe von Neubrandenburg’, p. 166?

Archiv d. Ver. d. Fr. d. Naturgesch. in Mecklenburg, Bd. XLV, p. 161—
179; 1891.

27. Scummr, F. — »Einige Bemerkungen über das baltische Obersilur in
Veranlassung der Arbeit des Prof. W. Dames über
die Schichtenfolge der Silurbildungen Gotlands”.

Bull. de V’Ac. Imp. d. Se. de St. Pétersbourg, N. S. If (XXXIV), 1692,

p. 381—400; also: Mél. géol. et paléont., tirés du Bull. etc, T. I, p. 119 —138.
28. Baruer, F. A. — »The Crinoidea of Gotland”. I.
Kongl. Svenska Vet.-Ak. Handl., XXV, no, 2, 1893.

29. Wivan, C. — » Ueber die Silurformation in Jemtland”.

Bull. of the geol. Inst. of the Univ. of Upsala f. 1893, I, p. 256—276;
Upsala, 1894.

30. Srorrey, E. —- »Die cambrischen und silurischen Geschiebe Schleswig-
Holsteins und thre Brachiopodenfauna”. 1. Geologischer
Theil.

Archiv f. Anthrop. u. Geol. Schleswig-Holsteins u. d. benachb. Gebiete,
I, 1, p. 85—136; 1895.
31. Couny, E. and Dreckn, W. — » Ueber Geschiebe aus Neu-Vorpommern
und Rügen)’. Erste Fortsetzung.
Sep.-Abdr. a. d. Mitth. d. naturw. Ver. f. Neuvorpommern und Rügen,
Jg. XXVIII, 1896.
32. Steeerr, L. — » Die versteinerungsführenden Sedimentgeschiebe im Glacial-

diluvium des nordwestlichen Sachsens”’.
Zeitschr, f, Naturwiss., Bd. 71, p. 37-188; 1898,

( 517 )

33. Scmapr, F. — »Ueber eine neue grosze Leperditia aus lithauischen Ge-
schieben’’.
Verhandl. d. k. russ. Min, Ges. zu St. Petersburg, 2e Ser., Bd. XXXVII,
Lief. 1, VI, p. 307—311; 1900.
34.-Cumetewski, C. — »Die Leperditien der obersilurischen Geschiebe des
Gouvernement Kowno und der Provinzen Ost- und
Westpreussen’’.

Schrift. d. phys.-oekon. Ges. zu Königsberg, Jg. 41, 1900, p. 1—38.

35. Hotmsrrém, L. — »Otto Torell’, Minnesteckning.
Geol. Fören. i Stockholm Förhandl., XXIII, H. 5; 1901. (Separat-Abdruck).
36. Jonker, H. G. — »Bijdragen tot de kennis der sedimentaire zwerfsteenen

in Nederland.

LI, De Hondsrug in de provincie Groningen.

1. Inleiding. Cambrische en ondersilurische zwerfsteenen’’.
Acad. Proefschrift, Groningen, 1904.

GRONINGEN, Min.-Geol. Instit., 31 December 1904.

Chemistry. — “On some phenomena, which can occur in the case of
partial miscibility of two liquids, one of them being anomalous,
specially water.” By J. J. van Laar. (Communicated by Prof.
H. A. LORENTZ).

1. In the second part of his Continuitiit (1900) *) Prof. vax DER
Waars has given the theory of the so called longitudinal plait on
the y-surface, and in the last Chapter (§ 12, p. 175 sequ.) he gives
moreover a special, ample discussion of this plait, in particular with
regard to anomalous components. It is shown there, that for the
appearance of certain complications, which can present themselves
at this plait, one of the two components must be anomalous *).

In the following pages I shall try to explain the appearance of the
different particular forms, which can present themselves, when one
of the components is associative, specially when this anomalous
component is water.

2. We begin to remember briefly the theory of the phenomenon
of partial miscibility for binary mixtures of normal substances.

It is well known, that the total thermodynamic potential is repre-
sented by

1) p. 41—45.
2) Also compare These Proceedings of Nov. 5, 1902,

(518)

Z = — Z(n,k) T (log T — 1) + Zln,le\) — T F(A (1,),) —
— f pee + pu + RT X(n, log n‚),
or ®
Li == 2(n,C.)\— | [ree — RT Yn, . log Zn, =p | + RTS sz).
e =n,

Differentiating subsequently at constant 7’ and p with respect to

n, and n,, we get:
nF 0
u, == ER log
ane ny =n;
OZ 0
=z Sg, as |
On, A 25 |

where C, and C, are pure functions a the iN eee represented by
n= — kT (log Lias Ta).
C, = — kT (log T — 1) + (e)o — Ta)

whereas the quantity @ is given by

@ = (pdw — RT Zn, log Sn, — po. 2 2 2 2)

e
The meaning of the different quantities 7,, (e,),, (1,),, ete. ete. is
supposed to be known.
We will substitute now the variables n, and n, by w, so that
n,=1—2, n, =x and 2n,=1. As w is, just as Z, a homogeneous
function of the jirst degree with respect to 7, and ,, we may write:

dw
m=e,—(o- x oe) + RE log (1 — x) |

av 5
(2)
nl Cen Di) ar log «x |

Now, when there is a plait on the Z-surface, the spinodal-curve,
that is to say its projection on the 7, 2-plane, will be given

Petia MZ
by the condition pee or also, p, being ae ‚ and
Fi as wv
k OZ du, Ô Ou, 0
I= , by — =0 or == (0)
f 2 ) Ow Ow
We therefore find for the equation of this curve in the 7'v-plane:
07a RT
D= == (),
Det l1—z
or
ir)

RT = «(1 — ae) erect (3)

( 519 )

If we use the equation of var per Waals:
Sn, RES va

Lam v—b vy?
then we obtain:
w = Xn, . RT log (» — b) +5 — — RT Ln, . log Zn, — pv *).

Supposing now, that in the case “of liquids the external pressure p
(or the vapour-tension} can be neglected with respect to the mole-
. a . u .
cular pression — , the equation of van DER Waars may be written:
v
a an, RT
oo ob

and the expression for @, when in the same manner pv is omitted

.

a .
by the side of — , passes into
v

PO oy ard a
oO Zn, KT log tm, log Xn, ,
la
or
= T a
o = 2n, . RT log +—,
vi v
that is to say into
“Gt
mn a
o = RT log —
/t v

2

when Yn, = 1. For — we find consequently :
av

2

Po GE RP 0? ; a
Ou? TEE Te ok Ow? a or’

by which the equation (3) of the projection of the locus of the points
of inflection on the 7, z-plane passes into

0° a 0? a
WSS ie Mlt
( Z) Fe 5) uy Or? v? |

or into

»

1 If D= fw), ten (par still gives a term ne | oe . But this term may be

v—h

regarded as independent of w, and so can be added to the temperature function Cj.
36
Proceedings Royal Acad. Amsterdam, Vol, VII.

( 520 )

n a 5 :
The term with /og— was introduced some time ago by VAN DER
v 5 5
Waats'); in the original theory this term was neglected, and so the

ik
equation (4) was simply R7’= aw (1—.) 43 (5)

a a
In consequence of the relations — = RT, pes where ac-
v v

cording to the variability of the liquid-volume v, the coefficients f
and y will still vary slowly with the temperature (fis the well
known factor of the vapour-tension, which may be put circa 7), we
can also write for (4):

Ske
mr TE Ow?
JSS Be (A)
0? log pe
1+.a4(1—z2) eae
i

We see, that only in the case, that the critical pressures of the two
components differ little, the term with Log p. can be omitted. This
will be also the case, when w is in the neighbourhood of 1 and 0.
But in all other cases if would be inaccurate to omit a priori the
designed term.

Further we write:

a = (l—«)? a, 4- 24 (le) a,, + 27 a,

v= (l—a) v, + av,
since for liquids at low temperatures ¢ can be supposed dependent on
wv in entirely the same manner ash = (1—v) 6, + «b,. The molecular
volumes 7, and v, must then be regarded, just as >, and 5,, as constant
or as slowly varying with the temperature *). We then find after some
reductions :

1) These Proceedings, in Ternary Systems, specially IV, p. 96—100. (June 12,
1902); see also July 13, 1904, p. 145 sequ.

2) If we substitute in the case of liquids v by b, and then write 6 = (1—.r) bj + «bz,
the difficulty arises, that in that way quantities of order v—b are neglected
against those of order 7, and the question would present itself, if this is only
upon very definite conditions no¢ in contradiction with omitting p by the side of

a

ye

(This observation was kindly made to me by Prof. Lorentz).

{ hope to escape this difficulty by not substituting » by 6, but by simply
supposing the volume v linearly variable with « in the case of liquids at
low temperatures; by writing therefore for 7, analogous to the expression for b,
b= (1—2) 1, Her. As 1 remarked, vj and #7 still vary slowly with the tempe-
rature, whereas bj and », of course would be perfectly constant. Now it

( 521 )
= (<)= = (a, Vg d =F dv, — 24,4210)
wv v

or — when ‘we suppose for normal components the relation of
BerTHELOT, viz. d,, =Wa,d,, as approximately exact:

0? 2
ia(<)= = @; VAS REE EEEN (53)

=

As the second member will be always positive, even if a,, might
be Va.a, *), the curve T'= f(x) will always turn its conver side
162 ’ v v
to the z-axis.

:
We will now determine EE log . With a,, =Wa,a, *) the expres-
av
sion for a becomes:
a= [(l—e) Wa, 4e Ya,]’,
so that
a (lez) Wa, + 2 Ya,

log — = 2 log
v (la) v, + ev,

Consequently we have:

5 eats F a ak a 1
will be better justified to subsitute a by fRT, than 5 (and afterwards = by
1

a F : ;
fRT, and De by fRT), where f will vary in the same manner as v with
2

temperature. For it is easy to ‘show, that the expression for the vapour-tension

DY

E ke A) y2 &| db P : :
for a single substance at low temperatures is log — — —— — (v is in
Pp RT v—b
l

the first two terms the liquid volume), whence we can deduce, in connexion with

an : De Te ziel Ee a 7

the empirical relation log" =f (7 — 1); where f is circa 7, that > fl
The error made by supposing 7 linearly variable with z, will certainly be
much smaller than by putting v =O. In that way errors of,at least 16°/) would

be made, since a will be nearly */, for liquids in the neighbourhood of the

melting-point.

The quantities 7; and v, can now also immediately be substituted by the expe-
rimentally determined values in the liquid state.

1) See van DER Waars, These Proceedings of Oct. 8, 1902, p. 294.

2) Although there is no sufficient reason for this relation, [ have supposed it
approximately exact, also because only in this case a simple expression could

a2

. 0 a
be obtained for — log —.
Cut g Vv

36

( 522 )

Ò ie gen eG Va, Va, UT
Ow ve Va v

es bg 5 = 2 | a | eo. o 02 (0)
7

a v a

and therefore

This expression can be reduced to a different form, and then
we find:
0? a 2

— log —==—, (v, Va,—, Va.) (0, Wat; Va.) + 20 (Waak

Ow? TE Gok
whence it appears, that the factor v, Wa, —v, Va, occurs in the

2 2
: 5 a 2 3 a
expression for — log — as well as in that for —| — |.
Ow? vy? Ou?

7
a a, ay EZ oe
Now when v,Va,=0,V 4, or —, = —,, when in other words the critical
v v
1 2
; : O2 a
wessures of the two components are equal, then — log — becomes — 0.
/ q On? e pe

5
But then simultaneously zl) will be = 0, and the whole longi-
tudinal plait will disappear, (at the same time the curve Te = /(«)
will then pass into a straight line).

We see therefore, that for occurrence of the phenomenon of partial
miscibility at attainable, that is too say at not to low temperatures, the
critical pressures of the two components must difjer as much as possible.

Now this is not the case for the greater part of normal substances,
and that is the explanation of the well known fact, that for mixtures
of normal substances the phenomenon of limitated miscibility has
been so very rarely found at the common temperatures.

When we substitute (5) and (6) in the equation (4), then we
find finally :

JE ON aR (Mi ye A ENA i te

where eu eter (00)

A= 2e (le) je Va Va) |

a

This would be a pure parabola, if v and 1 + A were indepen-
dent of

3. We will now determine the values of « and 7’ for the ‘critical

É ye Oe en Ee Of
point of miscibility.” For this the conditions aa =O andel
‘ ‘ Ou

( 523

combined must be satisfied, or — what is the same -— the conditions
du, dT
—=—0)0 ; —=J,
Ow da

as is obvious. Now from (7) follows, when 1 + A is supposed
independent of #, which will be certainly permitted, in consequence
of the small values of A in the ease of normal substances:

1 i a 2 (wv, Vat, Va)’ { 12 ve 32 Cal

R ie —— :
da IA | Dh vt

v,)}s

as v= v,-+ 2(v,—v,). This expression becomes = 0, when
(L—22) 1+re) — 3 ra (la) = 0,
VTV, B :
where „=S. ‘This’ yields:
rv? —2(r + 1)¢a4+1=0,

whence
1 EEE:
Be = rde 1) = Vri erde 1).
5

When r= 0, that is to say when v, =»,, then « =0,5. At all
events this will be approximately the case, if A should still be in
any way dependent on w«.

We will reduce now the equation (7) somewhat. With a, = 7 RTv,
and a, =f RT,v,, where T, and 7, are the critical temperatures of
the components, these equations pass, after substituting x, for «, into
RT, = 2e (le) ee SEA)

eas 3
Ore (: -f oe =)
1
VT we, VT)

JA == ve (1 = We) pater ls a ih ath a En
fe,” (: JE av My we =) IV 1 1 Hel a3 Vv pel
}

1

or with 7, = AT, and »v, = gv, into

px BVP oa
Te = 2 fv. (l — we ~- Tl + Be
F ®el Ü Ja SDE (1 + A) |
Le: b, = (8)
pij et

(
= je a
coe mie (let (V Ap—1) 2)? |
since (py — V4)? = 9 (Vo — Wvo)’.
We shall illustrate these equations by an example. In order to
find the critical point as high as possible, we will choose two

( 524 )

normal substances, of which the critical pressures differ as much as
possible. We take therefore ether and carbon disulphide. The critical
data are the following:

CS T, =b48R ip = 76 alm.

3

ether | Tr AGT" Te pr 30 atm:

3

E v ded
In order to determine p= —, we remark, that EE
UV; Y Pi
FR i . > ay, ” Tal a,
v, = —.—, as for instance — —=fRT, and —=yp,. We have there-
Y Ps vy VY,

fore:

OE i mencl 8s
. D .
that is to say p — Oa, where the proportion ee represented by a.
st ; 5
2
Now for the designed substances 6 = 0,852, a = 2,17, so that
we find p= 1,85. Since r= py —1, the equation for 2, passes into

1 Ss
a Gn er): te Te ORE)
and hence we find for «a, the value 0,29. Further y/4==0,923,
io = 1,36, f= 7, and so (8) becomes:

14 X 0,206 « 1,85 X 0,191
one Goan eS

548: (1 + Ad),

eC
Tis

or
1,019
ms
1,94
We have further:
0,723 0,0650
1,555 1,153

548: (1 + A.) = 288:(1 + Ad.

=

A. = 0,412 — 0,412 X 0,409 — 0,169,
so that we find for 1 + A, the value 1,17.

Hence 7, becomes 288: 1,17 = 246 = — 27°C.

The critical point of the chosen substances lies therefore still a
thirty degrees beneath the common zero of Celsius. And for the
greater majority of other normal substances we will find for 7; still
much smaller values — because the critical pressures will differ
there in most of the cases less than in the case of ether and CS,.

4. All that precedes now undergoes important modifications, when
one of the two components is anomalous, specially water, For in
the first place the critical pressure of the water is very high, not

( 525 )

less than 198 atm. so that it will differ much from the critical

pressures of most of the other substances. And in the second place the
value of v, is here so extraordinarily variable with the temperature.
Water is in this respect exceptional in Nature, and gives therefore
rise to very peculiar phenomena, which are not found with other
substances, or not in that degree. Alcohol e.g. is also an anomalous
substance, but neither is the variability of the molecular volume
there particularly great, nor the critical pressure particularly high.
We know, that the variation of the molecular volume finds its
cause in the decomposition of the double molecules with the temperature.
gradually grows smaller and smaller, the quantity

(eVa Va),
which principally determines the value of 7, will become greater
and greater. And the initial value of that quantity is in the case
of water as one of the components already higher than for mixtures
of normal substances. This is connected with the high critical

‚Because v,

pressure of water, being 198 atm., whence can be calculated, that
the critical pressure — if water continued to consist of only double
molecules — would yet still amount to circa 66 atm, i.e. higher than
that of most of the normal substances. [Of course the designed express-

Vartan

ion will inerease with decreasing values of v, only when tee aa
1 2
that is to say, when the critical pressure of the first component is
greater than that of the second. This condition will nearly always
be satisfied, when we assume water as the first component |.
As said, the decrease of v, is very considerable in the case of
water. I remember, that I found some years ago’), that for 18 Gr.

water v, = 19,78 cem., when all the molecules are double ; and only

— 11,54 eem. for 18 Gr., when all the molecules are single. When
therefore the temperature increases from nearly 90° C., where all
the molecules are double (supposing, that the water had not congealed
long before), to cirea 230° C., where all the molecules have become

single, then v, will diminish down to nearly *

|, of its original

value.
{In the same Memoir I showed, that in this fact lies also the
explication — qualitative as well as quantitative — of the well-

known phenomenon of maximum density at 4° C,|
Now the consequence of this variability of v, will be, that the
second member of (7) — we will represent it (divided by R) in

DZ. f. Ph. Chemie 34 (Jubelband vay ‘rt Horr), p. 1—16, specially p. 13,

( 526 )

the following by A — will be no longer a constant for a definite
value of ., but a funetion of temperature.

If we draw therefore (see fig. 1) the straight line OM, which
divides into halves the angle of coordinates (O7’ is the axis of
temperature, OA' that of the values of A’) — then for mixtures of
normal substances the point of intersection of the straight line K =
const, which runs consequently parallel with the 7-axis, with the
line OA will represent the temperature, corresponding in the Tyx-
projection of the spinodal curve with the chosen value of x. If this
were w,, then we should find in this manner 7’. That temperature will
be — as we have shown on the preceding pages — extremely low,

On the other hand, in the case of anomalous mixtures, that is
here: where one of the components is an associative substance, the
straight line AA’ will transform itself into two straight lines, joined
by a curve (see fig. 2). The first straight line corresponds then with
the temperatures, where all the molecules are double, that is therefore
in the case of water below — 90° C. ; the second straight line will
correspond with the temperatures, where all the molecules have become
single — so for water above 230°C. The joining curve will cor-
respond with the temperatures between — 90° C. and 230° C., where
the process of dissociation of the double molecules is going on.

Several cases can occur here, which presently we will briefly discuss.

5. We should now have to deduce an expression for R7' and
4, analogous to (7), but this time for the case that one of the
substances is anomalous. The required considerations and calculations
will not be reproduced here, however, because I shall do so in the
more ample Memoir, which will soon be published in the Archives
Teyler. We therefore will limit ourself to the communication of the

final result, viz.

RT = 22 (1—2) (: +

1—B \(v, Wa, vo, Va.) \
av |— — 28 (il \
=) : (44)

v
= Ar (le) 1 Em al SS
[N= 4 ob Te =n

(Reh) ide EN (10)

u” a

These expressions come in the place of the former expressions
(7). Of course they are somewhat more complicated, but they
have essentially the same form, as will be discussed amply in the

À IE

Wu ZERE

designed Memoir. It will only be remarked, that +n, = onm
ad Kd

where 3 is the degree of dissociation of the double molecules ; that
va ='/,(1—86)v, + Br, where v, is the molecular volume of the
double molecules, and 7, represents that of the single molecules;
and then onee more the relation «,,=— Wa,a, has been used, by
which again the calculation of 4 was practically possible.

The expression for « reduced, in consequence of @,=4a,, @,,=2d,,
Oo) Ores UO

0
a=(1—2)? a, 4+ a2’ a, + 2a(1— a) a,

That for 6 or v to v=(1—2)v,, av, where v,, has the

meaning as is indicated above. (The index O relates to the double

molecules, the index 1 to the single molecules of the associative

substance ; the index 2 relates to the second, normal substance).

As is already briefly indicated above, it will be principally the
factor (v, Wa,” Va)", on which the phenomenon, studied by us,
depends. The great variability of the quantity v,, with the temperature
is the only cause of all these peculiar phenomena of partial miscibility,
occurring in the case of mixtures, when one of the components is
anomalous, specially water.

That factor will increase more and more with the temperature,
because v,, decreases in consequence of the continual formation of
new single molecules from the dissociating double molecules

,
01

a
single molecule being much smaller than half a double molecule.
(compare § 4).

It is evident, that the denominator v
with the temperature, so that the value of the second member of
(10) will inerease still more. The variations of the other terms
have comparatively but little influence.

* (by v,,) will equally diminish

6. What will now be the different forms of the plait — i.e. in
the 7c representation — when the course of the curve A = /(7’)
(see § 4) is continually modified with the different components added
to the water? (We call attention to the fact, that A’ represents the
second member of (10), divided by #, and that the following figures
indicate therefore the graphical solution of the equation 7’ = A
with respect to 7’).

a. The case of normal substances has already been considered by
us. It is represented by fig. 1. The spinodal curve will have the
same form as in fig. 2.

b. In fig. 2 the straight part of the curve A= /(7), where A
has the initial value A, (all molecules are still double), intersects
the line OM in the point A; whereas the curved part, and the

( 528 )

second straight part, where A assumes the final value A, (all
molecules have become single), lie wholly on the right of OM. The
plait will consequently be identical with that of the preceding case
— only with this difference, that the point A lies below — 90°C.,
where the dissociation of the double molecules begins, so that this
point lies wholly beyond the region of attainable temperatures.

ce. As soon as the value of A’ increases a little, we get the case
of transition of fig.3. The curve A= f(T') touches now the line
OM in B,C, and from this moment the ¢solated plait will begin to
appear, extending itself above the just regarded normal plait, which
lies in unattainable depth. Here it is only two coinciding critical
points in the one point B,C.

d. When the value of A, is still a little greater, the case of fig.4
will present itself, where the line OJ is intersected, besides in A,
in still two other points B and C. The isolated plait above the
normal one is formed now, with two critical points, a lower one
in B and an upper one in C. Everywhere between B and C
K is > 7, just as below A, so that we are, in consequence of
0°Z ; : : Hee 5 :
rede in the unstabie region, i.e. within the spinodal line of
the plait.

This case — or the case of fig. 6 — is realised by a great
number of substances, also in the case of #vo anomalous substances *).

a. In some cases the wpper critical point is found, as in the

CH,

case of water and COG i}

(Rorumenp), and of H,O and #sobutyl-

alcohol (Aurexerew); propably also in the case of water and ether
(Kiopsm and Arexesew), of HO and CO (C,H,), (Rorumunp), of
H,O and ethyl-acetate (Arexerew), and of H,O and amyl-alcohol
(ALexesew), in which latter cases, however, the point C was not
reached. As to water and ether e.g., Kropper has already found,
that the values of « of the two coexisting liquid phases reapproach
each other, when the temperature is lowered. That is an indication
for the existence either of a lower critical point, lying still more
down or of a contraction as in fig. 6.

8. In other cases it is only the /ower critical point, that is observed,
as in the case of water and triethylamine (RoTHMUND), water and
diethylamine (Germ), and of water and g-collidine (RoTHMUND).

1) Many anomalous substances namely can be regarded as normal ones, because
the variation of v is so small; only in the case of water this variation is excep:
tionally great.

(529 )

According to these observations the first mixture has its critical
point (B) at nearly 18° C., the third at 6° C. *).

In the case of water and nicotine?) Hupson (Z. f. Ph. Ch. 47,
p. 113) has observed the complete isolated plait. But here a hydrate
is formed, being decomposed continually, when the temperature rises.
The theory of the phenomenon remains however formally the same:

everywhere, where a pretty considerable variaton in the value of

v presents itself — whatever should be the cause of it — the
existence of such a plait may be expected — as soon as the required

conditions are satisfied.

Still another example is found in mixtures of carbonic-acid and
nitrobenzol (Bicuner), which makes it probable, that CO, in iguid
state is an associative liquid. Indeed, there exist important reasons
in the thermal behaviour of that substance which would confirm
that supposition.

AreN has observed, that CH,Cl and pyridine mix in every pro-
portion, but that the combination, which is soon formed. is nearly
unmixable with both components. In this case again there is found a
lower critical point, for both plaits — i.e. for that, formed by CH,CI
and the combination, and for that, formed by pyridine and the
combination.

It is a matter of course that the existence of a lower critical
point necessarily determines that of an wpper one. With rise of
temperature the /iquid mixture approaches more and more to a
gaseous one, where of course miscibility in every proportion takes
place. (How the plait can transform itself there, and pass into the
transversal plait, lies entirely without the plan of this inquiry).

Inversely we can not always conclude from the existence of an

upper critical point to that of a lower one, because — even, when
the connodal curve begins to contract downward — the case of

fig. 6 can occur.

But this is certain, that when an wpper critical point is found
at ordinary temperatures, we have always to deal with the point C,
and not with A, the latter always lying (see fig. 2) in the ease of
mixtures of water and a normal (or anomalous) substance below
—90° C., and in the case of mixtures of two normal substances
(compare § 3) at most some thirty degrees below 0° C.

Nearly always there may therefore be expected the case of fig. 4,

1) Kuenen (Phil. Mag. [6] 6, p. 637—653 (1903)) could however not confirm
the existence of a lower critical point for diethylamine. In an earlier Memoir
Kuenen has found also a lower critical point for mixtures of C,H, and ethyl
isopropyl and butyl-alcohol,

( 530 )

or that of fig. 6, when partial miscibility presents itself. The normal
plait with the critical point in A will appear only in a great minority
of cases, and can be regarded as highly exceptional. So the mixtures
of water with phenol (ALExesmw), with succinitrile (SCHREINEMAKERS),

with aniline (ALEXEIEW), with dsohutylic-acid (id), ete, ete. — which
all present an upper critical point — will offer with great certainty

examples of the very general case of fig. 6 or of that of fig. 4.
e. Fig. 5 again represents a transitory case, where the value of

K, is still a little greater than in fig. 4. The two plaits — the
normal one and the isolated one — will coincide from this moment

into one continual plait.

f. This will be the case in fig. 6. lt is observed for mixtures
of water and secondary butyl-alcohol (Aurxesew). But, as we already
remarked above, many observations with an upper critical point

may belong just as well — whether the compositions of the two
coexisting phases approach each other at lower temperatures or
not — to this case as to that of fig. 4. The example mentioned

belongs with certainty to the class of fig. 6, because it is observed,
that the values of 7 after beginning to approach each other diverge
again at still lower temperatures.

Fig. 7 shows, that the contraction at D, where the curve A = f(T)
comes into the neighbourhood of the line O.J/, gradually vanishes, so
that the plait at last again will assume the normal form — only
with this difference, however, that the critical point C of our quasi-
normal plait will appear at higher temperatures than the critical
point A of the real normal plait.

Remark. It will be superfluous to remark, that the numerical
calculations by means of the formula (10) can be executed only
then, when the conditions are satisfied, on which that expression is
deduced. That will accordingly only be the case, when really p is to

. . a . .
be omitted against — (see $2), that is to say at temperatures, which
5

are not higher than circa half the eritical temperature (in the ordi-
nary meaning) of the mixture.

7. “The question rises now, what will be the conditions to be
satisfied, that the transitory cases of the figs. 3 and 5 may present
themselves. Here too we only communicate the results of the caleu-
lations, that we have made on this subject. We found namely, that
the isolated plait (fig. 4) is only possible, when the second (normal)
substance has a critical pressure between circa 35 and TO atm, and

J. J. VAN LAAR. “
one of them beir

J.J. VAN LAAR. “On some phenomena, which can occur in the case of partial miscibility of two liquids,
one of them being anomalous, specially water.”

K= Const.

| 0

)

Sig 2

Proceedings Royal Acad. Amsterdam. Vol. VII.

( 531 )

this nearly independent of the critical temperature of these substances
(provided that the latter is between '/,- and 1-time of that of the
water).

All normal?) substances, which possess a critical pressure above
+ 70 atm., mix in every proportion with water; all such sub-
stances, having a critical pressure below + 35 atm., will form a
continual plait (tig. 6).

To the first group of substances belong those with relatively sica//
molecular volume (many anorganic substances and salts); to the
second group those with relatively great molecular volume (many
organic substances).

0? log Pe
Ou
have taught, that this factor at higher temperatures, where 3 comes
into the neighbourhood of 1, can become very great, and also will
be pretty strongly variable with z. So I found for that factor for
B=1(7T= 230°) the values 2,57, 2,54, 2,25, 1,94 and 1,70, resp.
for v= 0,1, 0,2, 0,3, 0,4 and 0,5. But at such high temperatures
the deduced formulae are not longer exact, p being in that case

no longer to be neglected against 7/,.

However, for lower temperatures, where 8 approaches 0, 1 + 4

will not differ much from 1, and will be little dependent on wv. At

As to the factor 14+ A=1+4.24(1— 2) , the calculations

these temperatures — and for these temperatures the formulae are
deduced — 1+ 4 can, when not negleetable, yet be regarded as

a constant factor. So I found for 1+ 4 the values 1,08, 1,10, 1,10,
1,09 and 1,08, resp. for z — 0,1 unto 0,5.

Finally, 1 have applied the formula (10) for the case of triethyl-
amine and water, and found that, whenever the critical pressure,
viz. 30 atm., lies below the above designed limiting pressure of
35 atm., the appearance of a lower critical point at circa 18° C.
is not in contradiction with the given theory. It must not be
forgotten here, that when the temperature, where 3 is practically
—0, lies above — 90°C., the limit in question also will lie below
35 atm.

1) And as we have already seen above, also many anomalous substances, where
the variability of v is small.

( 532 )

Mathematics. — “The equations by which the locus of the principal
aves of a pencil of quadratic surfaces is determined” by
Mr. CARDINAAL.

1. The communication following here can be regarded as a con-
tinuation of the preceding one included in the Proceedings of Nov. 26
1904. It contains the analytical treatment of the problem, of which
a geometrical treatment is given there. It ought to have been con-
ducive to the finding of a surface of order nine; this has not been
effected on account of the calculations becoming too extensive;
however, the form of the final equation has been found.

2. In the first place the equation must be found of the cone of
axes of the concentric pencil of quadratic cones, at the same time
director cone of the locus of the axes of the pencil of surfaces. To
this end we regard the intersection of the two cones, determining
the pencil of cones, with the plane at infinity P, and besides the
isotropic circle situated in this plane; then we have the three
equations in rectangular cartesian coordinates :

A=a,, 2 + 4,,y° Haeg 2° + 2a,, vy + 2a,, 22 + 2a,, yz = 0,

B=b,, 2? + by + bea 2 + 2b,, vy + 2b,, wz + 2b,, ye = 0,

C=27 fy? -- 2? — 0:

Out of these equations we find that of the cone of axes in the
same way as we determine the Jacobian curve of a net of conics:
|
|

| A, B, Gi |
| A, 18 Gi | ==),
LAS op SBE SCA

|

where A,, A,, A,, ete. are the derivatives of A with respect to z, y, 2.
So the equation of the cone becomes

OF Ant Hay Has bur Hbiny + bn

by, v Ae bs 4) = bys

bie + bn 4 + ban

co)
a

n
a
>

|

ly Myf Gy, @ + As, Yi > Gs,
| ;
RS a,,% + Aya Y + Ass

a
x

Without harming the generality we can always assume that the
principal axes of one of the cones coincide with the axes of coordinates ;
from this ensues that we may put 6,,=06,,=6,,=0, by which
the equation of the cone is simplified.

3. After having found the equation of this cone we can pass to
the formation of the set of equations, by means of which is found
the equation of the locus of the axes,

( 533 )

The equation of the pencil of quadratic surfaces now becomes
ff led itn eerie ease AR)
where however A and 4 have a wider meaning than before, A being
ant? + a,,y* + a,, 2? + 2a,, ey + 2a,, wz + Za, ye +
+ 2a,,u + 2a,,y + 2a,,2+4,,,
and 5 being the same expression with the coefficients 5.

Let us now put the coordinates of the centre of the surface (1)
pqr and let us regard this centre as origin ©’ of a new system
of coordinates with axes parallel to the original ones. We then
arrive for surface (1) at an equation in #’, 4’, 2’, in which the terms
of order one are missing and those of order two possess the same
coefficients. The principal axes of this surface are given by the three
equations :

(a, a! J G4 y + a, 2) + À (b, a’ + Bis y' + 6,, 2’) + ka’ = 0,
(a,,0' + a,,y' Hans) + 4(b,, el +5,,y' + b,,2') + ky’ =0,
(a. T+ a3 y =e z) +4 (bs aw ban y + b,, z') hei:

As could be foreseen the elimination of 2 and # furnishes the
same equation as was already found for the cone of axes.

If we wish to form the equation with respect to the original
system of axes, we must put «’ =2—p, y’=y—q, 2 =2- Pr
and make use of the equations of condition for p, g, 7:

(a, ie Àb)p == (a4, a= Ab,,) q + (a; ia àl) r + dis oi DD = |
(41, Sin Ab,,)P + (ass + db) q + (4,3 + Ab) Te dan ie db, = |
!

(as aH Ab) p hi (a; in Ab) q si (an ie Ab55) r “= Cs 4 + abs, = 0.

By this substitution the equations assume the following form:
(aje Haay O52 + 4) HA (bit + Drag + bne Hb) + k(e-p) = 9,
(eo a Ayo = daz =F d,,) si À(be ie Day se boa? = ba) “5 k(y-q) = 0, (3)
(4,50 + day + 4552 + 4,4) + 4(0,,% + 0,,y + 65,2 + Das) + Kler) = 0,
or written shorter

A, + B,~+k(@—p)=0,
Ap ot see Ane t= (ie Q) k=" Ose eee weet ator eae (4)
A, + B, +k (2 —r)=0.

The surface S, is obtained by eliminating p, q, 7, k, à out of the
equations (2) and (4).

4. This elimination leads to extensive ealeulations as the variables
appear also as products two by two, We shall here point out the

( 534 )

general course by which at the same time the application in special
cases is rendered possible.
The equations (4) can be written as follows:

kp = A, + B, A + ka,
kg =A, + B, À ky,
kr = A, + Ba + ka

Let us multiply each of the equations (2) by # and replace the
values kp, kg, kr; we then obtain:

(411 zie db) (A, af B, d dr ka) si (41, ae Ab.) (A, IE Bà al ky) TE \

(4,,; + 40,;) (Ast B,A + kz) + ka, =F kb, A0;
or:

(A, + BAe + (a, + O11 (A, + B, 4) + (Gas + 4,24) (A, + BH

(a,, + 6,,4)(A, + B, 4)= 0.

We likewise find: ‚©

(A, + B, ak + (1g Hbo DA, HBD) + (yy + Baad) (A, + Baa) +

(a, a= bys A) (A, tr B, 4) = 0,
and finally :

(A, slp B, Ak + (ds ar bis a) (A, == Ab) + (4,5 aa bd (A, ain B,A) ae

(ass br ANA, + B,a)=09.- /

If we reduce these equations and if we regard / and 2 as variables,
we shall get as result three quadratic equations, out of which % and 4 can
be eliminated. As however these equations are linear in 4, the elimi-
nation of / can take place without any difficulty. By putting the
values of / in the first and second equations equal to those in the

third and the fourth we deduce from (5):
(ae bs) (Ay BB, a)\(ALS- BY) -f (a-0,, 4 (4, Beas
(a;,+5,, 2) (A, LB, 2) (A, 4B, D=(,,42,,4) (4, +B, Pe
(@,--1-b,, 2) (4, 4B, 2) (4, 4B) a) (atb 2) (4, 4B, NEEN)
and (6)
(an BED (AEB INA EBD Ela DD (A.B, (A, BREE
(asbl) (A, HB (ars Hb (A HBD (A HBD
(aasb NA BE (apd, Ay BI) (A. 4+ B.A).

When reduced these equations prove to be of order three in 2; we
?

can write them in an abridged form:

( 535 )

Mi ANP +PA+Q=09, |)
ME Ne Pago |

which give, according to the method of Brzour, the following resultant:

(7)

(MN) (MP') (MQ)
|
| (MP') (MO) (NP) * WD ISA . . ®)
| (4Q) (NQ) (PQ)

5. From this is evident that the form of the final equation is
found, but it is a very intricate one, as is proved from the values
of the coefficients, given here:
i508 5 Bache Lt BB bebt tr Bb BB;
Na OB Bea), Al Bebe ANB + 2, A 2. SB set
PB BN be ALB eb ANB Sa, Be Oh. ARB =
END he ARBs Da Br Bb Bi
Om A; B, :

Pb A Aa B Asa A Bra, A, BO brt
an pele, od B, GT Bett ope: dB
DAANEN AB Beb AA Br

At 2

3 A, 155 3

a,

Qa A, A. a, Ay a, A, A; — a, es —a,, A, A, —a,,A,A3.
M' =>... B, B, + },, B, B, + 6,, Bs — ba B, B, — bs By —b,, B, B,;
IN! Sá BB; 4 ay A, B, + 6,, A, B, + a, B, B, + 6,, A, B, +
bs A, B, + a, B;? + 26,, A, B, — as B, B, — bs pi B, —
bis A, B, — as B,? — 2b,, A, B, — a, En B, — b,, A, B, —
ne Al Tey

BE by AA eA, Biel a. AB, + bis AAS Aras, Ay By

2a 4, B, + 2a,, A, B, + bs As — b,, A, A, — 4,, 4 Ee Ei

Z = 2 Z Z == A
a,, B, 4, — 2a,, A, B, — b,, A,’ — bo, A, A; as. B, —
APEN:
fi ae 2
Opal AR ANS ar AA tran oA? ee fale al, ee val IS ha

6. With the aid of these expressions the equation of the locus
of the axes can be determined for each separate case, which was
the purpose of this paper; we shall conclude by giving a few
observations.

a. Even in the general case abridgement is possible in the operation.
If we assume that the axes of coordinates coincide with the principal
axes of one of the surfaces, e.g. of B—O, then 6,,=0, b,, — 0,

( 536 )

bis = 0; -b,,—0; %,,=0; 6,,=0, whilst also“, Br
simple forms and all coefficients except Q and Q’, are simplified.

At the same time this substitution shows that in equation (8) a
factor may be omitted; if namely we make use of the above named
values for bj, we shall find:

Vib ld 5 VNS 12

from this ensues that the first column of the determinant (8) is
divisible by £,. This divisibility is connected with the fact that the
equation of the locus of the axes must become of order nine, whilst
when developed the determinant (8) becomes of order twelve. So
when a complete operation is executed factors must disappear out of (8).

4. Out of the former geometrie treatment it is evident, that in
some cases the locus of the axes S, breaks up. As one of the special
cases appearing there the case of a cireular base curve of the pencil
was treated where S, broke up into a cubic surface and into a
surface of order six. The equations of the algebraic treatment of this
case become, when one chooses the plane XO Y as the plane that
is intersected according to a pencil of circles:

‚2 PPM 1) ET) 5 Or
A=a,, 27 + a,, y” + 2a,, vz He 2a,, ye 2Q,, 2 Arn

B= bss 2? + 26,, ez + 26,, ye + 26,,2 + 20,, 2 =0-

From these equations the simplified values for M, V.... can be
deduced.

(February 23, 1905).

KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM,

PROCEEDINGS OF THE MEETING

of Saturday February 25, 1905.

IEC

(Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige
Afdeeling van Zaterdag 25 Februari 1905, DI. XIII).

CGliOEN FERNS:

M. C. Dexuuyzen: “On the osmotic pressure of the blood and urine of fishes”. (Communicated
by Prof. C. A. PEKELHARING), p. 537.

H. ZWAARDEMAKER Cz.: “On the relative sensitiveness of the human ear for tones of different
pitch, measured by means of organ pipes”, p. 549.

H. W. Baxuuis Roozesoom and E. H. Bucuner: “Critical terminating points in three-phase
lines with solid phases in binary systems which present two liquid layers”, p. 556.

P. H. Scnoure: “On non-linear systems of spherical spaces touching one another”, p. 562.

JAN DE Vries: “On a special tetraedal complex”, p. 572.

Jan DE Vries: “On a group of complexes with rational cones of the complex”, p. 577.

M. W. Berserinck: “An obligative anaerobic fermentation sarcina”, p. 580.

H. A. Lorentz: “The motion of electrons in metallic bodies”, II., p. 588.

The following papers were read:

Physiology. — “On the osmotic pressure of the blood and urine of
fishes” By Dr. M. C. Deknuyzer at Utrecht. (Communicated
by Prof. C. A. PEKELHARING.)

(Communicated in the meeting of November 26, 1904).

Migratory fishes (eel, salmon, shad) move in a relatively short
time from sea-water, having an osmotie pressure of about 24 atmo-
spheres, into fresh water, in which this pressure is */, of an atmo-
sphere or less, without experiencing any harm. And this same transition
also takes place in the opposite direction. It is very mysterious how
such an emancipation from the laws of osmotic pressure is possible.
It is known in general outlines that bony fishes, as well in salt as
in fresh water, keep up an osmotic pressure in their blood which is
relatively independent of that of the surrounding medium. But it is

37
Proceedings Royal Acad. Amsterdam. Vol. VIL.

(538 )

unknown between what limits the organism regulates the percentage
of salt (for this is the principal factor) in the different fluids of the
body, or by what means it keeps up this percentage. It is certain
that bony fishes are in general stenohaline, i.e. that each species is
bound to an osmotic pressure of the water in which it lives, and
which must not vary too much and especially not too quickly. A
behaviour like that of the migratory fishes is exceptional. The Baltic
Sea which contains about all gradations between salt and fresh water
and the fauna of which has often been studied, furnishes a proof of
the statements made. Of every species of fish, found in the Baltie
Sea, I have traced the range of distribution, and the lists, for the
publication of which we have no room here, show that most fresh
water fish go some length into the brackish water and most sea-fish
sustain a certain diminution of the percentage of salt, but that cer-
tain limits are not exceeded.

If we want to penetrate into the mechanism of these physiological
phenomena it is of primary necessity, to know the osmotic pressure
of the blood of the various species of fish. Some determinations were
made by Borrazztr and Roper. About five years ago I began to take
part in these measurements under very unfavourable circumstances
with sea-fish that had been transported alive from Katwyk to Leyden.
These animals were mostly alive, in any case entirely fresh. The
results were not published until in the summer of 1904 they appeared
to be quite concordant (and sometimes fully to agree) with results
that had been obtained under more favourable conditions.

The excellent opportunity of obtaining live sea-fish in great variety,
afforded by the fishmarket at Bergen in Norway, induced me last
summer to take up the investigation again and to extend it. After
that determinations were made on fresh-water fish from the environs
of Utrecht and finally I was enabled through the kindness of Dr.
Kersert, director of the zoological garden and aquarium at Amster-
dam, to study sea and freshwater fish, among these species that are
difficult to procure. I wish to express here my indebtedness to
Dr. KeRBERT.

At Bergen the fish are offered for sale alive in a large num-
ber of open wooden troughs through which a vigorous current of
sea-water is passed, which, as I was assured, is pumped up from the
fjord at a great distance from the town. The fish are not entirely
normal, however; the catching, the lack of food, the transport, their
being handled by sellers and buyers, all harm the animals. Before
they come into the market tanks, namely, they swim in caufs, closely
packed together in the surface water of the harbour, which sometimes

(539 )

is considerably diluted by the rains. There are reasons for assuming
that these influences make themselves felt in the osmotic pressure of
the blood. An investigator who should stay for a long time at Ber-
gen, choose his material carefully, keep it for some time in aquaria
and note for each specimen everything that could have any influence,
would without doubt obtain more constant results than can be published
here. Still it would be more recommendable to accompany the fisher-
men and to collect blood and urine immediately after the catching.
The figures here given must be judged as one of the first attempts
in this almost unexplored region. Only during and by my investigation
I have become aware of the necessity of taking the condition of
health of the animals very much into account.

The specimens bought were conveyed to the biological station
either in pails of seawater or without this precaution, a distance of
twenty minutes. and placed there again in aquaria in which sea-
water circulated containing about 32°,°, salt, corresponding to a
freezing point of —1.731° to —1.742°. The fishes that showed
signs of debility were examined first, the others remained for some
hours and even for two or three days in a spacious aquarium
without special food. Many specimens proved to have still filled
stomachs and to lodge few parasites, others were in a less satis-
factory condition, but all these details were not recorded. The
quantity of blood furnished by each fish is relatively small and
varies as well with different species as with different individuals.
As a rule, for a determination of the freezing point the blood of
several specimens is required, since ten to fourteen cubic centimetres
must be put into the freezing tube. The fishes were washed in
tapwater, well wiped and their tails cut off with a pair of bone-
scissors. Sometimes it appeared to be necessary to make an incision
in the heart; in this case the gills were once more cleaned from
seawater with a dry towel.

Would it not be better to use serum? This does not appear to
be necessary to me, since the same sample of blood generally gives
the same freezing point in repeated measurements and later a serum |
separates which as a rule is not coloured red, even with Raja clavata
and Trygon pastinaca which are supercooled to — 2.7°. Also Hrpin
and Hampurcrr’) have found that it is not necessary to separate
the serum. ;

In order to diminish the quantity of blood necessary for a deter-
mination, which is desirable especially with small or rare species, I

1) Hampurcer, Osmotischer Druck und Ionenlehre in den medicinischen Wis-
senschaften. Wiesbaden 1902. IL. p. 453.

37%

( 540 )

have tried whether it is admissible to add some soft organs, spleen
or liver to the blood. With the blood of three specimens of a fresh-
water fish from Surinam, Mrythrinus unitaeniatus, Spix, the same
freezing point was found twice with the bulb of the thermometer

not quite immersed, — 0.577°; then the livers of two specimens
were added and — 0.60° was found. So improvement must not be

sought in this direction'). With the rabbit still more considerable
differences were obtained, probably on account of the conversion of
glycogen into glycose.

The determinations were made with an in many respects modified
BECKMANN apparatus, of which the description will be given later.
Here it may suffice to remark that if a supercooling of 0.5° is used
as well for the determination of the zero point as of the freezing
point, the figures obtained for pure salt solutions are very accurate.
For 1 °/, NaCl (4 gram NaCl dissolved in 100 grams of water, the
weights reduced to a vacuum) the apparatus gives — 0°.589, the
result of the ““Präzisionskryoskopie” (HAMBURGER |. e. pag. 96).
BECKMANN’s correction for increase of concentration with supercooling

a al
was always applied, amounting for 0.5 to 160 Or 50=05 (80
calories being the latent heat of melting ice). The temperature of
the cooling bath was — 2°.5, when necessary it was lowered to
— 2°.9 by strong stirring.

We will first deal with the results obtained with freshwater
animals. Let A be the freezing point in degrees centigrade, omitting
the — sign.

Freshwater bony fishes.

L
Perch, Perca fluviatilis L. ae 2 and 5 specimens from Utrecht.
Remark. The perch occurs in the whole Baltic sea as far as the Sound, Le.
in water containing to 12°/o, salt and with A = 0.64.
0.527 _ 1 spec. from Bergen.

Carp, Cyprinus carpio L. E 1 » the Amsterdam Aqua-
Cee 0.540 rium, lived in water of A =0,039.

Remark. The carp seldom penetrates into the Baltic sea, oftener into the Asow

and Caspian seas.

0.466 ) 1 resp. 3 spec. from Utrecht,
Tench, Tinca vulgaris Cuv. caught healthy in October and
0.514 ) kept some time in the caufs.

Remark. The tench goes from the Haffs and bays as far as Gothland, where
A is about 0.42.

1) By reducing the size of the freezing tube and by using a smaller BreckMAnn’s
thermometer of an old pattern, the quantity required can be reduced to 5 or 6 ce.
By means of salt-solutions it will be controlled how far the results need correction.

( 541 )

0.519 | Resp. 4, 5 and 3 freshly caught

a Ae ) 59¢, | spec. from Utrecht, Sept. andOct.,

it, SC EDE Le: 0.526 | some of them examined in a half
0.530 | dead condition.

Remark. The pike in the S.E. part of the Baltic Sea leaves the coast to fairly
great distances and is occasionally caught near Bohuslän, where 4 is at least
0.69. Goes even some distance into the Arctic sea }).

18 fine spec. from the Amsterdam
Aquarium, Nov, occurs in all
brackish bays of the Baltic Sea.

Rudd, Leuciscus erythrophthalmus L. 0.495

Bleak, Abramis blicca, Bloch. 0.497 12 fine spec. Aq. Amst. Nov.
Remark. In the Baltic sea the bleak occurs in the brackisn bays.
Trout, Salmo fario L. 0.567 1 fine spec. Ag. A/dam. Nov.

Remark. The trout is a freshwater fish which seldom occurs in the brackish
haffs, but belongs to the Saimonidae, a family of migratory fishes, fishes of the
sea-coast and freshwater. Nüssrin looks upon them as original seafish.

Waranga, Erythrinus unitaeniatus Spix. 0.577 3 fine spec. Ag. A/dam. Nov.
Remark. Surinamian freshwater fish, living in water of 20° C. By evaporation
during the collection of the blood the number is probably slightly too high. Also
the quantity of blood available (8 cc.) was somewhat too small. They belong to
the Characinidae, old genuine inhabitants of freshwater (Ostariophysi).

Average of 13 observations on freshwater bony fishes 0°.521.
Excluding the trout and waranga, the first as being a Salmonida,
the other because the observation is less reliable than the others and
because we have here a fish of which the somatic temperature is
higher than that of the others, we obtain an average value of 0°.512,
round which the 11 observations are pretty regularly grouped.

To these I can add still six measurements on other cold-blooded
freshwater vertebrates :

Lamprey, Petromyzon fluviatilis L.

0.473 ) Observations at Leyden, on six, resp.
five specimens in tolerably good con-

0.500 } dition, made in 1899.
16 Q
Frog, Rana esculenta L. Wire id. A fine spec, caught in the autumn.

37 spec., Sept. 1904, animals sent from
Berlin, kept a day ina terrarium with
a dish of water.

Salamandra maculosa. Laur. 9.479

Fr.water turtle, Emys europaea. Gray. 0.474 | En heated

Average of these: 0°.476 and of the above mentioned eleven
together with these six: 0°.499.

The freezing point of freshwater is about 0°.02. Borrazz’s Emys
lived in water of this freezing point. In the tanks of the Amsterdam

1) Paracky, Die Verbreitung d. Fische. Prag. 2e Aufl. 1895. p. 54.
2) R. Quiyton. L'eau de la Mer, milieu organique. Paris 1904, p. 441.

aquarium a somewhat brackish water circulates of A = 0.089 (ori-
ginally water from the river Vecht). The percentage of salt of the
lake of Geneva is given in Cart Voar’s Lehrbuch der Geologie 1.
p. 538 as 0.1574°/, which would point to a freezing point of only
0°.01. According to the figures collected by Dupors (see Verslagen
1900 p. 12 and 30) A in Lakes Wener and Wetter is still lower.

The osmotic pressure in atmospheres at 0° which we shall hence-
forth denote by P,, is obtained by multiplying A by the factor
12.08 according to Srentus') or 12.03 according to Jorissen *) and
hence is in freshwater of the order of '/, to */, atmosphere. In
such a medium the cold-blooded vertebrate animals, breathing mostly
through gills, maintain in their blood an osmotic pressure of six
atmospheres! With birds and mammals (see the table of HAMBURGER,
Le. I. p. 456) also a pretty constant freezing point of the blood
has been found. I have proposed to call the power of keeping ?,
at a certain level, albeit within certain limits ‘“‘zdeotony” *), a property
comparable with the homoiothermic power. That also the freshwater
bony fishes possess this ideotony can hardly be doubted from the
results communicated. The limits between which the figures of the
same species lie, are narrow, only in the tench the differences are
fairly considerable. The ideotony is mest conspicuous when the
agreement between the cold-blooded freshwater veriebrates among
each other and the great difference with P, of the surrounding medium
are remembered. One is led to the supposition that for these animals
which indeed are not closely related: Cyclostomes, Teleosteans,
Amphibians and a reptile, the ZP, of about 6 atmospheres is an
optimum. For warm-blooded animals there seems to be a tendency
to maintain P, at 6°/, to 7'/, atmospheres; A’s of 0.570 with man
and of 0.6 to 0.625 with mammals and birds are namely kept up
with great constancy.

The kidneys are the regulators. For the A of the urine of man
varies between 012 and 3 (Hampurerr Le. I. p. 317) when the
separately discharged portions are examined, whereas A for the whole
quantity of 24 hours varies from about 1.3 to 2.4. For normal man
Scnoutr *) found that A of the blood, provided digestion were eli-

1) Srenius. Ofversigt af Finska Vetenskaps-Societetens Förhandlingar 46. No 6.
1903—4.

2) W. P. Jorrssen. Physisch-chemisch onderzoek van zeewater. Chem. Weekbl.
le jaarg. No 49, p. 731. Sept. 1904.

3) M. CG. Dexuuyzen. Ergebnisse von osmotischen Studien, namentlich bei Knochen-
fischen, an der Biol. Stat. d. Berg. Museums. Bergens Museums Aarbog. 1904. No 8.

4) D. Senoure. Het physisch-chemisch onderzoek van menschelijk bloed in de
kliniek. Diss. Groningen. 1903.

( 543 )

minated, by taking the blood in the morning before breakfast, only
varies between 0.56 and 0.58.

We can only to a limited extent imagine why the percentage of
salt (for this is the chief point) of blood and lymph may only vary
between narrow limits. The globulines require a certain concentration
of “medium salts’ in order to remain in solution. If horse serum is
diluted with 1°/, volume of distilled water, a precipitate is already
formed, i.e. with a percentage of salt corresponding to A — 0.24.
Why an increased percentage of salt should be tnjurious is less
clear. Danger for precipitation of albumens would only occur with
much higher concentrations, at any rate with horse serum. Yet the
fact, found by Roprer’) that the blood and the somatic fluids
(pericardial and peritoneal) of rays and sharks are isotonic with
seawater but contain less salt, the deficiency being compensated by
the retention of 2 to 2.7°/, of urea, points to a strong need of the
organism of the Vertebrates to keep the percentage of salt below a
certain value. Grijs ®) has found that blood-cells are permeable for
urea so that this substance helps to bear the osmotic pressure against
the seawater but discharges the celis of a third of 23 to 24 atmos-
pheres. I have proposed le. to call this power of being isotonic
with respect to seawater but of taking away from the cells them-
selves part of the osmotic pressure “metisotony’’.

The blood of Teleosteans has a freezing point which differs considerably
from that of the seawater, in which they live. They possess ideotony
but the individual differences are greater than have been remarked
with the remaining veriebrates, so that it appears that they only
imperfectly possess the faculty of rendering their P, independent
of the surrounding medium. Before the figures are given, a summary
of A and P, of different seawaters may be inserted. The numbers
have been taken from M. Kyupsen’s Hydrographische Tabellen, from
Prrrerson’s Review of Swedish hydrographical research in the Baltic
and North seas and from Morsivs und Hninckr, Die Fische der Ostsee *).

1) Roprer. Sur la pression osmotique du sang et des liquides internes des
poissons sélaciens. Comptes rendus. Dec. 1900. p. 1008.

2) G. Gruss. Ueb. d. Einfluss gelöster Stoffe auf die rothen Blutzellen. Pflüger’s
Archiv. 63. 1896. p. 86.

5) M. Knupsey. Hydrogr. Tab. Kopenhagen 1901, Perrersson in Scottish geogra-
phical Magazine 1894. X.; Moregius u. Heineke. Fische d. Ostsee. Berlin 1883.

5
Sl 3
~ es
23 “2.
En
S33 oa
hed a? TS ce
2 Ea
ao
ee Sep
= & :
Ans BER
599.2
ES Ae
sS Qo
Een
aprd a6
2 Be
5
6 1.00478
/ 1.00559

8 1.00640
9 1.00721
10 1.00802
12 1.00963
20 1.01607

32 1.02571

55 1.02813

38 1.03055

Gadus morrhua L.

„ ”
„ ”
„ „
„ ”
” „
„ „
” „
„ „
” „
” ”

0.318

0.531

0:659

1.074

1.739

1.908

2.078

( 544 )

Po
in atmo-

spheres.

22.9

24.9

Gulf of Bothnia in summer.

) Surface water (till 60 M.) of the Baltic
sea proper between Rügen and Gothland
in summer. With 7.59/o) salt, M. and
H. assume the limit between brackish
and salt water.

/

sea south of the Danish isles in

Water of the shallow part of the Baltic
| summer.

The same in winter (Mogrprus and
HEINcKE).

Seawater in the fjord before Bergen,
8 M. below the surface in summer.

Northern Atlantic.

A Seawater, Gulf of Naples, in Nov.
1903, 2.105.

a Seawater as it circulates in the tanks
of the Amsterdam aquarium 2.085.

Bony fishes living in the sea.

Cod,

A
0.644

0.673

2 apparently normal spec. Aq. Amsterdam,
Noy. 1904.

1 spec. bought on Monday at Bergen,
remnant of the fish supply of the end of
the preceding week.

8 spec. from Katw. immediately conveyed

to Leyden.
8 spec. id.
” ” ”
13 »

(April 4 1900, an ample quantity of blood
had been taken from 13 fine spec.).
Bergen 1904, summer, 3 large spec.
Leyden, as above.

Bergen, , 5 2 spec.
” ” ” 3 ”
Leyden „ „ Gs
” ” ” 8 „

( 545 )

Remark. The cod penetrates very far into the Baltic Sea. We do not mean
to assert that individuals caught in the Bothnian gulf and not larger than 45—50
centimetres have swum in from the Atlantic. They may very well belong to local
races, propagating in the brackish water and which do not reach a greater length.
Near Stockholm the cod only reaches a length of 60 centimetres, in the Sound
80 to 90 cm. at the utmost, on the coast of Bohuslin 90 cm., but near the
Lofoden Islands even 140 to 150 em, Also the common mussel is much smaller
in the brackish water of the eastern parts of the Baltic sea than in the more salt
containing sea.

In the Gulf of Bothnia the percentage is in summer north of the Quarks
3— 40/0, 40.159 to 0.212, Py 1.9 to 2.54 atm., north of Stockholm resp. 5°/o9,
0°.265, 3.18 atm., at Stockholm 69, 0°.318, 4 atm., on the north coast of
Gothland 7°/9), 0°.37, 4.45 atm., and till Riigen—Schonen 7 to 8°/o9, 0°.424, 5 atm.
At Bohuslän the salinity of the surface is in summer 13/9, A= 09.69, but in
the depth North sea water occurs of 32 to 33%/, and 1.8°.

A
Gadus aeglefinus L. Haddock. 0.767 Leyden, summer, spec. were dead but fresh,

Remark. The haddock does not penetrate further than the Mecklenburg coast.

A
Gadus virens L., Coalfish, Green Cod 0.760 Bergen.
3 = 0.761 = 3 spec.
7 5 0.837 5 Ss
i 5 0.338 ‘

Remark. Gadus virens does not penetrate further than the bay of Kiel and is
rare there. In the fishmarkel at Bergen it is always supplied in large quantities
but generally in a bad condition, showing wounds and traces of having bled,
many specimens lie on their backs at the surface and breathe little. When a
purchase was made good specimens were selected, but I think it very probable
that the animals whose blood froze at — 0°.837 and — 0°.838 were abnormal.

Gadus merlangus, L. Whiting 0.760 Bergen. 14 spec.
Remark. The whiting enters the Baltic Sea with difficulty, about as far as
Bornholm. Only once it has been caught near Gothland.
A
Molva vulgaris, Flemm. Ling, 0.716 Bergen. 3 fine spec.
Remark. The ling no more than G. virens penetrates into the Baltic Sea.
A

Moiva byrkelange (Walb.), Trade Ling, 0.730 Bergen, 4 fine spec. were dead
but fresh.

Remark. Deep sea fish, not to be had alive. Had been caught at a depth of
400 metres. Does not come further than the Cattegat.

A
Motella tricirrata (Bloch), Whistler, 0,605 Bergen.
Remark. Motella tricirrata has only once been caught near Göteborg

( 546 )

A

Hippoglossus vulgaris, Flemm., Halibut, 0,671 Bergen. The specimen suffered
from a disease of the skin, had
lived long in the aquarium and
threatened to die.

Remark. The halibut does not wander into the Baltic sea further than Mecklenburg,

A
Pleuronectes platessa L., Plaice. 0.672 Bergen.
î 5 0675 Leyden.

Remark. The plaice goes as far as Stockholm. The Pleuronectides everywhere
show a tendency of penetrating into brackish or even fresh water. The flounder
has been found in the Moselle near Metz. Wicumann found species of flounder in
small mountain lakes of New Guinea.

A
Pleuronectes microcephalus Donovan, Lemon Dab 0.681 Bergen.

Remark. P. microcephalus very rarely comes as far as Eckernforde.

A
Labrus bergylta Ascan., Ballan Wrasse 6.694 Bergen. 3 spec.
E is 0.704 5 6, of which one very
ill, liver and intestine full of nematodes.
d J 0.708 ;

Remark. This Labrus is only seldom found in the western part of the Baltic
sea, where the freezing point is ™ summer about —0°.6, —1° in winter.

A
Labrus mixtus L. Blue lipfish, Striped Wrasse 0.681 Bergen. 4 spec.
: j os, Vee

Remark. L. mixtus (& red, 2 blue) seldom comes as far as the Sound.

A
Conger vulgaris Cuv. Conger-eel. 0.696 3 spec. Aquar. Amsterdam.
5 - 0.786 1 „ Bergen.

Remark. Only seldom caught in the Baltic sea, repeatedly in the lower course
of the Weser.

A
Salmo trutta L. Sea or Bull trout. 0.785 6 spec. Bergen, caught with the rod in
the fjord, in bad condition and partly dead.

Remark. The Sea trout is an anadromous migratory fish.
A
Labrax lupus Cuv. Bass, 0.720 1 spec. Aquar. Amsterdam.
Remark. Rare in the western part of the Baltic sea.

A
Trigla hirundo Bloch. The gurnard, 0.669 2 spec. Aquar. Amsterdam

Remark. Not often occurring in the western part of the Baltic sea.

(.547 )

A

Anarrhichas lupus L., Sea-wolf, 0.665 2 spec. Bergen. The fishermen use io beat
. Fy OGS lear > - out the teeth of these
4 ; OVOM 5 somewhat dangerous ani-

mals; in any case the
sea-wolves arrive at the
market alive but not in
a normal condition. They
react slowly and die when
they are too much
handled.

Remark. The sea-wolf penetrates at the utmost as far as the coast of Pomerania.

The average of these 38 observations is A = 0 .7245 or P, = 8.7
atmospheres. The figures are grouped pretty regularly round this:
13 between 0.600 and 0.700, 13 between 0.700 and 0.750, 12
between 0.750 and 0.850. The average lies fairly well at the same
distance from the two extreme values. By omitting the extreme values
0.605 for Motella, 0.808, 0.811, 837 and 0.838 for codfish and
G. virens, which latter pretty certainly are based on pathological
deviations, the average is only little shifted and becomes 0°.716.

The differences between the extreme values and the average are
relatively large, 0.120 and 0.118, about *, of the probable normal
value. If the 5 extreme figures are rejected, the deviations from the
new average, 0.716, are only 0.072 and 0.070. We found a similar
result with the freshwater fishes; only after rejecting the values for
waranga and trout we obtained an average of 0.499, differing only
0.041 and 0.035 from the extremes.

If we bear in mind that these fishes live in a medium of which
the osmotic pressure is 21 to 23 atmospheres or even more, no one
will object to ascribing zdeotony to these animals. But the considerable
oscillations in P,, which we noticed e. g. in the cod, give the impression
that the power of maintaining ?, at a certain level, is limited. And
one is involuntarily reminded of the oscillations in somatic temperature
which homoiothermie organisms show with many disturbances in
the general well-being.

There exist in literature still a few data concerning the freezing
point of the blood of bony fishes living in the sea. borrazzi’) found
with Charax Puntazzo Gm. — 1°.04 and —1°.035, with Serranus
gigas L. —1°.035 and — 1°.034, but these figures do not deserve

1) F. Borraza. La pression osmotique du sang des animaux marins. Arch. ital.
de biologie 28, p. 67, 1897.

too much confidence as he was wont to use a cooling bath of —12°*).
Neither could he apply the later published correction of BRCKMANN.
With Chelone imbricata L. (sea-turtle) he found —O’.61 and —0°.62.

Roper?) found in a Ganoid (sturgeon?) A = 0.76, in Lophius
piscatorius L. O°.68 and 0°.80, in Orthagoriscus mola I. 0°.80, in
the sea-turtle Phalassochelys corticata Rondelet 0°.602 and in a
mammal, living in the sea, the grampus Phocaena communis Less 0°.74.

The numbers obtained for the blood of the eel Anguilla vulgaris
Flemm. are very remarkable. With vigorous specimens I found
formerly at Leyden — 0.773°, at Bergen — 0.653°, at Utrecht
— 0.587°. Now the eel belongs to a family of tropical sea-fish ;
most species occur in the Dutch Indian Archipelago, they often go
into brackish water, others are deep-sea animals. Our common eel
excellently bears quick variations of the percentage of salt. Born in
the sea, it enters the river mouths as a young animal and remains
in fresh water until the time of propagation approaches. The eel
which is caught in the fresh or somewhat brackish waters of Frisia,
is put at Workum into caufs into which the seawater has free
entrance, goes to London and is sold on the Thames from these
caufs. The layer of slime, with which their skin is covered facilitates
this transition. Pavun Burr*) noticed that all the eels which he put
from the fresh water into the seawater himself, supported the sudden
change, whereas those which his assistant handled, all died. He used
a little net, whereas the assistant took them with his hand, held
them in a rough towel and in this way removed the layer of slime.

The eel shows in its osmotic pressure sometimes the type of a
seafish, sometimes it approaches that of a freshwater fish. The high
P, with the trout, as an original migratory fish, now also becomes
to some extent explainable.

Here a field of study lies open which may be urgently commended
to the Committee for the international investigation of the sea.

How do the marine bony fishes maintain in their blood a so
much lower osmotic pressure than exists in the seawater? Some
observations on the urine of the cod, sea-wolf and G. virens can
perhaps throw some light on this question. The 4 of the urine was
always lower, the osmotic pressure less than that of the blood.

1) G. Fano et F. Borrazzr, Sur la pression osmotique du serum en différentes
conditions de l’organisme. Arch. ital. de biol. 26, p. 46, 1896. See especially p. 47.

2) Hampurcer |. c. I. p. 466. The original article of Ropier in Travaux des
laboratoires d. 1. soc. sc. et station zoolog. d'Arcachon. 1899. p. 103, I have not
at my disposal.

3) P. Reanarp, La vie dans les eaux. p. 438. Paris. 1891,

( 549 )

With a large specimen of the sea-wolf, whose blood had given A
0.681°, the urine gave 0.631°. With other individuals I found 0.555°.
The urine taken from some twenty specimens of G. virens gave
4 0.630°. With the cod 0.652 and 0.619 have been stated.

It is very simple to take the urine. A sea-wolf, e.g. is taken
behind the gills and suddenly lifted from the seawater, the skin of
the belly is dried, while the assistant stands ready for collecting the
urine which often is ejected in a vigorous jet. By some pressure
on the belly a little more is obtained, but often the “bladder” (the
extended part of the ureters) is empty. Most animals gave little or
nothing and were given back to the seller so that a comparison of
4 of the hlood and urine was only possible in exceptional cases.
At Bergen I had for the three species that were studied, found not
a single figure for 4 that was lower for the blood than for the
urine. At Amsterdam, however, it has appeared that there also
occur specimens, the blood of which shows a still somewhat smaller
osmotic pressure than any of the urines (cod).

The remarkably low /P, of the secreted product of the kidneys
with marine Teleosteans certainly points to this: that these animals
do not keep the osmotic pressure in their blood 23—8.6=—14.4 atmo-
spheres lower because the kidneys so quickly eliminate the surplus
of salts taken in. The relative richness in water of the urine rather
points to these fishes resorbing from the sea-water in opposition to the
osmotic pressure, hence by using chemical energy, water or if one
prefers, a diluted solution of salt. But Rrexarp has stated (l.c. p. 391)
that certain freshwater fishes secrete from their gills soluble car-
bonates! About the mechanism of ideotony we are still in the dark.

Physiology. — “On the relative sensitiveness of the human ear for
tones of different pitch, measured by means of organ pipes.”

By Prof. H. ZWAARDEMAKER Cz.

(Communicated in the meeting of January 28, 1905.)

Almost simultaneously, but by different methods, the relative
sensitiveness of the human ear as depending on pitch, was investigated
by Max Wier!) and by F. H. Quix and myself’). The result of

1) Max Wren. Physik. Ztschr. IV p. 69. Pfliiger’s Archiv Bd. 97. p. 1. 1903.
2) ZWAARDEMAKER and Qurx. Ned. Tijdschr. v. Geneesk. 1901 II p. 1374: 1902
Il p. 417. and Engelmann’s Archiv. 1902 suppl. p. 367.

( 550 )

these parallel investigations were concordant in some respects, different
in others. They agree in this that:

1s*. there is only one maximum of sensitiveness ;

2nd, that this maximum lies at g*;

3rd, that the zone of fair sensitiveness extends from c’ to gq’.

4th that outside this region toward the limits of the scale the
‘sensitiveness diminishes very strongly.

They differ in this that:

1st. with Max Wier the sensitiveness still diverges very much
within the zone of fair sensitiveness, whereas with us it is of the
same order.

2nd. that the perceptible minimum for the most sensitive point is
with him 100.000.000 times smaller than with us.

In this state of affairs it seemed desirable once more to determine
the perceptible minima throughout the whole scale by an entirely
different method. Telephone as well as tuning-forks ought thereby
to be avoided. So we had recourse to wide roofed organ pipes of
which a wooden set of uniform pattern, extending from C to g*
was at our disposal which partly coincided with the well-known
Epr~Mann whistles and could be continued by the Galton whistle.

Some series of such experiments were made, partly on the heath
at Milligen, partly in the gallery of the university library at Utrecht,
partly in the sound-tight room of the physiological-laboratory. Since
the results, generally speaking, agree fairly well and a full account
of them will be published later, for the present only two series
taken under the simplest conditions, will be dealt with. These are:
a, the concluding series on the heath, 6, in the gallery. The arran-
gement, which was the same for both, will first be deseribed.

The organ pipe which serves as the source of sound, is mounted
vertically on a stand, near the floor, with as little contact as possible.
It is connected with a Hurcuinson spirometer. Close under the air-
room of the organ pipe and connected with this latter by a wide
opening, is a ligroine manometer. The manometer being bent
into an obtuse angle as little as */, mm. of waterpressure can be
read. The spirometer is now loaded with a little box containing
sand, so that it forces out the air very regularly and causes the
organ pipe to emit a soft sound without an audible frictional noise
and without partial tones. The air used is read off on the scale of
the spirometer and calculated per second by at the same time starting
a timing watch. The product of the volume of air, pressure and
acceleration of gravity (all in em.) then give the energy supplied
per second in ergs.

(551)

What part of this energy is converted into sound is unknown.
Wesster *) values the “efficiency” at 0,0013 to 0,0038 ; Rarrrian *)
on the other hand supposed in 1877 as a preliminary estimate, that
all was converted into sound (“supposing the whole energy of the
escaping air converted into sound and no dissipation on the way”).
The truth will probably lie between these two, since we have always
paid attention to clear and easy sounding. For such a case Max
Wien remarked in 1888: A loss of energy certainly takes place,
first on account of the fact that part of the air-current is not
converted into sound-waves at all, but is lost by the formation
of vortices, partly inside, partly outside the pipe. We shall see
later that this part is small only for a definite position of the
lip of the pipe and for a definite pressure. A second loss of
energy takes place by friction on the walls of the pipe and by
tremors imparted to them; a third on the way between source and
observer by friction on the floor, motion of the air (wind) and
viscosity of the air. This latter part especially is relatively large
with RayLxiGH, since by viscosity a loss of energy of + 22°/, took
place *).

If 22°/; is considered relatively much, we may assume that Max
Wien at that time supposed for the losses by other causes a similar
or smaller amount. But whatever the “efficiency” of the supplied
energy may have been, there is no reason for assuming that it has
been appreciably different for the different pipes. The wooden pipes
at any rate belonged to the same set of uniform pattern. So the
method suffices for comparative measurements.

While one observer read the scales of spirometer and manometer,
the other moved to the greatest distance at which the tone was just
heard and recognised (“Erkennungsschwelle”). This distance was
then later taken as the radius of a hemisphere through which the
energy of the sound spread.

A. Experiments on the heath at Milligen.

Perfectly level ground, trees only at 600 metres. Quiet, fine evening,
October 19, 1904. Acoustical observer F. H. Quix, optical observer
H. F. Minkema (See Table I).

B. Experiments in the gallery of the university library.

Afternoon of January 3, 1905. Acoustical observer H. Zwaarpr-
MAKER, optical observer H. F. Minkema. (See Table II).

1) A. G. Wesster Boltzmann’s Festschrift 1904 p. 870

2) RAYLEIGH Proc. Roy. Soc. vol 26 p. 248 1877.

3) M. Wies, Die Messung der Tonstärke, Inauguraldissertation. Berlin 1888 p. 45,

(552)

c g cl g} c2 g? c3 g3 ct

Minima perceptibilia in the course of the scale; the minimum for g*—‘

(absolute value of the chosen minimum: in 1902 0.79.10

8 Erg, in 1904 0.3240—8 Erg).

( 553 )

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Proceedings Royal Acad. Amsterdam. Vol. VII.

Now, if for the present we only take into account the energy
supplied and neglect the necessary loss of energy in the organ pipe
and in the air; if we further assume the validity of the theoretical
law of distances (extension over a hemisphere), we obtain the following.
results :

1. that the sensitiveness of our ear has only one maximum,
ving in the four times marked octave.
2. that there is a zone of fair sensitiveness, extending from gq’ to g’.

€

3. that outside this zone the sensitiveness diminishes very rapidly.

4. that in the zone of fair sensitiveness the perceptible minima
are of the same order.

5. that, for the most sensitive part of the scale the perceptible
minimum is 0,32 >< 13-8 ergs for Mr. Quix, 1,9 « 10-8 ergs for myself.

The true perceptible minimum for the most sensitive point of the
scale will of course lie lower. How much lower cannot be determined
for the present, but at any rate the perceptible minimum found with
organ pipes certainly remains a million times greater than that which
was calculated by Max Wien from his telephone experiments. The
minima, found on the heath and in the library, are in satisfactory
agreement, however, with the minimum which we formerly caleu-
lated for tuning-forks, using the data of TépLur and BOLTZMANN ').

Taking into account the “efficiency” of an organ pipe, found by
Weester (0,0013 and 0,0038), the perceptible minimum for the
most sensitive point of the scale becomes lower, namely 0,45 to
1,3. 10—-"" ergs, but it does not reach the amazingly small value of
Max Wren’s telephone experiments by a long way. Even if we
assume that one hears better at night in the profound silence of a
laboratory, than on the heath, not to mention an afternoon hour in
the library, yet this difference is by no means accounted for. But I
see no reason why the results of experiments made on perfectly
level ground, far from woods or buildings, which, according to
Max Wien’s former valuable investigations, fall perfectly under the
theoretical law of the distribution of sound, should deserve less
confidence than experiments with a telephone, which require very
complicated calculations.

1) Töprer u. BoLTZMANN. Ann. d. Physik u. Chemie Bd. 141 p. 321.

( 555 )

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( 556 )

Chemistry. — “Critical terminating points m three-phase lines with
solid phases in binary systems which present two liquid layers.”
By Prof. H. W. Baknuis RoozeBoom and Mr. B. H. Bücnner.

(Communicated in the meeting of January 28, 1905).

Up to the present only one critical terminating point has been
found in systems of one component, namely in the equilibrium
liquid-vapour. If this is represented by a p,¢-line this suddenly
terminates in the point where liquid and vapour have become
identical. There exist at the moment no well-founded reasons to
assume critical terminating points also in the equilibria solid-liquid
and solid-vapour. We will not further consider these here.

In systems of two components we get instead of a critical point
liquid-vapour, acritical
line A,A, which con-
nects the critical points
of the components. In
a p, t, #-diagram this
line runs in space,

P

here its p, projection
is only indicated. (AK,
and BK, are then the
equilibria-lines liquid-
vapour for the com-
ponents A and 2). If
there is a homogeneous mixing for all concentrations in the liquid

phase, this will then be the only critical line.

The recent researches of Smits!)
have shown how in some binary
P nap * mixtures the equilibrium line
x, for solid B with liquid and
A B vapour may meet this critical
line (it is necessary that the
t melting point of B should be
Fig. 2. situated higher than A). There
are then two such meeting-points p and q with the two parts op
and qg& of the three-phase line. This line therefore acquires two
critical terminating points owing to its meeting the critical line liquid-

vapour. Between p and q both lines cease to exist.

1) These Proceedings 1904,

(557)

Other cases of similar critical terminating points through the
meeting of three-phase lines in binary mixtures might be conceived
when no homogeneous mixing in all concentrations is possible in
the liquid phase and when, therefore, two liquid layers are possible.
It was of great importance for the knowledge of the conditions of
existence of solid phases at high temperatures and pressions to also trace
the critical terminating points in similar cases. If we indicate the two
liquid layers by ZL, and Z,, the vapour by G and the one or other
component in the solid state by S, then besides the three-phase line
SLG, two other three-phase lines SL,L, and L,L,G will be possible.
In order to trace the critical terminating points of these lines we
first consider the line L, L, G. Here
we first take exclusively the cases
where the p,z-diagram has the form
of Fig. 3, in which G, L, and J,
indicate pressure and concentration of
coexisting vapour with two liquid
layers. The three-phase pressure for
this system lies, therefore, between the
vapour pressure of liquid A and 5

Py and B):

Fig. 3. The p, ¢-line CM for the three-phase
equilibrium L, L,G is situated accordingly in Fig. 1 between AA,
and BA.

On elevation of temperature £, and L, may approach each other,
or recede. If the first happens and if they still coincide below the
line A,A,, for instance in J/, then the two layers become identical,
a case of which many instances have been found by ALExEsEW and
XOTHMUND. This point has been quite properly called a critical point.
This critical terminating point of the line for 1,/,G' must of course
necessarily be a point of intersection with a critical line. The said
critical line is the line DM in Fig. 1.

If the liquids Z, and Z, can be made to mix more completely
by an increase of pressure a mixing point D will be found to exist
in Fig. 3 at a sufficiently high pressure. If now L, L, approach each
other at a higher ¢, D will then be situated at a lower pressure
whilst the concentration in the mixing point may differ. In this way
is obtained the critical line DM for the phases L,Z,. Its one ter-
minating point lies at the meeting point J/ where G occurs together
with £,L, and where consequently the lowest possible pressure is

1) The other case where the three-phase pressure is higher than those two does
not lead to materially different results.

(558 )

attained. In the upper direction a terminating point would only be
conceivable in the case of the occurrence of a solid phase. If an
increase of pressure promotes the separation of the two layers, DM
would then run from the point J/ to the right instead of to the left.

In the case, therefore, in which the two liquid layers possess an
upper mixing point J/ which lies below A,4,, the critical lines
AOM, for GL and DM for LL, are quite independent of each other.

If, however, the upper mixing point is not yet reached below
KK, CM will continue up to a meeting point with A,A,. As the
phases occur in the order GL,L,, G will then become identical
with “, in O, Fig 4.

The critical line A,O is here at the same time broken off. From
a consideration of a series of p, v-diagrams for successive temperatures
we may, according to Mr. BücHrer, easily demonstrate graphically
that the other end of the eritieal line GL, which commences in K,
now amalgamates with the upper part of the critical line LL, the
lower continuation of which is not realisable owing to the absence
of J/. In this way is formed the critical line A,PD whose upper
part may eventually also run to the right.

A junction of the three-phase line GL,L, with the critical line
GL in O will, secondly, always take place when ZL, and L, diverge
by an elevation of temperature. This may frequently oceur with two
liquid layers which have a lower mixing point J. The three-phase
line GL,L, then terminates in O by intersection with the critical
line GL and in Jf by intersection with the critical line IP for
L,=L,. From the p, r-diagrams we may now again deduce that
the upper continuation of this line is not now realisable because the
lower part coincides with the second part of the critical line GL
which proceeds from A. In this way the critical line 12’ PK, originates.

Lately, KuerNeN has
found instances of the
cases represented by
Figs. 4 and 5.

The figures 1, 4 and
5 exhibit the three
main types of the
manner in which the
three-phase line meets
either the critical line
GL or the critical line
L, L, and then finds
1 or 2 terminating

points, also of the fusion of
parts of the one critical line
with those of the other.

In the second place we
will now consider the oecur-
rence of critical terminating
points on those three-phase
lines where one of the phases

Fig. 5. is solid. We may then limit
ourselves to the case where B occurs as a solid phase as no critical
phenomena can occur in presence of solid A.

In many cases where the line GL,L, proceeds to lower tempe-
ratures the solid phase will occur ina point Q. According to previous
research by myself, 4 three-phase lines then meet in the quadruple
point Q. If we take the ease of Fig. 1 we obtain in Fig. 6 QM
for GL,L,, QB for GL,S, EQ for GL,S, whilst QN indicates the
equilibrium of the solid phase S with the liquid layers L, /,. Similar
lines have been formerly
studied by me in cases
where hydrates of SO,
HBr, HCI occurred as
solid phases. For the com-
ponent B as solid phase
their courses will be ana-
logous, and like all ordi-
nary melting lines the
direction will diverge only

Fig. 6. a little from the vertical
either to the right or the left according to the volume differences
of the phases.

If the line runs to the
right and the critical
line MD to the left there
might be a possibility of
their meeting in a critical
terminating point N as
the two liquid layers
A, might here become ident-
ical in presence of solid B.
The chances that this will

Fig. 7. occur with an attainable
pressure only exist when Q and J/ do not differ too much in temperature,

N

( 560 )

Fig. 7 represents a similar meeting point MN for the case corre-
sponding with Fig.4. Far greater chance of attaining a critical ter-
minating point of the line 4,,5 is offered by the case of Fig 5
which would lead to Fig. 8. The line OQ is here supposed not to
proceed as far as the lower mixing point J/’ because the solid phase
occurs previously at Q. For this reason the downward continuation
of the critical line L, = L, is wanting.

If, however, the mixing
point J/’ should lie but very
little below Q (metastable)
the point of intersection WV
might be found at a compa-
ratively low pressure. Mr.
Bicuner has frequently no-
ticed a similar proximity of
M' and Q in systems of all

Fig. 8. kinds of organic substances
which on being dissolved in liquid CO, give rise to two layers.

Besides the three
cases fig. 6, 7, 8 in
which there exists a
critical — terminating
point of the line QM
a fourth type is pos-
sible. This occurs
when the liquid on the
line HQ already be-
comes critical with the
vapour before Q has
been reached, that is
beforethesecond liquid
Fig. 9. occurs In presence of

the solid phase. ZR then intersects the critical line liquid-vapour in a
critical point A which quite corresponds with the point p of Smits
(fig. 2). The line QO is now wanting, namely, below /è we notice
nothing of a second liquid. Mr. Bicuner has here again graphically
deduced that in this case the line QS, for the second series of liquids
in presence of solid B and vapour, fuses with QN to a single line
BQN where the one fluid phase has the character of vapour on the
lower part and of liquid on the upper part, whilst these however,
gradually pass into each other.

In this case, like in fig. 4 the lower part of the critical line GL

( 561 )

fuses with the upper part of the critical line £,L,, K,PN, and we
have here again the possibility that the line BQN also possesses a
critical terminating point N. This point bears some resemblance to
the second critical terminating point found by Smits in gq fig. 2. That
we are dealing in fig. 9 with a region of two liquid layers could
only be made plainly visible, if, owing to the non-appearance of the
solid phase, two liquids, in presence of vapour (metastable), occurred
below A. Otherwise it is only the peculiar course of BQN, which
shows that we have this type.

Mr. Bicuner has succeeded in finding a ease where this course
could be indicated (although MN remained unattainable). Fig. 10 gives

a correct representation of the p,¢lines HR and BQ for solid
diphenylamine in presence of solution in liquid CO, and in vapour
very rich in CO,, on ER, and in presence of a much more diphenyl-
amine-containing second liquid phase on BQ. The point PF is situated
only 0°.6 above the critical point of pure CO,, consequently diphenyl-
amine is but very sparingly soluble in CO, at that temperature.
Between 31°6 and 38°8 two liquid phases are not capable of
existing in presence of solid diphenylamine; above these we again
find the second three-phase line with two fluid phases now much

richer in diphenylamine. This line was determined up to 120 atm.
pressure. The significance of all the regions in which three-phase
lines are absent can only be expressed by a series of p, z-diagrams.

The above considerations foreshadow the possibility of enunciating
in general terms the conditions for the existence of a solid phase in
presence of one or two fluid ones, when traversing the region of
the critical phenomena of those latter ones, also for those binary
mixtures which in the liquid state are not miscible in all proportions.

Mathematics. — “On non-linear systems of spherical spaces touching
one another.” By Prof. P. H. Scnourr.

1. Before passing to our real investigation it is necessary to find
how many spherical spaces touch 7-1 spherical spaces given arbi-
trarily in the 7-dimensional space S,. And in its turn the answer to
this question demands a knowledge of the situation of the centres
of similitude of those given spherical spaces. So we start with a
study of these centres of similitude. To this end we represent the
spherical space, which is in S, again the locus of the points situated
at a distance 7 from the centre J/, by the symbol Sp, (M, 7).

2. Just as is the case with two circles lying in the same plane,
two spherical spaces Sp (J/,,7,) and Sp (M,,7,) lying in S, admit
of two centres of similitude on the line M, J/, connecting the centres,
an external one U, and an internal one /,,; through U, pass the
lines P, P, connecting the extremities P,, P, of direct parallel rays,
through /,, pass the lines P, P', connecting the extremities P,, pe
of opposite parallel rays.

Supposing that in S, a number of 7-+-1 spherical spaces Sp (Ml, 7),
(k=1,2,...,n-+1) is given arbitrarily, we shall now investigate
the situation of the (» +1), pairs of centres of similitude (U, 4)
with respect to each other. To this end we first notice that the
three pairs of centres of similitude of the three spherical spaces
Spun (Mi, ri), @= 1, 2,3) form the three pairs of opposite vertices
of a complete quadrilateral, each of the four triplets of points

(U. U. U). (Ge Fn La) (1 U» 1). Am Ls U.)
consisting of three points of a right line; we indicate these lines in
the given order by

12

hee RAE en a

. . . {
If we now further regard the n—1 pairs of lines (/,2), Ie) through

( 563 )

‘ er 2) (I

U. and the n —1 pairs of lines (/j,, /3;) through /,, — where
12 I b + ce 12

successively p assumes the 7 — 1 values 3,4, ...,”n-+1 —, we

see immediately that each space S,—-; through »n —1 lines / through

U. (or 7,,) —with all indices p differing mutually — contains one of
12 13 / o «

the two centres of similitude of each of the +1), pairs (U, Ly).

Thus a space S,—: through 7 —1 lines / through U, will contain

I 2 - 1
the point U, or the point /,,, according to the two lines / with
| Pq Pq S

p and g as third index being of the same kind or not; just the

reverse is found for a space S,—; through »—1 lines / through

I,,. As the choice of the lines / corresponds in both cases to n— 1

bifureations 2”—! of those spaces S,—; pass through each of the two

! I gs
points U,,, J,,. So the theorem holds good:

“We can indicate 2" spaces S,-;, each of which contains (n-+-1),
centres of similitude of a system of » +1 spherical spaces Sp, given
arbitrarily in S,, and namely one of each of the (n +1), pairs
(Tyr Lpg)”

We need not enter into further details about the situation of the
centres of similitude for the purpose we have here in view.

3. From the well known properties of the figure consisting of two
circles and their centres of similitude we read (fig. 1):

u

UP," : UP, rn: se r, | } Ou zae 7 WIE == Jit ‘ vol
EN eaeey tp IP,

1

5

IP; — IC, é zen

With the aid of these relations we can easily find the following

A

u

Fig. 1.
theorems, where for Sp, (M,,r,) and Sp,(M,,7,) we shall write the

B (1) > (2
abridged form Sp,’ and Spe.

( 564 )

: , (1) , (2 ‘ :
“The spherical spaces Sp, and Spy’ are homothetic and directly
similar with U, homothetie and inversely similar with / as centre

1 . . . er rn .
of similitude and + — as quotient of similitude. The points corre-

lis

; “ : : i
sponding to each other, P," and P, in the first case and P, and P,
in the second, are called homologous.”

“The spherical spaces Spi” and Sp correspond to each other
in an inversion with U as centre and UC“. UC, as positive power
and in an inversion with J as centre and / C: . IC, as negative
power. The points P,“ and P," corresponding to each other in the

first case and P, and P, in the second are called antihomodlogous.
And the two inversions appearing in these theorems shall furtheron
be indicated for shortness’ sake by the symbols U (4, 2) and J (4, 2).”

“Each spherical space Sp, through a pair of antihomologous points

Ie rant > (2 5
Pand ot Sp? and Spo cuts these spherical spaces at equal
angles. If the: spherical space Sp, through P, and P, touches the

° SHAD) . : B (2
spherical space Spo? in P,, it will touch the spherical space Spo
in P,. And these contacts will be of the same kind or not, according
to U or J being the centre of the antihomologous correspondence.”

In connection with the general theorem concerning the situation
of the centres of similitude the second and the third of these three
simple theorems form the foundation of a method of solving the
problem to construe a spherical space Sp, touching n-+-1 spherical
spaces Spo, Spe, oe, Spor) given arbitrarily in ,. As will
immediately be evident, to each of the 2” spaces 5"! through m1),
centres of similitude answers a pair of tangent spherical spaces Sp,

and the contact of one of these spherical spaces with Spt and Sp?

is of the same kind or not, according to the chosen space S,—1 con-

as, ‘Ave em ‘ Cl
taining of the centres of similitude U, J), of Sp? and Spx either

the first or the second. So 2! is the number of the theoretic solutions.
And if we indicate external contact by —+ and internal contact by
—, then the 2” pairs of solutions are indicated by the pairs of
completely opposite combinations of signs of the series consisting of

m+ 1 terms

(565 )
Ee had SF crt sE,

where the two solutions of a selfsame pair correspond in all signs
or differ in all signs.

The construction of the tangent spherical spaces proves the above
assumed concerning the number of the solutions and their connection
with the 2” spaces S,—. We give it here — to avoid prolixity —
for the case » =4 in a form, in which it is immediately transferable
to the case of an arbitrary ». It is:

5 4d 5 veal y (2 y (5 5
a. “If in S, the spherical spaces Sp4 ) Sp§ Bee Spy are given

: " sp Je .
arbitrarily, if d, is the space through the points

Te T he p
U» U» U 23) Us Li Ls Lis Ls Ls Is

and if P, is an entirely arbitrary point of Spy”, then the antihomo-
logous points P,, P,, P,, P, of P, in the inversions U (1, 2), U (4, 3),
11,4), 71,5) are to be determined and the spherical space Sp, (/)
through the five points P,, P,, P,, P,, P;.”

b. If e, is the plane of intersection of din with the radical space
of Spa” and Sp,(P), let us bring through ¢, two spaces touching
Spy and let Q, and Q,’ indicate the points of contact.”

c. Finally must be determined the pairs of points (Q,, Q,’),
(Qs Qs) (QQ), (Q@;, Q;") which are antihomologous to (Q,, Q,’)
in the inversions U (1, 2), U (41,3), 71,4), 7,5) and the spherical
spaces Sp,(Q) and Sp, (Q’) passing through the quintuples of points

Qi, Q;,..-,Q, and Q,’, Q,’,...Q,’. These spherical spaces Sp, (Q)
and Sp, (Q/) form one of the 2” pairs of solutions of the problem.”

The proof of this construction is plain. When P, moves over

y (1 . q EN 1 . . : 45
Sp” the power of each of the ten centres of similitude lying in d,,,
with respect to the spherical space Sp, (/) remains unchanged; con-
sequently the spherical spaces Sp,(2?) which are possible form a

5 5 15 ° .
pencil with d, as common radical space and ¢, is a common radical

plane of Sp? with each of the spherical spaces Sp, (P) of that
pencil. If now we choose for P one of the two points of contact

° (1 : 5
Q or Q of Spy’ with a space through ¢,, then this tangent space

( 566 )

must likewise touch the spherical space Sp,(Q) or Sp,(Q’) passing
through this point in the same point, ete.

We have now arrived at the first part of our investigation proper
concerning the system of the spherical spaces Sp, touching 7 spherical
spaces Sp, (Mz, rx), (= 1, 2,...m) given arbitrarily in S, and we

reduce the general case — following the way indicated by Rye for
our space — to a simpler one, in which the centres J/; of the n

spherical spaces which must be touched lie in a space Sn

The centres J/; of the n given spherical spaces Sp, (Mx, rj) deter-
mine a space S,—;, intersecting these spherical spaces according to
“central spherical spaces” Sp,—1 (Mr, 7), thus intersecting them at
right angles. Let O be the radical centre of these 2 spherical spaces
Spr: and 7? the power of this point with respect to the spherical
spaces Sp, provisionally supposed to be positive. Then the spherical
space Spr1(O0,7) lying in S,—1 intersects at right angles the n
spherical spaces Spy (My, 77), thus also the 7 spherical spaces
Sp (Mi, rj). So an inversion with an arbitrary point OQ’ of the
surface of the spherical space Sp,—1(0,7) as centre makes the 7
given spherical spaces Spn (Mr, 17) and the spherical space Sp (O, 7)
cutting them at right angles to pass into n new spherical spaces Sp’,
and a space SS,» cutting them at right angles. This special case
where the centres WM, of the nm spherical spaces which must be
touched lie in a space S,—2 shall be treated first.

5. If Sp", is a spherical space touching the new spherical
spaces Sp’, then this spherical space Sp", rotating round the space
Ss through the centres WV will touch in any position the 2
spherical spaces Sp’, and will thus form a singular infinite series of
tangent spherical spaces. In an arbitrary space .S,;—; through the
axial space S,-2 we find according to the results obtained above
2-1 pairs of spherical spaces Sp",—1, touching the central spherical
spaces Sp',-1— lying in S,—1— of the # spherical spaces Sp’, and as
a matter of course each of these pairs consists of two spherical spaces
Sp"n—1 lying symmetrically with respect to S,-2. As each of those
pairs by rotation leads up to a singular infinite series there are 2”—!
of such series. The spherical spaces of each of those series are
enveloped — compare my preceding communication on page 492 —
by an n-dimensional torus T,,,; their centres lie on a circle. And if
we confine ourselves to one of the 2%! series, we can extend the
system of the touched spherical spaces Sp', to a n—2-fold infinite
series by representing to ourselves all the spherical spaces described

( 567 )

out of the points of S,-2, in such a way that they touch one of
the spherical spaces of the singular infinite series, thus all the
spherical spaces of that series too.

6. If we now confine ourselves to a single series of the 27—!
singular infinite series we have found two systems of spherical spaces
possessing the remarkable property that each spherical space of
one touches all the spherical spaces of the other. Of these two
systems one is a singular infinite series of equally large spherical
spaces with a circle C(M,, 7,) having M, and 7, as centre and as
radius and lying in the plane «, as locus of centres, whilst the other
is an n—2-fold series with the space Sa perpendicular in WM, to
& as locus of centres. How do these two systems transform them-

selves if we apply to both — in order to return to our ” given
spherical. spaces Sp, — the inversion with O’ as centre and the

formerly used power?

To answer this question is made easy by the observation that the
n-dimensional figure consisting of the two systems S/,, Sy',~2 and
their inverse systems Sy, Sij,—2 have a plane of symmetry, the
plane o through 1/,, O' and the projection O” of O' on Ss. This
plane o forming the. plane of fig. 2 has with ¢, in common the
diameter m' parallel to O' O" of the cirele C'(M/,,7,) and is according
to that line m’ perpendicular to €,; so it is a plane of symmetry for
Sy. It has moreover with S,—2 the line J/,0" in common and is
according to that line « perpendicular to Sy,—2; so it is also a plane
of symmetry for Sys. And if it is a plane of symmetry for Sy’,
and Sy',—2, then it is so too for Sy, and Sy,—2, because it contains
the centre ©’ of the inversion.

We prove to begin with that the centres of the spherical spaces
of Sy, lie in a conic. To this end we regard in the plane of sym-
metry o (fig. 2) the points of intersection J/’, J/" with the cirele
C(M,,7,), the circle of section C(J/,,7,) with the spherical space
Sp'n(M,,7,) of Sy’, and the point O of the line J/'O', for which
MOM Or. Then point A of a, which is at an equal distance
-from O' and O, is the centre of a sphere Sp,(4,A0') with A) as
radius, intersecting Sp", (J/',7") and so all spherical spaces Sp", of
the singular infinite series at right angles. This sphere is transformed
by the inversion with O° as centre into a plane ¢ perpendicular to
O'A, intersecting 6 according to a line m normal to O'A; this plane
e must contain the centres of the spherical spaces of Sy, as it cuts
all those spherical spaces at right angles. And farther, when inverting,
the centre of a spherical space remains on the line connecting this

point with the centre O' of the inversion; so the oblique cone with
O' as vertex and circle C(M

r,) as base must contain the centres
of the spherical spaces of the series Sy,, and the locus of those
centres is the conic of intersection of this cone with the plane «.
Of this conie m is an axis of symmetry and the points M' and 1",

0?

becoming the centres of the inverted spherical spaces Sp", (J/', 1"),
Sp"n(M',r') are vertices. This conic is an ellipse 4, a parabola
P or an hyperbola H, according to none, one or two of the spherical
spaces Sp", of Sy', passing through 0’, i.e. according to O' lying
outside the two circles C(M!, 7!) and C(M"‚ 7), on one of those
circles or inside one of them. Of these three cases fig. 2 represents
the first and this will be furtheron exclusively under consideration.

If we suppose that the conic obtained is an ellipse # the inverse
spherical spaces Sp, (M', r') and Spa (JZ", r") of the spherical spaces
Sp", CM’, 7) and Sp", (MY, 7’) will touch every spherical space Sp, (MZ, 7)
of the system Sy,—2 in the same kind. From the triangle MZ M' M"

( 569 )

then ensues, if we represent the radii vectores M'M and J" M of M
with respect to the fixed points JM’ and M" by w and », that
“u—v=+(r'—r"). So the locus of the centres of the spherical
spaces Sp,(M,r) of the system Sy,—2 is the figure of revolution,
which is generated when the hyperbola H with M' and M" as foci
and + (r'—r") as half real axis rotates round m in the space S,—1
through SS, and O'. And because each spherical space Sp, of Sy,
touches the spherical spaces of Sy, having the vertices of the
hyperbola H as centres, those vertices of the hyperbola H/ are reversely
the foci of the ellipse £. Thus the theorem holds good:

“The spherical spaces Sp„ touching 2 spherical spaces Spn given
arbitrarily in S, form 2”-! singular infinite series. The spherical
spaces of any of those series are connected by this that they intersect
a definite spherical space Sp',® at right angles and that their centres
lie on a definite conic (A); the determining figures, the spherical
space Sp’, and the conic (A), change from series to series. To each
series corresponds as envelope of its spherical spaces a definite curved
space of order four, the n-dimensional eyclid of Dupin. And if we
confine ourselves to a single series, the system of n-given spherical
spaces Sp, can be extended to an n—2-fold infinite series of spherical
spaces Sp, connected by the fact that they cut another spherical
space Sp,\° at right angles and that their centres are situated on the
surface of a figure of revolution generated by the rotation of a conic
(K). These two conies (K) and (X’') lie in mutually perpendicular
planes in such a way that the foci of one are vertices of the other
and reversely.”

7. The inversion applied becomes impossible within the region of
reality when the common power of the radical centre O of the n
given spherical spaces Sp, with respect to those spherical spaces is
negative. In this case before inverting we can diminish the radii of
the m given spherical spaces by a common quantity in such a way
that the radius of one of those spherical spaces disappears. Then the
power of the radical centre O of the new spheres is certainly positive.
By operating now with the new system and after that, when the
system Sy, has been found, by adding the assumed quantity to the
radii of the spherical spaces of Sy,, we arrive at the desired aim.
As is evident we can even augment the radii of some of the
given spherical spaces by the radius of the spherical space that is
to become a spherical space reduced to a point if only the series of
the tangent spherical spaces is chosen so as to correspond to this.

39

Proceedings Royal Acad. Amsterdam. Vol. VII.

(570 )

8. Are there not also non-linear systems Sy; and Syn of
spherical spaces Sp, respectively K-fold and n——1-fold infinite
situated in JS, in such a way that each spherical space of one system
touches all the spherical spaces of the other?

This question must be answered in the affirmative as we shall
prove here analytically.

If in a space S, of S, the spaces St and Sj, which have but
the point O in common are perpendicular to each other in this point,
if OP is the normal in OV on Sa, OQ an arbitrary line through
O in S&, OR an arbitrary line through O in S,—,—1 and if we assume
(fig. 3) in the planes OPQ and OPR an ellipse (/) with the half
axes OA—=a, OB=b6 and an hyperbola (#7) with the half axes
OC=c=Va'—b?, OD=6, then by rotation of (#) round OP
in the space Sy — (OP, Sj) — when every point describes a spherical

: : 3 0)
space Sp a quadratic space of revolution Qh is generated, by
rotation of (H) round OP in the space Sr == (OP, S, ri) —
when every point describes a spherical space Sp: — a quadratic

: Oe ie
space of revolution qe, is generated.

Fig. 3.
If now FE and # are arbitrarily chosen points of those figures of
revolution the distance HH can easily be calculated. If namely we
use a rectangular system of coordinates with O as origin, OP as

axis ONX,, the plane OPE as plane OX,X,, the plane OPH as
plane O.X,X,, then the coordinates of the points E and H are
DEE SET CA WEU Oer ON
TS —GSEC OP, | a —— 0 7 a0 tos tz
and we find

oe tn =d

EH =asecrb — cocos p.

From this ensues that the spherical spaces Sp,(2,ccosg++e) and
Spn (H, a sec w-|- 9), where 9 represents an arbitrary constant, touch
each other and that this contact is an external one or an internal
one, according to ccosp + @ and asecy Ho having the same sign
or not. Thus the theorem holds good:

“If we describe out of each point £ of Qin with acos@ as z,_
a spherical space Sp, (2, ¢ cos y+ 9) and out of each point H of

Oo with csect as zr, a spherical space Sp, (H, a sec p + o)
where y represents an arbitrary constant and p and w assume all
possible values, then two systems Syr, Sy. of spherical spaces
Sp„ are generated with the property that each spherical space of

one system. touches all spherical spaces of the other.”

Both systems of spherical spaces are enveloped by the same curved
space of order four. If namely of a rectangular system of coordinates
with O as origin and OP as axis OX, the axes OX,, OX,,
OX,, - - . OXppo are situated in Spy, the axes OX,, OX 43.
ON, OX, in SS, rp, then the coordinates of two points E and

: kde (2) (2) : : :
H lying arbitrarily on Qi and Q,~; can be written in the form

E H
ct, =acosp ©, =csecw
zt, =bsing cosg, Deh)
rh x, — big Poos,
ee bsin P| sin @, cos P, jj
L, —bsingpsingp, ing, cos p, Tr)
wr bsingpsingp, sing, Eri = 0
SUN Pk; — 2 COS Pl-—1
Tra bsingpsing,sinp, 49 = 0
sin PE—2 SIN Pl 1
zis 0 Ert3 = btg sinds, cos wp,
aera Tha = btg W sin Wp, sinw, cos yp,
Di 0 In—1 = big wsin yp, sinw, aes
SIN Wn—k—3 COS Wn 9
Zn =0 En big sin, siny,....
sin Wz Sin Wnr—k—e

39%

and
(w,— acosp)? + (a,—b sing cosp‚)° Har? + (a, —b sing sing, cosp‚) +...
+ (tE+2 — bsing sing, sing, ... sinpp—s singz—1)? + a Fene Fn —
= (¢cos p + o)°
is the equation of the spherical space Sp,(E,ccosy+e). If we
write this equation in the form.
n
Zur +b—?=
gl
= 2 far, cos p Hb sing [w, cosp, +a, sing, cosp‚ Haya sin Py- « sin Pier Ì
and underneath the / equations formed out of it by differentiation
according to 9, ,,...ge—1, then addition of the £ + 1 equations, after
having squared them, furnishes us with

“ k+2
(Zi + 6? — 9%)? = 4[(aa, + co)? Hb (oe, + 2 m’)). . (

Jl Uik
And this same equation is obtained in the form

n

n
(= a? — b? — 0’)? = 4[(ex, + ag)? — 8? (#,? + = 2;7)],
LS

di

if we consider the spherical space of system Sy,

9. For a variable parameter @ equation (1) represents a system
of parallel n-dimensional cyclids of Durin. Here we can then ask
after the m numbers indicating successively how many of those eyelids
pass through a point or touch a line, a plane, a space, ete. In this
investigation the &£-+-(n—4—1)-, i.e. the n—1-fold congruence of
the right lines is in prominence, connecting an arbitrary point Z of

aya with an arbitrary point M of QS: the case of 7 = 3 has been
treated before in a small paper (“Prace matematyczno-fizyczno”’,
vol. 15, pages 83—85, 1904). And the more general case we do
not touch here.

Mathematics. — “On a special tetraedal complex.” By Prof. Jan

DE VRIES.

1. By the equation

Ze re. |

c

DAG
ar
a system of similar ellipsoids is indicated.
The normal in a point P, on the ellipsoid containing this point is

determined by

or also by

For its orthogonal coordinates of rays, i.e. the quantities

' ’ =
AE, PY Ds

0, Ve p= ea ae), p= ay ya! ;
we find
u— u (b°—ce?) (u'—u)
p= a a, etc. P,= IN Unen etc,

From this ensues that the co? normals of the system of ellipsoids
form a quadratic complex with the equation

nn nne a eee (3)
2. For the traces of the normal with YOZ and YOZ, we have
successively Te a 5? and u = — das so
c?—b? ca
zi zy ande Zn
c a

Now follows from
2": z' = (c?—8?) : (?—a?)
that the complex can be built up out of oe* linear congruences, of
which the directrices form two projective pencils of parallel rays
situated in NOZ and YOZ having the direction OX and OY. So
the complex is fetraedral and has as principal points O and the
points X,, Y,,Z, lying at infinity on the axes.

The trace of the ray of the complex with XOY is determined
by wu" =—c’. If we notice that the parameter w is proportional to
the distance of the point P indicated by it to the point P,, we see
that out of

!

u —ú arc.

mn DTE
uu b-—c?

the characteristic anharmonic relation of the complex is obtained,

namely
(PP peu Pp" 8) = (a°—c’) H (b?—c’).

3. The footpoints P, of the normals let down out of P, lie
evidently on the cubic curve

wu by c?z
t=, ee = STA ars eka hey | U (4)
ate b° Hv ctv

( 574 )

which passes through the points P,,0,X, ,V,,.2,,,(v=0, ©,—a*,-—b*,—c’),
and which is thus an orthogonal eubie hyperbola w*. Each of its
points P, determines an ellipsoid, for which P,P, is the normal
ins P.

Through a given point P, pass @' curves w'; their “foci” (Null-
punkte) P, are indicated by

aa) (af uti by (bi a) yn, esa, (CN) A ENG)

so they lie on the normal having P, as footpoint.

These curves are all situated on the surface determined by the
equations

(2 + ua, n= (0° + u)y, (te Wes

ZES

Pe

SS SS ar: ; = (6)
atv bv ety

or also by the equation obtained from these by elimination of u
and v

| ae) & @

En
|
©

NOS de B (7)

BNA ZZ
This same equation we obtain out of (3) if we express the
coordinates of rays in the coordinates of points. So the locus of

the curves w* passing through P, is the complex cone of P,.
Corresponding to this we find out of (6) for v= const. a right
line through P,, whilst w = const. indicates a curve w’ through P,.

4. Out of the preceding ensues that all bisecants of a curve o*
are rays of the complex. This is further contirmed by the caleulation
of the coordinates of rays of the bisecant (v, v'). We find out of (4)

aay (v' Et 2) 0) etc
(a? + v)(a* Hv)

b*c*y 2, (b° —c*) (v! -— v)
== = —— EEG

(0? Ho) (C° + v) (b Hv) (0? Hw)

from which ensues readily
appa + O’p ps + PaP =

(i

5. The planes of coordinates and the plane at infinity are the
principal planes of the complex. The complex cone of a point lying
in a principal plane must degenerate.

We truly find out of (7) for z,—=0 the planes z =O and

(a? — c*?)y,a@—(b? —c)a,y — (at —D*)ayy,. . « ~ (8)

In. connection with this the curve w* consists now of the hyperbola

dl} (at — bay bye ae y—=0 ... . (9)

( 575 )
and the right line cut by it
(a2 =e a= ala, tit (bi Seb ij « . (10)
lying in the plane (8).
If in (7) we substitute 29, ue, re for w,, y,, 2, and reduce it to
the form

a? GC — i) (vy — uz) +b? (2 — «) (Az — va) + ¢? 5 — v ius —Ay)=0,
Ox 9 9

then @ = furnishes the equation
(a? — ?) Auz + (b? — c?) uve + (ce? —a')vdy=0, . . (LI)
which represents the plane containing the normals with the direction
(4, u,v). The footpoints of these normals lie evidently on the right line
DELEN SUP = SOT 6 oe at a eee (UI)

6. We determine with respect to the ellipsoid (4,) the polar line
of the normal n, having P, as footpoint.

For an arbitrary point P' of that normal we find the polar plane
at a

4

=>

3 a

7 zj
20 — es

For all values of w' this plane passes through the line of inter-
section of the planes

Er SE = Boia ate! gatas)
a b c
et be EL)
a b c

This line of intersection is the required polar line. When /, changes
it displaces itself evidently parallel to itself.
Out of (13) and (14) we obtain the equation
(aise) nd (OS CUBE Fe

bt c' a}.

which becomes identical with the equation

rg — de lie

Ys 23
of a projecting plane of the normal n, with footpoint P,, if the
conditions are satisfied

(a*—b?*) y, 6? (a?—c’) z, ANP Ce
Eo (Gacy, | be 1
From this we can deduce that the polar line of the normal n,

with respect to the surface (/,) is again a normal n,; the footpoints

vi

, and P, are connected with each other by the involutory relations

‚=P ko: (a’—b’) (a’?—c’),
I= BK, : Wa?) Oe),
ANO s\(C*—a;)\(c? be).
By polarisation with respect to each of the ellipsoids the complex

is thus transformed in itself. This agrees with a well known property
of the tetraedal complex.

7. The footpoints of the normals are then arranged in an involutory
quadratic correspondence, which transforms a right line into a twisted
cubie, thus the tetraedal complex into a complex of twisted cubies
which all pass through the points O, X,, Y,, 7. Let us now regard
in general the transformation

Bie! == 07, | yy AEN ooo (15)

It substitutes for the ray of the complex indicated by (2) the

twisted curve

hj a a ; b? B: A Gn
(c? + u) 2, i

2 = ———_—_ Ve z

(EH), ST Ey ’

If we still put

2 2 2
a Y
— = 2, EE (16)
vy 9. sl
then this curve is indicated by
Vn 2 n3
Da a re b Yo. vie eae
atu b? Hu : ctu

So it is the curve w? belonging to the “focus” P,, which corre-
sponds in the transformation to the footpoint P, of the normal.

The complea of normals is thus transformed into the complea of
the curves ow’.

In connection with this the cone of the complex of 7, passes into
the locus of the curves @’ containing the point /,, thus ($ 3) into
the cone of the complex of P,. Indeed the equation (7) does not

change in form if we apply the relations (15) and (16).

8. If the vertex of the cone of the complex moves along the
right line / represented by
he As Pee anal ais Ae dt
TT eene ;

U ==

REED TE

then the cones form a system with index two represented by
LUO EN 0;

where

C= Saed), Ui Zar — a) (ey — Ys, 2)s
3 3
2 U = a} ET a5) (2, BY) eel) zie (x TA #,) @, Ul ale z)}.
3

The envelope of this system, at the same time the locus of the
conics of the complex having / as chord, has for equation
n= A T ne En
U =U, Y= Ui):
The eight nodes which this biquadratie surface of the comple.
must possess are the points of intersection of the surfaces
Te == iF —— 0} raes
R=, ii Ore D= 0".
For we have

ÒU___ ÀU, OU,

U + U, au, Ee
DE a 3 3

Ow ? 0a

El

=

Ol
so that — disappears for each of those eight points of intersection.

Ow

To these nodes evidently belong the points O, X,, Y,,
four other ones change their places with the right line /.

That / is double right line of the surface of the complex is
immediately proved by the substitution «= 7, + 4a, y= y, + ue,
Z=2,-+ v0; on account of 2,y — y,z =o (uz, — vy,) we see that
U then obtains the factor o°.

Ze; the

Mathematics. — “On a group of complexes with rational cones
of the complex.” By Prof. JAN pe Vries.

§ L. In a communication included in the Proceedings of May
1903 *), I have treated a group of complexes of rays possessing the
property that the cone of the complex of an arbitrary point is
rational. In the following a second group will be indicated with the
same particularity.

We consider a pencil (s) with vertex S in the plane o, and in a
second plane tr a system of rays [f], with index m (thus the system
of the tangents of a rational curve t,) and we suppose the rays ¢
to be projectively conjugate to the rays s. The transversals of homo-
logous rays form a complex, which will be investigated here.

Out of an arbitrary point P the pencil (S, 0) is projected on the
plane t in a pencil (S', ©), projective to [¢],. Together these systems
of rays generate a curve of order (n +1) having in S' an n-fold
point; for on an arbitrary ray s through S' lies outside S' the point

1) “On complexes of rays in relation to a rational skew curve.” VI, p. 12—17.

( 578 )
of intersection of s with the corresponding ray ¢; on the rays s’
conjugate to the m rays ¢ passing through JS’ this point of intersection
falls in S', so that the locus of the point (s', 7) must pass 1 times
through S'; the curve is therefore of order (n +1).
The cone (P) of the complea is of order (n +1) and of class 2n
and has an n-fold edge PS.

§ 2. If the point S’ lies on the envelope t, two of the n rays t
passing through S’ coincide, so also two of the tangent planes through
2S to the cone (P).

The locus of the points P for which two tangent planes through
the n-fold edge of the cone of the complex coincide is the cone = of
order 2(n—1) projecting the envelope t, out of S.

The 8(7— 2) cuspidal edges of © contain the points P, for which
three of the tangent planes of (P) coincide along the n-fold edge.
The 2 (27 — 2) (n—s) double edges of © form the locus of the points
P, for which two pairs of tangent planes of P coincide along PS.

The cone = is a part of the singular surface of the complex;
the remaining parts are planes.

To these belongs in the first place the plane o. Each right line of
o is cut by m rays ¢, can thus be regarded n times as ray of the

complex. Consequently 6 is an n-fold principal plane. In connection
with this the cone of the complex of a point P assumed in 6
degenerates into m planes coinciding with o and into the plane
through P and the right line ¢ corresponding to the ray s deter-
mined by P.

On the contrary r is single principal plane, for each of its right
lines rests on but one ray s. The cone of the complex of a point P
lying in r degenerates into t and into the planes through P and
the m rays s corresponding to the right lines ¢ through P.

Finally there are still (n +1) principal planes y,,(k=1 ton HD),
each connecting two homologous rays s,¢. For the points of the
line of intersection of o and + are arranged by the projective systems
(s) and [4], in a (1,7) correspondence; in each of the (n-+-1) points
of coincidence CC} two homologous rays meet. In connection with
this the cone of the complex of a point P assumed in one of these
principal planes degenerates into the combination of this principal
plane with a cone of order #, for of the projective systems (s’) and
[t|, lying in t two homologous rays coincide.

§ 3. The curve of the complex (a) in the arbitrary plane a is of
class (2 + 1) and has the line of intersection (o sr) as n-fold tangent;

(579)

so it is of order 2. Its points of contact with (¢ x) are determined
on (oa) by the nm rays s corresponding to the 7 rays ¢ through the
point (or 2).

If x passes through one of the 2(n— 1) points of intersection
of 5 with the envelope tr, two of the points of contact of (o ze) coincide.
Regarded as locus of points («) then consists of a curve of order
(2n —1) and the right line (62).

The planes containing curves of the complex for which two points
of contact of the multiple tangent coincide form 2(n—1) sheaves
having their vertices on the line of intersection of 6 and t.

If « passes through a ray s,, then (2) as envelope consists of a
pencil having its vertex in the trace of the homologous ray ¢, and
of the pencil (S, x) of which each ray belongs 7 times to the complex,
because it is intersected by 7 rays ¢. As locus of points (a) is here
the line connecting the vertices of the pencils counted 27 times.

If z contains a ray ¢, the envelope (2) consists of a pencil having
the trace S, of the homologous ray s, as vertex and of a curve of
class 7 for which (oa) is an (n—1)-fold tangent. As figure of order
2n the curve (x) breaks up into a curve of order 2(m—1), its (n —1)
fold tangent and the tangent which can moreover be drawn to it
out of S,.

If one brings a through one of the coincidences (;, then (a) breaks
up in the same way into a pencil with vertex C, and a curve of
class 7.

The complex possesses an n-fold principal point S and (n +1)
single principal points Cy.

§ 4. Let us now consider the surface of the complex A of an
arbitrary right line /, thus the envelope of the rays of the complex
resting on /. The rays in a plane a brought through / envelop a
curve (a) of order 2n ($ 3). If a is one of the 2» tangent planes
through / to the cone of the complex of the point P lying on J,
then two of the tangents drawn out of P to (a) coincide, so that
P is a point of (xr). So each point of / belongs to 2” curves of the
complex; consequently / is a 2n-fold right line of 4.

The surface of the complex is of order 4n.

In the planes connecting / with the principal points C‚ the curve
(a) breaks up into a curve of order 2 (7 —1) and two right lines.
This also takes place when z passes through one of the rays t
resting on /. In the plane through / and S the curve (a) degenerates
into a right line to be counted 27 times.

In each of the planes connecting / with the points of intersection

(580 )

of zr, and 6 the curve (a) consists of a curve of order (27 —1) and
a right line (§ 3).

On A lie besides the 2n-fold right line and a 2n-fold torsal right
line 6n single right lines more.

The plane o contains 2(n— 1) right lines of A and touches 4 in
the points of a curve of order (n-+-1), which is the locus of the
points where the curve of the complex (a) touches its „-fold tangent
(oz). For, if the ray s, resting on / corresponds to the ray ¢, cutting
o in 7, then one of the points of contact of the curve of the complex
of the plane (/7,) with o lies in the trace L, of /; consequently
the indicated points of contact lie on a curve of order (n +1). This
curve is generated by the pencils (Z,) and (S) arranged in a (1, 7)
correspondence ; so it has in S an n-fold point.

The plane t+ touches A according to a curve of order (n + 1)
which is the locus of the points of contact of the curves (2), in
planes x through /, with the traces (ar). This curve has an n-fold
point in the trace Z, of / on rt; the tangents in this multiple point
are the traces of the planes a cutting (or) on the m rays s conjugate
to the rays ¢ drawn out of L,.

The plane r has farthermore the envelope rt, in common with 4.
For, while a point P of the right line (ar) bears in general n
tangents of the curve of the complex (a) determined by the rays s
corresponding to the rays ¢ drawn through P, two of those
tangents coincide as soon as P lies on the envelope vr; then however
P belongs to the curve (a), thus to the surface of the complex 4.

Microbiology. — “An obligative anaerobic fermentation Sarcina.”
By Prof. M. W. Brrerrinck.

The following simple but yet delicate experiment gives rise to a
vigorous fermentation, caused by a sarcine, wherein microscopically
no other microbes are perceptible and which, when rightly performed,
can produce a real pure culture of this fermentation organism. The
simplicity of the experiment is the result of many previous invest-
igations, partly made conjointly with Dr. N. Gostines, which have
gradually rendered clear the conditions of life of the examined
microbe.

Bouillon with 8 to 10°/, glucose, or malt wort, is acidified with
phosphorie acid to an acidity of 8 ee. normal per 100 ce. of culture
liquid and introduced into a bottle, which is quite filled with it
and fitted with a tube to remove the gas. The infection is done

( 581 )

with an ample quantity’) of garden soil, from which the heaviest
and roughest portion has been removed, but in which so much
solid substance is left behind that in the nutrient liquid it forms
a muddy deposit from 5 to 7 or more millimeters thick. The
culture is effected in a thermostat at 37° C. After 12 hours already
the liquid is in a strong fermentation, which lasts from 24 to 36
hours, and whereby the surface is covered with a rough seum,
produced by gas bubbles mounting up from the depth. Whilst
the liquid itself remains wholly free from microbes, the micros-
copical image of the deposit shows a luxuriant, pure or almost
pure culture of a sarcine, of which the elementary cells measure
for the greater part about 3.5 uw, so that the species belongs to
the largest forms known, and the multicellular sarcine-packages are
easily visible to the naked eye. The cells are colorless and
transparent and the packages present irregular sides. Here and there,
but much less generally, a brownish intransparent form is seen,
with more regularly cubical packages of which the cells measure
2 to 2,5 uw.

The scum floating on the fermenting fluid consists of slime in
which the evolved gas remains for a time imprisoned. This slime
is produced by the outer side of the sarcine cells, whose walls
for the rest consist of cellulose, which becomes violet-blue by zinc-
chloride and jodine. This reaction was discovered in 1865 in the
stomacal sarcine .by SURINGAR®), who on this account argued the
vegetal nature of this organism, which fully corresponds to the
small-celled fermentation sarcine. The large-celled form more resembles
the figures which LINDNer*) gives of his Sarcina maxima, found, as
he expresses it, in “Buttersäuremaischen”, hence, in wort wherein
a spontaneous butyric fermentation. I am not, however, convinced
that both these forms do really belong to two different species of
sarcine, as it is well known that in this genus of microbes great
morphological differences may occur in the same species.

The gas is a mixture of about 75°/, carbonic acid and 25 °/,
hydrogen; methan is not present. Besides, a moderate quantity of acid
is formed, which for example, in a nutrient liquid with an acidity
of 6 ce. per 100, may mount to 12 cc, a percentage only found
back in the technical lactic fermentations. Furthermore a peculiar
odor originates, reminding of the ordinary lactic-acid fermentation, by

1) With Jittle soil fcr infection, the experiment becomes doubtful.
2) W. F. R. Surinear, De sarcine (Sarcina ventriculi Goopsir), pag. 7,
Leeuwarden 1865. Here very good figures are to be found.

8) Mikroskopische Betriebscontrolle in den Gärungsgewerben, 3e Aufl. p. 432, 1901.

(582 )

Lactobacillus. If, as is probable, this acid will prove to consist
entirely, or for the greater portion, of lactic acid, the fermentation
sarcine may be considered as the most differentiated lactic-acid ferment
hitherto known.

When using a sufficient quantity of soil for the infection, that is
a relatively great number of sarcines, which thereby, in the given
circumstances, may compete with advantage with, and conquer
all other microbes, the experiment described succeeds within very
wide limits. Thus the sarcine fermentation may i this case be
obtained as well in an open flask as in a closed bottle, whence it
follows that the sarcine can suffer a moderate quantity of oxygen;
and it will appear below, that a slight quantity is even wanted
under all circumstances. Notwithstanding this, the name of obligative
anaerobic remains applicable as the cultivation at full atmospheric
pressure is impossible. The acid may further be varied between 3
and 11 ec. normal phosphoric acid per 100 ce. The phosphoric acid
may be replaced by lactic and even by hydrochloric acid, if the
acidity of the latter is not taken higher than 6 to 7 ec. per 100 ce.,
but not by nitric acid.

Instead of glucose cane sugar may be used, but with milk sugar
and mannite the experiment does not succeed. As source of
nitrogen only peptone can be used, such as found in malt-wort or
bouillon; simpler nitrogen sources, like asparagin, ureum, ammonia
and saltpeter, are unfit for the nitrogen nutrition of the sarcine. The
limits of the temperature are wide and may vary between 28° C.
and 41° C.

Although the experiment may thus be modified in many respects,
the first deseribed arrangement is recommendable, as it is best adapted
to the optimum of the different conditions of life of the organism.

A property peculiarly important for this research is the readiness
with which the function of fermenting, that is the power of evolving
gas, gets lost under the influence of a secretion product, probably
the acid, and through which all transports with old material become
perfectly useless. Hence it is necessary to transport cultures still in
fermentation to insure the success of further „experiments.

That some aeration enhances the life-functions of this obligative
anaerobic and that’ access of a little air is even necessary in the long
run, is evident from the fact that the most vigorous fermentations are
obtained in a closed bottle, with the deposit got in an open flask,
whereas renewing of the nutrient liquid formed above the deposit
in a closed bottle will after few repetitions give rise to diminuation
or cessation of the fermentation.

(583 )

For the continuation of the culture by inoculating slight quan-
tities of material of a rough fermentation into the same nutrient
liquid, two precautions should be taken. First, the inoculation should
be done into the medium, freed from air by boiling, the bottle
being entirely filled with the hot liquid, so that on cooling no air
can dissolve, Second, an acidity of less than 7 proves not suffi-
cient, hence this should be 8 or 10 ec, as otherwise the lactic acid
ferments might prevail and supplant the sarcine.

From the necessity of expelling the air we see that the fermen-
tation sarcine undoubtedly belongs to the ordinary anaerobics, which,
considering the success of the rough accumulation experiment with
aeration, might perhaps not have been expected ; but the fact holds good
in the same way for the butyric acid ferment, generally accepted as
an obligative anaerobic, so that, also with respect to the fermentation
sarcine, there should be spoken of ‘‘microaerophily.” Further exami-
nation shows that in deep test-tubes with maltwort-agar, very easily
pure cultures may be obtained, whereby the sareine is recog-
nisable by the obvious size and the remarkably rapid development
of its colonies. On the other hand, on maltwort, or broth-bouillon-
glucose-agar-plates with or without acid at 37° C., with access
of air, no growth at all of the sarcine takes place, as might
be expected. Of course the packages can also be seen on the
plates without growing and be removed in a pure condition. When
we make use of little acid for the rough accumulation, colonies of
lactic acid ferments, belonging to the physiological genus Lacto-
bacillus, will develop on the plates at the air, which can grow
as well with as without air, but whose other life conditions corre-
spond to those of the sarcine. In this case the experiment shows at
the same time tbat everywhere in garden soil real lactic acid fer-
ments are present, whereof the proof had not been given until now.

When using much acid, for example 10 ce. or more normal
acid per 100 ce. of culture fluid, through which the vital functions
of the sarcine, such as rapidity of growth and the faculty of assimilating
oxygen, are lessened, certain alcohol ferments, proper to garden soil,
come to development, but they can, together with some of the other
impurifications of the rough accumulations, as moulds, JZucor and
Oidium, be checked and expelled by exclusion of air, hence, by
culture in closed bottles. To this end however, it is necessary to render
the conditions for the sarcine as favorable as possible and not allow
a temperature below 37° C.

The staying out of the butyric acid fermentation (caused by Granu-
lobacter saccharobutyricum), which so readily originates with exclusion

( 584 )

of air in glucose-bouillon and maltwort, is due to the acidity of
about 8 ce. or more, whereby this fermentation becomes impossible.

Although it is evident from the foregoing, that the growth of the
sareine is less inhibited by the acid than that of the lactobacilii and of the
butyric ferment, it may still be easily proved that already 7 ce. acid per
100 ce, are less favorable than 3 or 5 ce, also for the development of the
sarcine itself, so that the higher amount of acid in the accumulation
only serves to render competition with the said ferments possible.
If by timely transports into maltwort with more than 8 ce. phosphoric
acid, or by separation in solids, real pure cultures are at disposal, the
further transfers, with entire omission of the acid, show that then
also vigorous growth and fermentation may occur. We thus see how
wide the limits are of the life conditions of the sarcine, as soon as
competition with all other microbes is quite out of question.

The discovery of this certainly unexpected fermentation has sprung
from the working out of the general question which organisms of
the soil can develop in a sugar-containing culture fluid in presence
of an acid and with imperfect aeration. At temperatures of about
30° C. and lower, alcoholferments, Mucor racemosus and Oidium
prove to be the strongest, but then already a few sarcines are
observed. At about 40° C. most alcoholferments of garden soil,
besides Mucor and Oidium can no more compete with the sarcine
and the lactobacilli, which then become predominant. This being
fixed the last steps which led to the culture of the fermentation
sarcine alone, were the recognition of the obligative anaerobiosis,
and of the superiority of the resistence of the sarcine with respect
to anorganie acids compared with that of Lactobacillus and the
butyric ferments.

Above, already, I pointed to the perfect correspondence of the
small-celled form of the fermentation sarcine to the description which
SURINGAR gives of the stomacal sarcine, and I suppose that in the
cases of non-cultivable Sarcina ventriculi, of which, for instance,
pe Bary speaks’), there should really be thought of the ferment-
ation sarcine. This view is supported by different observations in
the older literature, cited by Surincar. But still more convincing is
my accumulation experiment, which proves that the conditions for
the existence of this sarcine are just of a nature to render its life in
the stomach possible.

It will be easy to obtain certainty thereabout by a repetition of

1) Vorlesungen über Bacterien, le Aufl. pg. 96, 1887.

( 585 )

this experiment, not with garden soil for infection material, but by
using the stomacal contents of such a case of stomacal sarcine. The
“not cultivability’ of pr Bary may mean the same as anaerobiosis,
for it is well known how difficult if is, even at the present time,
to cultivate anaerobies if the particulars of their life conditions are
not exactly known.

For the rest I do not doubt of the precision of FALKENHEM's *) and
Mievra’s *) observations, wno have seen aerobic colonies of micrococci
originate from stomacal sarcine. It is true that I for my part have
not succeeded in confirming this observation with regard to the
fermentation sarcine, but for other species of Sarcima 1 have, with
certainty, stated the transition into micrococci, and with various
anaerobies, although not belonging to the genus Surcina, I have seen
now and then colonies originate of facultative anaerobics, which in all
other respects, corresponded to the obligative anaerobics used for the
cultures. Therefore this modification also seems possible for some
individuals of the fermentation sarcine. Accumulation or transfer
experiments with stomacal contents will however only then give
positive results, if these are used when still in fermentation; with
long kept material nothing can be expected.

Already the older observers*) as SCHLOSSBERGER (1847), SIMON
(1849) and Cramer (1858) have tried, although in vain, by a kind of
accumulation experiments, to cultivate the stomacal sarcine, wherefore
they prepared, as nutrient liquid, artificial gastric juice with different
additions. Remarkable, and illustrating the biological views of
those days, is the fact, that for the infection they did not use
the stomacal contents themselves, but beer yeast, supposing, that the
sarcine might originate from the yeast cells, which somewhat resemble
it, and are always found in the stomach together with the sarcine
itself.

Physics. — “The motion of electrons in metallic bodies.” U. By
Prof. H. A. Lorentz.

(Communicated in the meeting of January 28, 1905).

§ 11. By a mode of reasoning similar to that used in the last §,
we may deduce a formula for the intensity ¢ of the current in a
closed thermo-electric circuit. For this purpose we have only to
suppose the ends P and Q, which consist, as has been said, of the

1) Archiv f. experiment. Pathologie und Pharmacologie. Bd. 10, pg. 339, 1885.
2) System der Bacterién. Bd. 2, pg. 259, 1900.
3) Cited from Surtnear (I. c.).

40
Proceedings Royal Acad. Amsterdam. Vol. VIL.

( 586 )

same metal and are kept at the same temperature, to be brought in
contact with each other. The potentials pp and pq will then become
equal, but the stream of electrons » will no longer be 0. We shall
have on the contrary, denoting by + the normal section, which may
slowly ‘change from point to point, as has already been observed,

=P 1°. . 4 Ua
Taking this into account and using (23), we get from (21) and (80)
dep Ih ah? md (1 m dlog A 1
lie Male hee (=) Sn ane

We shall integrate this along the circuit from P to Q. Since 7
has the same value everywhere and

Pp PQ" Vn=Vo : hp ho,

Q
m (ldlog A (de
= da — î ==);
ZO IE Che o=

P

we find

Here, the first term is reduced to the form (34), if we integrate
by parts. Hence, if we put

the result is
VR Gd
R

as was to be expected. Indeed, 6 being the coefficient of conductivity,
R is the resistance of the circuit.

$ 12. We shall now proceed to calculate the heat developed in a
circuit in which there is an electric current 7, or rather, supposing
each element of the wire to be kept at a constant temperature by
means of an external reservoir of heat, the amount of heat that is
transferred to such a reservoir per unit time. Let us consider to this
effect the part of the circuit lying between the sections whose positions
are determined by « and w+ dw and let wdt be the work done,
during the time dt, by the forces acting on the electrons in this
element. J)’ being the quantity of heat traversing a section per
unit time, we may write
d

(WE) de
DE
for the difference between the quantities of heat leaving the element
at one end and entering it at the other, and the production of heat
is given by

d
g=w— 7 (UILEN ar RE ss (BH)
da

In order to determine vr, we observe in the first place that the
work done, during the time df, by the force acting on a single
electron is

mX 8 dt

and that, by the formula (1), the element 2/2 contains

GS a6) dads
electrons having their velocity-pomts within the element dà of the
diagram of velocities. Taking together the forces acting on all these
particles, we find for their work

m X > dxdt. § f(S, 7,5) da ,
an expression that has yet to be integrated over the whole extent
of the diagram. On account of (4), the result becomes
mXv = dx dt,

so that, by (36)

miX

=== — av.

e
Now, the value of Y may be taken from (21). Substituting
.
v=

em

and using at the same time (23), we find
1 d log A d 1 e Z
X= — -| — == 6 o o (Ge
2h dx a du (;) ie mo ey

0 a,

mi{ 1 dlog A d (AN 1 39
ne acs) |e a

i= da.
PKP
The expression (22) may likewise be transformed by introducing
into it the value (38), or, what amounts to the same thing, the
value of

so that

if we put

and

2hAX a
h a a

Lv
that may be drawn from (21). One finds in this way
W—W.+W,,

( 588 )

if
Ww NDE Ee (40)
Eh
and
WV Spe EO a NEN
7, h exh

§ 18. The expression (37) for the amount of heat produced in
the element dv may now be divided into three parts.
The first of these

12
WI da
oz
; de 4
corresponds to Jounn’s law. Indeed — is the resistance of the part

of the circuit extending from (2) to (ez + da).
The second part

d
er (W, =) de

is entirely independent of the current, as appears from (40). It may

therefore be considered to be due to ordinary conduction of heat.

This is confirmed by comparing it with what has been said in $ 9.
It remains to consider the quantity of heat

d
7 di aA We =) de,

or, if (389) and (41) are taken into account,
mid log A
Sere agin
This expression, proportional to the current and changing its sign
if the latter is reversed, will lead us to formulae for the PrLtmr-
effect and the Tuomson-effect. Reduced to unit current, it becomes
: m dlog A j 3
Ca Zeh da EAN a (42)
a. 1 shall suppose in the first place that, between two sections
of the ecireuit, there is a gradual transition from the metal I to the
metal II], the temperature and consequently / being the same through-
out this part of the circuit. Then, reckoning « from the metal I
towards II, and integrating (42), I find for the heat produced at the
“place of contact” by a current of unit strength flowing from I

towards II,
m Ajj 200 Aq
lon - log 3
POI Noa SCM NCA

( 589 )

Hence, if we characterize the Prrrrer-effect by the absorption of
heat I, ;, taking place in this case,

K 9, a 4B | A7 2 a a Ni 43
== Nu == = —— Wg En . . . J
Suid FC aati. area mm oy,

Dh. In the second place, substituting again for / the value (14),
we shall apply (42) to a homogeneous part of the circuit. We have
then to consider loy A as a function of the temperature 7’, so that
we may write

1

2aT dlog A
Ln
for the heat developed between two points kept at the temperatures
T and 7+d7, if a current of unit strength flows from the first
point towards the latter. What Kervin has called the “specific heat
of electricity’ (THomson-effect) is thus seen to be represented by
2aT dlog A

de rds

(44)

u=—T—

$ 14. An important feature of the above results is their agreement
with those of the well known thermodynamic theory of thermo-
electric currents. This theory leads to the relations

— 7 d HH U 45
Ur — U = AT Pr 5 A NL deeds : 5 ( oD)
and

fed

SOE
Dd at Pers eyes 0)

1

T'

in which wy and wy are the specific heats of electricity in the
metals I and II, at the temperature 7, whereas / denotes what we
have calculated in § 10, viz. the electromotive force in a circuit
composed of these metals and whose junctions are kept at the
temperatures 7” and 7", the force being reckoned positive if it
tends to produce a current which flows from L towards I through
the first junction.

The values (44), (45) and (85) are easily seen to satisfy the
equations (45) and (46).

Instead of verifying this, we may as well infer directly from (42)
that our results agree with what is required by the laws of thermo-
dynamics. On account of the first of these we must have

2 Gi = — FF

and by the second

the sums in these formulae relating to all elements of the closed
circuit we have examined in $ 11. Now, by (42), these formulae
become

i da
2e h dz
P
and
Q

al; 1 dlog A et,

AT da laks
P

The first of these equations is identical with (34) and the second
holds because 47’ has everywhere the same value.

It must also be noticed that the formula (35) implies the existence
of a thermo-electric series and the well known law relating to it.
This follows at once from the fact that the value (85) may be
written as the difference of two integrals depending, for given
temperatures of the junctions, the one on the properties of the first
and the other on those of the second metal. Denoting by Illa third
metal, we may represent by /y 77, Fi, mi Fi, the electromotive
forces existing in circuits composed of the metals indicated by the
indices, the junctions having in all these cases the temperatures 7”
and 7" and the positive direction being such that it leads through
the junction at the first temperature from the metal indicated in
the first towards that indicated in the second place. Then it is seen
at once that

Epa + Prime Er Or EEEN

Strictly speaking there was no need to prove this, as it is a con-
sequence of the thermodynamic equations and our results agree
with these.

§ 15. In what precedes we have assumed a single kind of free
electrons. Indeed, many observations on other classes of phenomena
have shown the negative electrons to have a greater mobility than
the positive ones, so that one feels inclined to ask in the first place
to what extent the facts may be explained by a theory working
with only negative free electrons.

Now, in examining this point, we have first of all to consider the
absolute value of the electromotive force /’. If we suppose the tem-
peratures 7" and 7” to differ by one degree and if we neglect the

(591 )

variability of Nj and N7 in so small an interval, we may write
for (35)

The value of the first factor on the righthand side may be taken
from what, in § 9, we have deduced from the electrochemical
aequivalent of hydrogen’). We found for 7’= 291

Te
B LOP
é
so that
Nir 4
log —" — 0,00011 Fe.
Nr

In the case of bismuth and antimony, #9 amounts to 12000,
corresponding to
Nir Ni

(Qe — a,
Á Ni ’ Nz :

I see no difficulty in admitting this ratio between the number of
free electrons in two metals wide apart from each other in the
thermo-electric series *).

1) The numbers of that § contain an error which, however, has no influence
on the agreement that should be established by them. The value of 3 p and that of

7

deduced from the measurements of JaeGer and DiessetHorst are not 38 and

é

47, but
op == 38 x 10°
and
T
a ATCO
e

N
2) Let x be the mean value of log - " between the temperatures 7” and Tv.
Nr P

Then the equation (35) may be put in the form
9

Be 5 na(ir" — 7").

This may be expressed as follows: The work done by the electromotive force
9

in case one electron travels around the circuit is found if we multiply by 5 n

the increase of the mean kinetic energy of a gaseous molecule, due to an elevation
of temperature from 7” to 7",

(592)

The question now arises whether it will be possible to explain
all observations in the domain of thermo-electricity by means of
suitable assumptions concerning the number of free electrons. In order
to form an opinion on this point, I shall suppose the Peurier-effect
to be known, at one definite temperature 7, for all combinations
of some standard metal with other metals and the THomson-effect to
have been measured in all metals at all temperatures. Then, after
having chosen arbitrarily the number N, of free electrons in the
standard metal at 7, we may deduce from (48) the corresponding
values for the other conductors, and the equation (44) combined with
(13) and (14), will serve to determine, for all metals, the value of
N at any temperature we like. Now, the numbers obtained in this
way, all of which contain N, as an indeterminate factor, will suffice
to account for all other thermo-electric phenomena, at least if we
take for granted that these phenomena obey the laws deduced from
thermodynamics. Indeed, these laws leading to the relation

My, 17 + Wy, za + Mur = 9,

similar to (47), the values of V we have assumed will account not
only for the Perrwr-effect at the temperature 7, for all metals
combined with the standard metal, but also for the effect, at the
same temperature, for any combination. Finally, we see from (45)
that the value of 11/77 at any temperature may be found from that
corresponding to 7, if we know the THomson-effeect for all inter-
mediate temperatures and from (46) that the values of the electro-
motive force are determined by those of JI.

There is but one difficulty that might arise in this comparison of
theory with experimental results; it might be that the assumptions
we should have to make concerning the numbers NV would prove
incompatible with theoretical considerations of one kind or another
about the causes which determine the number of free electrons.

As to the conductivities for heat and electricity, it would always
be possible to obtain the right values from (24) and (27), provided
only we make appropriate assumptions concerning the length / of
the free path between two encounters *).

it must be noticed, however, that, whatever be the value of this

. ; k
length, the foregoing theory requires that the ratio pn shall be the

1) If the electric conductivity were inversely proportional to the absolute tempe-
rature, as it is approximately for some metals, and if we might neglect the varia-
tions of N, the formula (24) would require that 7 is inversely proportional to
VT. 1 am unable to explain why MN should vary in this way.

(593 )

same for all metals. The rather large deviations from this law have
led Drvpe to assume more than one kind of free electrons, an hypo-
thesis we shall have to discuss in a sequel to this paper. For the
moment I shall only observe that one reason for admitting the
existence not only of negative but also of positive free electrons
lies in the fact that the Hani-effect has not in all metals the same

direction.

(March 22, 1905).

KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM,

PROCEEDINGS OF THE MEETING
of Saturday March 25, 1905.

DC

(Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige
Afdeeling van Zaterdag 25 Maart 1905, DI. XIII).

OENE EEN MES:

R. Mrenaker: “On moments of inertia and moments of an arbitrary order in spaces of arbi-
trary high rank”: (Communicated by Prof. P. H. Scnoure), p. 596.

S. J. pe Lance: “On the branchings of the nerve-cells in repose and after fatigue”. (Com-
municated by Prof. C. Winkie), p. 599. (With one plate).

J. A. C. Oupemans: “A short account of the determination of the longitude of St. Denis (Island
of Réunion) executed in 1874”, p. 602.

W. Huiskamp: “On the presence of fibringlobulin in fibrinogensolutions”. (Communicated
by Prof. C. A. PEKELHARING), p. 610.

J. D. van per Waats: “The transformation of a branch plait into a main plait and vice
versa”, p. 621. (With one plate).

JAN DE Vries: “A group of algebraic complexes of rays”, p. 627.

Jan DE Vries: “On nets of algebraic plane curves’, p. 631.

Errata, p. 633.

The following papers were read :

Mathematics. — “On moments of inertia and moments of an
arbitrary order in spaces of arbitrary high rank.’ By Prof.
Dr. R. MexMke at Stuttgart. (Communicated by Prof. P. H.
SCHOUTE).

In the “Mathematische Annalen” Vol. 23 (1884) pages 143—151
I have pointed out a manner of calculating the moments of inertia,
leading easily and quickly to the purpose and being independent
of the number of dimensions. As an instance I chose the case of
a figure filled with a homogeneous matter in the space of (n—1)
dimensions, analogous to the tetrahedron, thus according to the well-
known expression of Mr. Scnourr a simplex S,. Without being
acquainted with this Mr. Scnoutr has lately treated this case in

41
Proceedings Royal Acad. Amsterdam. Vol. VII.

(596 )

another way in the “Rendiconti del Circolo Matematico di Palermo”,
Vol. XIX (1905) and has arrived at the same result. Instead of
contenting myself with the reference to these facts 1 wish to com-
municate how in the same way moments of any higher order than
the second can be found. It is true this problem has been prepared
in the above mentioned place, pages 146—147, for a simplex so far,
that but a slight step would have been necessary to bring about its
solution.

Let us presuppose a flat space of (n — 1) dimensions, a space of
“rank” (“Stufe”) 7 as GRASSMANN expressed it as early as 1844, or of
“point-value’ 2 as Prof. Scrovrr has said in his excellent textbook
on polydimensional geometry. The moment J/, of order v of an
arbitrary material figure belonging to this space with respect to a
space ME of rank (n—1) (thus n—2 dimensions) contained in the

same space is
i f r’ dm,

where 7 indicates the distance of a central point p in an element
of that figure from ZE, dm the mass of the element. According to
GRASSMANN however

7 =[Ep],
i.e. equal to the “outer” product of ZE and p, when we assign
both to # and p the numerical value 1, consequently

M, = { Eph a . Mes oes

I assume that rv is a positive integer. If » is an even number
and if the moment is to be calculated with respect to a space A of
a rank smaller than (n — 1), if thus it is e.g. a case of a moment
of inertia with respect to an axis (yp=2), then according to GRASSMANN

x? = [Ap| Ap],
where the symbol | denotes the “inner” multiplication, and we
arrive at
v

2

M,= | [Ap|Ap]adm. . TED)

The integrals appearing in a) and 6) can be evaluated by one
and the same integration, if we make use of the very useful notion
of the “gap-expressions’ introduced by GRASSMANN. If namely we

place the point p appearing in [Zp}’ or in [Ap | Ap]? symbolically

(597 )

outside the brackets and if with PEANo we indicate every gap thus
formed by +, we arrive at
v
r— |Help» resp. % == [As|As|?’. p
or
pe ahh
where the expression Zi (furnished with » gaps) is equal to |Z}
v

in the first case and to [A+lA+|? in the second. The expression Z
remaining constant in the integration it can be placed before the

symbol f of the integral, so that we get

M, = b.{p dm = Ip Hulsen hee arete “ate UL)

This has reduced our problem to the determination of the “point-

quantity of order v”
pe) = frr an (2)

belonging to the given material figure. (The vt power of a point
p we have to imagine as the v-fold point p. The algebraic product
of » different points is the total of these points, where on account
of the interchangeability of the factors of an algebraic product the
order of succession of the points is arbitrary. The sum of an arbitrary
number of such like quantities has primarily but a formal meaning,
but then it may be represented geometrically by a figure of order
vy, the analogon of the ellipsoid of inertia). The integral 2) is depen-
dent only on the form and the distribution of the mass of the given
material figure, and whilst when treating our problem in the usual
way with the aid of cartesian coordinates the space or A may
have a very disturbing influence upon the integration this influence
is here entirely done away with. Various other problems lead to a
similar integral as 2). If inter alia we wish to calculate the kinetic
energy 7’ of an (invariable or affinitely variable) continuously moving
system of masses for an arbitrary epoch, then

2 T = [v%dm,
where v denotes the velocity of a central point p in the element
: : 3 : Í;
dm. But v? is equal to the “inner” square of the vector — repre-
at

senting the velocity of p according to length and direction, i.e.
41*

(598)

dp dp
Ent

and when the symbol 4 denotes a certain affinity the momentaneous
system of velocities of the system of masses is indicated by

hence we have

D= [Mx | Ux], pO= fp'dm.

The evaluation of the static sum of the forces of inertia of an
arbitrary order called forth during the motion of the system of mass
at any epoch and the evaluation of the energies of higher species
inter alia considered by J. Somorr also lead to the integral p®.

It does not raise the slightest difficulty to find the integral p™ for
a simplex of constant denseness with the vertices a,, a,, .., Om
We can put

pA, a; = A, a; = © And
where all points inside the simplex are obtained, when to the
numerical quantities 2,, À,, .., 4, are given all positive values
compatible with the condition

EN a
If we think the simplex broken up into elements of the shape of
the parallelotop, i. e. of the (2—1) dimensional analogon to the
parallelepiped of our space, and with edges parallel to the edges of
the simplex starting from a,, then a slight calculation to be found
(l.e.) on page 147 gives us
dm = (n—1)! M da, da,... dan,

where J/ indicates the mass of the entire simplex. Hence we find
p”) = (n—1)! M I (A, a, + 2, a, +... + dna,)'da, da,... dy.
The polynomial theorem gives

=, vy!

CASAS eem An) = DRE

Jd Ara, ae. An”
Pl

with
DI Dare Dn == Oee dd > ?P, try, t---+m=—n.

On the other hand we find according to a wellknown theorem
of LiovviLLe under the above conditions for 4,, 4,,...4n:

(599 )

Deel ARD
fr Aer anal AsO A a duns ie A, — uae =! =
; WV, +r, +...+ 7, + nl)!
Abt pl vl... pl
aw; (» + n—l)!
hence
vl (nl)!
po) M oe WEN OL ab ore (5)
vp (» + n—1)
( gn } Mael
with

pv, t+ry+...+m=vr.

The sum to the right could evidently be arrived at out of

(a, ta, +... a)’ by developing it according to the polynomial
theorem and by suppressing all the polynomial coefficients. The factor
v! (n—1)!

(» + n—1)!

is nothing else but the reciprocal of the number of terms. By introducing
in 1) the obtained value of p”, we find, when the distance of the
vertex a; of the simplex from the space £ or A is indicated by y;,
of '
WG Eee M NS IND oP
(»-+n—1)!l mm
(vy, Hv, +..-4-m=?).

For »=2 I have deduced (l.c) the sum in 3) to a sum of (n+41)
respectively 7 squares, in other words I have substituted for the
simplex a system of (x + 1) resp. m single material points, which
is equivalent to it with respect to all questions connected with the
moments of inertia. For » > 2 a similar reduction seems to be less
easily effectible.

Stuttgart, March 1905.

Physiology. — “On the branchings of the nerve-cells in repose and
after fatigue.’ By Dr. S. J. pe Langer. (Communicated by
Prof. C. WINKLER.)

In the laboratory of Marrutas Duvar some experiments have been
made by MaroufrraN in order to ascertain whether it is possible
to demonstrate modifications in the dendrites of the ganglion-cells in
cases of sleep through fatigue. His results have been published in
the “Comptes Rendus de la Société de Biologie, 28 Févr. 1898” and
subsequently.

The animals he made use of for his experiments were mice, and
he proceeded in the following manner: For the space of an hour

( 600 )

together a mouse was driven to and fro in a cage, without granting
it any rest; after that the exhausted animal fell asleep or at any
rate remained perfectly quiet. The control-animal was kept in perfect
repose. Both animals were then killed, and small pieces of the
brain were immediately fixed after the method of Gone. He
obtained manifest results already when only feebly magnifying: the
collaterals of the dendrites have vanished, instead of these the
dendrites have globular tumefactions, retracted branchings which
seem to have loosened themselves from the neighbouring end-arbor-
isations.

MANOUELIAN writes:

“On pense, en présence de ces images, a celle d’une sangsue vue
comparativement dans l'état d’élongation et dans l'état de rétraction
en boule.”

Previous to these experiments, RaBr-RückHarpr had published a
theory on the amoeboid motion in the cells of the central nerve-
system, a theory not founded however on mieroscopical data. (Neurolog.
Centralblatt 1890, p. 199). The investigations of WiepersHeiM who
experimented on a living Crustacea, Leptodora hyalina, and those
of Prreens and others on the retina of Leuciscus rutilus, seemed to
confirm the conjectures of RABL-RücKHARD.

WiepersHem has been able to follow the motion of the processes
of the nerve-cells with the microscope and arrives equally at the
conclusion: “dasz die centrale Nervensubstanz nicht in starre For-
men gebannt, sondern dasz sie activer Bewegungen fahig ist.”

J. Demoor injected dogs with lethal dozes of morphia, and studied
a small piece of the cortex cerebri, which he extirpated before the
death of the animal. He too, and likewise SrrraNowska, after injecting
mice with ether, found similar changes as observed by MANovgLIAN:
the branchings having become smaller and shaped like a string
of beads.

Two american authors however, Frank and Wein, did not obtain
these results on animals under narcosis.

In order to obtain some certitude whether any differences might
in reality be observed, I tried a few experiments in the laboratory
of Professor WINKLER.

Firstly I did repeat the experiments of STRFANOWSKA and DEMOOR,
albeit the methods employed were not in every respect the same
as theirs.

The mice were brought under narcosis by means of chloroform
instead of ether: immediately after death they were decapitated, the
head was caught into a liquid, prepared after the method of Gore1

S. J. DE LANGE. “On the branchings of the nerve-cells in repose
and after fatigue.”

Nerve-cells of the cornu Ammonis from a mouse, exhausted by incessantly
running in a turning cage for four hours together.

Proceedings Royal Acad. Amsterdam. Vol. VII.

( 601 )

modified by Cox, whilst the brain was prepared directly in the liquid.
For the control another mouse, not having been put under narcosis,
was treated in the same manner. No differences whatever were to be
observed in the microscopical preparations, obtained by’ means of the
freezing-microtome.

Neither did I observe any differences in the case of mice, injected
in the manner used by Drmoor with repeated doses of morphia until
death ensued.

Thinking these results might have been impaired by the fact that
the animals were decapitated only after death, I next tried with the
utmost accuracy a repetition of the experiments of MANOUELIAN.

A mouse was put into a turning cage, being therefore constrained
to run incessantly, whilst the cage was kept in continual motion by
means of a small motor driven by water. The motion was continued
for four hours together, the animal experimented upon being therefore
perfectly exhausted. Meanwhile the control animal had been kept in
darkness, enveloped in wadding. The four hours having elapsed,
both animals were very quickly decapitated, the heads being caught
into the fixation-liquid, and the brain being further prepared in it.

After ten weeks the preparations were impregnated with celloidine
and section-series in frontal direction were made of both brains. In
this way it became possible to obtain a comparable material.

For further control another pair of mice was sacrificed, for the
purpose of demonstrating by means of the method of Nisst the
presence of the well-known modifications in the easily tinctured parts
of the protoplasma of the nerve-cells.

For whilst under normal conditions the elective tincturing part of
the protoplasma of the ganglion-cells is divided into small granula,
in case of fatigue these granula tend to dissolving more and more,
the tineturing of the cellular body thus becoming homogeneous.

These modifications are clearly to be observed in the ganglion-cells
of the exhausted animal experimented upon: the fatigue therefore
must have been exquisite.

The preparations, made after the method of Gone! modified by
Cox, offer however beautiful arborisations as well in the case of the
non-fatigued animal, as in that of the exhausted one used for the
experiment, the annexed photographical reproduction of the exhausted
animal presenting no trace of retracted branchings, or of globular
tumefactions, neither of being shaped like a string of beads.

I have therefore not succeeded in demonstrating after this method
modifications in the branching system of the nerve-cells of the cortex
cerebi, caused by intense fatigue.

( 602 )

Astronomy. — Prof. J. A. C. OupeMans presents as a first communi-
cation on his journey to Réunion for observing the transit of
Venus: “A short account of the determination of the longitude

of St. Denis, Island of Réunion), executed in 1874.”

In our ordinary meeting of October 30, 1875 I communicated
a few details on the state of the computation of the observations
at St. Denis on the transit of Venus of December 9, 1874. The
purport substantially was, that the computations had been carried
out as far as was possible at that moment.

Several circumstances, independent of my will, were the cause that
this state of things remained the same till the middle of last year,
and that the computations could not earlier be taken in hand again.

What I communicated then has been inserted in the Proceedings
of that meeting. Passing by all that refers to the heliometer
measures, which I hope to take up at some later time, I will only
mention the fact that the necessity was pointed out of determining
with precision the longitude of the place of observation.

For this purpose we, viz. Mr. Ernst VAN DE SANDE BAKHUYZEN,
Mr. Sorters and myself, have observed a number of occultations,
not so much of the brighter stars, announced in the Nautical Alma-
nac, as rather of fainter stars, of the 8 or 9t* magnitude, the posi-
tions of which were not yet known with precision at that time.
These had to be determined therefore by meridian observations; our
honoured president readily undertook the task of having these deter-
minations made at the observatory under his direction.

As a rule at least four determinations have been made of each star.
Though the added epochs show that this was done between the
years 1879 and 1884, it lasted a considerable time, till November
1901, before the reductions of those determinations had proceeded so
far that the results could be communicated to me.

At the same time my attention was called to the fact that most
of these stars had been since also observed at other observatories.

It thus became necessary, in my opinion, to look for all these
determinations in the several Annals and to reduce them to the same
epoch, (of course 1874), in order to make allowance for proper
motion, wherever necessary. In many cases it proved sufficient to
retain the Leiden determination unchanged.

But besides, the errors of the lunar tables, that is to say of
the positions published in the Nautical Almanac, had to be derived
from observations. For this purpose the observations at the meridian-

( 603 )

circles of Paris, Greenwich and Washington and those at the Alta-
zimuth of Greenwich have been used.

It is true that, in a remarkable paper, Znwestigation of corrections
to Hansen’s Tables of the moon, with tables for their application,
Newcoms brought together the corrections to be applied to the for-
mulae by means of which Hansen calculated his tables of the moon.
The paper contains the terms which had to be added according to
the state of science in that year, and also an empirical correction
determined by the most recent observations.

Moreover a table of corrections for 1874 was given, founded on
these data. But after having made a diagram representing, both the
corrections found by direct observation and those furnished by
Newcomp’s table, I came to the conclusion that the former was to
be preferred *).

As for the longitude of St. Denis, I will remark, that it has been
determined by the French naval officer Germain in 1867 and 1868
by means of 13 culminations of the first and 12 of the second limb.
In the Connaissance des Temps of 1871 a short report of that deter-
mination is to be found. Though the 25 results there given, agree
tolerably well, this kind of determinations is always liable to the
drawback that the difference in the constant error, made in observing
the culmination of the moon’s limb and of the comparison stars,
enters into the result, about thirty times magnified. There is no fear
of such an injurious influence in a determination of longitude by
occultations *). If the voyage to reach the isle of Réunion did not
last so long, and if the Indian Ocean were not so wild and bois-

1) The present state of science requires a correction of one of the tables of
Newcoms. He points out (page 9) that the parallactic equation of Hansen is founded
on the value 8”.916 of the solar parallax, whereas the value which he derived in
1867 from all the available materials is but S'.S4S, which is less by 0.068.
Further that later determinations require rather a diminution than an increase of
that number. At present 8.800 is generally adopted as being the most probable
value of the solar parallax, which is less than Newcomp’s value by 0'.648. The
parallactie correction of Newcoms must therefore be increased 1,7 fold; in other
words: three terms have to be added, viz. :

+ 0".67 sin D + 0".05 sin (Dg) — 07.09 sin (D + 9’),

where D represents the mean elongation of the moon from the sun, g the mean
anomaly of the moon and g’ that of the sun.

*) Newcomp says at the beginning of his paper above mentioned: “Determi-
“nations of longitude from moon occultations are found by experience to be
“subject to constant errors which it is difficult to determine and allow for. It
“was therefore a part of the policy of the American Commission to depend on
“occultations rather than upon culminations for the determinations of longitudes, etc.”

( 604 )

terous, these voyage would also present an occasion of determining
the longitude by transport of chronometers. Unfortunately the results
given by the different chronometers were so diverging as to be of no
value whatsoever.

The report above mentioned of GrrMain’s determination is accom-
panied by a plan showing his place of observation. We see from
this plan that west of the town the river St. Denis runs nearly in
a north-north-westerly direction towards the sea and that the place
of observation of GERMAIN was still on the west of the river.

A briek pillar, on which stood his transit instrument in 1867
and 1868, was still extant during our stay in 1874.

The result of Gerrmatn’s determination of longitude and latitude
was given by him as follows :

Longitude of the place of observation east of Paris 332™25s,7
Reduction to the flag-staff, east of the Barachois (7. e.

of the little creek which protects the sloops in landing) + 1 ,07
Longitude of the flag-staff east of Paris, (sic.:) . . 3"32™26s,8

Southern latitude of the place of observation deter-

mined by 4 northern and 3 southern stars . . . . 20°52! 2",0
Reducuon 10 the; lac stane se ee — 23 ,7
Southern latitude of the flag-staff. . . . . . . 20°51'38"3

Our observations of occultations took place at different points, the
relative position of which was accurately determined by Mr. Sorters.
Taking the difference of longitude of Paris and Greenwich
= 9m20:63 from the Nautical Almanac of 1874, (as given at that
time both in the C. d. T. and in the N. A), we got from the num-
bers just mentioned, for the flag-staff 3'41™47,43 east of Greenwich.
Corresponding therewith :

Place of observation : Long. E.of Gr. _ Latitude
1st on the ground of the harbour office 3'41™47s,32 — 20°51'40",6
Dd 8 +. ,, our dwelling house,

NG lank wesdue Conseil aes nneeee 48 11 46 1

3°¢ Near or in the pavilion of the helio-
meter on the battery

y 47 81 35 „3

The calculation of the longitude from the occultations has been
‘arried out on printed forms, arranged according to the method which
I developed in the Astronomische Nachrichten N°. 1763.

In this method the declination of the moon is taken from the

( 605 )

astronomical almanac, using an adopted longitude; the parallax is
then computed for that point of the moon’s limb, where the star
has disappeared and which therefore has the same right ascension
and declination as the star. We then have to add or to subtract two
terms to or from the right ascension of the star, to get that of the
moon’s centre, and finally we find from the almanac the Greenwich
time corresponding with that right ascension.

The longitude of the place of observation, then found, is the right
one, if it agrees with the adopted longitude. If it does not agree,
we have only to repeat a small part of the calculation with a
modified longitude of the place, to derive the true longitude from
the two differences.

This method corresponds with the method, which was customary in
the 18 century (which we find inter alia explained in the well known
treatise of BoHNENBERGER : Anleitung zur geografischen Ortshestimmung)
with this distinction that then the whole computation was carried
out in longitude and latitude, whereas we use the right ascension
and declination. Further, that for BoHNENBERGER c.s. there is no ques-
tion of any second hypothesis.

I will readily grant that Bessrr’s method of computing ecliptic
phenomena and thus also for the prediction of occultations and for
the calculation of the longitude from an observed occultation, is
justly considered to be the classic method. It is also the only
one explained in most of the textbooks. But it seemed to me that
the method indicated by myself is more expeditive and only in a
few cases inferior to that of BesseL in point of accuracy. The
drawback of this last method consists in the troublesome preparatory
calculations, which it requires. Any one may convince himself of
the truth of this statement by consulting the wellknown textbook of
Cuauvenet: A manual of spherical and practical Astronomy, Phila-
delphia 1874, vol I, p. 550°).

The horizontal equatorial parallax of the moon could be derived
from the Nautical Almanac, without any correction. As for the appa-
rent semidiameter of the moon, I myself made a determination of this
quantity, based on an elaborate investigation in 1859, (vid. Verslagen
en Mededeelingen der Natuurkundige Afdeeling, Vol. V1, p. 25 seqq.)

1) I have calculated a single example by this method ; the result differed only by
0s,1 from that obtained by the other method; in the first however 57 logarithms
had to be taken out, against 37 in the latter. Thinking the matter over, however,
I believe that the method of Besset will probably admit of a modification by which
this difference will be materially diminished. I hope shortly to investigate this more
thoroughly.

( 606 )

which furnished 0.27264 for the proportion of the mean moon’s semidia-
meter and that of the earth’s equator (at least this is the result of the
occultations discussed). After mature consideration, however, I now
adopted the value 0.2725 hor. equ. parallax + 0"04, This leads
approximately to the same value as when we take the sine of the
moon’s apparent semidiameter 0.272525 of the sine of the equatorial
horizontal parallax.

This factor is the mean of those which were derived from occul-
tations during total eclipses of the moon by Lupwie Srruve in 1888
and by J. Perers in 1895 (0.272535 and 0.272518). The Nautical
Almanac, which used both the semidiameter and the parallax as given
in the Tables of Hansen, gave a value larger by 1'4 to 1"6. This
difference has remained the same up to the present time.

About the observed occultations we may communicate the following
particulars. They were mostly observed by myself, partly with the
Fraunhofer telescope, (aperture 11 em.) mounted on a stand, which
Mr. Stoop of Amsterdam had kindly lent to the commission for the
observation of the transit of Venus, partly with the telescope of the
heliometer (aperture 7*/, em.). At a later epoch, when the assistance
of Mr. Ernst BAKHUYZEN was not so constantly required, as in the
beginning, for the experiments of Dr. Kaiser with the photoheliograph,
he also took part in the observation of the occultations, as also
did Mr. Sorrers in one case.

Altogether 35 disappearances and 4 reappearances were observed ;
but 12 disappearances and 1 reappearance had to be rejected. There
thus remained 23 disappearances and 3 reappearances, that is al-
together 26 observations, which furnished useful results.

The reason of the rejection lay partly in the fact that, already in
recording the observation, the remark “uncertain” had been added,
an addition due to the faintness of the star as it approached the
moon’s limb, or to passing clouds.

For another part the correction of the longitude determined by
GrrMAIn and adopted by myself, came out so extravagantly large
that some mistake or other seemed probable. There seemed to be
reason to suspect that a wrong star had been taken for the oeculted
one. In five of the cases I succeeded to find out the right star by
means of star catalogues, but in four other cases all my endeavours
proved in vain. Ultimately there remained five cases in which
the correction to the adopted longitude was found so considerable
(— 215, 20s, —28s, 24s and ++ 338), that there was no escape
from the conclusion that either a mistake, however improbable in

itself, had been committed in writing down the time, or that the

( 607 )

Results for the longitude of St. Denis-Réunion, (flag-staff), obtained by
occultations, without making a difference between disappearances
and reappearances.

1874
Sept. 19
» »
» »
» 99,
» »
» »
» »
» »
» 26
October 2
» 4
» 16
» »
» »
Die
Ds)
DD
De 5)
Aly}
De
DD
>» »
» 18
» 19
yo 5)
yo

Observer

Star, Name or
apparent place.

Disapp.
or reapp

Limb

AL
= Corr. 1G
Germain

eS
CB op eee eee Loe] Seo po oe 8 Se S
eee

Arg. Z. 223, No. 75
Cordoba III.1589
» XVIII.124
33 Capricorni

Arg. Z. 255, No. 27

yo » 82
yy. 8) » 34
Dy Sys) eh)

73 Piscium

53 Geminorum

G= = 9h0m39s, up
9 = + 22°57'38/,7

Arg. Z. 223, No. 47

ts = 18h6m41s,75

d = —28°0'56",8
Gould 24851

x= 19h2m35s.76
8 = —27°5417"'75

rg. Z. 241, No. 9
de ed 12
DN » 11
» » 239, »103
» » 247, » 99
x Capricorni

%2—= 21hIm4s.71
de

D |D\

S

SSSR STEE

+226 | 0 70
+6.64 | 0.74
+8.21 | 0.60
+4 .00 | 0.29
—6.10 | 0.50
—1.54 | 0.63
—1.51 | 0.89
—5.75 | 0.97
13.44 | 0.94
11.97 | 0.28
44.39 | 1.00
== 398) || 4500
+9.67 | 0.40
—5.99) | 0.95
43.84 | 0.515
4.65 | 0.49
—4,96 | 0.99
45.84 | 0.87
+5.89 | 0.19
—5.10 | 0.58
1.45 | 0.35
==4-73)| 0:62
—5.40 | 0.95
—2.62 | 0.98
—4,99 | 0.97
—8.95 | 0.94
18.30°
AD 16.85

— 18.305

—2.57 | —1.70 | 2.83
—4.09 | —3.30 | 10.56
—8.44 | —8.03 | 60.61

+33.93 | 25 m? = 450.96

—50.78 m2—= 18,04

—16.85 m= +4895
= — 05,92 + 05,99,

(608 )

Results for the longitude of St. Denis-Réunion, reappearances
and disappearances separately.

The 3 reappearances give: © G= 2.19
The total sum was:

Therefore the disappearances

separately give:

TGAL =-+ 7.58 therefore A L=-+ 35.462
18.305 —16.85

16.115 — 24.43 „ AL=-—1.516

G2

ND

Disapp,

—1.43
Reapp.
—0.35
2.19
40.93

0.60 | 56.80
0.29 1.84
0.50 | 10.49
0.65 0
0.89 0
0.97 | 17.35
1.00 9.71

0.87 | 47.13
0.19 | 9.07
0.58 | 7.32
0.35 | 2.50
0.62 | 6.39
0.95 | 12.48
0.98 | 4.19
0.97 | 7.07
0.94 | 51.89
on | ou
0.27 | 1.34
1.00 | 0.86

Mean :

+ 08,97,

N.B. As there is no reason to suppose that
a reappearance at the dark limb should be so
much more accurate than a disappearance at
the dark limb, I have combined them.

99 m*= 406.16

|

m= 18.06 |
Ss
m—=t4,34
(not used)
Together :
24m? = 408 .47
m? — 17.02
s
m =+ 413
m*
Teas a 1.056
ze
2m? = 2.31 Ho
| m? = 1.155 aoe
es
m= +£1.08

(not used)

v=+1'.03,
v=+2.79.
v=+1.49.

( 609 )

occultation had taken place at a point considerably elevated above
the rest of the limb. In the following lines we will only communicate
the results of those observations which have been retained.

We remark that the weights G, which have been added, were
taken equal to sin? w, 2w being the arc, of which the star would
describe the chord behind the disk of the moon, were this disk at rest;
(according to the notation of Crauverer this would become cos* w).
This quantity could be easily derived from the numbers occurring
in the computation.

The calculations have been all made in duplicate ; the first by
myself, the other by Mr. Kress, amanuensis at the observatory of
Utrecht.

We thus find:

Taking disappearances and reappearances together :

Correction to GurMarn’s longitude : — 05,92 + 03,99 (m. err.)
Treating them separately : 0 IAEA AO se)

We thus come to the conclusion that the oecultations observed by
us leave undeeided whether the longitude of St. Denis, according
to the determination of GerMAIN in 1867 and 1868, must be increased
or diminished; in other words they confirm his result.

Only one of these days I noticed, that since 1886 the Connatssance
des Temps gives a longitude for that place, which is larger by 152
or 18"; in the last column of the table of the geographical positions

ee ee Ghee Newcomb. N. — Merid.
= 7 = 7 : 7
Sept. 49, | 0.52 | —4.3 ORDEN EE: 40.43 | +4.6
3 221/, —0.51 1.9 =i | eee +0.02 —0.5
295/,, | —0.54 oad MAES EG 40.02 | —0.5
LEN NDE SE, 0:70 |) 5:8 40.03 | +
Octs 21cm — 0.794 “| EAT —0.95 | 40.6 DA it APA
MY), | —0.75 0.0 TT |e 200 —0.02 | 42.9
Gt SE OR NEEN —0.12 | +445
Wy Ah OS oad = 0:46 Oe 03) |) Foes
ASU ee On Soe A2 EIN ID LK
19 OREN 25 Oe Warn S50) We

s Li
Mean: —0.03 +1.3

( 610 )

we find: GERMAIN corr. 86; the reason for the correction is however
not stated. I have therefore written to Paris asking for information.

Moreover I will observe that the difference of longitude Paris—
Greenwich above used, must be increased according to the determination
executed by French and English observers in 1902. The result
obtained by the English observers was 9™20%,932 + 0°006; by the
French observers 9™20s,974 + 08008. Mean 9™208953. (Monthly
Notices of the R. A. 5. Jan. 1905).

Finally we subjoin a comparison of the corrections to the moon’s
ephemeris of the Naut. Alm. of 1874, furnished by the meridian
observations on the one hand, and by NewcomB’s formulae on the other.

It might be worth while to ascertain, whether the agreement of
the results is improved, if we adopt the corrections according to
NEWCOMB.

As for the meridian observations, some have been made at other
observatories (Leiden, Pulkowa ete.). I hope to investigate this more
closely ; it is not probable however that the result will be greatly altered.

A last remark in conclusion. According to the “Post en Telegraaf-
gids” the isle of Mauritius is already connected telegraphically with
Kurope. There is reason therefore to expect that the same will
shortly be the case for Réunion also. In that case the “Bureau des
Longitudes” will no doubt endeavour to obtain a telegraphically
determined longitude of St. Denis.

Utrecht: 1905 March 24.

Physiology. — “On the presence of sibringlobulin in fibrinogen
solutions.” By Dr. W. Huiskamer. (Communicated by Prof.
C. A. PEKELHARING).

After Hammarsten had proved that in fibrinogensolutions, which
had been coagulated either by heat to 55° or by means of fibrin-
ferment, a proteid, afterwards called fibringlobulin, appears which
coagulates at 64°, there existed several possibilities with regard to
the formation or appearance of this proteid.

Firstly the original fibrinogensolution might already have contained
the fibringlobulin as an admixture; in the second place it was possible
that at the heat-coagulation or by means of fibrinferment, the fibrin-
molecule was disintegrated, and that in such a way that an insoluble
substance, fibrin, is formed, along with a soluble one, fibringlobulin;
and lastly the fibringlobulin might perhaps be an altered fibrinogen,
which has remained in solution, a sort of soluble fibrin.

(611 )

Against the first of these possibilities HAMMARSTEN ') has raised
serious objections, and by his later researches he came more and
more to the conviction that fibringlobulin must be a somewhat
changed soluble fibrin.

A research of CALUGAREANU ?) was the occasion for experiments to
be made in this direction. The author inter alia demonstrates that
natriumfluoride, in strong concentration, greatly increases the effect
quantities of fibrinferment. CALUGAREANU prepared horseoxalateplasma,
which contained a quantity of fibrinferment so small, that the plasma
remained fluid for a considerable time.

If this plasma was mixed with natrium fluoride to a quantity of
about 3°/,, either by addition of a saturated solution of NaFl or also
of finely powdered NaFl, then there ensued an almost immediate
coagulation. That the formed precipitate really is fibrin, CALCGAREANU
derives from the fact, that it is like fibrin insoluble in diluted salt
solutions. Further CanuGarranu discovered that horseoxalateplasma,
if it was only perfectly free from ferment, did not coagulate by
addition even of several volumina 3°/, Na Fl. When therefore no.
ferment is present the natriumfluoride remains inactive, from which
CALCGAREANV concludes that the Na Fl exercises its influence on the
fibrinferment but not on the fibrinogen.

When the experiments of CALUGAREANU were repeated I obtained
results which partly differed from his.

It namely appeared that perfectly fermentfree solutions containing
fibrinogen gave a precipitate with natriumfluoride; this precipitate
is in case horsefibrinogen is used gelatinous and in consequence
reminds one more or less of coagulation; if however oxenfibrinogen
or oxenbloodplasma is used, the precipitate is flocculent and does
therefore not, outwardly at least, resemble coagulation.

In the second place it appeared that the precipitate formed by
NaFl could be easily dissolved, when treated properly, and that
these solutions coagulated with fibrinferment.

Some experements I will describe here in detail.

A rabbit was injected in the vena jugularis with 65 eem. leech-
extract, next the blood out of the Carotis was received in a centri-
fugalglass covered with paraffine and the corpuscles where centri-
fugalised off. Plasma in this way prepared contains no ferment as
PEKELHARING *) has demonstrated; the plasma, meant here, remained

1) Pflügers Archiv., Bd. 22, p. 431.

*) Arch. internat. de Physiol. Vol. IL, p. 12.

5) Untersuchungen über das Fibrinferment. Verhand. Kon. Akad. van Wet.
Amsterdam 1892,

42
Proceedings Royal Acad. Amsterdam. Vol. VIL

(612)

fluid for a number of days, as long as it was kept, yet by addition
of three times the volume of saturated natriumfluoridesolution a
flocculent precipitate was slowly formed; a fibrinogensolution prepared
from the plasma could also be precipitated by the addition of
saturated natrium fluoride solution; by saturating with solid natrium
fluoride a precipitate ensued immediately.

Other experiments were taken with horsefibrinogen. The fibrinogen
solutions used, which were prepared by three times precipitating
with salt from oxalateplasma showed even after being preserved
for several days, no trace of clotting; by addition of Ca Cl, no clotting
was caused either at 37° or at the temperature of the room. In such
a fibrinogen solution a thick precipitate is then immediately formed’) by
addition of the double volume of saturated natrium fluoride solution ;
this gelatinous precipitate can be easily wound round a glass rod and
in this state be taken out of the liquid for further research. The
precipitate washed by water showed the following properties. It did
not dissolve perceptibly at the temperature of the room in 8——5°/, salt,
„more easily the solving succeeded in this way, at the temperature
of the body, or still better at 40—45°. On cooling, the precipitate
does not return. The surest way to obtain a complete solution is to
make use of */,,°/, ammonia as solvent; if the precipitate is divided
with a glass rod, rather concentrated solutions can easily be prepared
in this way. Such a solution can after addition of salt, to a quantity
of 3
when the concentration of the solution was very great, a part of the

5°/, be neutralised without a precipitate forming anew (only

dissolved substance precipitated often again after some time; this preci-
pitate was solved however at 37°). Such a solution may be preci-
pitated again in the same way, with the double volume saturated
natrium fluoride solution and may be dissolved in */,,"/, ammonia. Such
neutral solutions containing 3—5°/, salt and prepared by being once
or twice precipitated with Na Fl possessed all the properties of fibri-
nogen; by addition of an equal volume saturated salt solution, a
ereat precipitate was formed; acetic acid caused a precipitate soluble
in excess; the coagulation temperature was at 54°; the solutions
coagulated quickly and completely with fibrinferment for which I

mention the following experiments as example.

5 eem. fibrinogen solution of 0.842°/,--1 cem. fibrinferment solution;
the coagulation begins at 37° after half an hour; the tube further
coagulates completely.

5 ecm. of the same fibrinogen solution + 5 drops of oxenblood-

1) The mixture contains then not much more than 3 0/, Na Fl.

( 613 )

serum, the coagulation begins (at 37°) after ten minutes; after an
hour a solid clot was formed.

Placed at 37° a tube with 5 eem. of the same solution, without
ferment for control, remained perfectly fluid.

The above mentioned experiments were now repeated with horse-
oxalateplasma, which was not perfectly free from ferment, as was
obvious from the partial clotting of the received blood; the results
were in general the same; the precipitate obtained with Na Fl dis-
solved only with somewhat more difficulty; the solution of this
precipitate meanwhile possessed the properties of a tibrinogen solution
and coagulated with fibrinferment.

Further experiments were taken with fermentfree oxenfibrinogen
prepared after the method of HaAMMARSTEN. It was stated that to
precipitate this fibrinogen with NaF] more natrium fluoride solution
was needed than for horsefibrinogen. The flocculent precipitate obtained
with NaFl dissolved at 37° more easily in a diluted salt solution
than the horsefibrinogen precipitated with NaFl; on the contrary less
easily in */,,"/, ammonia; rather great quantities dissolved already at the
temperature of the room in 8—5°/, NaCl. The coagulation temperature
of the neutral solution, containing about 3°/, salt was at 53—54°.
addition of acetic acid caused a precipitate which dissolved in excess ;
by half saturating with NaCl the fibrinogen could be precipitated.
That the solution coagulates with fibrinferment appears from the
following experiment.

5 eem. of the solution in 3°/, NaCl + 5 drops of oxenbloodserum.
Complete clotting after two hours.

Although it might seem after the above mentioned experiments
that the fibrinogen remains unaltered on being precipitated with
Na Fl, a closer inquiry brings to light a remarkable alteration. If
namely a solution of fibrinogen precipitated with Na Fl is heated to
55—58", very little fibringlobulin is found in the liquid filtered off
from the coagulum; if the fibrinogen is precipitated twice with
vatrium fluoride, no or only few traces of fibringlobulin can be
obtained from the solution as appears from the following experiments.

I. A solution of fibrinogen prepared after the method of Ham-
MARSTEN was partly precipitated twice with Na Fl; the last precipitate
was dissolved in */,,°/, ammonia and the solution was neutralised
after addition of salt; 8 eem. of this solution, which contained
0.445°/, fibrinogen were heated for five minutes to 55—60°, then
it was filtered; the clear filtrate was heated to 72°, by which only
a small opalescence ensued, which did not increase perceptibly after
the liquid had been made slightly acid and afterwards boiled.

( 614 )

For comparison the fibrinogen from 8 cem. of that part of the
fibrinogen solution which had not been prepared with NaFl, was in
the same way first coagulated and afterwards the fibringlobulin in
the filtrate; although the fibrinogen solution used for this experiment
contained 0.565°/, fibrinogen and so had been but little more con-
centrated than the solution prepared with Na FI the quantity fibrin-
elobulin found was remarkably larger, as there was formed abundance
of floeculent precipitate by heating to 70°.

Another experiment gave the following results.

II. The solution of the fibrinogen not precipitated with Na Fl con-
tained 0,634°/, fibrinogen, that of the fibrinogen precipitated with
Na Fl 0,452°/,, after this the fibrinogen being removed from the two
solutions by heating to 55—58° and by filtering off of the coagulum,
5 eem. of each of the filtrates were mixed with 1'/, cem. of a
saturated solution of picric acid. In the filtrate of the fibrinogen treated
with Na Fl there was formed only an opalescence which after some
time passed into a very slight precipitate; in the filtrate of the fibri-
nogen not precipitated with Na Fl there was immediately a con-
siderable flocculent precipitate.

Ill. A solution of oxenfibrinogen was precipitated by four times
the volume of a saturated natrium fluoride solution; afterwards the
precipitate was centrifugalised off, washed with water and dissolved in
4°/, salt; the solution contained 0,232°/, fibrinogen; after its having
been removed by heating to 55—58° and filtering off of the coagulum
the filtrate remained perfectly clear on being boiled, and so contained
no fibringlobulin, although the original fibrinogensolution had been
precipitated with Na Fi only once.

So it appears that by means of natrium fluoride fibrinogensolutions
may be obtained, which by heating produce no fibringlobulin or
only traces of it.

This confirms the opinion that the fibringlobulin was present
already in the original, not heated fibrinogensolution either combined
with fibrinogen or simply as admixture, and that consequently fibrin-
globulin is not formed by alteration of the fibrinogen during the
heating ; in the last case it could not be explained why the fibrinogen,
prepared with Na Fl should not be altered as well in the same degree
by heating in fibringlobulin. If however the fibringlobulin was present
already in the fibrinogensolution, every thing may be explained in
this way that on being treated with Na FI the fibringlobulin passes
into the filtrate at any rate for the greater part, while the fibrinogen
proper precipitates. The possibility that the fibringlobulin does not
precipitate also appears from the following experiment. In a solution

( 615 )

of horsefibrinogen prepared after the method of HAMMARSTEN the
fibrinogen was coagulated by heating to 60° and filtered off; to the
filtrate was added a double volume of saturated natrium fluoride
solution; the liquid remained perfectly clear.

The question whether the fibringlobulin passes into the filtrate
when the fibrinogen is precipitated with Na Fl cannot be answered
immediately by examining the filtrate, while the fibrinogen with
Na Fl does not precipitate completely, so a certain quantity of fibri-
nogen exists still in the filtrate, and when, after heating, fibringlobulin
is still found, the possibility exists, that all this fibringlobulin proceeds
from the quantity of fibrinogen present in the filtrate; only the quan-
titative research can decide here; if on precipitating with Na Fl the
fibringlobulin passes into the filtrate it must be possible to prepare
from this filtrate nearly as much fibringlobulin as from the original
fibrinogensolution. As the fibrinogen precipitated with Na FI is not
perfectly free from fibringlobulin, an accurate agreement is not to be
expected. In the first place I subjoin the results of such an experiment.

a) 100 eem. of a pure horsefibrinogensolution, prepared after
HamMarsten’s method were precipitated with 200 ccm. saturated
natrium fiuoride solution. The precipitate was taken with a glass rod
out of the liquid, pressed out firmly, dried to constant weight and
weighed, the substance was burnt carefully, the weight of the ashfree
substance proved to be 0,2485 gram. After the precipitate obtained
with Na Fl had been removed a clear liquid remained, which was
neutralised with some drops of diluted acetic acid, as the reaction
of the solution of NaFl used was faintly alkaline, which mostly is the
case. The liquid (285 eem.) was heated afterwards for a quarter of
an hour in a waterbath to 55—60°; the coagulated fibrinogen was
filtered off on a weighed, ashfree filter, with a diluted saltsolution
and after that washed with water, dried to constant weight- and
weighed together with the filter; the filter and the substance was
carefully burnt. It proved, that 0.2262 gram ashfree fibrinogen had
been present on the filter; this quantity was obtained from 285 eem;
so in the original 300 eem. there would have been found 0.2381 gram.

In order to determine the quantity of fibringlobulin 250 eem. liquid
filtered off of the coagulated fibrinogen were heated during a quarter
of an hour to 67—69° in a waterbath. The liquid remained perfectly
clear till 64°; to obtain a coagulation as perfect as possible 5 eem.
1°/, of a sulphas cupri solution were added as soon as the liquid
became turbid; by this the coagulum became roughly floceulent and
could easily be filtered off. The weight of the filtered fibringlobulin
was afterwards determined in the same way as was done with the

( 616 )

coagulated fibrinogen and amounted to 0.1141 gram ashfree sub-
stance; so from 300 eem. filtrate would have been gained 0.1369
gram. In the liquid filtered off from the fibringlobulin no proteid
could be demonstrated.

Db). For comparison it was determined how much fibringlobulin
the fibriogen solution used gave without treatment with Na FI.
Therefore 100 eem. of this solution was again mixed with 200 ccm.
4°/,°/, salt through which volume and salt quantity in this experiment
was made equal with that of experiment «). By heating for a quarter
of an hour to 55— 60° the fibrinogen was coagulated and was
treated further as mentioned above; the weight of the fibrinogen
amounted to 0,4548 gram ashfree substance. 250 cem. of the liquid
filtered off from the coagulated fibrinogen were heated for a quarter
of an hour to 67-—69 ; the liquid remained perfectly clear till 64°,
just as in experiment a); here also 5 eem. 1°/, Cu 50, were added
as soon as the first turbidness became visible.

The coagulated fibringlobulin was filtered off and treated as men-
tioned above; the weight of the fibringlobulin amounted to 0,1354
gram; in the liquid filtered off from the coagulated fibringlobulin no
proteid could be shown.

Taking together the results of these experiments we find, that in
experiment qa) after the removal of the precipitate obtained with
Na FI, 0,2881 gram fibrinogen and 0,1369 gram fibringlobulin were
present; and in experiment 5) 0,4548 gram fibrinogen and 0,1625
gram fibringlobulin. So the quantity of fibrinogen was in experiment

52 ;
a) reduced to 100 through precipitation with Na Fl while the quantity

-

85
of fibringlobulin only showed reduction to 100: Consequently there

must have passed a considerable quantity of fibringlobulin into
the filtrate after precipitation with Na Fl. The difference of 0,0256
gram between the quantities of fibringlobulin, found in both experi-
ments, must be attributed, apart from any errors of determination
to the fact that the fibrinogen which was not precipitated with Na Fl
is not perfectly free from fibringlobulin; the weight of this precipitate
amounted to 0,2435 gram; if we abstract from this 0,0256 gram as
being fibringlobulin, this precipitate contained to 100 mg. at 55°
coagulable fibrinogen 11,7 mg. fibringlobulin; in experiment 6) 35,7
mg. fibringlobulin was found to 100 mg. at 55° coagulable fibrinogen
and in experiment a) after removal of the precipitate obtained with
Na Fl 57,5 mg. fibringlobulin to 100 mg. at 55° coagulable fibrinogen.
By precipitating with Na Fl the fibrinogensolution was consequently

( 617 )

divided into a precipitate, which contained relatively little and a
filtrate which contained relatively much fibringlobulin.

In experiment a) about half of the fibrinogen was precipitated
with NaFl; the liquid poured off from this precipitate was clear;
if however such a liquid is left standing for some time it becomes
turbid and a new precipitate has formed itself after 24 hours, in
the filtrate of this precipitate a new turbidness forms again ete.,
till at last after some days all the fibrinogen has precipitated. It
may me expected after the above mentioned experiments, that, as
more fibrinogen precipitates, relatively (that is to say with regard
to the quantity of fibrinogen which was not precipitated) more
fibringlobulin will be present in the filtrate; this supposition is
confirmed by the two following experiments.

1. 100 eem. horsefibrinogensolution of 0.642 °/, were precipitated
with 200 cem. saturated natrium fluoride solution ; the precipitate was
removed with a glass rod and the liquid remained standing after
that twice 24 hours. When the fibrinogen, precipitated after that
time, also was removed by filtering, the quantity of fibrinogen and
fibringlobulin in 250 eem. filtrate was determined in the same way
as in the above mentioned experiment a). It appeared that in these
250 eem. 0.0742 gram fibrinogen and 0.1113 gram fibringlobulin
were present.

2. 100cem. of the same fibrinogensolution were precipitated with
200 eem. saturated natrium fluoride solution; after removal of the pre-
cipitate the liquid remained standing for eight days; putridity did
not occur from this on account of the quantity of Na Fl, the new
formed precipitate was filtered off, the filtrate became again turbid
and after 24 hours a slight precipitate had again formed, that was
filtered off. The filtrate was neutralised with a few drops of diluted
acetic acid; by heating of the neutral liquid to 55—60° there followed
only an exceedingly slight opalescence; the fibrinogen was therefore
precipitated almost completely by the Na Fl; when the opaline liquid
was filtered a considerable flocculent precipitate was formed by
heating the filtrate to 67—69°.

While in experiment a) after the removal of the precipitate obtained
with NaFl still 1°/, times more fibrinogen than fibringlobulin was
present in the filtrate, the analogous filtrate in experiment 1) con-
tained only *, times as much fibrinogen as fibringlobulin, while in
experiment 2) with a considerable quantity of fibringlobulin only
a small quantity of fibrinogen was present.

The results of the above described experiments lead to the con-
clusion that at the coagulation of the fibrinogen, the fibringlobulin

( 618 )

does not proceed from the fibrinogen, but that this proteid was
already present in the fibrinogensolution, for it could not be explained,
that on one hand, the fibrinogen precipitated with Na Fl produces
no or but little fibringlobulin, and that on the other hand the liquid
filtered off from this precipitate contains fibringlobulin in such greater
quantity.

It here is necessary to discuss still a few objections that might be
raised against this conclusion.

Firstly — on account of the fact that the fibrinogen precipitated
with Nakl, dissolves with more difficulty in diluted saltsolution than
the usual fibrinogen, and that the solution does not produce any
fibringlobulin by heating — it might be asked whether the substance
precipitated with NaF l might not be a kind of soluble fibrin, as for
instance the “fibrine conerete pure” described by Denis. The latter
also principally dissolved in diluted saltsolution at 40°; while the
dissolution went very slowly at the temperature of the room. Against
the opinion that the substance precipitated with NaFl is a soluble
fibrin speaks first the coagulation temperature which was found by
Denis for the dissolved “fibrine conerète pure” at 60—65°, while in
every case it is not higher than 55° for the substance precipitated
with Nak]. The strongest argument against the opinion that this last
is fibrin, namely the power of this substance to clot with fibrinferment,
I have already stated several times; if we further take into consider-
ation that the fibrinogen prepared with Nall behaves with respect
to acetic acid, half saturation with salt ete. quite as common fibri-
nogen, the opinion that this substance is fibrin may be considered
as having been refuted.

As to the slight solubility of the fibrinogen precipitated with NaF]
and in diluted salt solution, this peculiarity may be explained in
this way, that on being heated with NaFl it forms a slight soluble
fluorine-compound of the fibrinogen, which dissolves only very slowly
in saltsolution; by the great abundance of chlorine-ions then present,
this dissolving will probably be accompanied by an exchange of the
fluorine by chlorine. It is still rendered more probable that a fluorine-
compound is formed, when we consider that the slight quantity of the
natrium fluoride solution cannot be put on one line with the precipitating
of proteid by the saturating of the solution with a neutral salt.

It might be imagined that the fibrinogen, it is true, is precipitated
as such by natrium fluoride, but that also (especially as natrium fluoride
solutions usually react slightly alkaline) part of the fibrinogen is
changed into fibringlobulin; by which the presence of fibringlobulin
in the filtrate would be explained.

( 619 )

Apart from this that then it would not be explained why the
fibrinogen precipitated with NaFl does not produce any fibringlo-
bulin by heating, it would have to be expected according to this
view that, if the fibrinogen were precipitated with Nall for the
second or third time also a part of it would be changed into fibrin-
globulin, which ought to be found in the filtrate. This however is
not the case; under these circumstances only very little or no fibrin-
globulin is found in the filtrate.

So, when it should be assumed, that the fibringlobulin is present
in the fibrinogen solutions beforehand already, the question remains,
whether this proteid is combined with the fibrinogen or must be
considered as a simple admixture.

For a compound plead some experiments of HAMMARSTEN ‘), in
which is demonstrated that from concentrated fibrinogen solutions
„after heating to 56—60° and filtering off of the coagulum, relatively
less fibringlobulin is obtained than from the same solutions after
their having been diluted. If the fibringlobulin were only an admixture
it would be expected that the relation between the quantities of
fibrinogen and fibringlobulin would always be the same; on the
other hand, if the fibringlobulin is combined with the fibrinogen the
results of HAMMARSTEN could be explained thus, that in diluted fibri-
nogen solutions the fibringlobulin is more easily disintegrated. To a
compound also points the fact, that when a fibrinogen solution is
precipitated for the first time with NaFl a not inconsiderable quantity
of fibringlobulin is precipitated also.

Against a compound speaks however that by precipitating with
NaF the fibringlobulin passes into the filtrate, at least for the greater
part, for it is difficult to believe, that addition of alkali salt, as
NaFl up to a quantity of about 3°/, would have for its result a
splitting off of fibringlobulin. The following observation may perhaps
give some light.

100 eem. of horsefibrinogensolution were precipitated with the double
volume of saturated natrium fluoride solution; the solution of NaFl
used reacted almost neutral by exception ; with litmuspaper the alkaline
reaction was hardly perceptible. Part of this natrium fluoride solution
was now made weakly alkaline by addition of 0.8 eem. normal
sodium hydrat with 200 eem. of the natrium fluoride solution ; with
this 100 cem. of the same fibrinogen solution were precipitated twice
in the same way.

From the precipitates obtained with neutral and with alkaline
NaF! two fibrinogen solutions of equal concentration were prepared.

1) Loe. cit.

( 620 )

The fibrinogen was in both cases coagulated and filtered off by
heating to 55—60°. The filtrate of the fibrinogen prepared with slight
alkaline natrium fluoride solution gave a slight precipitate by heating
to 70° or by addition of picric acid, while the precipitate of fibrin-
globulin in the other filtrate was clearly greater, perhaps twice or
three times.

From this it would follow that the supposed splitting off of
fibringlobulin is not brought about by Na Fl but by the alkaline
reaction of the natrium fluoride solutions ; for this disintegration however
exceedingly small quantities of alkali are already sufficient, for also
that fibrinogen solution which was prepared with almost neutral
Na Fl produced much less fibringlobulin than a fibrinogen solution
of the same concentration, not prepared with Na FJ]. The supposition
that water also, in particular at a rising temperature could bring
about the splitting off of fibringlobulin is obvious; if this is the
case there would be present in a fibrinogen solution a compound of
fibringlobulin with fibrinogen, which is disintegrated more or less
by hydrolysis and this idea is, as appears to me, most easily recon-
ciled with the facts. The disintegration will in this case with raised
temperature e.g. at 55—60’ be rather complete; from diluted solutions
relatively more fibringlobulin may however be obtained than from
concentrated solutions, because in the first case the disintegration
will be more complete owing to the greater excess of water. That
not all the fibringlobulin passes into the filtrate by the precipitation
with Na Fl, becomes clear if only a partly hydrolytic disintegration
is accepted.

If the fibringlobulin is mixed simply with the fibrinogen in conse-
quence of hydrolysis, be it then for a part only, it cannot be expected,
— with a view to this, that by half saturation with salt as is usual
with the preparation of fibrinogen, no complete precipitation of the
fibringlobulin takes place, — that in every fibrinogensolution the
relation between the quantities of fibrinogen and fibringlobulin will
be the same; this may perhaps lead to the explanation of some
observations of HAMMARSTEN *) from which it appeared that fibrinogen-
solutions prepared from different plasma produce, it is true, relatively
different quantities of fibringlobulin, that however a diluted solution
does not always produce relatively more fibringlobulin than a con-
centrated solution.

In conclusion I will diseuss some facts here, relating to clotting
by means of ferment.

1) loc. cit. p. 456.

( 624 )

Of the indentity of the fibringlobulin which is obtained by the
coagulation by ferment and that which is found in filtrate after the
heat-coagulation of the fibrinogen, there is no doubt, on account of
the conformity in composition, coagulation temperature ete. When
however it must be assumed that the fibringlobulin is already before-
hand present in the fibrinogensolutions, then for the present falls
away every ground to assume that by the clotting by ferment the
fibringlobulin should be formed still in another way e.g. by trans-
formation of fibrinogen, the more so, as the quantity of fibringlo-
bulin which is obtained by clotting with ferment certainly is com-
paratively not larger than that which can be prepared by heating
from a fibrinogensolution. HAMMARSTEN ') found, it is true, that in
weak alkaline solutions relatively little fibrin was formed by ferment
and so relatively much proteid remained dissolved; this may partly
be explained by the fact that the fibringlobulin was disintregrated
very completely by the alkaline reaction, partly also, as HAMMARSTEN
himself observes, by the fact, that under these circumstances part of
the fibrin remained dissolved as “soluble fibrin”.

From the fact that a solution of fibrinogen, from which the fibrin-
globulin is removed by means of Nall, clots with fibrinferment, must
be deduced that by removal of the fibringlobulin the fibrinogen
proper is not, as might be expected from the formula given by
SCHMIEDEBERG and defended a short time ago by HeuBNer *) changed
into fibrin, and that in general the fibringlobulin does not play a
considerable part in the clotting. So the clottingprocess must consist
in an alteration of the fibrinogenmolecule itself. That fibringlobulin
is present in the serum of coagulated fibrinogensolutions can be
easily explained from this, that fibringlobulin was found already in
free condition in greater or smaller quantities in the fibrinogensolu-
tion, so the supposition, that the ferment causes a splitting off of
fibringlobulin is superfluous as may be deduced from this.

Physics. — “The transformation of a branch plait into a main
plait and vice versa.” By Prof. J. D. van per Waars.

If for a binary mixture the temperature is raised above the critical
temperature of one of the components, the y-surface has a plait,
which does not occupy the whole breadth from «=0O to «=1,
but which is closed on the side of the component for which 7%,
lies below the chosen value of 7. In normal cases such a plait

1) Pfliigers Archiv, Bd. 30, p. 479.
2) Arch. f. exp. Pathol. u. Pharmakol. Bd. 49, p. 229.

( 622 )

which is closed on one side, does not present any special particul-
arities, and starting from the open side a bitangent plane may be
rolled regularly over the binodal curve as far as the plaitpoint.
There are, however, also cases where we meet with complications,
and already in my “Théorie moléculaire’ 1 have allowed in my
description of the y-surface, for the possibility of the existence of a
branch plait by the side of the main plait. If two plaits exist
simultaneously over a very great range of temperature, we may
properly speak of a transverse plait and a longitudinal plait, and
the non-miscibility in the liquid state may be ascribed to the long-
itudinal plait. But if these two plaits occur only over a small range
of temperature, it is better to speak of a main plait and a branch
plait; I have chosen these names, because really in such cases
one of the plaits may be considered as main plait, and the other
only as branch plait. But, what has not been observed as yet,
the circumstance may occur, that at a certain temperature these
two plaits reverse their parts. What was a branch plait, becomes
a main plait, and the main plait is reduced to a branch plait.

In saying this I have chiefly in view the description of the
modifications to which the »-surface is subjected with change of
the value of 7, to account for the observations of KueNeN on the
critical phenomena of mixtures of ethane and some alcohols.

These mixtures have, for a value of 7’ only little greater than
T;. of ethane, a plait on the y-surface with a continuous course
without any complication. But with rise of 7’, besides the plaitpoint
on the ethane side, a new plaitpoint appears lying more to the side of
the alcohol. So from this temperature 7, we may speak of a three-
phase-pressure. With further rise of 7’ the new plait extends, and
at a certain higher value of 7’ = 7,, the first plaitpoint disappears.
Then the three-phase-pressure vanishes, and from that moment the
plait has resumed its simple form. Between 7, and 7, we have,
therefore, a plait with two plaitpoints. If referring to a plait we
speak of a base and a top, we have between 7, and 7, a plait
with one base and two tops. Beyond the limits of 7’ equal to 7,
and 7’,, the plait has only one base and one top. But whereas just
above 7, the top which has newly appeared, extends but little
beyond the binodal curve of the original plait, at a higher value of
T this top will extend further; the top on the ethane side contracts,
and disappears altogether at 7, and as we shall show, disappears
as a branch plait.

As therefore the plait appearing at 7’, is originally a branch
plait, a transformation must take place with increasing value of 7’

( 623 )

which converts this branch plait into a main one. On the other
hand that part of the plait, which at 7, was situated in the neigh-
bourhood of the existing plaitpoint lying below 7, and which was
then a main plait, must have been reduced to a branch plait for
values of 7’ slightly below 7.

That the distinction between a main plait and a branch plait
is not arbitrary, but essential, appears when we determine which
of the two tops which occur between 7, and 7, belongs to the
base of the plait, and when this is ascertained, examine in what
way the binodal curve of the other top must be completed.

So the question is, when the bi-tangent plane is rolled over the
binodal curve from the base part of the plait, which of the two
occurring tops will be reached by continued rolling.

If we consult fig. 1, it is easily seen that a rolling tangent plane
which comes from the right side, and which has reached the two
points of contact A’ and A”, has obtained a new point of contact in
A, lying on the same isobar and in this way has become a plane
touching in three points. At the assumed temperature we have there-
fore a three-phase-pressure. In this case there are two tops of a
plait viz. P and Q. But there cannot be any doubt as to which
of these two tops belongs to the base part lying right of A’ A",
If viz. we continue to roll the tangent plane when it has the
line A A' as nodal line, the binodal line on the side of the small
volumes between the points A” and A is completed by the curve
A" BCA, the configuration A'B'CA' giving on the other hand the
completion on the side of the larger volumes. This harmonizes
with the diagram in my Théorie Moléculaire. (Cont. II p. 23).
So when continuing to roll we reach P as top of the plait. We
are therefore justified in considering the part of plait A'PA as
belonging to the main plait. There lies, however, on and by the
side of the main plait, a second configuration, of which AQA" is a
part. If a rolling tangent plane is moved over it, starting from Q,
the binodal curve described in this way does not end in the points
A and A", but if the plane has reached those points and has there-
fore again assumed the position of the three-phase-triangle, we may
roll it continuously further till it has reached a point of the spinodal
curve. This curve is denoted by D in fig. 1. The binodal curve
under consideration has then obtained a minimum pressure; the
conjugate point D' is then a cusp‘).

1) For a proof of these and similar properties consult Cont. Il, fig. 3. Further
the very important papers of Korrewee on the theory of plaits.

( 624 )

When the plane is rolled further the binodal curve passes the part
DE' on the left side and the part D'# on the right side, where the
spinodal curve is again met with. For this part there is a maximum
pressure, while there is now a cusp in #’. And finally this plait,
which has its top in Q, is closed by the portion /’ RE of its binodal
curve. If we consider also unstable phases as realisable, states between
KF! and R coexist with conjugate ones between Hand RF on this part.
The point A closes this branch plait as unrealisable plaitpoint.

There is not the slightest doubt that for the above mentioned
mixtures of ethane and alcohol just above 7, the newly appearing
plaitpoint Q on the alcohol side leads to the diagram of fig. 1 and
that Q is then the top of a branch plait. If the points A and A" are
still very close together, then the distances from these points to points
of the spinodal line must, a fortiori, be extremely small, and we
have justly assumed that the tangent plane in A’ A” when rolled
further, passes through the spinodal curve on the side of A”.

That on the other hand at temperatures just below 7’, the plait
the top of which is P, must be considered as a branch plait, is
beyond doubt for the same reasons. Above 7’, namely, only the top
QQ is found, and the whole plait does not present the slightest com-
plication. Only with decrease of temperature below 7, an extremely
small bulging out appears in the beginning at P (i. e. in the position
which that point has at that temperature) and the same reasons
which led us to consider the point Q as top of a branch plait
just above 7’, must lead us now to consider Pas top of a branch
plait. Fig. 3 represents the binodal lines in this case. Only we have
assumed there that the temperature has fallen already so much below
T,, that the branch plait has got such an extension, that at first
sight it is not to be distinguished from a part of a main plait.

Both in fig. 1 and in fig. 5 there is asymmetry between the two
binodal curves of the tops P and Q. But when 7’ is gradually
changed from 7, to 7, or vice versa, fig. 1 will gradually pass into
fig. 3 or vice versa. This transition requires a value of 7, at which
the asymmetry between the two tops P and Q has vanished. What
the shape of the binodal curves must be at the transition temperature,
is represented in fig. 2. Then we have one plait with one base, but
with two heads.

If we compare fig. 1 and fig. 2, the only difference is that the
points B’ and £' have coincided, which involves that the node
belonging to B' and that belonging to //’, so the points B and Z,
also coincide. From fig. 2 we derive fig. 1 by separating again the
parts which have run together, at the points which have coincided,

VAN DER WAALS, ©

p, VAN DER WAALS. “The transformation of a branch plait into a main plait and vice versa.”

r

Fig 5

Proceedings Royal Acad. Amsterdam. Vol. VIL.

( 625 )

and which is denoted by B' and £', and by doing the same with
the point B ZE. In the same way fig. 2 leads to fig. 3. But the way
in which this separation must take place is different for these two
transitions. What happens in one case in the left-hand point, takes
place in the other case in the right-hand point.

The coinciding of the points B' and LH’ is represented in fig. 2
on the spinodal curve; also the coinciding of the points 5 and
E. The spinodal line is namely the curve which is denoted thus
— — — —, and which runs through the points BD L'PDRCQ BE.
That the coinciding must take place on the spinodal curve might be
anticipated from the characteristic which we have used to distinguish
between main plait and branch plait. We had to consider Q as
top of a branch plait, if the rolling tangent plane, arrived at the
position A’ A", reached the spinodal curve on the side of A” when
rolled further, so in the space lying within the top Q. On the
contrary P was the top of a branch plait when this happened on
the other side. For the case that there is symmetry between the
two tops P and Q, the meeting of the spinodal curve must take
place on both sides simultaneously. But we might also have taken
as criterion for the main plait, that the main plait is such a plait
for which the points Band £' are separated *). The comparison of
these two criteria leads to the fact that the coincidence of the points
B' and £" must take place on the spinodal curve. But as long as
the two tops P and Q are present, whatever the character of these
tops may be, there is a third plaitpoint, viz. the point F, belonging
to a composition of the binary mixture which lies between the com-
positions belonging to the points P and Q.

In the figs. 4, 5 and 6 the complete (p, #) curves have been given
for the coexisting phases. Fig. 4 for a temperature which is little
higher than 7, and at which Q is still the top of the branch
plait, and fig. 6 for temperatures below 7’, at which P is still the
top of the branch plait. Fig. 5 represents the transition temperature.
I may assume as known that the differential equation for this (p,)
curve is:

075

Cee Ci DR Te) ER Ad et a ee dE)
Òz pT

Whenever that the (p,v) curve has a point in common with the
; es
spinodal curve
02 >T

= 0), p is a maximum or a minimum. This

1) Cf. Wiskundige opgaven enz. IVde deel, 5de stuk, Vraagstuk CXXXIX, where it
is also demonstrated, that the branches of the binodal curve which touch in B'E,
have the same curvature. Also the conjugate ones, which touch in BE.

( 626 )

is the case in the plaitpoints, but also in the other points, in which
a phase coexisting with an other, passes through the spinodal curve.
In fig. 5 there must therefore be maxima or minima at P, Q, BW,
BE, D, C, R. If from the differential equation we calculate the

2

dp
value of Ee for the points B'E' and BE, it appears, that for the
at,

two branches which meet, this value is the same there. If we
differentiate equation (a), we get:

dp dp d(v,,) ( | 025 dv dp 05 dz, z,)
ne en (Ei. — ,
Dn deden de 6 : (0x,*p7 de’, de, Òz oT de,

dp eS . REMI 4:

—— and being O, this equation is simplified to:
da, Oz 3
dp ) 0%
Var == (Li?
de? (5 : Ors? PT
5
The quantities v,,, (w‚—,) and Gel are the same for the
mi pT
Pp . f
two branches, and so also ——~,. In fig. 5 this has not been fulfilled
Uk,

in the tracing of the branches in the neighbourhood of the points
Bi. Better in the neighbourhood of the points BE. Also in the
eusps an inaccuracy in the proper curvature of the branches may
be detected here and there. But the figs. should be considered as
only schematical. The properties that the two curves in fig. 2 which
touch have the same curvature, and that this is also the case with

the two eurves which touch in fig. 5, are of course closely allied.

dp Op Op dv

Pelt)

da 0a Jor Ov Je pdx
d*p 0*p

dp dv dp dv\? Op do
pach pat (elen ln) tan
da? Òm?.r Ow Ov \ da Ov? \ da Ove 7 da

follows for two curves, passing through the same point, and for
adhere fared ee ek andi en | whiel
which, therefore, - —, ze and — is the same, and which
"Ox? Ox Ov Ov? Ov ay

From

and

;

dr d
touch in that point, and for which also (=) is therefore the same, that
at
: ae. f ‚dp :
the equality of EE involves also the equality of — and vice versa.
ax at

Kortrwne’s thesis, which has also been proved by Krvyver, might
therefore also be proved by the method followed here.

( 627 )

Mathematics. — “A group of algebraic complexes of rays”. By

Prof. JAN DE VRIES.

§ 1. Supposing the rays a of a pencil (A, @) to be projective to
the curves 6" of order n, passing through 7? fixed points, By, of the
plane 8, we shall regard the complex of the rays resting on homolo-
gous lines. For n =1 we evidently find the tetrahedral complex.

Out of any point P we project (A,a@) on 8 in a pencil (4’, 8),
generating with the pencil (47) a curve c’+!. So we have a complex
of order (n +1).

Evidently the curve emt! does not change when the point P is
moved along the right line AA’; so the intersections of the op” cones
of the complex (7?) with the plane 8 belong to a system o*. It is
easy to see that they form a net.

For, if such a curve ct! is to contain the point X and if Die
is the eurve through Bj, and X, and ax the ray conjugate to it
through A, the point A’ must be situated on the right line connecting
X with the trace of ax on the plane 8. In like manner a
second point through which c”tt must pass, gives a second right
line containing A’. The curve et! being determined as soon as A!
is found, one curve ct! can be brought through two arbitrary
points of 9.

On the right line @ the given pencils determine a (1, 7)-corre-
spondence; its (n +1) coincidences C, are situated on each c”t!.
So the net has (n° +7 +1) fixed base-points *).

§ 2. When A’ moves along a right line a’ situated in 8 and
cutting the plane a in S, the curve ct! will always have to pass
through the 7 points Dj, which @ has in common with the curve
bn conjugate to the ray AS. It then passes through (n + 1)* fixed
points, so it describes a pencil comprised in the net.

To the 387° nodes of curves belonging to that pencil must be
counted the # points of intersection of ¢8 with that c” passing through
the points Bj, and Dy. Hence a’ contains, besides |S, (3 2? — 7) points
A’ for which the corresponding curve c+! possesses a node.

If A’ coincides with one of the base-points 4; then the projective
pencils (A) and (6") generate a c+! possessing in that point B a
node. According to a well known property B is equivalent to
two of the nodes appearing in the pencil (c*+!) which is formed

1) To determine this particular net one can choose arbitrarily but 4 2(z-++-3)— 1
points B and three points C.
43
Proceedings Royal Acad. Amsterdam. Vol. VII.

( 628 )

when A’ is made to move along a right line @ drawn through B.
From this ensues in connection with the preceding :
The locus of the vertices of cones of complea possessing a nodal
edge is a cone A of order n (38n—1) having A as verter and
passing twice through each edge Ab.

§ 3. If P moves along the plane @ then the cone of the complex
(P) consists of the plane @ and a cone of order n cut by @ along
the right lines ACh. So a is a principal plane and at the same time
part of the singular surface.

The plane @ belongs to this too. For, if P lies in 8 then the rays
connecting P with the points of the ray a corresponding to the
curve $" drawn through P belong to the complex. All the remaining
rays of the complex through P lie in 8. So 8 is an n-fold principal
plane and the singular surface consists of a simple plane, an n-fold
plane and a cone A of order n(3n — 1).

The complex possesses (77 + n + 2) single principal points, namely
the point A, the n? points Bj and the (n +1) points Ct

$ 4. The nodes of curves c’ belonging to a net lie as is known
on a curve Hf of order 3 (p— 1) the Hessian of the net, passing
twice through each base-point of the net. This property can be
demonstrated in the following way.

We assume arbitrarily a right line /anda point M. The er touching
lin L, cuts ML in (p— 1) points Q more. As the curves passing
through J/ form a pencil, so that 2(p— l) of them touch /, the
locus of Q passes 2(p— 1) times through M; so it is of order
3(p—1). Through each of its points of intersection S with / one
ce passes having with each of the right lines 7 and MS two points
in common coinciding in S; so S is a node of this cp.

Consequently the locus of the nodes is a curve of order 3 (p — 1).

If / passes through a base point B of the net then the pencil
determined by J cuts in on / an involution of order (p — 1). This
furnishing 2 (p — 2) coincidences L, the locus of Q is now of order
(Bp — 5) only. So B represents for each right line drawn through
that point two points of intersection with the locus of the nodes,
consequently it is a node of that curve.

If / touches in B, the curve c,? having a node in B, and if one
chooses J/ arbitrarily on this curve, then the curves of the pencil
determined by M have in B, a fixed tangent and 2, is one of the
coincidences of the involution of order (p — 1). The locus of the
nodes has now in B, three coinciding points in common with /;
consequently it has in 5, the same tangents as c‚!.

For the net N+! of the curves c’+! lying in the plane 3 the
locus of the nodes H breaks up into the right line «3 and a curve
of order (Bn —1). For, «8 forms with each curve J” a degenerated
curve c*t!,

The locus of the nodal edges of the cones of the complex is a
cone with verter A of order (8n—1) having the n? right lines
AB, as nodal edges.

§ 5. The tangents in the nodes of a net Ne envelop a curve Z
of class 3(p—1)(2p —3) '), the curve of Zeurnen. It breaks up
for the net N+! indicated above; for, the tangents to the curves
br in their points of intersection with @ envelop a curve, which
must be a part of the curve Z The pencil (/”) is projective to the
pencil of its polar curves p”~! with respect to a point V; the points
of intersection of homologous curves form a curve of order (2n—1);
in each of its points of intersection S with ag a curve 4” is touched
by OS; so these tangents envelop a curve Z' of class (2n — 1).

So for N+! the curve of ZeurHeN consists of the envelope Z/
and a curve Z" of class 3n (An —1)— (2n —1)=(8n—1) (An —1).

The pairs of tangents in the nodes of the genuine curves of
N+! determine on a right line / a symmetrie correspondence with
characteristic number (22 — 1) (Bn — 1). To the coincidences belong
the points of intersection S of / with the curve H; to such a point
S are conjugated (2n — 1) (8n —1)— 2 points distinct from S; so
S is a double coincidence. The remaining 4 (n — 1) (Bn — 1) co-
incidences evidently originate from cuspidal tangents.

The locus of the vertices of cones of the complex, possessing a
cuspidal edge consists of 4 (n— 1) (Bn — 1) edges of the cone A.

A general net of order (7+ 1) contains 12 (7 —1)n cuspidal
curves, thus 4(2—1) more;- therefore each of the 2(mn—1)
figures consisting of the right line «ap and a curve 4” touching it
is equivalent to two curves c’t+! with eusp. Evidently the nodes of
these figures form with the point C, the section of «8 with the
curve H.

§ 6. On the traces of a plane a with the planes « and @ the
pencils (a) and (/") determine two series of points in (/, 1)-corre-
spondence; the envelope of the right lines connecting homologous
points is evidently a curve ef class (7+ 1) touching ez in its point
of intersection with the ray a conjugate to the curve 6" through

1) This has been indicated in a remarkable way by Dr. W. Bouwman (Ueber
den Ort der Beriihrungspunkte von Strahlenbüscheln und Curvenbiischeln, N.
Archief voor Wiskunde, 2nd series, vol. IV, p. 264).

( 630 )

the point agr, whilst it touches Ba in its points of intersection with
the curve 6," for which the corresponding ray passes through afz.

The curve of the complex of the plane x has the right line Ba
for n-fold tangent, so it is rational.

If the curve 5,* touches the intersection Ba, then the multiple
tangent is at the same time inflectional tangent.

We now pay attention to the tangents 7 out of the point S= a’
to the curve 4* corresponding to a. The envelope of these tangents
has the right line «8 as multiple tangent; its points of contact are
the 2(n—1) coincidences of the involution, determined by the
pencil (6") on eg. As S evidently sends out n(n — 1) right lines r
the indicated envelope is of class (n — 1) (mn + 2).

The planes containing a curve of the complex of which the n-fold
tangent is at the same time injlectional tangent envelop a plane
curve of class (n— 1) (n + 2).

$ 7. The curve (a) can break up in three different ways.

First the point egt may correspond to itself, so that (a) breaks
up into a pencil and into a curve of class n. This evidently takes
place when ar passes through one of the principal points Cp.

Secondly the involution on gr may break up, so that all its groups
contain a fixed point; then also a pencil of rays of the complex
separates itself. This will take place, when 2 passes through one
of the principal points Ay.

Thirdly the curve a may contain the principal point A. Then the
curve 4 corresponding to the ray a — aa determines on ga the
vertices of 7 pencils, whilst also A is the vertex of a pencil. The
curve a is then replaced by (+ 1) pencils.

In a plane through ag, thus through all principal points Cy, the
curve (a) consists of course also of (n + 1) pencils.

A break up into two pencils with a curve of class (n — 1) takes
place when the plane a contains two principal points B, or a point
B, and a point Cy.

§ 8. To obtain an analytical representation of the complex we
can start from the equations
Da) 5 wv, J Aw, = 0;
Dai) 5 a + ab” ==" (5
Here at and & are homogeneous functions of #2, 2, a, of

order 7.
For the points of intersection Y and Y of a ray of the complex

( 631 )
with @ and 8 we find
By Py 3 —= Uy Pig — Ve * Pas,
Yr? Pre = Ya Pas = Ys * Pas

After substitution, and elimination of 2, we find an equation of
the form

Pos (A Pia H As Pas + %s Poa)” =P is (0, Pig + Os Pas + Do Poa)
by which the exponent between brackets reminds us that we must
think here of a symbolical raising to a power.
If in pru= ee y,—#, ye we put the coordinate x, equal to zero,
we find for the intersection of the cone of the complex of Y on
the equation

(Ys Ca — YoU) (4, 2) + a, %, + a, &,)” = (y, U — Yi, Xs) (b, 7, Hb, «4-6, wo)”,
or shorter

5 . . je Nm
NEN bn — Yet, a+ Io (2, ar — a, 7) = 0.

This proves anew, that the intersections of the cones of the complex
form a net.

Mathematics. — “On nets of algebraic plane curves”. By Prof.
JAN DE VRIES.

If a net of curves of order m is represented by an equation in
homogeneous coordinates

gr az + yy be + ys cr = 0
to the curve indicated by a system of values y,:y,:7, is conjugated
the point Y having y,, y,, y, as coordinates and reversely.

A homogeneous linear relation between the parameters vj then
indicates a right line as locus of Y, corresponding to a pencil com-
prised in the net.

To the Hessian, H, passing through the nodes of the curves belonging
to the net, a curve (2) corresponds of which the order is easy to
determine. For, the pencil represented by an arbitrary right line /y
has 3(n—1)* nodes. So for the order n” of (Y) we find n"=3(n—1)?.

If one of the curves of a pencil has a node in one of the base-
points, it is equivalent to two of the 3(n—1)* curves with node
belonging to the pencil. Then the image /y touches the curve (Y)
and reversely.

Let us suppose that the met has 6 fixed points, then H passes

( 632 )
twice through each of those base-points; so it has with the neteurve

” . . a . . r . . .
cy indicated by a definite point Y yet (nn'—6) single points in
common; here 2’=8(n—1) represents the order of H. The curve

er having a node in D, determines with cy a pencil represented by
a tangent of the curve (Y). From this ensues that the class of (Y)
is indicated by hk” = 3n (n—1) — 20.

The genus g" of this curve is also easy to find. As the points of
(Y) are conjugated one to one to the points of H these curves have
the same genus. So we have

= } (n'—1) (n'— 2) — b = £ (8n—A4) (8n—5) — b.

We shall now seek the number of nodes and the number of cusps
of (Y). These numbers dé” and x" satisfy the relations

2d'+3x n' (n"— 1) — k",
J" 4x! = 4 (n'—1) (n"— 2) —g".
From this ensues after some reduction
d' = 2(n—1) (n—2) (3n?—3n—11) +3,
x! = 12 (n—1) (n—2).
The curve (Y) has nodes in the points VY, which are images of

the curves cz possessing a node in a base-point of the net. For, to
each right line through a point Yp a pencil corresponds, in which

cp must be counted for two curves with node.

Each of the remaining nodes of (Y) is the image of a curve «r,
possessing two nodes.

So a net N” contains * (n—1) (n—2) (8n*—3n—11) curves with
two nodes.

To a cusp of ()’) will correspond a curve replacing in each pencil
to which it belongs two curves with node. According to a well-
known property that curve itself must have a cusp. For a definite
pencil its cusp is one of the base-points; this pencil has for image
the tangent in the corresponding cusp of (1’).

So a net Nr contains 12 (n—1) (n—2) curves with a cusp.

The two properties proved here are generally indicated only fora
net consisting of polar curves of a et We have now found that
they hold good for every net, independent of the appearance of fixed

points 5.

We can now easily determine the class z of the envelope Z of
the nodal tangents of the net.

Through an arbitrary point P of a right line / pass z of these

( 633 )

tangents. If we add the second tangent in the corresponding node to
each of these tangents, these new set of z tangents intersects the
right line / in z points P'. The coincidences of the correspondence
(P,P') are of two kinds. They may originate in the first place
from cuspidal tangents, in the second place from the points of inter-
section of / with the curve H; each of these latter points of inter-
section however is to be regarded as a double coincidence. Thus
22 = 12 (n—1) (n—2) + 6 (n—1) = 6 (n—1) (2n—32).

The curve of ZEUTHEN is of class 3(n—1)(2n

3).

ER RAT AY

Page 504, line 13, for members read member.
, 004, ,, 15, ,, not wanting read wanting.
» 009, ,, 24, ,, blewish read bluish.

(April 19, 1905).

KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM,

PROCEEDINGS OF THE MEETING
of Saturday April 22, 1905.

(Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige
Afdeeling van Zaterdag 22 April 1905, Dl. XIII).

GONE Nr WS:

J. J. van Laar: “On the different forms and transformations of the boundary-curves in the
case of partial miscibility of two liquids”. (Communicated by Prof. H. W. Baxuvuis RoozEnoom),
p. 636. (With one plate).

J.J. van Laar: “An exact expression for the course of the spinodal curves and of their
plaitpoints for all temperatures, in the case of mixtures of normal substances”. (Communicated
by Prof. H. A. Lorentz), p. 646.

F. M. Jarcer: “On miscibility in the solid aggregate condition and isomorphy with carbon
compounds”, (Communicated by Prof. H. W. Baxuvis RoozrBoom), p. 658. (With one plate).

F. M. Jarcer: “On Orthonitrobenzyltoluidine”. (Communicated by Prof. A. F. HorLEMAN), p. 666.

F. M. Jarcer: “On position-isomcrie Dichloronitrobenzenes”. (Communicated by Prof. A. F.
HorrrMaN), p. 668.

H. Kameruiscu Onnes and W. Heuser: “On the measurement of very low temperatures,
V. The expunsioncvefficient of Jena- and Thüringer glass between + 16° and — 182° C.” p. 674.
(With one plate).

H. A. Lorentz: “The motion of electrons in metallic bodies”, ILI, p. 684.

H. G. Jonker: “Contributions to the knowledge of the sedimentary boulders in the Netherlands.
I. The Hondsrug in the province of Groningen. 2. Uppersilurian boulders. 2nd Communication:
Boulders of the age of the Eastern Baltic Zones H and I’. (Communicated by Prof. K. Martin),
p- 692.

Ernst pe Vries: “Note on the Ganglion vomeronasale”. (Communicsted by Prof. T. Prace),
p- 704. (With one plate).

J. W. van Bissruek: “Note on the Innervation of the Trunkmyotome”. (Communicated by
Prof. T. Prace), p. 708. (With one plate).

JAN DE Vries: “On linear systems of algebraic plane curves”, p. 711.

Jan DE Vries: “Some characteristic numbers of an aigebraic surface”, p. 716.

K. Bes: “The equation of order nine representing the locus of the principal axes of a pencil
of quadratic surfaces”. (Communicated by Prof. J. Carpinaar), p. 721.

Pu. Konnstamm: “A formula for the osmotic pressure in concentrated solutions whose vapour
follows the gas laws”. (Communicated by Prof. J. D. van per Waars), p. 723.

Pu. Konnstamm: “Kinetic derivation of van ’r Horr’s law for the osmotic pressure in a
dilute solution”. (Communicated by Prof. J. D. van per Waars), p. 729.

Pu. Korysramu: “Osmotic pressure and thermodynamic potential”. (Communicated by Prof.
J. D. van DER Waars), p. 741.

J. Weeper: “Approximate formulae of a high degree of accuracy for the relations of the
triangles in the determination of an elliptic orbit from three observations”. (Communicated by
Prof. Il. G. van DE SANDE BAKHUYZEN', p. 752.

A. W. Visser: “A few observations on autocatalysis and the transformation of j-hydroxyacids,
with and without addition of other acids conceived as an ion-reaction”. (Communicated by
Prof. H. J.;HAMBURGER), p. 760.

Artuur W. Gray: “Application of the baroscope of the determination of the densities of gases
and vapours’. (Preliminary Notice). (Communicated by Prof. H. KAMERLINGE ONNES), p. 770.

The following papers were read:

44
Proceedings Royal Acad. Amsterdam. Vol. VII.

( 636 )

Chemistry. — “On the different forms and transformations of
the boundary-curves in the case of partial miscibility of two
liquids.” By J. J. van Laar. (Communicated by Prof. H. W.
BakKnuis RoozrBoom.

(Communicated in the meeting of March 25, 1905).

1. In a preceding communication) I showed, that when one of
the two components of a binary mixture is anomalous, the 7, z-
representation of the spinodal curve, and consequently also that of
the connodal curve, the so-called saturation- or boundary-curve
w=f(T), can assume different forms, which are indicated there. It
depends principally only on the value of the critical pressure of the
normal component, with regard to that of the anomalous component,
which of the different forms may occur with a definite system of
substances.

An aflirmation of the theory, developed by me, that is to say
of the cases and transformations deduced by me from the general
equations, is found in the circumstance, that these cases and trans-
formations may be realised im quite the same succession with one and
the same. substance, and this by external pressure. In the same way
as with dijferent normal substances as second component the form
drawn in fig. 7 Le., presents itself at relatively low critical pressures
(with regard to that of the anomalous component), and that of fig. 2
le. at relatively high critical pressures — the form of fig. 7 may be
realised at relatively low external pressure, and that of fig. 2 at
relatively high external pressure, whereas at intermediate pressures
all the transitional cases again will return in just the same succession.

2. For that purpose we but have to look at the p,7-diagram
of the eritical curve for ethane and methylalcohol, as projected
by KueNeN *) in consequence of his experimental determinations (com-
pare fig. 1). We see, namely, immediately from the form of the curve,
departing from C, (the higher critical temperature, that of CH,OH),
which indicates the pressures, at which for different temperatures
the two coexistent phases coincide, and above which we have
consequently perfect homogeneity, that according to the value of the
pressure one critical point « may occur (at the pressures 1 and 2),
two viz. a and b,c (at 8), three, viz. a, b and c (at 4), again two, that
is to say (a,b) and c (at 5), and finally again one, viz. c (at 6). (also
compare fig. 2).

1) These Proceedings of 28 Jan. 1905.

2) Phil. Mag. (6) 6, 637—653, specially p. 641 (1903).

( 637 )

All this is rendered still more conspicuous, when we project a
space-representation, in connexion with fig. 1 and of some successive
p, x-sections. In fig.1 D, and PD, represent the vapourpressure-
curves of the two components; AM is the threephasepressure-curve,
which terminates abruptly in M, where the gaseous phase 8 coin-
cides with the liquid phase 1 (which consists for the greater part
of ethane), because it meets there the critical curve C,C,, that is
to say the curve of the plaitpoints P. Beyond JM there is coexi-
stence only between the jlwid phase 3,1 and the phase 2, which
consists principally of alcohol. It is the equilibrium between these
latter phases, of which in fig. 2 the 7, r-representation is projected
at different pressures. (The dotted houndary-curve O corresponds with a
pressure inferior to the critical pressure of the second component, and
superior to that of the first one). The 7. x-representation of fig. 3
corresponds, at the (variable) threephase-pressure, with the threephase-
equilibrium unto J/. In fig. 4 the indicated space-representation is
drawn, which will be clear now without the least difficulty *). For
the different higher pressures the corresponding 7’, x-sections are
drawn in that representation.

Remark. From C, (see fig. 1) to the maximum at 126°, where
a and 6 coincide, and also from the minimum at 26°, where 6 and
e coincide, to the lowest temperatures, increase of pressure will
lower the critical temperature Q, and these critical points will be
upper critical points in the 7’, 2-sections at constant pressure (see
fig. 2). On the other hand, from the maximum at 126° to the
minimum at 26° increase of pressure will raise the critical tempe-
rature, and the designed critical points will be Lower critical points.

That increase of pressure favours the mixing, as is clear from
fig.1 and from the p,z-representations — as well in the case of
an upper critical point, as of that of a lower one — is also in
agreement with the 7’ 2-representation of fig. 2. For in the case of
an upper critical point (see also fig. 5) a point A, situated within
the boundary-curve will come — when the pressure is increased, by

a)

! aT . A
which 7, will be removed to the lower point 7’), (for aan negative, as

Pp
we saw above) — without the new boundary-curve. And the same
; ; si ke dT
will be obviously the case for a lower critical point, where a
jo

is positive.

1) This space-representation (without the 7, w-sections) has been already pro-
jected independently by Mr. Bicuyer; but is not inserted in his communication.
(These Proceedings of 28 Jan. 1905).

44%

( 638 )

To the considered type also belongs SO, + H,O, C,H, + H,0,
and equally ether and water. This latter mixture only with this diffe-
rence, that the composition of the vapour-phase is here continually
between that of the two liquid phases (see. fig. 3%). Kuenen *) found,
that at 201° the vapour-phase coincides with that liquid phase, which
consists for the greater part of ether. The threephase-pressure is then
52 atm. (At C, we have 7, =195°, p, = 38 atm.).

The p, z-diagrams would now show a maximum-vapourpressure,
if the two liquid phases 1 and 2 could become identical. In
connexion with this the threephase-pressure will be higher (here only
some mM) than the vapourpressures of each of the components, and
it follows immediately from fig. 1, that the critical curve C,C,, or
rather CM, will at first run back from C;, that is to say will present
a minimum critical temperature. In the ease of C,H, + CH,OH, where
the composition of the vapour-phase is without that of the liquid
layers, the threephase-pressure will always be between the vapour-
pressures of the components.

3. Now, as to the representation of the so-called transversal- and
longitudinal plait on the y-surface at different temperatures (in its
projection on the v, z-surface) in the case of C,H, + CH,OH, it will
be obvious, that the critical point Q, considered above, of the longi-
tudinal plait always lies at the side of the small volumes. For
increase of pressure finally favours (see above) the mixing.

The successive transformations of the transversal- and of the longi-
tudinal plaits are further represented schematically, in agreement with
the p,-sections, in fig.6. The longitudinal plait, occurring here,
is regarded by van per WAALS *) and this equally in the ease to
as a transformed transversal plait. Many

be considered presently
questions however, connected with these plaits, lose — as has been
remarked already by van pur Waars ®) — much of their weight, and
become of secondary interest, as soon as we succeed im connecting with
other properties of the components of the mixture the often so com-
plicated transformations, which may occur at the different plaits.
And to do this an attempt is made in my preceding communication.
There I showed, that the ordinary theory of the association is capable
of representing the different possible forms of the boundary-curves quali-
tatively, and in many cases even quantitatively.

4. We will now consider the second of the three principal types,

1) Z. f. Ph. Ch. 28, 342—365, specially p. 352 (1899).
2) These Proceedings 7, p. 467 (1899).
8) Id. 25 Oct. 1902, p. 399.

( 639 )

indicated in a recent communication of BAkHvis RoozrBoom and
Bicuner !), the first of which is amply considered above.

Fig. 7 gives the p,7-representation of it; fig. 8 the 7, r-diagram
of some sections at constant pressure (the dotted boundary-curves are
relative to pressures between that of J/" and C,, and to that below M').

The series of p,v-diagrams, and also the space-representation (also
projected already by Bicnner) are omitted here. We find that case
with mixtures of ethane and ethyl-, propyl-, isopropyl- and normal
butylalcohol, all examined by Kurnen *). Also triethylamine + water,
and some other mixtures *) with a lower critical point (this lies at
18°,3 C in the last mentioned case) belong to that type. Indeed, it
is obvious from fig.8 and from the p‚r-diagrams, that JL’ is at
present a lower critical point, contrary to M in fig. 1, which was an
upper critical point. For, whereas in M (fig. 1) the threephase-pressure
ends, this pressure commences in M'. Farther however, in JZ", the
gaseous phase coincides again with the first liquid phase (rich in ethane)
(because the threephasepressure-curve J/M" anew meets the critical
curve CC), after which the further course is the same as with the
preceding type. The 7’,x-representation with variable threephase-
pressure, that is to say between J/’ and J/" (fig. 9a), is in agreement
with it.

It appears that, as little as with the preceding type, there exists
here a properly-said upper critical point. For in both cases 1 and 3
coincide, when the threephase-pressure comes to an end, and not 1 and 2
(see fig. 3a and 9a). The vapour-phase becomes identical with the
upper liquid layer and vanishes, but then there remain still two
phases, the lower liquid layer 2 and the fluid phase 1,3. These
however always pass into each other with further increase of the
temperature, as is plainly indicated by the space-representations
(see e.g. fig. 4), at the vapourpressure-curve of the second component
at we, =,= 1 (also compare the dotted boundary-curves in fig. 2
and 8). So, if we begin with a mixture of a definite composition,
then with increase of temperature we come finally beyond the bourdary-
curve 1,3 at the moment, that the liquid layer 2 has entirely vanished.
So we have demonstrated more exactly at the same time what I have
said in my preceding communication, namely that the existence of
a lower critical point involves necessarily that of a higher upper one.
It is however not, as we have seen, a critical point proper.

1) Id. 28 Jan. 1905, p. 531—537.
2) Z. f. Ph. Ch. 28, p. 358—363 (1899).

8) Among others €-collidine, and the bases of the pyridine- and chinoline-series
with H,0.

( 640 )

From the series of p,a-diagrams we might still see, that the
boundary-curve of the two liquid phases with the plaitpoint Q below
the temperature, where 1 and 2 coincide (in M’), does not come
within the boundary-curve vapour-liquid, as Kuenen *) thinks, but has
entirely vanished. In fact, there is no reason to suppose, that in M'
decrease of pressure should cause again permanent separation (if
that state were realisable), as apparently Kurnen thought that he had
“undoubtedly” to expect with mixtures of triethylamine and water *).
What he has observed in another case with propane and methylalcohol®),
must be ascribed in my opinion to this, that the expansion just above
the (upper) critical point has caused the temperature to fall a little,
so that he came on the (metastable) part of the two liquid curves,
which lies below the threephase-pressure. But when the cusp was
reached, the metastable equilibrium became immediately stable, and
by further expansion homogeneous liquid and vapour reappeared.

Because increase of pressure in this case too favours the mixing, as
appears from fig. 7 and from the p, z-diagrams, the plait on the y-
surface will have turned its plaitpoint Q also to the side of the small
volumes. Fig. 10 gives a schematical representation of the successive
transformations of the two plaits, or rather of the transversal plait,
for we can regard again with van DER Waats the longitudinal plait
as a transformed transversal one.

In fig. 7 we see, that increase of pressure raises the critical points
Q, at least in the beginning, if the curve M'C, should present
a maximum; and from the p, a-diagrams, that these critical points
will be again in that case /ower critical points, just as in fig. 1 between
the minimum and the maximum in the curve of the critical points Q.

Equally in the case of the second general type the threephase-
pressure may be either between the vapourpressures of the two
components, as in the case of C,H, and the mentioned alcohols, in
which case the composition of the vapour-phase will be not between
that of the liquids — or may be /zgher than that of the components.
Then there is again a mazimum-vapourpressure after the coinciding
of the two liquid phases below the lower critical point, and the
composition of the vapourphase is between that of the liquid layers *).

1) Phil. Mag. 1. c. p. 645.

2) Id. p. 652.

3) Id. p. 646.

4) That at the coinciding of the two liquid phases 1 and 2 in M’ the vapour-
phase does not necessarily coincide with 1 and 2, has been already remarked by
Kueyen, and still earlier has been deduced by me theoretically for an analogous case

( 641 )

(see fig. 95). It is a matter of course, that in fig. 7 the critical curve
C.C, or rather C,J/", again turns back in that case, consequently
presents a minimum; equally the critical curve M'C, will not
seemingly eut CC, between C, and M" in this case, but remain on
the left of C,. This case occurs e.g. with mixtures of water and
triethylamine, where in consequence of the almost complete unmis-
cibility of the iwo components above the lower critical point the
threephase-pressure will be but very little smaller than the sum of
the vapourpressures of the two components. So KurreEN found, that
at 93° C. the threephase-pressure was 142,6 cm., whereas the vapour-
pressures of triethylamine and water were resp. nearly 86 c.m. and
58,6 em., consequently together 144,6 em.

5. It is worth remarking, that the region of the threephase-
pressure continually shrinks the more we ascend to higher alcohols
(Kurnen). In the case of C,H,-+C,H,OH the temperatures in
M' and M" were resp. 31°,9 (46 atm.) and 40°,7 (55 atm);
in that of C,H, + C,H,OH these are 38°,7 and 41°,7; in that of
C,H, + isopropylalcohol ? and + 44°; in that of #-butylalcohol
38°,1 and 39°,8 (55 atm.). Finally with isoamylaleohol three phases
were no longer realisable, so that the critical points Q coincide
there with the critical points P, one continuous critical curve being
formed from C, to C, (General type III).

In agreement with this is the fact, that the anomaly of the
alcohols decreases, as these are higher. In fact, we approach then
more and more the ease of mixtures of normal substances, where
only at very low temperatures (see my preceding communication)
a formation of two liquid layers can present itself. KurreN found
indeed, that C,H, + ether mix in all proportions, whereas C,H,-++H,O
again present a threephase-equilibrium.

The influence of the fact that the alcohol is higher is also sensible
in the case of mixtures of CS, and different aleohols. So we find for
the upper critical points of CS, with CH,OH, C,H,OH, C,H,OH and
C,H,OH successively +40°,5 (Rorumonp), —10°,6 (Kuenen) —52° (K.)
and —80° (K.). :

Equally the influence of the hydrocarbon was examined by Kuenen.
While, as we saw above, the separation between type I and II in
the case of C,H, + different alcohols was between CH,OH and

(equilibrium between two solid phases and one liquid phase). This will obviously
also be the case with an upper critical point, as appeared from the experiments

of Scureinemakers with water and phenol. We will return to this question
in § 8.

( 642 )

C,H,OH, and that between IL and III between x-butyl- and isoamyl-
alcohol, the separation between I and HI in the case of CH,OH +
different hydrocarbons is lying now between C,H, and C,H,. Propane
and the following hydrocarbons + CH,OH belong consequently, just
as C,H, + isoamylaleohol, and the greater part of the mixtures with
an upper critical point, examined by Gururm, ALEXEJEW and ROTHMUND,
to the third general type, which we will briefly consider now.

6. The third general type is principally characterised by this,
that the threephasepressure-curve meets vo longer the critical curve
C,C,, but has come to an end already before (fig. 11). This third
type may proceed either from the second type (see fig. 7), the
threephasepressure-curve J/' M/" shrinking more and more, and
finally vanishing, as is the case with the transition from C,H,4C,H,OH
to C,H, + ©,H,,OH (see above) — or from the first type, when AM
ends already before C, C,. In the first case (e.g. with C,H,+C,H,,OH,
C,H, + ether) there exists no threephase-equilibrium at all — or it
should be at very low temperatures, which even may be expected
according to the theory (see my preceding communication), so that
the question arises, whether also in the case of fig. 7 there exists
at low temperatures a new threephasepressure-curve, and we will
return to that question at the end of this communication — in the
second case there exists a threephase-pressure from the beginning,
which vanishes at a definite temperature (wpper critical point). To
this latter case belong the mixtures of CH,OH with C,H,, ete.,
H,O + CO,, CS, + different alcohols (KunnEN), and also the greater
part of the mixtures formerly examined (see above).

But in the case of this third type there exists still another diffe-
rence. Firstly the threephasepressure-curve again may lie either
between or without the vapourpressure-curves of the two components,
with all the consequences, connected with it in the p, a-diagrams,
ete. (see above). As to the mixtures of CH,OH with C,H,, C,H,,,
C,H,,, with all these is found (Kurnen, l.c), that the threephase-
pressure is higher than the vapourpressures of the hydrocarbons, con-
trary to C,H, + CH,OH, which belongs to type I, where the three-
phasepressure is lower than that the vapourpressure of C,H,. Equally
with phenol and water (SCHREINEMAKERS, Vv. D. Lee), H,O + CS,
(RuenavLt) — where, according to the exceedingly small miscibility,
the threephase-pressure is again a little smaller than the sum of the
vapourpressures of the components — H,O + Br, (Bakuuts RoozeBoom),
H,O + isobutylalcohol (Korowarow), CS, CH,OH and C,H,OH
(Kueren), H,O + aniline (Kuen), ete. we find everywhere the three=

( 643 )

phase-pressure higher than the vapourpressures of each of the com-
ponents. Only of H,O + SO, (Baxnuis RoozeBoom), and of some
systems more (S + Xylol and Toluol, CO, + H,O) we know with
certainty, that the threephase-pressure is between the vapourpressures
of the two components.

There exists, however, still another, important difference. Wheras in
the case of type I (fig. 1) the eritical curve QC, presents alternately

ry

dT C .
positive and negative —, and in that of type II (fig. 7) — is of
dp dp

course positive in the beginning in M/' (indeed, the point Q just
appears in M') — in the case of type III the initial course of MQ may
be as well to the left as to the right.

a ee
Is this course to the /eft, that is to say is = negative, then —
ap

just as in fig. 1 between C, and the maximum and between the
minimum and the lowest temperatures — increase of pressure will
lower again the critical temperature in the case of these upper eritical
points, and the plait on the y-surface in its v, v-projection will again have
turned the plaitpoint Q to the side of the smal/ volumes. (This is
equally the case with C,H, + isoamylalcohol, where no threephase-
equilibrium could be stated, but where the plaitpoint Q, which has
become here identical with P, has removed strongly to the «x-axis,
just as in the case of mixtures of C,H, and the /ower alcohols). We
find this e.g. with C,H, + CH,OH.

But when the initial course of MQ is to the right, as in the case
of C,H,, + CH,OH, C,H,, + CH,OH, and of phenol and water, then
increase of pressure will raise the point Q. and the mentioned plait
will now have turned the plaitpoint Q for the first time to the side
of the large volumes.

The question, whether the longitudinal plait, as in the case of phenol
and water, will present still a second plaitpoint at very small volumes,
consequently at very high pressures -— in other terms, whether the
coexistent liquid phases, after diverging initially, will reapproach
afterwards in composition, has not yet been answered theoretically
with certainty. It however appears to me, that where in the
case of C,H, + CH,OH the plait has turned the plaitpoint Q to
the side of the small volumes, whereas C,H,, and C,H,,, equally
with CH,OH, have turned this point to the side of the large
volumes, there must exist a continuous transition between the two
kinds of longitudinal plaits, and that also the latter (as long as it
has not yet detached itself from the liquid curve of the transversal
plait, that is to say below the upper critical point) must be regarded

( 644 )

as an appendix of the transversal plait. Only when the longitudinal
plait has detached itself entirely from the liquid curve of the trans-
versal plait above the critical temperature of mixing, it can be
regarded in my opinion as a separate plait by the side of the
transversal one. This is in full agreement which what we find e. g.
for the boundary-curves in the 7Zx-representation (see fig. 2).

As long as the two parts of the boundary-curve, for instance N°. 2,
are not yet separated, we can hardly speak of two boundary-curves:
it remains one continuous boundary-curve; only beyond the transitional
case N°. 3, e. g. N°.4, we have a right to speak of two isolated
boundary-curves.

As to the values of the different critical temperatures, we still
mention, that with C,H,-+-CH,OH the upper critical point was
found at 21° C., with C,H,, + id. at 19°,5, and with C,H,, + id.
at cirea 40°. With C,H,, + C,H,OH the latter temperature immediately
falls down to — 65°.

7. Resuming all that precedes, we have the following summary.
(p‚ designs the threephase-pressure, p, and p, the vapour-pressures
of the two components).

C,H, + CH,OH

Type I |, between p, and p, | C,H, + H,O

(fie. 4) SO, + H,O
ore) |

Drs Pp, and ip, Ether + H,O

C,H, + C,H,OH, C,H,OH,
ps between p, and p, nn

Type I iso-id., n. C,H,OH.
(fig. 7).
lp, >p, and p, Triethylamine + H,O
ee | ae OR
Type a OENE
(ie 11) 4 Op: and CAH En id. (dear +)

C,H,, + C,H,OH
|» >p. and p, { H,O + Phenol (%/ar +)

H,O + Aniline; id + isobutylalcohol.
Br, + H,O ;CS, + H,O, CH,OH, C,H,OH,
\ C,H,OH, C,H,OH.

( 645 )

8. We saw above, that when the composition of the vapour-phase
is between that of the two liquids — which is the case, when the
threephase-pressure is higher than the vapourpressures of each of the
components — there must be a mazimum-vapourpressure after the
coinciding of the liquid phases 1 and 2. That maximum may however
still be present before the coinciding of these liquid phases, which
is connected with the fact, that this maximum, which appeared origi-
nally at lower temperatures as a minimum (see fig. 12) in the meta-
stable region, has become gradually a maximum, and has moved
outwards before the coinciding of 1 and 2. The vapour-phase 3,
which was lying at lower temperatures always between 1 and 2,
as to its composition, remains not necessarily between these till the
moment of coinciding of 1 and 2, as was thought formerly, but
may have come outwards long before (see also fig. 95). It would be
very accidental on the contrary, when 3 coincided in the same time
as 1 and 2 to one phase. In the case of phenol and water SCHREINEMAKERS
has in fact shown experimentally this moving outwards’).

In what manner the moving outwards takes place, has first been
clearly shown and considered quantitatively by me?) in a series
of figures, and this in the case of coexistence of two solid solutions
and one liquid phase, whereas we have here — what of course is
quite the same’) — the coexistence of two liquid and one gaseous
phase. The figures 9—I4, drawn in the indicated communication
(which refer to meltingeurves, and consequently are Zr-repre-
sentations) are to be turned upside down, and the figs. 12, reproduced
here, are obtained (fig. 12 of the mentioned communication is omitted).
It will be remembered, that the case, which is realised with respect to
liquid-vapour with phenol and water, is realised with respect to solid-
liquid with Ag NO, + NaNO, (only the maximum of fig. 145 Le. at
D has been already vanished there).

Some months after the publication of my communication KUENEN *)
came independently of me also to entirely the same view. What
is described on the pages 471 and 472 of his communication,
is quite identical with that, which I have described and represented
on the pages 184—186 of the designed communication.

Z. f. Ph. Ch. 35, p. 462—470 (1900).

2) K. A. v. W. 27 June 1903.

8) The calculations were based on the equation of state of van per Waars, so
that the results of it are a fortiori applicable in the case of two liquid phases
and one gaseous phase.

4) K. A. v. W. 31 Oct. 1903.

( 646 )

9. Now, that we have sufficiently characterised the three general types,
and have brought some harmony into the multiplicity of the phenomena,
the question arises, whether there is a still farther synthesis, a still higher
unity. More than once the occasion presented itself in the treatment of
the different general types to remark striking agreements and continuous
transitions, often accompanied with great differences. Equally the
fact, that with a higher alcohol or a higher hydrocarbon, suddenly a
quite different type often appears, must certainly draw attention in
a high degree. All this induces us to look for the one fundamental
type, of which the three types, treated above, are special cases.

Also the analytical consideration of the question suggests that idea
to us. Indeed, the coexistence of two liquid phases and one gaseous
phase, or of two liquid phases, or finally of one liquid phase and
one gaseous phase, is determined by one and the same equation of
state, and it must consequently always be possible to reduce all the
different cases, which may occur to #vo fundamental proportions :
that of the critical temperatures and that of the critical pressures of
the two components — entirely in the same way as I have formerly
deduced al/ the different types in the case of mixed crystals, where
appear two solid phases by the side of one liquid phase, from two
fundamental proportions: that of the meltingtemperatures and that
of the latent heats of melting of the two components.

In a following communication it will be shown theoretically, that
the three types may be deduced from the ordinary equation of state
of Prof. vaN per Waals, even in the case of normal components. In
connexion with this we must not forget, that in the neighbourhood
of the eritical points of each of the components the influence of
anomaly vanishes nearly always. In the case of C,H, + H,O for
instance the water will be in the neighbourhood of 365° C. already
normal long before, and in the neighbourhood of 32° C. the liquid
phase, which consists nearly entirely of ethane, will contain the water
in such a dilute state, that this will be passed for the greater part
into the state of simple molecules.

Chemistry. — “An exact expression for the course of the spinodal
curves and of their plaitpoints for all temperatures, in the
case of mixtures of normal substances.” By J. J. van Laar.
(Communicated by Prof H. A. Lorentz).

(Communicated in the meeting of March 25, 1905.)
1. It is well-known, that the points of the ¢-surface, corresponding
to points of the spinodal curve on the y-surface, are given by the
simple relation

i
\
Ke
t

EET ES
V re a

a
ON
> Pel <a .
nar Liat = &
» . _

ee ee en in el il
7 nn
aia ek re

gh hd oa 7

és

ee en

3
t
fb

J.J. VAN LAAR “On the different forms and transformations of the boundary-curves in the cise of partial miscibility of two liquids.”

Ze MT Lore

Css CHOW AOM are hid

Fig 3 Fig. 3a,

Fig 5 Fig. 6

Fig. 4

Nile 7) (La tm)

GAL + GM OA Zi

Fig. 8. Fig. 9a Fig. 96. Fig. 10 Fig. 1. Fig. 12a.

Fig. 126. Fig. 12c. Fig, 12d. Fig. 12e Fig. 12f Fig. 13a Fig 136 Fig, 13e.

Proceedings Royal Acad, Amsterdam. Vol VI.

( 647 )

075 cae
Ow? Ps T y

Ors OW Op
Ox? Ov? <i Ordu
stead of the thermodynamic potential the free energy is used, and
not w,p and 7, but z, v and 7’ are the independent variables.

As we have further in the case of normal components e. g

2
which corresponds with the condition — )= 0,1) when in-

0g 0g
CS 5 Art
n, Hij
On, 05 ; AA
we have also nes Cann and the above-mentioned condition may
av U
be replaced by
Òu, 2
Sl
Ow

Now
fi dw a
fu, =C,— (e — 2 =) + RT log (le),
Hij

where C, is a pure function of the temperature, whereas w is

given by
o =| pdv — pr.

Ou,
The condition — = 0 is therefore identical with
Ue
do RT ò
Ue —
Ou? l—2 ;
or
io)
RP a (Va) eee en ate se OT
(la) (1)
from which I also started in my preceding communication ’).
2
rr . . . w
Now the difficulty arises, that the exact calculation of NE leads
av

to rather complicated expressions, so that vaN DER Waars contented
himself most times with approximations. These consisted in this, that
in the liquid state at sufficiently low temperatures 1st p was omitted

5 . a € . .
by the side of —, 2nd terms of order v—6 were neglected against
5

those of order v.
Starting namely from the equation of state of vAN DER WAALS

1) Compare van per Waats, Cont. II, p. 137.
1) These Proc. 28 Jan. 1905.

( 648 )

(» SE 5) BRD
5

where 6 will be regarded as independent of v and 7, then we
find for @:

wo = RT log (v—b) + oe = PU wen ENEN)
5
: REE a 5 :
If we write now — | Al for v—b, and omit p, then we obtain:
pt the

in which vaN per Waats further wrote 4 for v, whereas for illustrating
B a fs . . lanl a 5
several properties 5 was brought in connexion with 7, and = with pa.
) )

This is consequently a complete set of approximations, and
with good reason Prof. Lormnrz remarked to me, that in such
cases we must be carefull, whether these approximations are
not in contradiction, and up to what temperatures the results,
dw
Ow
Van per Waats himself considered therefore the deduced expression
merely as a more or less rough approximation, but which is at all

z ; dw 0 (a 0 fa
events better than the former expression — =—|{— |= —[—}],

deduced with the above-mentioned expression for —, can be used.

Ow Ox

a fp
where the term with — log — was omitted.
U v

Now, I showed in my preceding communication, that at low tem-
peratures, and in the case of normal substances, where the critical
pressures rarely differ much, this omitted term has in the greater
part of cases a very small value, and is of entirely the same order as
v—b

v

Only at higher temperatures the term has a large value, but then

Ow

the deduced expression for Er is not exact enough by far, for then
de

, which is constantly neglected.

; : a : v—b
neither p can be neglected against —, nor terms of order
v

can

be omitted in that case.
The matter is consequently this: at sufficiently low temperatures

( 649 )

dw 0 fa
the former simple expression eau) may be safely used, at
& wv

least in the case of normal substances; but at higher temperatures

0 a
equally the new expression with the term =a log — will be insufficient.
U v

4 0% 070

And we want a more accurate expression for Ae and = the
U Hij

more, when we — specially with respect to the course of the plait-

point-curve also wish to know anything about the course of the
spinodal curves from the lowest temperatures to the highest critical
temperature.

I therefore tried to solve that problem; I was the more encou-
raged to do so, as soon it appeared to me, that the entirely accurate
expressions are not so complicated as was expected. On the contrary,
the often occurring fact presented itself here, that the exact expres-
sion is relatively more simply than the approximated one.

2. If we write the equation (2) in the form

ys = + RT log (v—b) — p (v—b) — pb,

Vv

then we obtain:

dw 0 = Ra O(v—b) db
de Ox\v/) | ( =O P Ow P de :

\

N ARAL a
Now meee 2 =

consequently we find further:

yp?

dw 1 da a Ov a Ov a db db
ae eee et eo ee
Ox v de v°Òr v 0x v? de de

or
dw 1 da út ld 5
= jn WEER eh
Ow v dx f v? ) dx’ 6)
Ov
where 5, Wpears no more.
v

If we write now:
a=(l—2)’? a, + 2x (1—2) a,, + 2 a,,

and if we put «,, = Wa, a, by which the calculations and the results
are simplified in some way, without affecting much the exactness of
these results’), then we have:

1) J am convinced, that the expression aj = a @ 1s exact in the case of nor-
mal substances. At all events the inaccuracy, which results from this supposition,
will certainly not be greater than that of the equation of state used.

( 650 )

a = ((l—2) Va, + 2 Va)’.
Further we admit for 4 the ordinary linear relation
b = (1—2) b, + ab, .

The suppositions, on which the following calculations are based,
are consequently the following.

1st. the equation of state of van per Waats, with 5 independent
of v and 7.

2.4. the ordinary suppositions about a and 6.

3rd, the special supposition a,, = Wad.

From the expressions for a and 5 used results:

fi da
= = 2 (a — #) Va, + & va) (Va, Ty Va,) =2 Va. (Va, = Va)
Pa
dx? = (Wa, — Y4,)
db d°b
—=),—), ; —=0.
dau dax?

If we did not put a,,=—=Wa,a,, then we should have found
d'a

em) so only somewhat less simple.

Òw

Ou?

3. We will now calculate
For (3) we can write:

da Wa a
"== ae ( Va, a Wa) Tm (» =F 5) (6,— b,),

on

so that we nN when for shortness’ sake « is written for

Va,—Va,, and 8 for b,—b,:
dw 2 ae i
PE =— (Ma, — Ma)? — — (Wa, ape
2Va ; 2a Ov
ee (b, Or b,) ae (Va, ay Va.) TEE a —
v v av

207 Zag Wa zE (= 2a z) Ov
? En

v v v* v? Ox
2 ( Ov
st | Fes ae Va ze “(0 Va En a) v
v v v v Oz
dv

Consequently we must calculate an
vu

( 651 )

a
From the equation of state (» + 5) (v — b) = RT we deduce:
5

a\/òv dh b) GG da _2adv =,
(» cv JN =) ca ee v? de ede) ”

yielding

or also

Ow 2af, (v—b)? (4)
Tei oz
a a Se ae ay ey OO
Substituting this in the last equation for apie We obtain:
av
o_O, — 2/0 5)
ears: fae wie) UREA ee

a (r-5 de ee ce)
f f RT x : A oe ;

: da = ;
since = —2a a nt 2 =f. Further treatment yields after im-
wv L

portant simplification :

= ae le te ((D
Ou? v = 22/,,(v—b)? 6)

Comparing this entirely exact expression with that, deduced in my
. . w=
former communication, where p and —— were neglected, we see
5
that the exact expression (5) is already simpler than the approx-
imated one, which may easily be written down by means of the

: : 0? (a 0? a
expressions for and log —, deduced there.
dx? \v Ov oe

4. Consequently equation (4) passes into
2x (Lr) (av—Bp Ya)?
u? Fe 2af, (v—b)? y
RT +

Ra =

Proceedings Royal Acad. Amsterdam. Vol. VIL.

( 652 )

that is to say into
5 (v—b)? 2x (1—2)

RT a — = = ; (av—B pa)’,
v Be uv
or into
ar
B == = E (1 — x) (av — By a)? Ha (wv — |

Now av—PBYa=a(v—}b) + ab — Bya
=a(v— b) + a(b, + B) — B (Wa, + aa)
= a(v—b)4+ (ab, —BY a, )=a(v—b) (baba).
Therefore we obtain (compare also vaN Der Waars, Cont. IL,
p. 45):

9 2
gdb zalen a,— ba) + av — Hy -f- cbr} eG)
rr

being, with the above mentioned suppositions, the sought, quite
general expression for 7’=/(v,x), by which for each given tem-
perature the v, v-projeetion of the spinodal curve is entirely determined.
We may also construct a ‘spinodal surface” 7’= fw, x), and im-
mediately deduce from the subsequent sections 7’= const. the forms
of the spinodal curves of the transversal- and longitudinal plaits, and
this in just the same v, x-representation as is used by vaN pur WAALS
for the projection of the spinodal curves of the surfaces p=/(7v,x)

for different values of 7.

5. The equation (6) gives rise to some results, which may be
deduced from it without further calculation.

Ist. Is vb, that is to say, is the limit of volume 6, reached at
any value of w, then (6) reduces to the equation of the boundary-
curve, lying in the v,a-plane :

29

I = = MUD NA Aere en 5 > (Ee)

viz. the same expression, which was formerly found for small values
of » by means of the approximating method.

It is obvious at present, that only for v — b the expression (6a)
holds rigorously good. In every other case terms with »—4 must be
added. But it also results from the found expression (6), that as
long as terms with v1 —6 may be neglected, the formula (6%) gives
approximately the projection of the spinodal curve on the 7’, v-plane,
without it being necessary to take into account the corrective-term with

i Z 3 ; 5
— log —, indicated by van per Waars. In a former communication

( 653 )

I showed already, that this correction-term is small in the case of
normal substances, about of order » — 5.

As the second member of the expression (6%) is always positive,
even when a,, should be CVa, a,, the longitudinal plait on the
y-surface (for it is obvious, that in the neighbourhood of v=d
the spinodal curve belongs to the longitudinal plait, which can
be regarded as a prominence of the transversal plait) will always
close itself above a definite temperature at the side of the small
volumes.

This temperature 7, is the plaitpoint-temperature, corresponding to

ak

Ax

(6%); it is given by (6%, in connexion with the expression for ==0);

deduced from it, yielding for the plaitpoint after elimination of 7
the value

CI

1 eneen ees
[on verter et |,

b,—b

where r= ———. (compare vaN per Waars, Cont. IL, and also my

1
preceding communication, p. 579). Only when 5, = 4, (r = 0), ze will
be ='/,. In each other case x, will be removed to the side of the
smallest molecular volume.

Just at 7, the closing will take place at the limit of volume
v=b(«=a-); for values of 7’< 7, the longitudinal plait will
remain wiclosed up to the smallest volumes. For in that case (compare
the representation in space) a section 7=const. will cut the boundary-
curve (6%), lying in the boundary-plane v = %, in a straight line.

This temperature 7’, may consequently be regarded in any respect
as a third critical temperature. For above that temperature a for-
mation of two liquid layers will never present itself at values of v in
the neighbourhood of 5, that is to say at very high pressures; just
in the same manner as above the ordinary critical temperatures of
the single substances can never appear a liquid phase in presence
of a gaseous one.

nd Is v=o, then for each value of x, 7’ will be =O, that
is to say, the equation (6) cannot be satisfied in that case. The plait
will consequently never extend to v=o.

3. Is «=O or 1, then (6) passes into the two boundary-curves,
lying in the two limiting 7’ v-planes, viz.

Lae SAG . 2a, i
R1 re he ban RE 3 (v — b,)?.

With »>— 3+, (resp. 3b,) these two curves yield duly:
45*

( 654 )

RE 8 a, RT 8 A, (6%)
v = — 3 v == 5 5 aca koe are )
2 TREO a yale
which is again a good test of the exactness of our formula,
deduced above.

These two critical points are at the same time plaitpoints of the

Ov

(transversal) plait, for it can easily be shown, that (5 ) and also

x

oT :
(5) will be there = 0.
Ou p

Before deducing the equation of the plaitpoint-curve, I shall first
point out, that the second member of (6) is always positive, as
consisting of the sum of two essential positive terms, so that the
Tv, v-surface possesses nowhere points beneath the v, z-plane, which
of course cannot occur, because 7’ cannot be negative. Further,
that from (67) and (6%) results, that as to the limiting-curve (6%),
there will be found 7’—O for «=O and x=1, and as to the
limiting-curves (6), 7’ assumes again the value 0, as well for v — d,
(resp. b,), as for v =o.

Since the values of 42/7, and #/,, can be very different, according
to different substances, the surface (6) will also present very different
forms. Generally a greater value of 6 corresponds with a greater
value of 7, and in that case the surface has the form, as is indicated
in the figure. It is manifest already at superficial consideration, that
this form will be pretty complicated.

6. We will now determine from (6) the locus of the plaitpoints.

.

( 655 )

This may be found by combination of the two conditions

Ou, —=0 ; ois — 0
Ow i : Ow? DT je

of of Ov =o ;
eestor (ear aan an ie | (7)

when / represents the second member of (6). Indeed, this second

leading to

member has in all points of the spinodal curve on the y-surface the
same value, so that we have, by passing along an element of that curve :
of
Ou
But in the plaitpoint we may regard an element of the spinodal
curve also as an element of the connodal curve, that is to say as the

oF
de +—~—dv=0,
Ov

line which joins the two tangent-points of a double tangent-plane,
when the tangent-points have approached each other to an infinitely
small distance. And as in these two tangent-points the pressure has
the same value, the latter does not vary, when at the plaitpoint we
pass along the considered element of the spinodal curve. Consequently

] Oe:
n= a,
1 ae he z

which yields immediately equation (7).

we have:

For shortness, we will write in the following >,/a, — b‚Wa, = 2,
by which the second member of (6) passes into

f==

2 KE (1—2) |= +a | +a oy | ;

Ov -
The value of 5 will be found from (4), viz.
JpT k

L
‘

2
7

2aVa (v—b)?

ES Es IRL py?
Ow pT are 1 2 &/..(v—b)?

RT 0

And since the denominator of this expression cannot become o,
(7) passes into

2 @/,(v—b)?\ Of Za Va (v—b)*) Of 4
1 — ——— 2 Sie 4 (7
( RT 0 Je T (8 RT 4 Ip pes rou

Now we have:

( 656 )

RU? in 6?(1— 2.x) —2a(1—«a)@ap—2a(v—b) B+ 2ay/a(v—b)?

df 3
EAD: = = 2a(1—.w)Ga + 2a(v—b) RAE
v 5

r . iQ Py 2 . € 6 nT / Br
where 9 is written for a + a(v—b), and p for */, vf.
The equation (7a) becomes consequently :

er

2aVa (p-b) ) Sp

e ai {|e c- 2x) + 2ay La (v- (or | = us

RT ot Sv
ade | z_ Zea (v- a 24), DIT
2 la (1—« a (wb = a =U),

+ 2 ja (1-a) Ga + a(v-6))) | ee RI
The expression between { | is obviously :
2V a), ea) 2Va/,, CEs,

— = (av—BYa) —

RT RT +

29
as av —BYa=a2+a(v—b)=6. Further we have R7=-—, in

EM

consequence of (6), so that we obtain:
a(v—by?
ae

1 v(vu—b)? ) 3
| 6? (1—2z) + 2a a (v —b) | = je SE ~ ! | 2 =

26 as a Ë (Le) Ga Ha cf = ou

And since p — a (v—b)? = a (1—a) 6’, we have, after multiplication
with go:

ale ; a(1—-a:)4? + a(v-b)?
a(1-x)6?| 6?(1-2.) 4+ 2aY a(v-b)* |—3—— -—

=== agaat

v

— 26 Wa (v-b)? za Oa Ha | =

In this expression the underlined terms vanish. And for
By —aYa.v(v —b)? may be written:

Ba (le) 6? — Va (v—b)? (av—B Va) = Ba ( (le) 6? — Ya(v—by? 6

so that we obtain, after dividing by 6, and multiplying by v:

z (1-2) 6 |e v — 3e (1-2) el + Wa (vb) |- 2av(v-b) + 32a (1-7) 6? —

rani 3 va Ba (1-2) 6 a 3a | = 0,

or finally:

,

| 1 Saal 1— 2.2) v—32 (1—2) d +Ya(v—b)? |» (L_a) HOB a)

| + a(v—b) ea | == hae alte cor (8)
where 6 — B /a may be substituted by ar — 23 Va.
This is consequently the sought equation of the 7, v-projection of

the locus of all the plaitpoints, which can appear on the y-surfaces
at different values of 7. Combined with (6), we find the points of
the surface, represented by (6), which satisfy the plaitpoint-condition,
that is to say the equation of the plaitpoint-eurve as space-curve.
Equation (6) may be written:

a

nt = [ras Hate | seth TG)
5

where thus 6 =a + a(r — b), and 7 =+), Va, — b, Va.
For + = (8) passes into
(L — 2x) b — 3a (1 — ee) B=0,
er 1 >
yielding «, = |e +1) —Vr+trt i], as we have deduced
5:
already above (in § 5) for that limiting-case.

To conelude, we remark, that the sections for constant volume of
the surface, given by (6), on/y extend down to 7 = 0 («=O and 1)
for v=b. For all volumes >> 5, 7 will assume for «= 0 and 1, as
: : : oes _ 2a(v — Bb)?
is obvious from (6), a finite value, viz. — EDS The 7'‚r-boundary-
curve suddenly ends then at the Z-axis at the designed value of 7
(also compare the space-representation).

The proper discussion of the equations (6) and (8) must be
reserved for a separate communication. It will appear then, that the
different forms of the spinodal- and plaitpoint-curves, which occur
specially in the case of anomalous substances, are already possible
in the case of normai substances, provided the proportion of the
two critical temperatures 7/7 be sufficiently large. The spinodal
curves, given by (6), will appear easily calculable, and as to the
course of the plaitpoint-curve (there are two, independent of each
other), some conclusions will be deduced in a simple way,

It will also appear, which indeed results already from (6), that
at least with respect to the

the longitudinal- and the transversal plaits
spinodal curves (compare also vaN per Waars, Cont. HU, p. 175) —
are no separate plaits, but one single plait, of which the plaitpoint
is lying, according to the different circumstances, either on the side
of the small volumes, or somewhere else,

( 658 )

Chemistry. — “On miscibility in the solid aggregate condition and
isomorphy with carbon compounds” [First communication].
By Dr. F. M. Janenr. (Communicated by Prof. H. W. Baknurs
RoozrBoou).
(Communicated in the meeting of March 25, 1905).

Since the discovery of isomorphy by MrrsenerrieH the power of
isomorphous compounds to form, on being mixed, a homogeneous
solid phase of gradually-varying composition has been experimentally
demonstrated in numerous cases.

In recent years several investigators have started theories as to the
course of the melting curves likely to be exhibited by such mixing-
series, and in conjunction with the theory of the equilibrium of
phases and with the aid of thermodynamical developments, a fairly
clear idea has been formed of the special cases which may be
expected to occur with substances of the said kind.

On the other hand, it is not permissible to draw conclusions as
to existing isomorphy, solely on account of the course of the melting
curve or the solubility lines. Since the introduction in chemistry
of the idea of “solid solutions”, many cases have already been
pointed out where amorphous or even crystallised solid solutions
exist of substances which bear either no or an unknown erystallo-
nomic relation to each other. We have only to think for instance
of amorphous glasses and on the other hand of the cristalline mix-
tures of ferric-chloride and sal-ammoniac. The difficulty is felt in a
particularly striking manner in the chemistry of the carbon compounds ;
not only do we know continuous series of crystalline mixtures
between morphotropously allied carbon-containing derivatives, as in
Murumann’s terephthalic-acid derivatives, but such mixing even in
the erystalline condition, has also been observed in the case of
organie molecules which have little or nothing in common.

Brunt and his collaborators, who have made a long series of eryos-
copic determinations conclude that the most dissimilar organic sub-
stances may yield “mixed crystals” and “solid solutions” of whose
erystallonomie relation not only nothing as a rule is known, but of
which the erystallographer will think the chances of isomorphy but
very small.

In any ease the relation existing between ““erystallonomie form-
relation’ and “miscibility: is as yet quite unknown. If substances
are isomorphous, that is if crystalline phases possess regulated mole-
cular structures, which may be assumed to be formed from each
other by a slight deformation, such phases may jointly yield a homo-
geneous mixing-phase of variable composition and their relations

( 659 )

based on the equilibrium of phases will take the course indicated by
theory. But the reverse is by no means the case and the question
as to the existence of “isomorphy”” can only be satisfactorily solved
by a erystallonomie investigation.

The problem has a particular interest in so far as it relates to the
determination of the limits in which morphotropously-related kinds
of molecules may exhibit such a miscibility. For the word “isomorphy”’
relates to a number of special cases in a series of much more general
phenomena of crystallonomy, namely, to those which show the rela-
tion between the chemical constitution of the substances which have
substitution-relations, with their innate erystalline form, which pheno-
mena are expressed by the name of morphotropy. If the chemical
relation of such substitution derivatives is confined within certain
limits such morphotropous substances may become “isomorphotro-
pous’ and will then be able to combine with each other in a
limited or may be unlimited proportion. And if the relation of such
substances has become so intense that a nearly identical property
must be attributed to their crystal-structures on account of a// their
physical properties, such isomorphotropous substances actually become
“isomorphous” and mixing is then always possible.

From the above it follows that the idea of “isomorphy’’ admits
of a certain gradation; only the crystallonomer can determine in
eacli case the degree of “isomorphy” by measuring the size indi-
cated by the parameters of the molecular structure in the cases to
be compared and particularly by studying the analogy in the cohesion-
phenomena of the crystal-phase. As the differences in the values of
the said molecular structure-parameters become smaller and a more
complete similarity in the directions of cleavage and nature of surface
of the similar limiting planes is found, a more complete isomorphy
exists and the probability of a complete miscibility in the crystallised
condition is at the same time enhanced.

It cannot be doubted that in the cases investigated by Brunt there
may be instances of such rea/ isomorphy and the following research
may even prove this fact. But it must also not be lost sight of that
many cases of miscible substances supposed to be instances of ““iso-
morphy” are only cases of isomorphotropy or even only of morpho-
tropy within relatively large limits.

All this renders it highly desirable to undertake an exhaustive
investigation of organic substances as to their miscibility in the solid
condition, coupled with that of their crystalline form so as to elucidate
the matter. The following research is a first communication on this
subject.

( 660 )

I. Nitro- and Nitroso-Derivatives of the Benzene-Series.

In consequence of some crystallographic facts, I intended long ago
to make a special study of morphotrophic action of NO, and NO
substitution in organie molecules. The matter became still more
important to me by the observation of the transformation of o- Nitro-
benzaldehyde into erystallised o-Nitrobenzoic acid under the influence
of light, and by a recent treatise of Brunt and Carrecart (Gazz.
Chim. It. (1904) 34. II, 246) who determined this formation of solid
solutions according to cryoscopic determinations. These investigators
arrive at the following conclusions :

1. As a rule aromatic nitrosoderivatives may form solid solutions
with the corresponding „itroderivatives.

2. In those solid solutions, also in the liquid ones which havea
green colour the nitroso-compounds have the simple molecular size.

It was particularly the first conclusion, which T wanted to submit
to a further investigation.

a. Paranitrodiethylaniline and Paranitrosodiethylaniline.

First of all I have extended the investigations paranitrodiethyl-
aniline by Senraur and myself‘). As I could only get proper erystals
of the nitroso-derivative from ethyl-acetate + ligroine it was necessary
to obtain the erystals of the /tro-devivative from the same solvent
in order to get strictly comparable preparations.

1. p-Nitrodiethylaniline.

C,H,.(NO,).N(C,H,), ; meltingpoint : 73°,6.

Crystals from ethyl-acetate + ligroïne. (Fig. 1).

Fig. 1.

=F 0
fl i a DES
ne on ; Af}
g bere
Sy Ar SJ) <<
fr Nn — f
| =n A
n' a mt 04
k 2
Dn
p-Nitrodiethylaniline, from

ethyl-alcohol.

p-Nitrodiethylaniline, from
ethyl-acetate.

1) Jarcer, Z. f. Kryst. 40. 127. (1905); cf ibid. 11, 105, Ref.

F. M. JAEGER. “On miscibility in the solid aggregate condition and
isomorphy with carbon compounds.” (1st Communication).

Figs 3:

Pleochroism of p-Nitro-diaethyl-aniline.

{100}

Fig. 5:

Pleochroism of p-Nitroso-diaethyl-aniline

Proceedings Royal Acad. Amsterdam. Vol. VII.

( 661 )

Brownish-yellow crystals, short prismatic or somewhat extended
towards the /axis and flattened towards {100} with a beautiful
reddish-violet reflexion on {LOL}.

Monoclino-prismatic.

lo) Res AOR sil pallets
B — 80°34’.

The measurements are identical with those formerly given by
me. The habitus of the erystals is, however, different from those
previously obtained from alcohol (Fig. 2): ¢ and give ideal
reflexes; a very good, but often somewhat curved; ¢ alone has a
violet reflexion ; the reflex is coloured light-lilac. Etch-figures on {100!
corresponded with the indicated symmetry.

Very completely cleavable parallel {O01}; fairly so towards {010}
with conchoidal fracture plane.

Optical behaviour. The optical behaviour of the compound is very
interesting. (Fig. 3).

First of all the crystals are strongly pleoehroie ; on {001}, sulphur-
yellow and blood red; the inflexion indicates the direction of the
vibration of the polarised rays; on {O10} yellow and orange; on
{100} yellow and orange-yellow. On {O01} are seen in convergent
light two red absorption hyperboles like the opening arms of an
axial cross.

On {100} an eccentric axial image is visible, the axial angle is very
small so that there is apparently present the image of a monaxial crystal
with a crossing of the axial planes for red and green light. The axes for
the red lie in a plane | b-axis, for the green in one | c-axis. The angle
for the red rays is smaller than that of the green rays. The double
refraction is positive; the first diagonal inclines to the normal 77
the plane of symmetry ; on {100} there are, therefore, at the same
time an inclined and a horizontal dispersion.

The angle of inclination is somewhat larger than in the ease of
the nitroso-derivative.

2. p-Nitrosodiethylaniline.

Some time ago') I made some preliminary communications on

paranitrisodiethylaniline C,H,(NO) N(C,H,), melting point 82°,2. I
(4) * (1)

have since succeeded in obtaining this compound in a form more

1) PF. M. Jarcer. Ueber morphotropische Beziehungen hei den in der Amino-
Gruppe substituirten Nitro-Anilinen. Z. f. Kryst. 40 (1904) 112—146.

( 662 )

suitable for investigation, so as to be able to make a clear comparison
with the corresponding para-nitvo-derivative.

The said crystals had the appearance of small emerald green
rectangular plates, which were most readily obtained from acetone.
They are very poor in combination forms and only exhibit @ = {100}
predominating, m — {110} and 5 = {0104 whilst « = {100} is generally
present in a rudimentary condition. They were determined as mono-
clino-prismatie with 2 = about 85°53" and a:b —10166:1; on
{100} an interfering image is visible with slight inclination to the

normal on that plane; the axial angle is small, the dispersion an
average of o<v round the sharp bissectrix, the double refraction
is positive.

More accurate investigations have, as will be shown, practically
confirmed these data. I obtained the improved material from a mixture
of ethyl-acetate and ligroine, in which the compound was dissolved
on warming. On slow evaporation small rectangularly bounded,
thicker plates or also emerald green prisms are formed, which on some
planes possess a splendid violet reflexion. They exhibit a particular
lustre and but rarely a curvature of {100}; therefore very accurate
determinations could be executed.

The symmetry is monoclino-prismatic: the axial relation:

a2 bse = 1201738 teal 96d
me ay

Forms observed: a = {100}, broadest
developed of all and well reflecting, some-
times a little curved; m={110; and 6 {010},
about equally developed, but 6 generally a
little broader, although sometimes mueh
narrower than m and much sharper reflec-
ting with light-violet reflex; ¢ — {001}, with
ideal reflexion also with a violet reflex
image ; 7 = {102}, lustrous, mostly narrower,
sometimes a trifle better developed and
sharply reflecting; the reflexes are often
violet, mostly, however, colourless or yel-
lowish; s = {101}, broad but very dull and
only approximately measurable ; {J .0. 13},
very narrow in vicinal form has been

observed in a few cases.

p-Nitroso-diethylaniline.

The habitus of the crystals is elongated
along the vertical axis with flattening towards {100}.

( 663 )

Calculated .
Gc (EO —
Qn — OO EN Is —-
c:r = (001) : (102) =*45 59'/, ie
r:a —(102):(100)— 48 19 48°177/,'
DR (100) (4101) 267 39 20829
Se —— (101) (O01) == 59) 4. 59 14
Gm (OO (LOST SL 5 87 0
m:b =(110): (010) = 44 35 44 54/,

mar — (110): (102) — 6244 62 92/,
ns — (10): (OLS 52 0 (arca) 51. 5

The crystals cleave very completely towards {O01} with a lustrous
separation plane; incompletely towards {010} with a conchoidal fracture.
Etch figures were not obtained.

Optical behaviour. The compound exhibits interesting optical
properties.

First of all the splendid, violet reflexion of the planes {O01},
{010} and {102}, which is wanting on the other planes. This reflexion
is not due to a streaking of these planes. If we cleave a crystal
along c or 4 the plane of separation also has that reflexion and a
streak made with the crystals on porcelain also exhibits the same
phenomenon. The light reflexes of the goniometer lamp on these
planes are coloured a beautiful pale lilac; on the other planes white,
sometimes yellowish.

Further, the compound exhibits on {100}, {O10} and {O10} the
pleochroie behaviour as seen in Fig. 5; the inflexion again indicates
the direction of the vibration of the two polarised rays, which arrive
along the normal on the respective plane. On {100} a difference is
only observable with thicker crystals, on {O01} the colours are light-
green and dark, somewhat bluish-green; on {O10} the difference is
most pronounced, namely light-green and dark violet; the latter colour
is indeed, as I noticed, no surface reflexion but the colour of the
phase in transmitted light. On {001} feeble absorption bundles of
hyperbolic form are observable resembling an axial cross opening
when the table is turned.

The optical axial plane is nearly horizontal; the acute bissectrix
makes a small angle with the normal on {100}. In convergent light
a very fine axial image is visible with a small axial angle and an
average dispersion of rhombic character; the dispersion of the said
diagonal is e <r, the double refraction is positive.

Calling the erystallonomie axes respectively a,b,c the optical

( 664 )

orientation of the elasticity-ellipsoid is therefore in the same order:
y,8,@ in which «>> g> y. The double refraction is feeble.

On {100} and {O01} orientated extinction ; on {010} the very small
inclined angle of the elasticity axis with the vertical axis could not
be sharply defined on account of the strong absorption; it is not
distinguishable from 90°.

The specific gravity of the crystals was found to be 1.240 at 15°;
the equivalent volume is, therefore, 143.53 and the topical axes become:
4: Ww = 4,2368 : 4,1623 : 8,1626.

The complete isomorphy of p-Nitrodiethylaniline and p-Nitro-
sodiethylaniline is therefore firmly established. The properties of both
substances are given here by way of a comparison.

p-Nitrodiethylaniline : p-Nitrosodiethylaniline :
Monoclino-prismatie. _ Monoclino-prismatic.

a:6:¢ =1,0342: 1 USS Deer te OM Sree 21,9611.
yi: @ = 4,4276 :4,2807 :8,4710. | pip: @ = 4,2363 :4,1623 : 8.1626.

di | d = 1,240
y = 158,36 | v = 143,53
Angular values : | Angular values:
(110) : (100) = 45° 34’ (110) : (100) = 45° 254’
(100) : (402) = 51° 13’ (100) : (102) = 48° 174’
(100) : (001) = 80° 34’ (100) : (001) = 85° 43’
In ethyl-acetate + ligroine the In ethyl-acetate + ligroine the

habitus is flattened towards {100} habitus is flattened towards {100}
elongated towards the c-axis; some- and elongated towards the c-axis.
times towards the D-axis.
Very completely cleavable towards | Very completely cleavable towards
{001}, fairly so towards {010}, with | {O01}, fairly so towards {010} with

a conchoidal fracture. conchoidal fracture.
On {001} violet reflexion. On {O01}, and on {O10}, {102}
violet reflexion.
Optical orientation : 7, By a. Optical orientation: y, 8, 4,
Double refraction, positive. ~ Double refraction, positive.
On {100 a but little-inclined On $100) a but little-inclined

axial image with small axial angle; axial image with small axial image;
axial plane parallel the /-axis for axial plane parallel the b-axis;
the red, parallel the c-axis for the rhombic dispersion : 9 Sv.
green rays; dispersion: @ << v.

Colour : brownish-yellow. Colour: emerald green.
Strongly pleochroic: blood red- Strongly pleochroic: violet-pale
orange-yellow. ereen-bluishgreen.

On ¢ absorptionbundles. |___On c absorptionbundles.
i p

( 665 )

There is not the least doubt that the two substances possess a
quite analogous structure ; cleavability, optical orientation whilst the
nature of the surface of the crystalplanes is quite in agreement.

From mixed solutions of the two components are formed small
ereenish-black mixed crystals with a vivid steel-blue reflexion.

As generally happens in the case of most isomorphous mixtures, the
crystallisation power is considerably smaller than with each of the
components separately. Under the microscope such mixed crystals
consist of thin olive-green little plates, which on their predominating
plane show little or no pleochroism. In convergent light a splendid
interfering image may be observed: slight inclination to the normal
on the horizontal plane, elliptical rings, and small axial angle, larger
however than in the two components. The double refraction is
positive: the dispersion has a rhombic character and shows: Ome

From the last motherliquors are deposited mixed crystals of a
lighter shade representing silky needles as those above with less
surface reflexion. Otherwise they are optical continuations of the
above described mixed crystals. From mixtures of the two components
in a melted condition these mixed crystals depose on the sides of
the testtube in a fine steel-blue lustrous condition.

The behaviour of the two isomorphous substances in the liquid
condition is elucidated by ihe investigation of the melting curve of
binary mixtures. On account of the dark colour of the fusion, the
course of the solidification curve was traced by the graphical method;
the determinations were made as usual in the van Eyk apparatus.
It should be noticed that all these fusions solidify to solid phases,
which also exhibit a splendid violet or blue reflexion.

The nitroderivative has a greater latent heat of fusion than the
nitrosoderivative ; in both cases the calorie effect was, however, very
readily observable in the solidification. The lower solidifying line
can by no means be determined so sharply as the upper one.

It was found that:

A mixture of 100"/, of p-Nitro and 0°/, p-Nitroso-derivative
melted at 73°,6.

A mixture of 85,14°/, p-Nitro- and 14,86 °/, p-Nitroso-derivative
commences to solidify at 75°,2 and completely solidifies at 74°,9.

A mixture of 72,5°/, of p-Nitro- and 27,5 °/, p-Nitroso-derivative
commences to solidify at 76°,2 and completely solidifies at 75°,9.

A mixture of 54,4 °/, of p-Nitro- and 45,6 °/, of p-Nitroso-derivative
commences to solidify at 77°,6 and completely solidifies at 77°,3.

A mixture of 38,64 "/, of p-Nitro- and 61,36 "/, p-Nitroso-derivative

750%
9.

commences to solidify at 78°,2 and completely solidifies at 77

( 666 )

A mixture of 10,0°/, p-Nitro- and 90,0 °/, of p-Nitroso-derivative
commences to solidify at 80°,8 and completely solidifies at 80°,6.

A mixture of 0°/, of p-Nitro- and 100°/, of p-Nitroso-derivative
melted at 82°,2.

The composition is given in molecule-percents.

0

In fig. 6 the course of the melting curve is represented graphically
and the double line for the initial and final solidifying points is
shown. It will be seen that the character of the line points to a
continuous series of mixed crystals; the average temperature-interval
between initial and final solidification amounts to about 0°,5.

400 90° 80 70 60 50 40 300 MKO
Fig. 6.
The result of the research reveals the complete isomorphy of

p-Nitro-dethylanaline and p-Nitroso-diethylaniline and also their
complete miscibility in the solid state.

Chemistry. — “On Orthonitrobenzyltoluidine’. By Dr. F. M. JARGER.
(Communicated by Prof. A. F. HOLLEMAN).
(Communicated in the meeiing of March 25, 1905).

Some time ago the o-Nitrobenzyl derivate of para-toluidine, (melting
point 72°C.) was investigated by NORDENSKJÖLD, who described it as
being tetragonal with the parameter-relation a:c = 1:0,6230; the
compound exhibits only one combination-form, namely {111} and is
optically monaxial: positive. (Bull. Geol. Instit. Upsala, (1892), 84,
also Ref. Zeits. f. Kryst. 24, 147).

( 667 )

For comparison I have investigated the o-Mitrobenzyl-derivative
of orthotoluidine *).

Recrystallised from acetone in which the compound (m. p. 96°) is
very soluble, the substance forms very large, transparent, pale yellow
or rather pale greenish-yellow crystals possessing a strong lustre and
assuming a more brownish tinge on prolonged exposure to the air.
Elongated, prismatic needles are also occasionally obtained.

The first-named crystals are nearly isometrically developed and
possess many combination forms; they admitted very well of accurate

measurements.

Rhombic-bipyramidal.

The parameters are:
a@:6:¢ =0,8552 ; 1 : 056138.

o-Nitro Benzyl-o-Toluidine.

Forms observed: g = {021},
strongly predominating and lus-
trous; 0—={211}, broadly developed
and yielding sharp _ reflexes;
7={101}, well formed and lustrous;
c = {001}, narrower but well
reflecting; 6 — {010}, dull some-
times present with only a single
plane; mostly a little broader than
c, but also somewhat smaller ;

=)

s = {201}, narrow and unsuitable

for measurement; the symbol has been deduced from the zone-relation.

The habitus is mostly thick-prismatie along the a-axis; particularly
the smaller crystals possess a very regular form.
Measured : Calculated :

b:

: 0

== OO1)
(OOH)
q= (010
i (aol
n=
== (dt
== (Al:
:0= (021
p= (O2
: 0 = (010)

: (021) =*50° 50' =
: (101) =*35 40 En
VOA EE 8) 39° 10
: (101) = 27 20 27 10
: (217) = 65 8 65 18
: (001) = 57 26 bad
(211) = 38 46 38 40
: (211) = 53 167/, 53 18
:(101) = 59 1! 59 8
: (2441) = 70 38 70 40

Completely cleavable parallel {021}, distinctly towards {211}.

1) Ber. d. d. Chem. Ges. 25. 3582.

46

Proceedings Royal Acad. Amsterdam. Vol. VII.

( 668 )

In oil of cloves as immersion liquid the situation of the elasticity
directions on the planes of {O10}, {021} and {001} orientated normally
in regard of the a-axis. The optical axial plane is {100}; the first
diagonal stands perpendicularly on {010}. On the planes of {021} a
brightly coloured interferential image is visible in convergent polarised
light; extraordinarily strong dispersion of a rhombic character with
0 >v around the first bissectrix. In oil of cloves the apparent axial
angle amounts to about 49° for the red and 46° for the green vays.

The oil caused on {021} little solution-figures, which had the form
of isosceles trapezia; they agree with the indicated symmetry of
the crystals.

The specifie gravity is 1,278, at 15°, the equivalent-volume is
189,28, and the topical axes are:

Y:W:Ww = 60875: 7,1175 ; 4,3688

Although differing from NorDENSKJOLD’s para-derivative in symmetry,
the analogy of the two isomers is still distinctly recognisable in the
value of the relation 6: c.

o- Nitrobenzyl-para- Toluidine: a:b:e = 1,000 : 1: 0,6280.
o- Nitrobenzyl-ortho-Toluidine : a:b: e = 0.8552 : 1 : 0,6138.

The difference in position of the methyl- and amino-group with
regard to each other therefore causes chiefly only a variation of the
crystal parameters in one direction.

Chemistry. — “On position-isomeric Dichloronitrobenzenes.” By
Dr. F. M. Janeer. (Communicated by Prof. A. F. HOLLEMAN).

(Communicated in the meeting of March 25, 1905).

Of the six theoretically-possible dicbloronitrobenzenes, which 1
received” some time ago for investigation from Prof. HoLiunman, |
succeeded in obtaining four in such a measurable form that their
erystallographieal determination could be satisfactorily undertaken.

Notwithstanding the great power of crystallisation of most of them,
the preparation of properly developed crystals is a troublesome and
very tedious matter. This is partly due to the very great solubility
in most of the organic solvents, which in connection with the low
melting points of these compounds often causes a not inconsiderable
supersaturation. During the spontaneous crystallisation, which then
takes place, no well-formed individuals, but erystal-aggregates are
formed, which are difficult of investigation. In addition, the peculiar
softness of the crystals causes most of them to exhibit curved planes
and considerable geometrical deviations. Again, owing to the heat

( 669 )

of the source of light during the measurement the erystals soon
beeome a dull surface, so that the inaccuracy of the measurements
is still further increased by the less sharp limitation of the signal
reflexes.

Of the substances examined the ortho-dichloroderivatives are both
rhombic, the mefa-derivatives probably all monoclinic and the para-
dichloroderivative triclinic; only the geometrically well-defined sub-
stances of this series are described here in detail. In crystalline
form they show comparatively little resemblance to each other,
chiefly in consequence of the considerable deformation of the molecule
owing to the mutual attraction of the Cl-atoms and of the (NO,)-group.

Fig. 1. a. 1-2-Dichloro-3-Nitro-Benzene.
C,H,.Cl.Cl.(NO,) ; melting point : 61° a 62° C.
(1) @) (3)

This compound erystallises from a mixture of
ethyl-acetate and ether and also from glacial
| acetic acid, on very slow evaporation of the solvent,
| NE en in colourless silky needles, which are limited by
IE small, lustrous pyramidal planes (Fig. 1).

I
i
|

ed
|
|

: li il i Rhombic-bipyramidal.
el 14 a:b:¢ = 0,6472 : 1 : 0,2780.

Forms observed : a= {100} and 6 = {010} equally
ie: strongly developed and both very lustrous;
EENS ii] p= {230}, m==fl10}, n= {430}; the latter form
Py edi is the smallest of the three and reflects less sharply

than p and m; « sometimes shows a delicate
streak parallel with o:a; 0 ={133! lustrous,

1-2-Dichloro-3-Nitro. Yielding good reflexes.

Benzene. The vertical zone is, geometrically very well
constructed. The angular values observed in different erystals differ

but ineonsiderably from the average values.

Measured: Calculated:
aop = (1.00) (230) 444° 97 —

ORO (IE: (133) ko Es =
Pan dd Op 2 11° 14’
met (AO) (430) ==) 7 Tk (eral
ee (Len ONO} 2 27 35!/,
Dor @00) TAS INS 58246 Sd
6:30 (OLO: 403) = 74:36 74 34
0:0 5

USS wd 15 46

Readily cleavable along 0. On m and p right-angled little etch-
figures are visible in cassia-oil, which correspond with the indicated
symmetry. In the vertical zone the direction of the optical elasticity
axis is orientated on all the planes. Anaxial image was not observed.

The specifie gravity of the needles as determined by means of a
solution of mercuric-potassium-iodide was 1,721 at 14°. The equi-
valent-volume is therefore 111.56 and the topical axes become;

y:W:w — 5,5190 : 85272 : 2,3706..

b. 1-3-Dichloro-2-Nitro-Benzene.

CAH Cla Cig(NOs
(1) (3) (2)
Fig. 2a

1-3-Dichloro-2-Nitro-Benzene.

’

melting point: 71° C.

The compound ery-
stallises from carbon
disulphide in large,
colourless, thin plates
of parallelogram shape
or also in smaller thick
crystals as shown in
figs. 2a and 25. The
crystals are often
opaque and difficult
to measure ; sometimes, how-
ever, they are more lustrous
and very clear.

Monoclino-prismatic.

a:b :c=0,6696 : 1 :0,4149.
p= STROE

Forms observed : a =}1L00}
generally strongly predomi-
nating and always sharply

reflecting; g = {011}, lustrous and either quite as narrow as o or
else the broadest developed of all, so that the crystals appear short-
prismatie towards the clino-axis; 0 = {111}, generally small, mostly
streaked parallel with a:o and reflecting rather dullisly ; 6 = {010},
very small and often only present in a rudimentary form.

Measured : Calculated :
a:o = (100): (441) =* 58°44’ —
gag (OLO —
arg OO OIS ES Ae —
ong (OLD =S 127 55 128°25/
Oni — (ela) Or) — PAS Ae
a= b= (100) (010) = 90 0
o:0=(111): (111) = 38 38 (about) 38 16

A distinet cleavability was not observed.

The erystals deposited from acetone, which were very large but
dull, show a predominance of a over 4; they are much elongated
along the vertical axis and further possess a form which is probably
{233} with (233) : (100) = 67°33’, calculated 67°24’. On a there is
diagonal extinction; the optical axial plane is {010}. One optical axis
descends almost perpendicularly on a.

The specific gravity is 1,603, at 17°, the equivalent volume 119,77.

Topical axes: 7: wy: w = 5,0596 : 7,5561 : 3,1350.

Although the parameter-relation a: and the angle 8 in this isomer
are comparable with those of the 1-2-3-derivative :

1-3- Dichloro-2-Nitro- Benzene: a:b = 0,6696:1; 8 = 87° 52’
1-2- Dichloro-3-Nitro-Benzene: a:b = 0,6472:1; @=90°.

ent
their crystalline forms are still rather different; the relation — of
c

the latter substance is about 1°/, that of the first derivative.
ce. 1-3-Dichloro-5-Nitro- Benzene.
C,H;.Cl.Cl.(NO,) ; melting point: 65° C.
(l) 5

In alcohol or glacial acetie acid, in which solvents the compound
exhibits a remarkably great crystallisation power, there are generally
formed very long, flat columns of considerable thickness, or also
right-angled or obtusely truncated pale-sherry coloured small thin
plates. Owing to the great softness of the substance and its great
plasticity, the erystals are in most cases so ill-formed and distorted
that measurements become impossible. With very slow evaporation we
sometimes get better formed crystals although they are very poor in
planes. They have a peculiar odour resembling nitrobenzene.

Monoclino-prismatic.

a:6=0,5940:1 ; B= 58? 43’.

Forms observed: « = {100}, broad and very lustrous; 5 = {010},
narrower and less lustrous; it is often absent altogether; m — {110},
narrow and c= {O01}, small but very reflecting; the habitus is
elongated along the c-axis and then flattened {100}.

Measured: Calculated:
beam (OLO GLO) * 63.5

GG (100) (O01) =" 58543 ==

a:m= (100): 110) = 2658 26°55!
m:m = (110): (110) = 126 15 126 10
me = (410), (OO — 62 25

GOS

h:¢ =(010):(001) = 8957 90 0

The crystals are completely cleavable along {010}, readily so
along {O01}.

On $100} extinction occurs on orientation; on {O10} under 28° with
regard to the vertical side. The optical axial plane is {010}; at the
border of the vision-sphere an optical axis is visible on {100}; the
axial angle is small. The direction of the vertical axis is here the
axis of the greatest elasticity.

On $100} etch exerescences were observable with a circumference
of isorceles trapezia, whose angular points appear to be connected by
straight lines with a point situated in the centre; this point lies
nearer to the smallest than to the largest of the two parallel sides
of the trapezium. They agree with the indicated symmetry.

Fig. 3: The specitie gravity is 1,692, at 14°C.;
the equivalent volume is, therefore, 113,4.

d. 1-4- Dichloro-2-Nitro-Benzene.

C,H,.Cl.Cl.(NO,) ; Melting point: 54°,5 C.
4) (1 2

In most solvents this substance shows
a very great crystallisation power, but
measurable crystals are but rarely
obtainable, as most of the individuals
exhibit important geometrical deviations
on account of the great softness of the
material and often possess curved and
very dull planes.

Some time ago the crystal form was
incompletely determined by Bopnwie; he

1-4-Dichloro-2-Nitro-Benzene. investigated crystals deposited from car-
bon disulphide but did not succeed in obtaining combinations admit-

ting of a complete determination of the crystal parameters (Zeits. f.
Kryst. 1. 589; Ann. Ch. Phys. (4). 15. 257).

From acetone I always obtained the largest crystals, sometimes
some centimetres in length; they are quite of the prismatic type of
the crystals investigated by Boprwie and possess in addition a lateral
prism; they exhibit, however, such considerable deviations and are
generally so opaque that an accurate measurement is out of the
question.

I sueceeded best by erystallisation from ethyl-acetate mixed with
a little carbon-tetrachloride; the pale sherry coloured crystals flattened
towards {100} so obtained, are very well formed and admit of accurate
measurement.

( 673 )

Triclino-pinacoidal.

a:b:¢=0,8072-: 1 : 0.8239

Ai TE sist n= SPAN
hldk Bik D= il 5
C= 58° 35! n= (HOE

Forms observed: «a — {100}, predominant, well-reflecting, better
than 6 = {010}, which form is also narrower; ¢ = {OO}, very lustrous
and well developed ; #7 = {110}, narrow but well-reflecting ; ¢ = {011},
narrow very lustrous; = 101}, somewhat broader and yielding
good reflexes.

The crystals are flattened along a and elongated in the direction
of the c-axis.

Measured : Caleulated :

mer (LOO) (O01j— * Gb: B/L —-
G20) = (100) =(O10) = 191 95 ==
b:e = (010): (001) = *100 27 =
a:r = (100): (401) =* 50 12'/, En
Giga (OO. (O11), = * 45,39 —
q:6 =(011):(010)= 5444 54°48’
c:r =(001):(101)— 64 40, 64 40"/,
m:b = (410):(010) = 75 23 75 19
m:a = (110):(100) = 46 2 46 6
P01): (01d) == 5150 51.35
m:r = (110):(101)= 65 36 65 22
m:q —(110):(011) = 62 54 63 3

The crysials are very completely cleavable towards {001}; th
plane of cleavage is very lustrous.

On {100} obtuse-angular extinction; its amount is small, only about
7°40' in regard to the vertical side; in convergent light a dark
hyperbole is notieed on this plane.

The specific gravity of the crystals is 1,696 at 12° C.; the equi-
valent volume is, therefore, 113,20.

The topical axes are y:wW:w = 4,8484 : 6,0065 : 55,1422.

( 674 )

Physics. — H. KaAMrRLINGH Onnes and W. Hruse. “On the measu-

rement of very low temperatures. V. The expansion coefficient

of Jena and Thüringer glass between + 16° and — 182° C.”

Communieation N°. 85 from the Physical Laboratory at Leiden.
(Communicated in the Meeting of June 27, 1903)

§ 1. At Leiden the hydrogen thermometer (cf. Comm. N°. 27
May ’96) is taken as the standard for very low temperatures.
To reach the degree of accuracy otherwise obtainable with this, it is
necessary to know the expansion coefficient of Jena glass 16™ to
about 1°/,. Hence we have determined the two coefficients in the
quadratic formula assumed for the linear expansion of glass below
0°’ C. At the same time we have, in precisely the same circumstances
made a similar determination for the Thüringer glass, from which
the piezometers mentioned in Comm. N°. 50 (June 99), N°. 69 (April 01),
and N°. 70 (May ’O1) were made, in order to be able to calculate
and apply the correction for expansion to the results attained with
these piezometers.

Some time previously we made measurements on expansion coeffi-
cients, among others on platinum. The value for this metal was
required for the reduction, from the measurements mentioned in
Comm. N°. 77 (Febr. ’02), of the galvanic resistance at low temperatures.

But the results which we have lately obtained for the two above
mentioned kinds of glass appear to us to be the first that are worth
to be published; the final reduction of the measurements named
above was postponed till the required accuracy was reached. However
the measurements on platinnm must be repeated.

Although the field of measurements at low temperatures is hardly
touched, still we consider that in this field preliminary and approx-
imate values are worth little. In the majority of cases approximate
values of this kind can be obtained by extrapolation, and thus only
those determinations which are accurate enough to allow a judgment
on the question whether such an extrapolation is allowed or not,
are really of use in advancing our knowledge. We have hence
arranged our observations on the expansion coefficient so as to reach
an accuracy of 200:

For general the investigation of expansion at low temperatures it
will be required to determine on the one hand the linear coefficient
of solids and on the other the absolute coefficients for those substances,
which remain liquid to very low temperatures, e. g. pentane, in such
an hydrostatic manner as DuLone and Prrit’s (improved by RrGNAULT).
The determination of the relative expansion of the liquid chosen can

( 675 )

then serve as a control and as the starting point for further measure-
ments. The present investigation forms the first part of this general
program and gives the linear expansion of glass with an accuracy
which suffices for our present purpose. From the description of our
measurements it will be seen that with practically the same apparatus
and in nearly the same way it will be possible to determine the
absolute expansion of pentane.

§ 2. We have determined the two coefficients a and 4 in the
formula for the linear expansion = L, (1+ at + bt’), for the
two varieties of glass from three observations for each. These
were made at ordinary temperature, at about — 90°, C. and at about
— 180° C., by measuring directly and at the same time the lengths
of the rods of the two substances.

The rods were drawn out at each end to a fine point which could
be accurately observed with a microscope. At the bottom and top, the
two rods project out of a vertically placed cylindrical vessel. The

bath is closed at the lower end and is filled with a liquefied gas giving
the required temperature. Care is taken that the points shall be kept
as nearly as possible at the temperature of the surrounding air, and
also that the air between the points and the objective of the microscope
shall be at the same temperature. The lengths are then read directly
against a scale by a cathetometer arranged as a vertical comparator.

Although this arrangement gives a convenient method for the deter-
mination of length it necessitates a considerable difference in temperature
between the middle and the ends of the rods. To correct for this, use is
made of the method employed in Comm. N° 83 (Febr. ’03) for the deter-
mination of the corrections along a piezometer or thermometer stem.
This depends upon the use of a uniform platinum wire wound uniformly
round the rod. Its use depends upon the assumption, that the change of
resistance of a wire wound in this manner is nearly proportional to the
mean change of temperature of the rod. This will be further considered in § 4.

After this general view we may consider certain details.

1st. The glass rods were about 1 m. long and had diameters of
5 mm.'). Round these 0.1 mm. thick platinum wires were wound
spirally and soldered to brass rings A, B, C, D (PLL fig. 1.) which
were tightened by screws.

Between B and C, the part which was immersed in liquified gases,
there were 140 turns with a pitch of about 0.5 em. Between A and B
or C and D where the temperature changes rapidly there were 25
and 40 turns respectively with a pitch of 0.25 em. Care is taken

1) A platinum tube provided with glass ends similar to those described above
was used for the determinations on platinum.

( 676 )

that the pitch remains constant in each section A to 5, B to C, or
C to D. At A, B, C, and D platinum wires a, b, c, d, e, f,g, and
h about 15 em. long and 0.5 mm. thick are soldered in pairs. At
the other ends they are connected to copper wires. In order to pre-
vent faults in insulation the spirally wound wires lay in shellae they
were also covered with a layer of tissue paper for purposes of pro-
tection. The portions A to B and C to D were enveloped in succes-
sive layers of fishglue and writing paper to about a thickness of 0.25 em.,
in order that the distribution of temperature should be as even as
possible along the rod. This protection was found to be proof against the
action of either liquid nitrous oxide or oxygen. To allow of contraction
on cooling the paper layers were only pasted together at both ends.

2d. The cylindrical vacuum jacket. The bath for the liquid gases
has the form of a tubular vacuum glass. Usually vacuum glasses
are made so that there is but one edge connecting the cooled and
uncooled walls. When it is necessary to remove liquid at the bottom of a
vacuum glass the lower surfaces are connected by a spiral tube. However
we required something quite different i.e. a double-walled tube open at
both ends and capable of holding a rubber stopper in one. Ifsuch a vacuum
tube were made by blowing simply together inner and outer walls
it would certainly crack when cooled, owing to the different expan-
sion of the outer and inner walls. Also it did not appear to be pos-
sible to make the outer wall sufficiently elastic by blowing several

spherical portions in it (see fig. 1).

EE Awe eee ee Eee
TI
Fig. 1.

Hence the outer wall was divided by a thin brass case V,, PI. I,
which allows a compression or expansion of 2 mm. This copper
box was inserted by platinising and coppering the two glass surfaces
and then soldering them to the copper box. The vacuum tube thus
produced was silvered and evacuated in the usual manner. In the
first arrangement the top was left clear in order to allow of the observation
of the surface of the liquid. In later arrangements we preferred a float,
Such tubes with compound elastic walls appeared to be suitable for our
purpose and will probably also be found to be useful for the solution of
various other problems. An example of how easily tensions arise which
cause such glass apparatus to crack, was found when the rubber
stopper at the bottom was pushed in too far. On admitting the liquid
oxygen the rubber became hard before it had reached the temperature of
the liquid, which temperature the glass immediately above had reached

On the measurement of very

V. The expansion coefficient of Jena and Thüringer

H. KAMERLINGH ONNES and W. HEUSE.
glass between + 16° and — 182° C.

low temperatures.

Proceedings Royal Acad. Amsterdam. Vol. VII.

( 677 )

already, and the lower rim cracked off. Later we made the connec-
tion tube and the stopper more elastic (cf. H Pl. I fig. 2) by inserting
between them a collar formed of several layers of paper glued together
at the borders. In this way a closure was obtained which was
perfectly tight, a quite necessary item, for otherwise the escaping
liquid streams past the reading points as a cold vapour, which
disturbs the uniform distribution of temperature supposed to exist
in the ends of the rods and obtained by continually blowing air
on to the points which is necessary also for keeping them dry.
At the top, the rods are supported sideways so that no strain
is caused in them. They are protected from the cold vapours which
arise from the bath. From the front and side elevation of the
upper end, Fig. 1, the arrangement of paper used for this protection
can be clearly understood, and the course of the vapour can be
followed as it streams over the wall of the bath through channels
of cardboard. This arrangement has moreover the advantage, that
the outer surface of the vacuum vessel is also cooled. This is of
great importance in the beginning. The cold gas and cooled air are
so conveyed away by various paper screens, that they do not come
into the neighbourhood of the cathetometer or the standard scale, and
also that air at the ordinary temperature remains between these and
the points. At the commencement the liquefied gas is introduced in drops
through an opening in the cork at the upper end, and afterwards
carefully in small quantities. When the bath is once full, fresh liquid
is continually added in small quantities to keep the level at the
same height. The liquids used were nitrous oxide and oxygen obtained
in the manner described in Comm. No. 14 (Dec. °94) and No. 51
(Sept. 99). In both cases considerable purity was aimed at, in consequence
the temperature of the bath did not change during the measurements.
There is no doubt that the temperatures at the top and the bottom
of the bath were not the same but this introduced no difficulty
since in the calculation only the mean temperature as determined
by the platinum resistance was required.

3, The comparator (cathetometer and scale). We used the
instruments which are described in Comm. No. 60 (Sept. ’00). The
scale was very carefully enveloped in wool and paper to protect it
from changes of temperature. Its temperature was read by two
thermometers divided into */,, and symmetrically placed above and
below, while the room temperature was maintained as constant as
possible. The telescopes were provided with the microscope objec-
tives which had been used for the measurements on the viscosity
of liquid methyl chloride (Comm. No. 2, Febr. ’91) and which

( 678 )

can be used at a distance of 10 cms. In this case one revolu-
tion of the head (divided into 100 parts) of the micrometer screw
(ef. Comm. No. 60 § 15) was equivalent to 60 to 70 u. The levels
on the telescopes were carefully calibrated; at the distance used, one
division on the levels corresponded to from 4 to 6 w and the uncer-
tainty in reading was less than 0.2 division or about 1 u. After each
setting, 80 seconds was allowed to elapse before reading and former
measurements have shown that this is sufficient for the attainment of
equilibrium.

The field of view of the microscopes was also investigated by
measuring at various points a '/, mm. scale, but no irregularity could
be found.

4th, Measurement of resistance. The doubled conducting wires
a, 6, c ete. at the ends of each measuring wire AB, etc. (cf. Pl. I)
were lead to eight cups of mereury for each rod, which cups could
be connected in pairs to the wires from the Wueartsronr bridge.
sy measuring

w,=a+AB+b
w,=—ce+ AB4+d

wadde
I= bd
the resistance of the wire AB
w,+w,—w,—w,
Win -
9
i

can be determined *). The galvanometer with reading scale (see
Comm. N° 25, April ’96) had a resistance of 6 w and a sensitiveness
of 2.5 > 107. Thermoelectric forces in the circuit of copper leads,
platinum leads and platinum resistances are unavoidable, they were,
however, onty small and could be eliminated.

§ 8. Survey of a determination. A complete determination com-
prises focussing the microscopes, referring to the standard scale,
and reading the thermometers, as well as the various determinations
of resistance between A and Bb, B and C, C and D.

In the following table all the readings for the determination of length
of the Jena rod in liquid oxygen are given. Column A contains the
readings of the micrometer heads, 6 the corresponding positions of
the levels, C the nearest division on the standard scale, D and H#
the micrometer and level readings for this and / the temperatures.

1) In our case the influence of the shunt between A and B, C and D was so
small that it could be neglected and then ws + w4 could be determined at once.

( 679 )

TABLE I. JENA GLASS.

| |
95/5 “03 | A po 23 Ge Ai D 8 F |
| | | |
1430! | | | |
(Point below | 97.82 | 6.1 | | | 16.14
| a | | | “146 34.93 | 5.2
| Millimeter | | 447 90-07 Bd
| |
(Point above | 49.44 Gelb | | 16.70
|
| Re | | 127 | 33.33 | 5.8 | |
| Millimeter | | 4498 17.44 58 |
| | A |
Point below RSS (620) 16.54
| |
| „ above | 19.47 | 6.0 | 16.80
| 1445! | | |

The readings on the micrometer head are now reduced to a
standard position of the level and the temperature readings are
corrected. This gives the following.

TABLE II. JENA GLASS

25/5 703 A’ ile | Cay EAD
| | |
1730 |
Point below | 27.81 | | 16.37
| ‘ an |
Wi = 34.30
Millimeter 7 | 90.45 |
| | |
‚Point above | 49.43 | 16.63
=e : 1127 33.35
Millimeter 1198 | 17:16 |
\Point below 27.83 | 16.47
|
4, above 19.47 | 16.73
|
145 |
Point below 116.458
| „ above 1127859
time 137,5 Length 1011.4C1

Nothing new was in the method used for the determination of resistance.
It is hence only necessary to give the final results, as the means of
the various measurements reduced to the same time.

To calculate the temperature we have used the following preliminary
formula, obtained in the measurements described in Comm. N°. 77

( 680 )

TABLE III. JENA GLASS.

ate

WAB

WBC

uCD

|

TABLE IV.

1°

0°

A B Ce eed BE F G Eon NE |
Dato] Mesa | Teme | ip zo | | So
| |
20 V.| 4410 | 15.58 |1012.594 |1012.587 | top 6.66 | 6.29
63 | 595 588 | middle 36.04 | 33.95 | 16.03
„69 „503 ‚587 | bottom 10.66 | 10.47
22 V 17.74 |4011, 834 |1011.865 | top 5.10 | 6.29 n=
BMS 5.61
17.82 „886 868 | middle 22.15 | 33.95 |- 87.87
18.00 844 .880 | bottom 6.98 10.17 DE
78.1
22 V| 520 | 18.32 10i4.827 1011-868 | top 5.01 | 6.29 en
18.44 ‚815 858 | middle 22.43 | 33.95 |- 87.87 a
bottom 6.91 10.47 n=
82.0
23 V.| 445 | 16.68 |1012.567 1012.579 | top 668 | 6.99
16.68 573 .585 | middle 36.09 | 33.95 | 46.44
bottom 10 72 10 17
25 V.| 12410 | 46.08 1G11.408 1011.409 | top 482 | 6.29 Nez
„13 Alt | _ m3 | middle -8.77 | 33.95 |-482.99 =
47 | „406 „409 | bottom 4.37 10.17 DE
51.0
25 VJ 445 | 46.38 011.407 HOM .414 | top 4.68 | 6.99 =
55 401 aal | mite 8.77 | 93.98 eol
bottom 4.34 10.47 i
| 52.8
26 V.| 340 | 17.30 |1012.565 |1012.588 | top 6.70 | 6.29
49 567 594 | middle 36.12 | 33.95 | 16.64
| bottom 10.66 10.17
ss gl ae

( 681 )

with platinum wire of the same kind as that used in the present
instance

wi = w, (1 + 0.003864 t — 0.0,103 #°)
thus to = — 182°.99.

The calculation of the temperatures of the projecting portions from
the values wag and wep will be described in § 4.

In the following table the final results*) for all the determinations
are given, the standard scale at 16° C. being taken as the reference
length. Column / thus contains the values for the rod lengths reduced
to this reference. We have used as the expansion coefficient of
brass between 16° and 17° the value 17.8 10-6. Column / refers
to the ends, and its contents will be considered in § 4.

TABLE V. THURINGER GLASS.

|
| 4 | Bele Ce Ne peel a. | Fr G een
— ! ! | —=
| | |
20 V.| 245 | 415.12 1013.407 |1013.091 | top 6.47 | 6.12
|
| 44| 408 | 098 | middle 36.53 | 34.53 | 45.08
| | | | |
| | | | | | bottom 10.21 | 9.68
| | | | | | |
oa v. |. 12:30 | 47.08 1012.24 1012263 | top 459 | 6.42 la
| | | | 26.6
| 33 | 238 | .262.| middle 22.55 | 34.53 |-87.71 |
| | 31 | 239 | .963 | bottom 6.51 | 9.68 | i=
| | | | 35.4
| | 35 | _ .240 „264 | |
| | |
23 V. 1140 | 16.68 103.086 1013.098 | top 6.52 | 6.12
|
| | .68| .088 | 100 | middle 36.70 | 34.53 | 16.36
| | |
| | | bottom 10.23 | 9.68
| | | |
| | | |
25 V. | 3120 | 17.04 011.744 |1011.763 | top 3.81 | 6.12 yee
| | | 1.0
| ANS „168 | middle 8.95 | 34.53 |-482.79|
| |
| EAO 720 .761 | bottom 5.29 | 9.68 i=
| | | 25.9
| | 25 „738 „160 |
| | |
i | |
26 V. | 11450 | 16.56 1013.095 013.105 | top 6.46 | 6.42 |
67} .098 | .A40-| middle 36.60 | 34.53 | 45.61 | |
| bottom 10.48 | 9.68
|

1) The numerical values are slightly different from the values given in the original
Dutch paper according to a new and more exact calculation. The final results
for the dilatation given in the original are quoted § 6 footnote.

( 682 )

§ 4. Discussion of the measurements. In $ 2 we have already
remarked that the mean temperature of the platinum wire, wound
round the portion BC of the rods, which is at the temperature

of the bath, may, with sufficient accuracy, be put as

equal to the mean temperature of that portion of the rod

asf itself. Throughout this length, the differences of temperature

or the length over which they are found, are on the whole

small, so that only the mean temperature comes into

account. Further consideration is however necessary in

respect to the relation of the temperatures of the ends
AB and CD and the resistances determined.

Fig. 2. Let us suppose that the level of the liquid reaches to a
position 4, fig. 2, and hence that the upper portion of AB is outside
the liquid. We may suppose that, for the length 4, the rod has the
temperature of the bath. The resistance of the wire between B and 2
is then w, = w, (1 + pt + qi’) where ¢ is the temperature of the bath.

Also we may suppose that at A, which was damp but just free
from ice, the temperature was about 0° C. Further let us suppose that
between 2 and £ the temperature gradient is linear, in other words
that the external conduction may be neglected in comparison with the
internal conduction of the glass. There is every reason to assume
that this was true to the first approximation, since the glass rods were
well enclosed in paper the conductivity of which is about */,,, of that
of glass. Then, neglecting the conduction of the platinum wire, itself
the resistance of an element of the wire between À and ZL is wdz,

nlb

where w= w, (L + pt, + giz’) and the whole resistance f dr.
CI

Further for « between O and 4, t‚=t,, between 4 and J,

Om

i= EE (@—a) and for c= lj t; = 0, so that
k 8 A :
Wan = Wap, 7 CEE (6) Sp
a aie ek i) ENE
es anes z } allt “u— ies
a) (AB) > aie Er (a q =e v

From this 2, the only unknown, can be obtained. One of the
most unfavourable cases, that for the upper end of the Jena glass
rod in N,O, shows when calculated that the linear form for the
resistance can be employed in our measurements without difficulty,
in place of the quadratic form. We found 2= 8.4 em. with the

( 683)

quadratic and 2 == 9.0 em. with the linear formula. The uncertainty
thus introduced into the determination of length, is less than Ly.

In order to determine the influence of various suppositions with
regard to the distribution of temperature in the rod, we have
calculated the change in length which would be produced, if the
temperature was — 87° C. from O to 4 and O° from 4 to Z, in
place of the distribution assumed above. The change was hardly
0.1 uw and thus lies within the degree of accuracy. However an
important control indispensable for more accurate determinations
would be obtained by measurements on a rod with similar ends
AB and CD, but where LC was only a few centimeters long ).

To apply generally the method of this section for the determination
of mean temperature it may be necessary to subdivide the portion
of at variable temperature AB into more parts while for each of
these separate portions the resistance would have to be found. In
our case this would have been an unnecessary complication.

§ 5. Influence of errors. These can be fully considered by the
RTs ni
i ek

The accuracy of the cathetometer reading can be put at 2u (the
whole contraction being 1200 u). This gives da = 2 x 108. For the
mean temperature of the portion BC the error is certainly less than
0.5 deg. C, whence da = 1.5 >< 10-8, and for that of the ends we
found 1a. Hence a greater uncertainty than da = 4 1078 is not to
be expected. Although the division of this error between @ and }
cannot well be made, it is certain that an error in the temperature
determination has by far the greatest influence on 5.

aid of « —=

§ 6. Final results. For the observed lengths Livo, at the tempe-
rature £vo, in nitrous oxide, Zo, in oxygen, and Lig at ordinary
temperature we have the three equations

Lino, = (Lee, + Ai + 4s) (1 + ato, 4- bvo) +
1 1
a Tes, sai oe (Zs in ID. Kees i) (: ae 5 atNo, + 3 beo.)

and two analogous ones for Lio, and Zico, with Liz, = 840 mM.,
Li, =97, L,, =59 for Jenaglass, and Lao, = 834, Lj, — 96, L,, — 60,
for Thüringerglass. For Lyc¢, (the length of the part LC in the figure
at 0’ C.), Li, Le, (that of the parts CD and AB in the figure) are
assumed approximate values; the exact values Zy, and Ls, to be

1) For Jenaglass in oxygen we found a negative value of a, we made therefore
the calculation on another supposition viz. that from A in the direction of B the
rod has the temperature 0’ over a length of A! cm. (ef. Table IV).

47

Proceedings Royal Acad. Amsterdam. Vol. VII.

( 684 )

ascribed accordingly to the lengths at 0° C. of the points projecting
beyond A and PD follow from the equations. These equations give
La, + Ls, = 16.587 and Ly, + Ls, = 23.095 for Jena and Thüringer
glass respectively, and further;
GD, (Lt at = bt)
V=Vi(1+4,t+h,0)
Jena glass 16™ (a — 7.74, 10 ®,b = 0.00882. 10 °

k= 28:21. 105 k;=0:02655 2055

(TAOS mdr (0) rey
| 9.15 10-§,b = 0.0119 10-6
Thüringer glass (n°. 50) ;

( k, =27.45 10-6, k, = 0.0357 10-6

The value found for Jenaglass 16"! differs much from that obtained
by Wiese and Börrcuer®) and from those obtained afterwards by
TrieseN and Scueer ®) for temperatures between 0° and 100°.

Physics. — “The motion of electrons in metallic bodies, UI.” By
Prof. H. A. Lorentz.

(Communicated in the meeling of March 25, 1905).

§ 16. We may now proceed to examine the consequences to
which we are led if we assume fwo kinds of free electrons, positive
and negative ones. We shall distinguish the quantities relating to
these by the indices 1 and 2; e.g. N, and NV, will be the numbers
of electrons per unit of volume, m, and m, their masses, oh and oh

ofl, te

the mean squares of their velocities. For simplicity’s sake, all elec-

trons of the same sign will be supposed to be equal, even if con-

tained in different metals. As to the charges, these will be taken to
have the same absolute value for a// particles, so that

i mre ora (45)

Our new assumption makes only a slight difference in the formula
for the electric conductivity ; we have only to apply to both kinds
of electrons the considerations by which we have formerly found the
equation (21). Let a homogeneous metallic bar, having the same
temperature throughout, be acted on in the direction of its length
by an electric force EE; then, just as in $ 8, we have for each kind
of electrons

1) In the original was given
Jena 161! «@=7.78 b=0.0090
Thüringer n° 50 a=9.10 b=0.0120.
2) Wiee und Börrener. Z. f. Inst. k. 10, pg. 234. 1890.
3) Turesen und Scneer, Wiss. Abt. der Ph. techn. Reichsanstalt. Bd. ILS. 129. 1895,

( 685 )

AnlAe „
Shen

The electric current per unit of area of the normal section, is the
sum of the currents due to the positive and the negative particles.
We may therefore represent it by

Al, A,e,? bs 4al, Ae E
hm, dh,m,

and we may write for the coefficient of conductivity

==

OEROL ON Gyo eo (4:00)
if
An l Ae? dt UR Aven:
— 1 Lara te = 2 #73
: dh, m, ; core dh, m,
or (cf. § 8)
We Ue INE CE POR ie INEGI
i= ne Men TT. (50
: 3a al i 3a aT B

These latter quantities may be called the partial conductivities due
to the two kinds of electrons.

$ 17. In all the other problems that have been treated in the prece-
ding parts of this paper, we now encounter a serious difficulty. If
either the nature of the metal or the temperature changes from one
section of the cireuit to the next, we can still easily conceive a state
of things in which there is nowhere a continual increase of positive
or negative electric charge; this requires only that the fofa/ electric
current be O for every section of an open circuit and that it have the
same intensity for every section of a closed one. But, unless we
introduce rather artificial hypotheses, it will in general be found
impossible to make each partial current, i.e. the current due to each
kind of electrons considered by itself, have the same property. The
consequence will be that the number of positive as well as that of
negative electrons will increase in some places and diminish in others,
the change being the same for the two kinds, so that we may speak
of an accumulation of “neutral electricity” in some points and of a
diminution of the quantity of neutral electricity in others. Now,
supposing all observable properties to remain stationary, as indeed
they may, we must of necessity suppose that a volume-element
of the metal contains at each instant the same number of really
free electrons. This may be brought about in two ways. We may
in the first place imagine that all electrons above the normal number
that are introduced into the element are immediately caught by the
metallic atoms and fixed to them, and that, on the other hand, in

47%

( 686 )

those places from which electrons are carried away by the two cur-
rents, the loss is supplied by a new production of free electrons.
This hypothesis would imply a state of the circuit that is not, strietly
speaking, stationary and which I shall call “quasi-stationary”. Moreover,
we should be obliged to suppose that the production of free electrons
or the accumulation of these particles in the metallic atoms could
go on for a considerable length of time without making itself any-
ways felt.

In the second place we may conceive each element of volume to
contain not only free positive and negative electrons, but, in addition
to these, a certain number of particles, consisting of a positive and
a negative electron combined. Then, the number of free electrons
might be kept constant by a decomposition or a building up of such
particles and we could arrive at a really stationary state by imagining
a diffusion of this “compound electricity” between different parts of
the circuit.

§ 18. The mathematical treatment of our problems is much sim-
plified by the introduction of two auxiliary quantities.
In general, in a non-homogeneous part of the circuit, the accele-
ration Y will be composed of the part Y,,, represented by (30), and
e : - . > Sy
the part — /, corresponding to the electric force /. The formula (21)
m

for the flow of a swarm of electrons may therefore be replaced by

2 1 2hAdV 2heA _ dA A dh
yp—=— al| — | — - + # ——|+2——]}. . (oi)
3 h? m de m de) hè dz

7
Hy

This will be 0, if the electric force # has a certain particular
value, which I shall denote by E and which is given by

E 1 dV m dlog A i md (1 (52
nn — ZE oe EE
e da cle Zhe de edx\h )

For any other value of the electric force the flow of electrons

will be
4 eA

NS

En)
hm
and if, in order to obtain the corresponding electric current, we mul-
tiply this expression by e, we shall find the product of 4 — E by
the coeflicient of conductivity, in so far as it depends on the kind
of electrons considered.
Substituting in (52) the value (14) and applying the result to the
positive and the negative electrons separately, we find

( 687 )

E 1 dV, 2 aT dlog A, 4 a dT

a en da je Sd 8 e‚ dx
(53)

fae legis 2 aT dlog A, Aerde

ES e, de nes da She, de

If the area of a normal section is again denoted by >, the inten-
sities of the partial currents are given by

tO (Ht), On (Ea eMart | (OA)
and that of the total current, on account of (49), by
(= tt, = (@ Bo, 0) TE ce (85)
Putting
6, Ox,
oer ’ Tat

0,

é : 1 5 : 3 0.
i, =), + — (©, — EE) >, =d + *(E, — E,) =.
5 5
It appears from these formulae that, whenever E, differs from E,,
the partial currents 7, and 7, will not be proportional to the conduc-

tivities 6, and o,.

§ 19. The above results lead immediately to an equation determining
the electromotive force /’ in an open circuit composed of different
metals, between which there is a gradual transition ($ 6) and which
is kept in all its parts at the same temperature. Let P and Q be
the ends of the cireuit and let us reckon 2 along the circuit in the
direction from P towards Q.

The condition for a stationary or a quasi-stationary state is got
by putting 7=0 in (55). Representing the potential by , so that

we get

de 7) 6

Se ei Meee. 2 86)
da Oo 0

and finally, taking into account the values (53), in which we now

dT
have == 0, and integrating from P to Q,
av
Q Q
1 he deo SLR
PDP =P = = y == igs
Pa Pp ey 6 dr , ey o dx =

Je P

( 688 )

Q Q
2 af (6, dlog A, 2aT (0, dlog A,
= A EE ig et ee ene
da: 33 de

O ed 6

Je Mig

At the same time the intensities of the partial currents are given by
_ 9,5, oO,

= : 0
i= 7 (E,—E) = , #= = (EE) =.

These values, which are equal with opposite signs, will in general
vary along the circuit, so that, even in this simple case, we cannot
avoid the complications I have pointed out in $ 17. Nor can the
difficulty be easily overcome. Indeed, we can hardly admit that the
state of two pieces of different metal, in contact with each other
and kept at a uniform temperature is not truly stationary. If, in
order to escape this hypothesis, we have recourse to the considera-
tions I presented at the end of § 17, we must suppose the neutral
electricity to be continually built up in some parts of the system
and to be decomposed in other parts. The first phenomenon will be
accompanied by a production and the second by a consumption of
heat. That these effects should take place in a system whose state
is stationary and in which there are no differences of temperature,
is however in contradiction with the second law of thermodynamics.

The only way out of the difficulty, if we do not wish to confine
ourselves to one kind of free electrons, seems to be the assumption
that there is no accumulation of neutral electricity at all, 1. e. that

7, and 7, are simultaneously 0. This would require that E, = E,,

2

or in virtue of (53)
] dV, 2 al d log Ae : 1 dV, 2 a ak d log A,

: Nee ee - 58
En da 3 2 da (Op da 5 GC da: ( )
Since e, = —e,, we might further conelude that
2, 1 log (A, A.) L(V. V.
z on ged core: a) zi ae ra Vv) — 0.
3 da da

which means that
3 ye re Id Al
log tt) nn ar A) == th (IE)

ought to have the same value in all parts of the circuit. We should
therefore have to regard this expression as a funetion of the tempe-
rature, independent of the nature of the metal *).

If we suppose the contact of two metals to have no influence on
the number of free electrons in their interior, we must understand
by A, and A, in the above equation quantities characteristic for each

1) Cf, Drupe, Annalen der Physik, 1 (1900), p. 591.

( 689 )

metal and having, for a given temperature, determinate values, whether
the body be or not in contact with another metal.
By the assumption E, = E,, (56) simplifies into

and (57) becomes

ien Paid A,pP
Bq — Gp = (Ve — Via) +5 lon (F |=

1 1 1Q
1 2 al A
== — (Vp mr Vg) = ba ( 2), > (59)
x es e A,a

a formula which is easily seen to imply the law of the tension-series.

§ 20. The question now arises, whether, with a view to simplifying
the theory of the thermo-electric current, we shall be allowed to con-
sider E, and E, as equal, not only in the junctions, but also in the
homogeneous parts of the circuit, in which the differences of tempe-
rature come into play. This seems very improbable. Indeed, supposing
for the sake of simplicity WV, and JV’, to be, for a given metal, inde-

T r

dV,

dV, :
== Wand —)
ar ar

pendent of 7, so that in a homogeneous conductor

we find from (53), putting E, = E,,

2 aT dlog A, 4 a dT 2 aT’ dlog A, 4 a dT

= i

3 ey dz ; 3 ë de = 33 e dr 3 es: dx :
or, since e‚ = (Fe
r d log (A, Ay) Bt ae =
da da:

which can hardly be true. It would imply that the product A, A, is
inversely proportional to the fourth power of the absolute temperature
and this would require in its turn, as may be seen by means of (13)
and (14), that the product N, N, should be inversely proportional to
T itself.

We are therefore forced to admit inequality of E, and E,. Now,
it may be shown that, whatever be the difficulties which then arise
in other questions, the theory of the electromotive force remains
nearly as simple as it was before. For an open circuit we have
again to put #5 0; hence, the formula (56) will still hold, as may

SA : dp Ek
be inferred from (55), if we replace Z bv — —. The equation for
det

the electromotive force becomes therefore

Sel

7 = 7 | = 1 3

P= Pal p= a = (o, EH, + 6, B,) da .. ttl)
P

In the case of a closed circuit, which we get by making the points

P and Q coincide, we shall integrate (55) along the circuit after

3 ; da - dep 7
having multiplied that equation by — and replaced / by — —. The
a 5

=
— av

intensity 7 being everywhere the same, the result takes the form

> de
lef RE en

This is the mathematical expression of Oum’s law.

§ 21. It must further be noticed that the equation (60) agrees
with the law of the thermo-electric series. This may be shown as
follows. If we suppose the temperature to be the same throughout
a junction, we may easily infer from what has been said in § 19
that the part of the integral corresponding to such a part of the
circuit can be represented as the difference of two quantities, which
are both functions of the temperature, but of which one depends
solely on the nature of the first metal and the other on that of the
second. Considering next a homogeneous part of the circuit between

two junctions, we may remark that in this E, and E, have
R ende ; 0, 0, 2 ; :
the form / (7) a and that the rations — and — are functions of the
da oO 0

temperature. We may therefore write for the corresponding part
of (60)
7e
| y (L) at.
2

This integral, which is to be taken between the temperatures 7” and
7" of the junctions, may be considered as the difference of the values,
for T= T7"' and 7 = 7", of a certain quantity depending on the
nature of the metal.

Combining these results, we see that the electromotive force in a
given circuit is entirely determined by the temperatures of the
junctions, and that, if there are two of these between the metals
[and //, the electromotive force #7; we have examined in §10c¢
may still be represented by an equation of the form

Fimo (2) = $1 2) = Sol) Sa 7);
the function $/(7’) relating to the first, and the function $/,(7’) to

(369i1>)

the second meta!. The law of the thermo-electric series may imme-
diately be inferred from this formula. However, in order to obtain
this result, it has been necessary to adopt the hypothesis expressed
by (58).

I shall terminate this discussion by indicating the way in which
our formulae have to be modified, if, in the direction of the circuit,
the electrons are acted on not only by the electric force caused by
the differences of potential, but also by some other force proportio-
nal to their charge and whose line-integral along the circuit is not 0.
Let us denote this force, per unit charge, by /, and let us write for

its line-integral
| Eede = F,.

This latter quantity might be called the “external electromotive
force” acting on the cireuit. Now, in the formulae (54), we must
replace EL by E+ L.. Consequently, (55) becomes

ND SJN D= IDS
ij | 1 1 af

2

and treating this equation in the same way as we have done (55),

we find instead of (61)
ide elle
eens Ee

$ 22. I shall not enter on a discussion of the conduction of heat,
the Perrier-effect and the THomson-effect.
In the theory which admits two kinds of free electrons, all ques-

tions relating to these phenomena become so complicated that I
believe we had better in the first place examine more closely the
Harr-effect and allied phenomena. Perhaps it will be found advisable,
after all, to confine ourselves to one kind of free electrons, a course
in favour of which we may also adduce the results that have been
found concerning the masses of the electrons. These tend to show
that the positive charges are always fixed to the ponderable atoms,
the negative ones only being free in the spaces between the molecules.
If however a study of the Harr-effect should prove the necessity
of operating with both positive and negative free electrons, we shall
be obliged to face all the difficulties attending this assumption.

Geology. — “Contributions to the knowledge of the sedimentary
boulders in the Netherlands. 1. The Hondsrug in the province
of Groningen. 2. Upper Silurian boulders. Second communi-
cation: Boulders of the age of the Eastern Baltic zones H and
I’ By Dr. H. G. Jonker. (Communicated by Prof. K. Martin).

Hi.

Besides the Borealis-limestone, described in my preceding commn-
nication (33) and on which I am going to touch later on, boulders
with Pentamerus-remains near Groningen are rare. I can mention
but three pieces here, in two of which the species is not to be made
out, while in the third, found in the “Noorderbegraafplaats” in Gro-
ningen, Pentamerus estonus Eicuw. occurs. Nor is this determination
beyond doubt and especially the possibility of its being Pentamerus
oblongus Sow. can in my opinion not be excluded, as indeed in out-
ward appearance the latter corresponds almost perfectly with the
former (12, p. 81 and 3, T. XVIII, f. 4°). As however, the latter
form in Gothland has no doubt to be looked upon partly as the
real P. estonus EKicuw. (27, p. 98), nothing can be said for certain
about its origin, as the rock, a weathered, yellow limestone does
not give sufficient indications for it. | mention this boulder however
for completeness’ sake.

With regard to the Borealis-limestone I wish to add, that after
all I did find an almost complete specimen of Pentamerus borealis
Eicuw., in the Groningen museum, evidently from a Groningen
boulder. The correspondence with the specimens from Weissenfeld,
mentioned before, is however not very great, the top of the ventral
valve in our specimen being much more curved and thus agreeing
more with E1cuwanp’s deseription.

A close investigation removing the existing confusion with regard
to the Upper Silurian Pentamerus-species is really most desirable.

31. Clathrodictyon-limestone.

White limestone, sometimes having a more or less light-yellowish-
gray tinge. At the surface and in cavities the colour is rather yellow.
It is always crystalline and the very irregular fractured surfaces
show a peculiar fatty silk-gloss, which is most characteristic of them.

If the colour becomes a little darker, as is sometimes the case, the

gloss remains preserved. The roek is a real Stromatopora-limestone,
which may be distinctly perceived in some pieces, as they consist of
slightly curved, concentric layers the surface of which is covered
with small knobby mamelons (25, Pl. XVII, f. 14), which make it
more than probable that we have to do here with

Clathrodictyon variolare Rosen sp.

Its structure, however, is not easily traced on account of the erystal-
line character of the stone.

This species of boulder further contains real fossils only in the
form of peculiar conical cavities, mostly slightly bent towards the
point. On the inside they are invariably set with annular edges,
which on an average are lying a little more than 1 mm. from each
other in specimens of an average size. The cavity is often completely
filled up with crystalline calcite bright as water. Its rather thick
wall presents on the outside small irregularly running lines of growth.
Frieprich Scumipt, Akademiker in St. Petersburg, whom I sent a
piece of this limestone, was kind enough to inform me that these
cavities originate from Cornulites sp. (1, T. 26, f. 5—8), a fossil of
the [-zone in Oesel, frequently occurring near St. Johannis.

These boulders are by no means rare near Groningen as appears
from the following list:

“Noorderbegraafplaats”’, Groningen 6
“Boteringesingel”’, 55 ti
“Nieuwe Kijk in ‘t Jatstraat”, 5 1
“Nieuwe Veelading”’, a 3
“Schietbaan’’, - Al
Behind the “Sterrebosch’’, és 1

6 1
Café “the Passage’, Helpman il
Villa “Edzes” near Haren dl
The “Huis de Wolf” near Haren 1
“Klein-Zwitserland” near Harendermolen _ 1

About the occurrence of the mentioned species of Stromatopora
NicHoLsoN records it from Borkholm and Worms in the Borkholm
stratum in Esthonia, but he has especially found them frequently in
the Estonus-zone there, chiefly near Kattentack. (25, p. 151). He
does not record it from Gothland, though this fact is not sufficient
altogether to exclude its occurrence there. Moreover LINDSTRÖM

( 694 )

mentions three other species of this genus (16, p. 22). Among my
material for comparison is a specimen from Klein-Ruhde, to the
west of Kattentack in the H-zone in Esthonia. This roek is some-
what darker, more grayish; but yet examples are to be found among
our pieces which perfectly resemble it, so that the correspondence
may really be called striking. The deseribed Cornulites do not occur
in it which it is true cannot surprise us im a piece of so small
dimensions (7 X 6 >< 2 c.M.).

Finally I wish to state that in a boulder of stromatopora-limestone
in Gothland, I found analogous Cornulites-cavities, which petrogra-
pbically does not altogether agree with our pieces. The place where
it is found is immediately to the north of Högklint, on the field
(not in the beach). But this fossil is of little importance for the
further determination of the age of the rock, as most likely various
species will be implied in the name of Cornulites serpularius SCHLOTH.
which is usually given.

Taking everything into consideration, it seems possible (perhaps
even probable) to me that this Clathrodictyon-limestone comes from
the H-zone in Esthonia or from its western continuation.

In connection with this must be said that among the very nume-
rous stromatoporae of the Hondsrug (of which speeifie determinations
are hardly ever possible) two occur which from their characteristic

astrorhizae may be called :
Stromatopora discoidea LonsD. sp... .. 25, Pl. XXIV, f. 2.

Both pieces, found in the “Noorderbegraafplaats” and in the “Violen-
straat’’ in Groningen, consist of fine-grained crystalline (stromatopora-)
limestone ; the former is all over white and therefore closely resem-
bles Clathrodictyon-limestone, the latter is rather grayish and also
partially weathered, which fact decreases the correspondence.

This species, very common in Wenlock limestone from England,
also occurs in the neighbourhood of Wisby in Gothland. Nicnonson
calls those Gothland specimens however usually highly mineralised
(25, p. 191), which with my material from Gothland corresponds
but to this extent that this fossil occurs only as a not always very
thick crystalline crust in marl or marly limestone. LinpstR6M records
it only from 4 (16, p. 22), his youngest zone of the Upper Silurian
of Gothland (/, Dams). Contrary to this I allege to have found a
specimen (it is true somewhat differing in a smaller number of
astrorhizae) in the calcareous marl immediately to the north of
Hogklint, occurring there as firm rock; this fossil comes from a

( 695 )

stratum about 1 M. above the beach. This petrographical and strati-
graphical occurrence is, it seems to me, hardly to be referred to the
age A; the other specimen in the Groningen museum supports my
observation only to the length of containing marly remains still
distinetly to be seen. The place where it was found is, however, not
further indicated.

Our Groningen fossils have upon the whole but little in common
with these Gothland pieces; meanwhile this fossil also oceurs in
Esthonia near Klein-Ruhde in the Estonus-zone. That is why these
two pieces have been mentioned here though no further data can
be brought forward to prove their origin from these Eastern Baltic
regions through want of material for Comparison.

I.

Boulders which correspond in age with the Lower Oesel zone in
the Eastern Balticum, are not rare near Groningen. BONNEMA already
pointed it out some years ago (31); this short essay, however, has
more of a palaeontological character, so that I wish to complement
these communications and enter into further particulars.

32. Baltica-limestone.

In an unweathered state rather hard, tough, fine-grained-erystal-
line limestones of a bright-gray or light-brownish-gray colour. Some
pieces are almost impalpable; some parts are coloured bluish-gray
on the inside, so that the rock may originally have had that colour.
Through weathering the bright-gray tinge passes into light-yellowish-
gray ; the uneven fractured surfaces then are very often covered with
sallow-yellowish and brown spots. Crystalline calcite rarely occurs.
The limestone is rather pure, but a little marly and hardly ever
slightly dolomitic. Real dolomites are not among them. Stratification
is imperceptible. The dimensions of the pieces found amount to 25 cm.

Fossils are not present in great numbers, chiefly Ostracoda, among
which Leperditia-shells are the most important. Whilst bright-brown
in the unweathered rock, the valves which sometimes occur frequently
in a single piece, have become nearly white by weathering. As is
often the case with the younger Leperditia-limestones, which are to
be described later on, this limestone is not unfrequently connected
with Stromatopora-limestone; the fossils to be mentioned below,
however, never occur in it. Besides these large Ostracoda-remains,
small Beyrichia- and Primitia-valves are also frequently found but

( 696 )

they become only distinctly visible through weathering. The fossil
fauna consists of the following species.

Leperditia baltica His. sp.

Strophomena rhomboidalis Wiek. sp.

Strophomena sp.

Atrypa reticularis L.

Meristella sp.

Encrinurus punctatus Wane.

Zaphrenthis conulus Linpstr. ...... . . 28, p. 32, T. VI, f. 65—68.
Orthoceras sp.

Murchisonia sp.

Tentaculites sp.

Primitia seminulum Jonus........... 14,p.413, Pl. XIV, f. 14.
Primitia mundula Jones. . . . 23 T. XXX, f. 5—7; 18, p. 375, Pl. XVI.
Beyrichia. Jonesy Born. „ss sed pio eee

Beyrichia spinigera Bou... ... . . 23, p. 501, T. XXXT, f. 19—20.

The first mentioned Leperditia-species is present in all pieces; all
other fossils, however, occur either few and far between or in a
single piece, excepting the small ostracoda. I have however not taken
much pains to increase the number of species of them (for the greater
part already mentioned by BonneMA), because their stratigraphical
value is still but tritling nowadays. Then to determine age and origin,
we can restrict ourselves to the communication where and in which
strata occurs the type-fossil of this group, Leperditia baltica Hus. sp.
(after which in accordance with the names of Phaseolus-limestone
and Grandis-limestone, generally in use, I have called these limestones).

First of all, however, the number of the pieces found here and
the special places where they were found, be given here:

“Noorderbegraafplaats”’, Groningen 7
“Boteringesingel”’, is 4
““Noorderbinnensingel”’, 55 5
“Violenstraat”, A 1
“Nieuwe Boteringestraat”’, 5 1
“Nieuwe Kijk-in ’t Jatstraat” 5 2
“Nieuwe Veelading”, 5 al

3 1
“Old Collection” 3
Helpman 2
“Hilehestede”, Helpman 1
Between Helpman and Haren 2
Harendermolen 1

( 697 )

So in all 831 pieces. The number found is presumably much larger,
because I have only mentioned here the boulders which beyond any
doubt belong to this group; among the numerous limestones with
Leperditia-remains which cannot be specifically determined there
will no doubt be a number of this age.

Leperditia baltica Ws. sp.

Literature: 1869. Kotmopin, 2, p. 13, f. 2—8.
1873. Scampr, 4, p. 15—17, f. 49
L876. Roemer, o, 0.1958 7.
1878. Martin, 6, p. 45.

1880. Koxmopiy, 8, p. 154.

1883. Scum, 10, p. 11—13, T.L f. 1—3s.

1884. Kiesow, 11, p. 275, T. IV, f. 4.

1885. Rrurerú, 13, p. 26, no. 226.

1888. Linpstrém, 16, p. 5, no. 25.

1890. Kimsow, 19, p. 89—91, T. XXIII, f. 14—16.
1890. Scumpr, 20, p. 255.

1890. Dames, 21, p. 1125.

1891. Krause, 22, p. 5, 7.

1891. Krause, 23, p. 488, T. XXIX, f. 1
1891. Scumipt, 24, p. 123.

1895. Srorrey, 27, p. 109.

1898. Bonnema, 29, p. 452.

1900. CranreLewskKi, 30, p. 17—20, 33; T. I, f. 17—20.
1900. Bonnema, 31, p. 138—140.

21.

oO.

From the literature about this fossil, cited above, which as regards
the later years is rather complete, it appears that for a long time
a certain confusion and uncertainty about the limits of the species
have existed, which have been removed but a few years ago.
Besides the real 4. baltica His., characterized by the comb-shaped
striae on the inverted plate of the left valve (L. pectinata Scumipr) —
which characteristic may be distinctly perceived in twenty of the
boulders from here —, Scumipr had also described another species :
EL. Kiehwaldt Scum. Boxxema has proved that both species have to
be united (31); at nearly the same time this has also been observed
by CraierewskKr The latter, however, distinguishes besides the typical
form two other varieties :

L. baltica, var. Lichwaldi Scumit
x nd … formosa CHMIEL.

These two varieties are present among our boulders, var. Lichwaldi

( 698 )

not unfrequently, var. formosa less often. But the characteristics of
these varieties are by no means conspicuous, so that there are
specimens which partake of the nature both of these varieties and
the real species, as CHMIBLEWSKI himself too has perceived. In
accordance with this is the fact that these varieties are practically
of no stratigraphical importance; it is on these grounds I have
thought it allowable to combine all these forms in one species under
the name of Leperditia baltica His. sp.

It has been frequently found in boulders. Kinsow deseribes it from
“weisslich-grauen Mergelkalk” of Langenau, from “ziemlich verwit-
terter und in Folge dessen gelblich gefärbter Kalk mit zahlreichen
Schalen der Leperditia baltica His. (Ff. Scrap); daneben finden
sich Knerinurus punctatus, Alrypa reticularis, und einige schlecht
erhaltene Beyrichien, u.s.w.’ from Zoppot-Olivaer Walde, also in
West-Prussia. The first stone corresponds perfectly with limestone
from Langers in the N.B. of Gothland, the second shows much
correspondence with the occurrence of Oesterby near Slite. Therefore
he refers these pieces to Gothland. (Of the co-occurrence of L. baltica
His. sp. and “. Hisingert Scum., which question I treated of in my
previous communication (33, p. 560), he is afterwards not quite
sure — 19, p. 90). In his excellent, already frequently cited treatise
CrumieLewski briefly deseribes six boulders in which he has found
L. baltica in Kurland, Kowno, East and West-Prussia. Most cor-
responding with our boulders seems to be his: “hellbräunlich-grauer,
deutlich krystallinischer, wenig thoniger, fester, unebenbriichiger
Kalkstein mit Anerinurus punctatus (80, p. 33),” from Kowno. He
does not give a decided opinion about the origin.

Farther to the west this species is still recorded from Brandenburg
by Rramrú and Krauss, mostly together with fossils, which also
occur in our boulders and from limestones which, so far as can be
gathered from the short descriptions, correspond in some respects
with ours. STOLLEY describes also various of those limestones from
Sleswick-Holstein among which “ein gelber Kalk enhält neben
L. baltica His, Atrypa reticularis L. und Enerinurus punctatus
WAHLENBERG” is again conspicuous. From Groningen our species
was already recorded in 1878 by Martin, from Kloosterholt after-
wards also by Bonnema (29, p. 452).

From these statements about the erratic occurrence of this species,
it appears sufficiently, that it has spread from Kurland and Kowno
to the Netherlands though nowhere, it is true, large numbers of such
boulders have been met with. In the Seandinavian-baltie area it is
found in different places in solid rock:

( 699 )

ist. In Malmö near Christiania which is not very important
to us;

20d, In Gothland, where Scumipt describes its occurrence as follows :

“Das grosse Centralmergelgebiet von Follingbo bis Slite und Farö,
das bald aus reinen Mergeln, bald aus Mergeln mit Kalken wech-
selnd besteht, wird neben andern Fossilien besonders durch die
urspriingliche Leperditia baltica His. mit kammförmiger Zeichnung
wif dem Umsehlag der linken Schale characterisirt, die einerseits
auch bis zu den Mergeln von Westergarn vordringt und andererseits
sich vielfach auch in den oberen Kalken der Wisby-Region findet,
so bei Heideby und Martebo. Auf F rö bei Lansa kommt sie zusam-
men mit Zaphrentis conulus Lanpstr., Strophomena imbrea VERN. u. a.
im Kalk vor, weehselnd mit Megalomus-banken.” (20, p. 255). These
places belong to Scumipt’s middle zone; besides KorMopin records it
from Oestergarn and Hammarudd near Kräklingbo (8, p. 134), where
no doubt younger strata are found. In these two places I have
been seeking for a long time, but failed to find it. According to
LanpsrröM : b—e.

3, In Oesel this species is a type-fossil of the Lower Oesel-
zone I. For a long time it was only known from dolomite from
Kiddemetz (var. Lichwaldi) but has later also been found in lime-
stones in the peninsula of Taggamois on the N. W. coast, thus
verifying Scumipr’s prediction. Only there this zone consists of erys-
talline limestone ; everywhere else of dolomite or marl (9, p. 46—49).

With regard to the origin of these boulders whose age has now
been determined, the following remarks may be stated. First of all
the fact that Skine cannot be thought of, as Leperditia baltica does
not occur there. In general the marly character of the rock found
in Gothland, argues against the possibility of its originating there ; no
doubt we have only to think of the north eastern part of the island.
Though indeed our boulders do not make the least impression of
originating in marly strata, it does not say so very much, because
in Gothland the limestome with ZL. baltica cannot everywhere be
decidedly looked upon as being limestone from marl. The question
then about their origin is not to be solved without extensive material
for comparison, which I do not possess; only a single piece of lime-
stone from Slite does not correspond with our boulders. Tis lime-
stone from Slite is differently coloured and also much more crystalline
and betrays by marl-remains and a small coneretion of little pyrite-
crystals its origin from marl. Now as regards Oesel, from this region,
too, I have but a single piece with ZL. baltica for comparison. It is
from Kiro, immediately to the south of Taggamois, and corresponds

48

Proceedings Royal Acad. Amsterdam. Vol. VIL.

( 700 )

much more with our boulders. It is however but a badly preserved,
weathered piece, so that it is not very important.

Taking all this into consideration the origin of our boulders is
probably to be found between Oesel and Gothland, where there is
every reason to assume that along the line Far6—Taggamois lime-
stones of the age of the I-zone have been developed.

More or less closely allied to this Baltica-limestone are different
boulders which for their fossil contents may best be referred to the
Lower Oesel zone:

a. Yellowish-gray limestones with:

Proetus concinnus Datm., var. Osiliensis Scum. 26, T. IV, f. 1—9.
Calymmene tuberculata BRÜNN. 26, LT
Cyphaspis elegantula Lov. sp. 7, TEM

Encrinurus punctatus W Ans.
Strophomena rhomboidalis Wiuck. sp.
Orthis sp.

They closely resemble some pieces of Baltica-limestone and most
likely neither differ very much from the latter in age. Without
tracing their occurrence in particulars here the following list shows
sufficiently why they are mentioned here:

Gotland (16) Oesel (26)

P. concinnus Daum., (var. Osiliensis ScHM.) (c—e) T
C. tuberculala Brinn. ej I
C. elegantula Lov. sp. c if
Eight pieces of this limestone are from the following places:

“Boteringesingel”’, Groningen 2

“Noorderbinnensingel”’, __,, 2

“Nieuwe Veelading”, Pe 1

EN il

“Hilghestede’’, Helpman 1

al

”

Again the tract between Gothland and Oesel must be looked upon
as the place of origin by reason of perfectly similar considerations
as mentioned in dealing with the Baltica-limestone.

b. Perhaps two limestone-rocks also belong to this with
Bumastes barriensis Murcu.
found in the “Nieuwe Veelading” and the “Schietbaan” in Groningen,

( 701 )

while Horm records this fossil from the Eastern-balticum from /
(15, p. 37), Linpstrém from 46—A in Gothland (16, p. 4, N°. 64).

ce. Thirdly various limestones with
Encrinurus punctatus WAuLB.
may be mentioned here. These ‘Encrinurus-limestones” are not
further to be determined in age on account of the want of other
adequate fossils. Some corals, Favosites and Halysites, together with
which they sometimes oceur, can be of no use for that purpose.

d. Among the great number of corals from the Groningen
Hondsrug there are no doubt many of the age of the Lower Oesel
zone e.g. Thecia Swinderenana Gorpr and others. However I do not
intend to occupy myself with this question, but later on I shall deal
with these together with the other corals, whose age is hardly ever
to be determined between narrow limits, under the heading “Coral-
limestone.”

e. Finally I wish just to make mention of a single piece of
dark-greenish-gray calcareous marl, which contains numerous pygidia
and head-shields of a Calymmene-species. This boulder found in
the “Boteringesingel” in Groningen suggests the marly stratum of
St. Johannis of the /-zone in Oesel, but also corresponds fully with
marls from different places in Gothland. About the origin, then,
nothing can be said. Probably we have to do here with Remen%’s
“Griinlichgrauer Calymenekalk”. (13, p. 27).

Here ends the enumeration of the boulders of the age Z. Be it
only added that this zone may possibly be well represented among
the very manifold dolomites of Groningen. These, however, but
seldom contain fossils and on account of this admit of no distinctly
separated groups. At the end of the description of the Upper-Silurian
boulders, I hope to be able to communicate some particulars about this.

LITERATURE.

1. Murcnisoy, R. J. — »The silurian system’.
London, 1839.
2. Kormopin, L. — »Bidrag till kinnedomen om Sverges siluriska Os-
tracoder”’.

Inaug.-Dissert., Upsala, 1869,
3. Davipson, TH. — »A monograph of the British fossil Brachiopoda.
IIT, Devonian and silurian species”.
London, 1&864—’71.

48*

er)

~I

ive)

10.

13.

14,

fe

18.

( 702 )

Scumipr, F. — »Miscellanea Silurica. — J. Ueber die russischen siluri-
schen Leperditien mit Hinzuziehung einiger Arten aus
den Nachbarlindern”’.

Mém. de Ac. Imp. d. Se. de St. Pétersbourg, Vile Sér., T. XXI, no. 2; 1873.

Roemer, F. — »Lethaea geognostica. I. Lethaea palaeozoica”’. Atlas.

Stuttgart, 1876.

Marvin, K. — » Niederlindische und nordwestdeutsche Sedimentdrgeschiebe,
thre Uebereinstimmung, gemeinschaftliche Herkunft und
Petrefacten’’.

Leiden, 1878.
ANGELIN, N. P. — »Palaeontologia scandinavica’’.
Stockholm, 1878.

Kotmopry, L. — » Ostracoda Silurica Gothlandiae’’.

Ofvers. af Kongl. Svensk. Vet.-Akad. Wörhandl. 1879, no. 9, p. 133 —139 ; 1 $S0.

. Scumipt, F. — »Revision der ostbaltischen silurischen Trilobiten nebst

geognostischer Uebersicht des ostbaltischen Silurgebiets”.
Abtheilung I.
Mém. de PAec. Imp. d. Sc. de St. Pétersbourg, Te Sér., T. XXX, no. 1; 18S1.
Scumipt, F. — »Miscellanea Silurica. — HI. 1. Nachtrag zur Mono-
graphie der russischen Silurischen Leperditien’’.
Mém. de PAc. Imp. d. Se. de St. Pétersbourg, Vile Sér., T. XNXI, no. 5 ; 1883.

. Kirsow, J. — » Ueber silurische und devonische Geschiebe Westpreussens’’.

Schrift. d. naturf. Ges. in Danzig, N. F., VI, 1, p. 205—300; 1884.

. Roemer, F. — »Lethaea erratica’’.

Palaeont. Abhand!., herausg. v. W. Dames u. B. Kayser, Tl, 5; 1885.
Remetb, A, — »Katalog der beim internationalen Geologen-Congress zu
Berlin ausgestellten Geschiebesammlung’’.

Berlin, 1885.
Rurerr Jones, T. and
Hout, H. B. — »Notes on the Palaeozoic Bivalved Entomostraca. —
NX/. On some Silurian Genera and Species”.
Ann. and Magaz. of Nat. Ilist., 5 Ser., Vol. XVII, p. 403—414; 1S86,

. Hotm, G. — »Revision der ostbaltischen silurischen Trilobiten”’.

Abtheilung IIT.
Mém. de PAe. Imp. d. Se. de St. Pétersbourg, Te Sér., 1. XXXIL, no. 8; 1886
Linpstrrém, G. — »List of the fossil faunas of Sweden.
LILI. Upper Silurian”.
Stockholm, 1888.
Kresow, J. — » Ueber gotländische Beyrichien” .
Zeitschr. d. deutsch. geol. Ges., XL, p. 1—16; 1888.
Ruperr Jones, T. — »Notes on the palaeozoic Bivalved Entomostraca.
XXVIII. On some North-American (Canadian)
Species”.
Ann. and Magaz. of Nat. Hist, 6 Ser., Vol. III, p. 373—387; 1SS9.

. Kresow, J. — »Beitrag zur Kenntniss der in westpreussischen Silur-
; EP

geschieben gefundenen Ostracoden’’.
Jahrb. d. k. pr. geol. Landesanst. u,s.w. f. 1889, p. SY—103; 1890.

( 703 )

. Scumipr, F. — »Bemerkungen über die Schichtenfolge des Silur auf
Gotland”.
Neues Jahrbuch, 1890, [L, p. 249 —266.
. Dames, W. — »Ueber die Schichtenfolge der Silurbildungen Gotlands und

ihre Beziehungen zu obersilurischen Geschieben Nord-
dentschlands’’.
Sitz -Ber. d. k. pr. Ak. d. Wiss. zu Berlin, 30 Oct. 1890 ; Bl. XLIT, p. 1111—1129.

22. Krause, A. — »Die Ostrakoden der silurischen Diluvialgeschiebe”’.
Wiss. leilage z Progr. d. Luisenstädtischen Oberrealschule zu Berlin, Ostern 1891,
23. Krause, A. — »Batrag zur Kenntniss der Ostrakodenfauna in silurischen
Diluvialgeschieben.”
Zeitschr. d. deutsch. geol. Ges. XLIIL, p. 488~521; 1891.
24. Scmapr, F. — »KHinige Bemerkungen über das baltische Obersilur in Ver-
anlassung der Arbeit des Prof. W. Dames über die
Schichtenfolge der Silurbildungen Gotlands’’.
Bull. d. PAe Imp. d. Se. de St. Pétersbourg, N.S. IL(XNXIV), 1892, p, 381—
400; and: Mél. géol. et pa'éont., tirés du Bull. ete, T. 1, p. LI9 — 38.
25. Nicnorson, H. A. — »A monograph of the British Stromatoporoids”’.
The Palaeoutographieal Society, London, 1886—12.
26. Scumipt, F. — » Revision der ostbaltischen silurischen Trilobiten.”’.
Abtheilung IV.
Mém. de YAec. Imp. d. Se. de St. Pétersbourg, Te Sér., T. XLII, no. 5; 1894.
27. Srorzey, B. — »Die cambrischen und silurischen Geschiebe Schleswig-
Holsteins und ihre Brachiopodenfauna. I. Geologischer
Theil”.
Arch. f. Anthrop. u. Geol. Schleswiz-Holsteins u.s.w., I, 1, p. 35—136 ; [895
28. Linpstriém, G. — » Beschreibung einiger obersilurischer Korallen aus der
Insel Gotland”.
Bihang till kh. Svenska Vet.-Akad. land, XXI, Afd. IV, no. 7; 1896
29. Bonnema, J. H. — »De sedimentaire zwerfblokken van Kloosterholt
(Heiligerlee).
Versl. v. d gew. Verg. d. Wis- en Nut. Afd. d. Kon. Ak. v. Wet. v. 29 Jan.
1898, dl. VI, p. 445 453.
30. CumreLewski, C. — »Die Leperditien der obersilurischen Geschiebe des
Gouvernement Kowno und der Provinzen Ost- und
Westpreussen’’.
Schrift. d. phys-oekon. Ges. zu Kénigsberg, Jz. 41, p. 1—38 ; 1900.
31. Bonnema, J. H. — »Leperditia baltica His. sp., hare identiteit met Leper-
ditia Eichwaldi Fr. v. Schm. en haar voorkomen in
Groninger diluviale zwerf blokken’.
Versl. v. d. gew. Verg. d. Wis- en Nut. Afd. J. Kon. Ak. v. Wet v. 30 Jun:
1900, dl. IX, p. 138—140.
32. Jonker, H. G. — »Bijdragen tot de kennis der sedimentaire zwerfsteenen

in Nederland.
I. De Hondsrug in de provincie Groningen.
1. Inleiding. Cambrische en ondersilurische zwerf-
steenen’.
Acad. Proefschrift, Groningen, 1904.

( 704 )

33. Jonker, H. G. — »Bijdragen tot de kennis der sedimentaire zwerfsteenen
in Nederland.
TI, De Hondsrug in de provincie Groningen.
2. Bovensilurische zwerfsteenen.
Eerste mededeeling: Zwerfsteenen van den ouderdom
der oostbaltische zone G”.

Versl. v. d. gew. Verg. d. Wis- en Nat. Afd. d. Kon. Ak. v. W. v. 28 Jan.
1905, dl. XU, 2. p. 548—565.

Groningen, Min.-Geol. Institute, April 4, 1905.

Anatomy. — “Note on the Ganglion vomeronasale.” By E. pr Vrins.
(Communicated by Prof. T. Pracr.)

The deseription and drawings given in this note derive from a
wellpreserved human embryo. This embryo was fixed in a ten
percent solution of formaldehyde. After fixation the greatest diameter
was 55 mm. Precise information as to the probable age of this
embryo was not to be obtained, but the dimension of the embryo in
connexion with the fact, that the corpus callosum was not yet
formed, makes it probable that the age of the embryo may be
estimated between 2'/, and 3 months.

After the embryo being hardened in alcohol the head was cut off
along the base of the crane and imbedded in paraffin; a complete
series of frontal sections of 10 u was made. A slight deviation from
the frontal plane existed, so that the top of the right hemisphere
first appeared in these sections. The greatest part was stained in
haematoxylin and eosine in the usual manner, the rest of the sections
with haematoxylin only, in slightly different ways.

A description is given of the right hemisphere, — which in the
microscopical sections corresponds with the left one —, concerning
only that part which has a closer relation to the rhinencephalon.
This description is illustrated by four drawings of successive sections
and by two semi-diagrammatic figures.

These figures (Fig. V, VI) are a projection of the olfactory lobe
on a sagittal plane and constructed from the series of sections.
Because the plane upon which the projection is performed is sagittal,
only these curvatures of the olfactory lobe are seen, which have a
component in that direction. The lines in these drawings denoted
from I to IV indicate the place of the four sections marked with
a corresponding roman number.

The olfactory lobe, as seen in this stage of development, forms

( 705 )

a hollow outgrowth from the base of the hemisphere vesicle. On
the external surface of the lateral wall of the hemisphere, the lobe
is limited by a shallow sulcus, the fissura rhinica. This suleus runs
in a fronto-occipital direction (fig. I, I, HI PF. rh). On the external
surface of the mesial wall of the hemisphere vesicle the olfactory
lobe is bordered by a very broad suleus which in the beginning
runs also in a fronto-occipital direction but bends afterwards more
vertically. This suleus is the fissura prima of His and only to be
seen in the first two figures (fig. I, IL F. pr.).

Bordered by these two grooves the olfactory lobe shows a double
curvature from lateral to mesial and slightly from behind forwards.
The anterior cornu of the lateral ventricle forms a prolongation in
the olfactory lobe reaching into the top of the bulb. This cavity
shows the same curvatures as the lobe, which can partly be seen
from the diagrammatic figure V. In its general feature and apart
from its curvatures this cavity of the olfactory lobe has the shape
of a funnel, the mouth turned to the lateral ventricle the tube to
the top of the olfactory bulb.

A close relation between the form of the external and internal
surfaces of the hemisphere vesicle does not exist. The internal surface
of the lateral wall is thickened by the appearance of the corpus
striatum. This thickening of the wall begins wellmarked at some
distance (2mm) from the top of the hemisphere vesicle; a prolonga-
tion of this thickening, described by His as the “Crus epirhinicum”,
which, along the top of the hemisphere unites the striatum with the
rhinencephalon does not seem to exist. The ventral edge of the striatum
is also clearly marked by a prominent crest, the crista ventralis
corporis striati; (fig. I, Cr. v. str.) which is bordered by a deep sulcus
(fig. I, S. v. str.). This suleus on the internal surface of the vesicle
does not agree in all respects with the fissura rhinica on the external
surface.

The ventral edge of the striatum first proceeds in a fronto-occi-
pital direction and then turns more ventrally over the posterior wall
of the funnellike outgrowth of the rhinencephalon. By its typical
configuration it is easy to follow this ventral edge of the striatum
till it goes over in an analogous formation belonging to the rhinen-
cephalon.

This formation of the rhinencephalon appears as a thickening of
the internal surface of the mesial wall of the hemisphere vesicle. It
begins pretty well marked a little more distant from the top of the
hemisphere than the striatum. Dorsally and ventrally this thickening
is limited by a deep groove, the suleus rhinencephali dorsalis and

( 706 )

ventralis (fig. I and H, 8. rh. d. and S. rh. v.). The ventral edge
of this thickened part of the mesial wall forms a prominence, which
goes over in a crest, the erista ventralis rhinencephali.

This crest first runs in fronto-occipital direction and then turns
more ventrally over tne posterior wall of the funnellike outgrowth
of the rhinencephalon where it goes over continuously in the same
formation proceeding from the stratium. This is clearly seen in figure II
(Cr. v.) where the ventral erista is seen on the posterior wall of
the depression of the rhinencephalon cut in a very oblique direction.

The line described by this ventral border of the corpus striatum
and thickened part of the rhinencephalon has, looked at as a whole,
the form of a horseshoe with its top directed to the occipital pole
of the brain and meantime turned ventrally, while its opening is
turned to the frontal pole of the hemisphere vesicle. The connection
of rhinencephalon and striatum, which lies initially in the base of the
brain comes with the outgrowth of the rhinencephalon partly on the
posterior wall which borders the cavity, that proceeds in the olfact-
ory lobe. This connection between striatum and rhineneephalon is
therefore a primary one.

The olfactory bulb in this stage of development of the rhinence-
phalon is limited by a circular groove, the suleus circularis bulbi
(Fig. I, H, Se. b), which deeply cuts in on the frontal pole of the
bulb, becomes more flat on both sides and is seen as a round
shallow groove at the posterior pole of the bulb (Fig. V S. ce. b.).
The top of the bulb is turned to the mesial side and in a slightly
forward direction, while the form of the bulb can be seen in the
diagrammatic drawing figure V.

The nerves which belong to the formation of the rhinencephalon
are of two different kinds, and leave the brain at two different
places. The first kind of nerves proceed from the top of the olfactory
bulb. They are easily recognised by the fact, that their nuclei are
small and not very numerous, so that the fundamental substance in
which they are imbedded is distinctly seen.

These nerves split up into very small tracts in the neighbourhood
of the mucous membrane of the nose, where they seem to end. These
nerves, which contain the olfactory nervefibres do not have any
connection with the ganglion olfactorium. They all pass along this
ganglion.

The second place from where the nervefibres proceed is given by
the mesial part of the suleus circularis bulbi. These nervefibres can
be differentiated from the olfactoryfibres by the fact, that their nuclei
are a little larger, and more numerous than the nuclei of the

ERNST DE VRIES: “Note on the Ganglion vomeronasale.”

Proceedings Royal Acad. Amsterdam. Vol. VII.

( 707 )

olfactory nervefibres; the fundamental substance in which these nuclei
are imbedded deeply stains with eosine. Where these fibres leave
the brain, the superficial layer of the hemisphere vesicle becomes
richer in elongated nuclei (A vn. fig. I, II, HI, IV and VI). They
form four bundles (Rd. N. vn. tig. Ill, IV and VI), which all
converge into the ganglion olfactorium (G. vn. fig. HI, IV, VI). The
nerves leaving this ganglion are arranged in five bundles (Fig. VI
Nn. vn) which all went to the mesial side along the cartilagineous
septum nasi (Fig. 1 Sp.m.n.). Unfortunately the course of these nerves
could be no further traced out, the head being cut off too close along
the base of the brain. In a second human embryo however, of the
same age, which was not so well preserved, it was possible to find
back the same relations and to see, that all these nervefibres run
exclusively to the organon vomeronasale (RuyscH, Jacopson). In the
whole course of these nerves ganglioncells are seen. The so called
ganglion olfactorium has therefore no connection with the olfactory
nervefibres but is the sensorial ganglion belonging to the organon
vomeronasale.

In figure VI is given a semidiagrammatic drawing constructed
from the sections where the whole apparatus belonging to the organon
vomeronasale is projected upon a sagittal plane. The ganglion vomero-
nasale (G. vn.) is seen in the niveau of the top of the olfactory bulb
from which proceed to the periphery the nervi vomeronasales (Nn.
Vn.) and to the centrum the so called internal olfactory roots (Rd. N. vn.).
which enter the brain in a large triangleshaped zone, the area
vomeronasalis (A. vn.)

In the guinea pig these relations are slightly different. From the
organon vomeronasale proceed two nervestrands, which at a short
distance and still in the submucosa of the nose have each a ganglion.
This paired ganglion vomeronasale has two roots which very soon
unite and intermingle with the nervi olfactorii, with which they
perforate the lamina cribrosa. Arrived at the base of the brain they
enter the olfactory bulb over a large area, reaching from the sulcus
circularis bulbi at the mesial side to some distance from the same
sulcus at the lateral side of the olfactory bulb.

Probably the same relations occur through the whole series of
vertebrate animals. Though the existence of an organon vomeronasale
can be doubted in anamnia, it seems very probable, that the
nerve described by Locy (Anat. Anz. 1905, Heft 2 and 3) in
Selachii is identical with the nerve of the organon vomeronasale
as described here.

From the preceding description it is obvious, that we have

( 708 )

to consider the organon vomeronasale as a special senseorgan of
which the function is unknown, while the duality seen in the
central tracts belonging to the rhinencephalon finds its source in the
anatomical independence between the system of the olfactory nerves
and the system of nerves belonging to the organon vomeronasale.

My thanks to Prof. J. W. LANGELAAN under whose direction these
researches were made.

Anatomy. — “Note on the Innervation of the Trunkmyotome”.
By J. W. vay Bissenick. (From the Anatomical Institute at
Leiden). (Communicated by Prof. T. Prace).

These researches form a sequel to professor LANGELAAN’S first
communication “On the Form of the Trunkmyotome”*), and were
performed under his direction in the anatomical institute at Leiden.

The aim of this research was to know if one single spinal nerve
innervates only one single myotome.

The method followed, existed in dissecting a spinal nerve and to
see if the different territories to which the nervestrands can be
followed, belonged to one and the same myotome. To this purpose
an Acanthias or a Mustelus was cut through along the mid-sagittal
plane and treated with a one tenth percent solution of osmie acid.
The nerves stained black and were easy to follow with the naked
eye or with a magnifier.

As a first result it was found, that all nerves passed through the
connective tissue laying between the myotomes; therefore a minute
dissection of this tissue was necessary.

The myotome itself is covered by a very thin layer of a fibrous
tissue which constitutes a perimysium. This perimysium extends
between the muscular fibres of the myotome forming an endomysium.
It affords a continuous investment for every muscular fibre and
forms in this way a frame for the muscular tissue. Where this mus-
cular tissue is broken off the framework is continuous and enables
us to recognize parts of the myotome belonging together. The
myotomes covered by their perimysium are separated by a coarser
and denser fibrous tissue. This intermyotomal tissue forms lamellae
which have only a very loose connection with the perimysium, so that
it is possible to dissect these lamellae as discrete formations. These
intermyotomal septa pass over in the fibrous tissue of the skin and
form a continuous formation with the latter. Where the myotome

1) Proc. K. Akad. W. Amsterdam 28 May 1904.

( 709 )

has a simple form, this line of insertion coincides with the border
of the myotome; where the myotome is elongated in a peak, this
line of insertion crosses this peak.

Figure I reproduces the external surface of the myotome extended
in a plane. The black line indicates the transition of the inter-
myotomal septum in the skin; where the myotome is elongated in
a peak, it has distended the septum, because the line of transition is
fixed upon the: skin. The peak is covered by this distended part of
the septum, and as far as the peak is adjacent to the skin, this
part of the corium is doubled by this triangular sheath. Whereas on
the line of transition the passage of the intermyotomal septum into
the corium is a direct one, this is not the case with this adjacent
part of the septum, which is only loosely connected with the corium
by means of some fibres of connective tissue. This makes it possible
to dissect these triangular slips from the corium.

In the same way as the myotomes, the triangular slips of the
intermyotomal tissue overlap. In concordance with the direction of
the peaks it is seen, that slips belonging to peaks directed towards
the caudal end of the body cover each other, so, that the more
caudal slip covers the more cranial one. If the peak is directed
cranially the mode of overlapping is reversed, the more cranial slip
being uppermost. Figure II reproduces the intermyotomal tissue as
far as this formation is adjacent to the skin.

On the mesial side the intermyotomal septum goes over in the
connective tissue which covers the axial skeleton and beyond this
forms a lamella between the left and right half of the dorsal mus-
culature. Ventrally the same formation goes over in the fascia trans-
versa covering the abdominal cavity.

Figure III gives the line of passage of the intermyotomal septum.
As can be seen there are two places where the muscular tissue is
broken off, the myotome becoming thinner from outside to inside.
The lamellae, where the muscular tissue is interrupted, cover each
other and in this way two strong continuous septa are formed.
The distance over which the muscular tissue is discontinuous in the
neighbourhood of the sagittal plane amounts to four myotomes in
the first septum and to three in the second. In agreement with this,
the lamellae are built up resp. by four and by three sheaths of
intermyotomal tissue. The dotted fields in figure III belong therefore
together, forming one myotome, as can easily be verified by dissecting
the myotome.

Each spinal nerve springs from the cord with two roots, which
separately leave the spinal canal through two foramina (AR and

( 710 )

PR fig. IV). When they have quitted the canal each root separates
into two filaments, one of these filaments is ascending (Asc. f.) and
one is descending (Dese. f. fig. VI). Both ascending -root filaments
unite to form a nerve, the internal branch of the posterior division
(fig. VD, the filaments of which pass over in the intermyotomal
septum at the places indicated by 3 D—5 D fig. IV, and leave the
septum to go over in the skin at the places indicated in the same
way in fig. V. Before these filaments go over into the corium
they each give off a small twig innervating the distended part of the
intermyotomal septum, which is adjacent to the skin.

Before the two ascending rootfilaments join, they each give off a
small branch, which also unite to form a small nerve, the first external
branch of the posterior divisions (fig. VI) entering the septum at
2D fig. IV and leaving the septum to pass over in the skin at the
corresponding place of fig. V.

Both descending rootfilaments before joining each give off a
small branch, which form together a small nerve, the second
external branch of the posterior division (fig. VI), which enters and
leaves the intermyotomal septum at the places indicated by 1D in
fig. IV and V.

The nerves described, all together, innervate the dorsal part of the
myotome and the intermyotomal septum, and form the posterior
primary division of the spinal nerve.

The descending rootfilaments also unite to form a nerve which pretty
soon divides into two branches, one of these innervating the lateral
part of the myotome and the intermyotomal septum ; the other is, the
continuation of the maintrunk, crosses the lateral part of the myotome
and innervates the ventral part of the myotome and the intermyotomal
septum. The branch innervating the lateral part of the myotome
divides into two branches, an external and internal braneh of the
lateral division (fig. VI). The external branch splits up into two filaments
one of which is recurrent (recurrent br. fig, VI) and innervates the
top of the lateral part of the myotome. The external branch enters
the septum at £ 1.2. fig. IV and leaves the septum at 1, 24
fig. V. The internal branch gives off several branches passing over
in the skin at 3 L—6 LZ fig. V.

The branch innervating the ventral part of the myotome and the
intermyotomal septum shows the same arrangement as the branch
for the lateral part of the myotome. It divides into two branches
one being the external branch of the anterior division, the other the
internal branch (fig. VI). The external branch passes over in the
septum at V 1.2. fig. IV, splits up into two smaller branches of

ptum

(VL
vena
reach
nd V.
lexus

tomal
f the
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Curven

J. W. VAN BISSELICK: “Note on the Innervation of the Trunkmyotome.”

Fig. |. External surface of the myotome extended
in a plane

The blackline indicates the transition of the inter-
myotomal septum in the skin. Nat. Size

Il. Form of the intermyotomal tissue adjacent to tre corium. Nat. size Fig. III. Mesial surface of the myotome, with the line
of transition of the intermyotomal septum. Nat. Size

Fig. VI. Spinal nerve of Acanthias with omission
of the rami sympathici. Nat. size

Fig. V. External surface of the myotome, with the Places
where the nervefilaments leave the septum. Nat. size

Fig. IV. Mesial surface of the myotome, projected on a sa-
gittal plane, with the places where the nervebranches enter
the intermyotomal septum. Nat. size

Proceedings Royal Acad. Amsterdam. Vol. Vil

€ Cit )

which one is recurrent (recurrent br. fig. VI) and leaves the septum
to go over into the skin at 1 V and 2 V. fig. V.

The internal branch can be followed up to the vena lateralis (VZ,
fig. IV) and then goes over in a loose plexus. On its way to the vena
lateralis the internal branch gives off several filaments, which reach
the skin through the intermyotomal septum 3 V—6V fig. IV and V.
Before passing over into the skin these filaments form a loose plexus
covering the most ventral part of the myotome.

The roots and mainbranches of the spinal nerve have a submyotomal
position and are not bound in their course by the form of the
myotome; these branches on the contrary, which go over into the
septum to reach the skin, are in their course fixed by the form of
the myotome. The final course of the branches in the corium was
not traced out with enough accuracy to give results here.

The descriptions given in this note only apply to that region of
the trunk which is situated between the thoracie and first dorsal fin.

Conclusions :

I. One single spinal nerve only innervates one single myotome
and the intermyotomal tissue through which the nerves pass.

IL. The roots and mainbranches of the spinal nerve have a sub-
myotomal position; the branches never perforate a myotome, but
run always in the intermyotomal septum to the skin. In general
they are to be found between the perimysium and the intermyotomal
septum.

III. The spinal nerve shows a primary division into three parts, a
posterior, lateral and anterior division in agreement with the diffe-
rentation of the myotome in a dorsal, lateral and ventral part.

IV. All larger branches are mixed nerves containing elements of
the anterior and posterior roots. 5

Mathematics. — “On linear systems of algebraic plane curves”.
By Prof. JAN pe Vries.

$ 1. The points of contact of the tangents out of a point O to
the curves ce" of a pencil lie on a curve *—! which I shall call the
tangential curve of QO. It is a special case of a curve indicated by
Cremona’). By Emm Werr?), Guccta*) and W. Bouwman‘) it has
been applied when proving the properties of pencils and nets.

1) Cremona—Currze, Einleitung in eine geometrische Theorie der ebenen Curven
(1865) p. 119.

°) Sitzungsberichte der Akademie in Wien, LXI, 82.
5) Rendiconti del Circolo matematico di Palermo (1895), IX, 1.
4) Nieuw Archief voor Wiskunde (1900), IV, 258.

(712)

If a linear system (c”);, of ook curves c* is given, we can consider
the locus of the points P41, where a curve of that system has a
(k-+ 1)-pointed contact with a right line, passing through the fixed
point O.

To determine the order g(#) of the locus (P+!) I consider the
curves (c"), having in the points P of the right line 7 a k-pointed
contact with the corresponding right line OP. Each ray OP cuts
the curve individualized by P moreover in (n — 4) points Q. Each point
of intersection of / with the locus of the points Q being evidently
a point P41, the locus (Q) is a curve of order ¢(f).

The curves of (c”); passing through O form a system (c”),-1. The
order of the locus of the points P‚ where a c” of this latter system
has a k-pointed contact with OP is evidently indicated by g(& — 1).
So on / lie g(k—1) points P for which one of the corresponding
points Q coincides with QO; in other words the locus (Q) passes
g(k —1) times through O, so it is of order g(& — 1) + (n — 2).

To determine ¢(/) we have now the recurrent relation

g(k) = elk — 1) + (n— 2A).
From this we deduce
g(k) =p (1) + 4 (A — 1) (Qn -- k — 2).
Here g(1) represents the order of the tangential curve, thus (22 — 1).
So we find

pk) = 4 (& + 1) Qn —k).

The locus of the points where a curve c*‚ belonging to a k-fold
infinite linear system has a (k + 1)-pointed contact with a right line
passing through a fired point O is a curve of order 3 (k + 1) (2n — B,
on which O is a &k(k-+1)-fold point.

For (c")¢ determines on a right line 7 through O an involution of
order n and rank &. The number of (4 + 1)-fold elements of this in-
volution amounts to (&—+1)(n — £); that is at the same time the
number of points Pp, lying on r. Consequently O is an $4(k + 4)-
fold point on (Pi).

§ 2. Each ray 7 through a fixed point O is touched by 2 (2 —1)
curves c” of a pencil (c”); the points of contact 7’ are the double points
of the involution determined by (c") on +. The curves c” indicated by
these points 7’ intersect 7 moreover in 2(n—1)(n— 2) points S.
When 7 rotates round O the points |S will describe a curve which
I shall call the satellite curve of O.

This curve passes (n-+-41)(n—- 2) times through O; for if r -
coincides with one of the tangents out of Q to c” passing through

( 713 )

O one of the points S lies in O. So the curve (S) ús of order
(n +1) (n — 2) + 2 (n — 1) (n — 2) = (rn — 2) Bn — DV.

If B is a base-point of (cn), then only 2 (n — 2) points 7’ (the double
points of an /*—!) lie on OB outside O and B. So OB touches in
B the tangential curve of O whilst it is (m — 2)-fold tangent of ($).

Each of the 2 (n — 2) eurves c* touching OB projects a point Sin B.
So each base-point is a 2 (2 — 2)-fold point of the satellite curve.

The common points of the tangential curve @"—! and the satellite
curve sr?) form four groups.

First there are (n + 1) (n — 2) united in O.

Secondly 2 (n — 2) lie in each base-point B.

Thirdly the two curves touch each other at each inflectional point
sending its tangent through 0.

‘Fourthly they cut each other in the points of contact of each
double tangent passing through O.

Now the inflectional tangents of a pencil envelop a curve of class
dn (n — 2). *)

So the number of points of contact of inflectional tangents through
O amounts to
(n — 2) (Bn — 1) (2n — 1) — (n — 2) (n + 1) — 2 (n—2) n? —6n (n—2) =

= 4n (n — 2) (n — 3).

The double tangents of the curves c* belonging to a pencil envelop

a curve of class 2n(n — 2) (n — 3).

§ 3. Following Emi. Weyr’) we consider the curve ct! gene-
rated by the pencil (c") with the pencil projectively conjugate to it
of the tangents in a base-point B. As each c” cuts its tangent more-
over in (2 — 2) points, B is a threefold point of the c’+!. From this
ensues easily that through B can be drawn (n + 4) (n — 3) tangents
to etl, As many double tangents of the pencil (ct) have one of
their points of contact in B.

We shall now consider the satellite curve of B. On each ray r
through B lie 2(m— 2) points of contact 7, so 2 (n — 2) (n — 8)
points S. If # coincides with one of the double tangents just mentioned,
one of the points S lies in B. So B is an (n+ 4) (n — 3)-fold point
on (S) and the order of (S) proves to be equal to (n + 4) (n — 3) +
2 (n -— 2) (n — 3) = 3n (n — 8).

The tangential curve of B has in B a threefold point; for a ray

1) For this is the number of tangents of pe

of OB can be drawn through O.
°) Sitzungsberichte der Akademie in Wien LXI, 82.

which besides the 7? tangents

(714)

through B bears but (2n—4) points 7, whilst the curve ¢ is of
order (22 — 1).

Of the common points of "—! and s*"@—®*) there are 3 (n + 4) (n — 3)
lying in B, 2 (2 — 3) in each of the remaining (n° — 1) base-points and
two in each of the inflectional points sending their tangent through B.

The number of those inflectional tangents is 3” (n — 2) —9, as
each of the three inflectional tangents, having their inflectional point
in B, must be counted three times. This is evident when we consider
a curve of (c*), where a base-point can lie only on inflectional
tangents for which it is inflectional point itself. This number amounts
to three, whilst the class of the envelope of the inflectional tangents
is nine.

So we find for the number of the points of contact, not lying in
B, of double tangents out of B
37(n—8) (2n—1) —3 (nd) (n—3) —2 (n—8) (n?>—1) —6 (n—8) (n41) =

=4(n—3)nm—J4An+1).

So B lies on 2 (n — 4) (n — 3) (n + 1) double tangents. This num-
ber is 2 ( 3) (n +4) less than the number of double tangents out
of an arbitrary point. The ( 3) (n+ 4) double tangents having
one of its points of contact in B must thus be counted twice.

The envelope of the double tangents has in each base-point an
(n + 4) (n — 3)-fold point.

§ 4. The locus of the points of contact D of the double tangents
of (ct) evidently passes (n + 4) (7 — 3)-times through each base-point
($ 3). An arbitrary c” having on its double tangents 7 (7% — 2) (n? — 9)
points of contact D, the curve D and c” intersect each other in
n° (n + 4) (n — 8) + n(n — 2) (n° —-9) points. Consequently the locus
of the points of contact D is a curve of order (n—8)(2n?-+-5n—6). ')

We shall now consider the locus of the points IV in which a
cr is intersected by its double tangents.

As each base-point B lies on 2 (m — 4) (n 3) 2 + 1) double tan-
gents ($ 3) the curve W passes with as many branches ith B.
So it has with an arbitrary c* in common 22° (n— 4) (n—9) (n-+1) +
+ 42 (n — 2) (n? — 9) (n —4) points. From this ensues that the curve
(W) is of order 4 (72 — 4) (n — 3) (5n? + 5n — 6).

The curves (D) and (W) have outside the base-points a number of
points in common equal to

4 (n — 4) (n — 3)? (5n? + 5n — 6) (An? + Sn — 6) —
— 2n? (n — 4) 3)? (n + 1) (n + 4).

1) See P. H. Scuoure, Wiskundige opgaven, II, 307.

From this ensues :
In a pencil (c*)

4 (n — 4) (n — 3)? 10n* + Bn! — 21n? — 80n + 20)

curves have an injlectional point of which the tangent touches the
curve in one other point more.

§ 5. The locus of the inflectional points / of (ct) has a threefold
point in each base-point and a node in each of the 3 (n — 1)’ nodes
of the pencil, out of which we immediately find that tbe curve (/)
is of order 6 (7 — 1) and of class 6 (n — 2) (4 n -— 3) *).

Let us now deduce the order of the locus of the points V deter-
mined by a e# on its inflectional tangents.

As a base-point / lies on 3 (n — 3) (n + 1) inflectional tangents
the curve (WV) passes with as many branches through B. So with
an arbitrary c* it has 37° (2 — 3) (n + 1) + 3n (n — 2)(n— 3) points
in common.

Consequently (V) is a curve of order 3(7— 3) (n? + 2n—2). Now
the curves (/) and (J”) have besides the base-points a number of
points in common represented by

18 (n — 1) (n — 3) (n° + An — 2) — In' (n — 3) (n + J).

These points can only have risen from the coincidence of inflectional
points with one of the points they have in common with the c” under
consideration, thus from tangents with fourpointed contact. Such an
undulation point, being equivalent to two inflectional points, is point
of contact for (J) and (WV) from which ensues:

5 ene)
A pe a wm . » 5
A pencil (c’) contains a (n

3) (n° + n? — 8n +4) curves with an

undulation point.

§ 6. Let a threefold infinite linear system of curves c* be given.

The ect osculating a right line / in the point P cuts the ray OP
drawn through the arbitrary point O moreover in (n—1) points Q.

The curves of (c"), passing through O form a net (er), determining
on / the groups of an involution /,”. The latter having 3 (2 — 2)
threefold elements, the locus (Q) passes 3 (n — 2)-times through O,
so it is of order (4 — 7).

Each of its points of intersection A with / is evidently a node on
a curve of (ct), with / and OK for tangents.

Each right line is nodal tangent for (An —7) curves of the system.

From this ensues that the locus of the nodes K sending one of

2) See Bosek, Casopis (Prague), XI, 283.

49
Proceedings Royal Acad, Amsterdam. Vol. VII.

( 716 )

their tangents through the point M chosen arbitrarily is a curve of
order (4m — 5); for M is a node of a c", so it lies on two branches
of (K).

Each point A’ of the arbitrary right line / is a node of a curve
belonging to (c”),. The points of intersection Mand M' of the tangents
in AK with the right line m chosen arbitrarily are pairs of a sym-
metric correspondence with characteristic number (47 — 5). To the
coincidences belongs the point of intersection J/, of / and m, and
twice even, because the c”, having in that point a node, furnishes two
points J/,' coinciding with J/,. The remaining coincidences originate
from tangents in cusps. From this ensues:

The locus of the cusps of a threefold infinite linear system of
curves of order n is a curve of order 4 (An — 3).

Mathematics. — “Some characteristic numbers of an algebraic
surface.’ By Prof. Jan pm Vrins.

In tbe following paper we shall show how by easy reasoning
we can find an amount of the characteristic numbers of a general
surface of order 7'). To this end we shall make use of scrolls
formed by principal tangents or double tangents.

§ 1. First I consider the scroll A of the principal tangents a of
which the points of contact A lie in a given plane a. The curve —
«along which « cuts the surface og" is evidently nodal curve of
A. The tangents in the 87 (7% — 2) inflectional points of «* being
principal tangents of g”, the seroll A has 37 (2 — 2) right lines and
the curve @ to be counted twice in common with 0”, so if is a
scroll of order 7 (Bn — 4).

The two principal tangents « and «im a point of e@ have each three
points in common with 9g”; consequently a” belongs six times to the
section of A and 4”. These surfaces have moreover a twisted curve
of order 1? (82n—4)—6n in common containing the Su(n— 2) w— 3)
points where gr is cut by the principal tangents a situated in «. In
each of the remaining (lin — 24) points of intersection of this
curve with « the surface o” has four coinciding points of inter-

section in common with a. From this ensues:
The locus of the points in which p*_ possesses a fourpoited tangent
(fleenodal line) 7s « twisted curve of order n (1ln—24).

1) We find the indicated numbers in Satmon-Fiepier, “Analytische Geometrie
des Raumes”, dritte Auflage, IL, p. 622—644, and in Scuupert, “Kalkül der abzählenden

Geometrie”’, p. 236.

CMe)

§ 2. I now determine the order of the scroll B formed by the
principal tangents cutting $? in points 6 of the plane @.

Out of each point B of the section 8" start (n— 3) (n? + 2)
principal tangents; this number indicates at the same time the
number of sheets of B which cut each other along gr". The inflect-
ional tangents lying in ~” evidently belong (— 8)-times to the
indicated scroll. So its order is equal to
3) (n? + 2) + 3n (n — 2) (n — 8) = n(n — 1) (n — 3) (n + 4).

According to § 1 (38n*—4n—6) principal tangents have their
point of contact A on a” and one of their points of intersection
B on Br. So this number indicates the order of the curve along
which @* is osculated by B. Beside this curve of contact and the
manyfold curve #8” the surfaces ¢" and B have still in common the
locus of the points B’ which determine the principal tangents AB
moreover on 9”. This curve (B) is of order n° (n—1) (n—3) (n-+-4) —
—3n (Bn? — An — 6) — n(n — 3) (nV? + 2) =n (n — 2) (n—A4) (n° ++5n-+8).

n (n

§ 8. To find how often the point A coincides with one of the
(1 — 4) points B’, I shall project the pairs of points (A, B) out of
a right line /. The planes through / are arranged in this way in a
correspondence with the characteristic numbers 7 (82?—4 n—6) (n—4)
and 7 (n — 2) (n — 4) (n? + 5n-+8). Each right line a resting on /
evidently contains (7—4) pairs (A, B), so it furnishes an (7 — 4)-fold
coincidence. The remaining coincidences originate from coincidences
A= B. Now n (8n?—4 n—6) (n—4) + n (n—2) (n—A4) (n? +5 nt3)

n (n—1) (n—38) (n-++-4) (n—4) = n (n—4) (6n?+2n—24). So this is
the number of fourpointed tangents which cut 9” in a point B of pr.

The points of intersection of &” with its fourpointed tangents form
a curve of order 2n (n — 4) (Bn? + n — 12).

If / is the order of the scroll of the fourpointed tangents then it
is evident that we have the relation
nf = dn (lln —24) + 2n(n — 4) (Bn? + n—12) = 2n? (n— 3) (Bn—2).

The fourpointed tangents form a scroll of order 2n(n—3)(8n—2).

If we make the point of contact /’ of a fourpointed tangent to
correspond to the (7 — 4) points G which that tangent has still in
common with 9”, a system of pairs of points (/’, G) is formed, of
which the number of coincidences can be determined again with the
aid of the correspondence in which they arrange the planes through
an axis /. By the way indicated above we find for this number :
n(1in—24) (n—4)+2n(n—A) (3n?-+-n—12 )—2n(n—28) (8n—2) (n—4)=
n(n— 4) (85n—60).

The surface P* possesses 5n (i — 4) (In —12) fivepointed tangents.

49*

(785)

$ 4. Returning to the scroll B ($ 2) I consider the points of
intersection of the twisted curve (5') with the plane @. Each point
of intersection of ¢” with an inflectional tangent lying in @ can, be
regarded as the point B, each one of the remaining (7 —4) as a
point 5'. Hence the curve (B) meets 3u (n — 2) (n — 3) (n — 4)-times
Br on the inflectional tangents of 3. In each of the remaming points
of intersection of (B) with 3 we find that &* is touched by a right
line having elsewhere three coinciding points in common with 9”. Such
a right line is called by me a tangent 423, A being its point of
osculation, B its point of contact.

The points of contact of the tangents tas form a curve of order

n(n — 2) (n — 4) (n° + 2n + 12).

$ 5. In each point C of the curve y” according to which 9” is
cut by the plane y I shall regard the (n— 3) (n + 2) tangents ¢ which
touch pr moreover in a point €’. On the seroll C of the double
3) (n+2)

sheets meet. Each double tangent situated in y representing two right

tangents c the curve y” is a manyfold curve in which (7

lines of C the order of this seroll is equal to

n(n—)(n + 2) + n(n — 2) (n — 3) (21+ 3) or n(n —3) (n° + An — 4).
The surfaces g and C touch each other along the locus (C’) of
the two points of contact. Of this curve the plane y contains the
points of contact of the right lines ec lying in y besides the points
C=C", where a right line ¢ is a fourpointed tangent. So the order
of (C’) is n (n—2) (n?—9) + n (lln—24) or n(n? — 2n? + 2n — 6).
Besides the curve (€’) to be counted twice and the curve y” to
be counted 2(n— 3) (n + 2)-times C and ¢ have moreover in
common the locus of the points S determined by the double tangents
c on $”. The curve (S) is of order n° (n — 3) (n° + An — 4) —
2n(n?—2n? + 2n—6) — 2n(n—3) (n-+2) or n(n—4) (n° +- n?—4n—6).
To the points of (S) lying in y belong the points of intersection
of y" with its double tangents c. As each of the two points of contact
of ¢ can be regarded as point C these points of intersection ‚S must
be counted twice. The remaining 2 (7 — 4) (n° + n° — An — 6) —
n(n —- 2) (n? — 9) zw — 4) points S lying in y are apparently points

of osculation of the tangents 423. So from this ensues:
The points of osculation of the principal tangents touching @*
moreover elsewhere form a curve of order n (n — 4) (38n? + 5n— 24).
The curves (A) and (4) formed by the points of osculation and
the points of contact of the tangents 43 have the points of contact of
the fivepointed tangents in common. Taking this into account we
find (by again projecting out of an axis /) for the order of the

( 719 )

seroll of the right lines #3 the expression 1 (”—2)(n—4)(n? +-2n-4-12)+
+ n (n—A4) (82? +5n—24 , (a —4) (Tn—12).

The principal tangents of @* whieh moreover touch the surface
form a scroll of order n(n — 3) (n — 4) (nr? + 6n — 4).

§ 6. The double tangents ¢ cutting ¢° in points D of the plane d
form a scroll D, on which the section d” of 9" with dis a manyfold
curve bearing } (7 n° + n+ 12)°) sheets. As moreover
every double tangent of & belongs to (7 — 4) different points D the

order of D is equal to

3) (n—4) n° dn?) + bn en n—_3) (n-++3) (n—4) =
n (n—1) (n+-2) (n—3) (n—4).

According to § 5 n(n — 4) (n° + n° — dn — 6) double tangents c
have one of their points of contact C in a given plane y and at the
same time one of their points of contact D in the plane d. So this
number indicates the order of the curve along which D and 9»
touch each other. If we take the manyfold curve dr into consideration,
it is evident that the points D’ which the right lines of D have in
common with g” besides the points of contact C and the points of
intersection lying in d, form a twisted curve (D’) the order of

which in equal to
nd Wet) (n—8) (n—4) — An (n—A4) (n?-+-n?—4n—6) —
ie (n 4) n° dn) = $n (n—2) (n—A) (n—5) (An* H5n3).

This curve evidently cuts d (n—4) (7—5)-times on each double

tangent of dé”. In each of its remaining points of intersection with d
the surface @” is touched by a right line, which is tangent to the
surface in two more points. From this ensues :

The points of contact C of the threefold tangents of o form a
curve (C) of order $n (n—2) (n—A) (n—5S) (n?+5n+12).

$ 7. On each right line c of the scroll D lie ( points D’
which can be arranged in 4 (n—5) (n—6) pairs D', D". If these pairs
of points are projected out of an axis / by pairs of planes 2’, 4", these
form asymmetric system, the characteristic number of which is $ 2(m—2)
(u—4) (n—S) (2n?+5n-+3) (n—6). Each right line ¢ cutting / deter-
mines a plane 4 evidently representing (n—5) (n— 6) coincidences
a an

A’ = 2". The remaining coincidences of the system (4) originate from
coine eee. D = D", thus from threefold tangents d. As however

1) In Cremona—Currze, Theorie der Oberfldchen, page 66 we find the expression
} (n—3) (n—4) (n?+n—2) by mistake for the number of double tangents cutting
g” in one of its points,

( 720 )

each of the three points of contact of a right line d can be formed
when D' coincides with YD" the number of threefold tangents cutting
g” on the curve dr is but the third part of the number of the indi-
cated coincidences of (4), thus equal to

+n (n—4) (n—5) (n—6) §(n —2) (2n?+-5n+8) — (n—1) (n+2) (n—3)} =
En (n—A4) (n—5) (n—6) (n?+3n?—2n—12).

This is at the same time the order of the curve (D) formed by
the points D which the threefold tangents d have still in common
with o”.

Now we can also find the order z of the seroll (d). This scroll
being touched by ©” in the points of (C) and being cut in the points
(D) we have namely

ne = n (n—2) (n —4) (n—S) (n?-++-5n-+12) +
+n (n—A4) (nN—5) (n—6) (n?+3n?—2n—12).

Out of this we find
The threefold tangents of a” form a scroll the order of which is
zn (n—3) n—A4) (n—5) (n?-+-3n—2) *).

§ 8. To find the degree of the spinodal curve I consider the pairs
of principal tangents a, a’ of which the common point of contact A
lies in the plane «. If two rays s and s of a pencil (S, 6) are con-
jugate to each other, when they rest on two right lines a and a’,
then in (S,6) a symmetric correspondence with characteristic number
n (dn —4) is formed. The coincidences can be brought to three groups.

First « and a’ can cut the same ray s; their plane of connection
is then tangential plane, their point of intersection A lies on the
polar surface of ‚S. Such a ray s coincides with two of the rays s'
conjugate to it. So the first group contains 7” (# — 1) double coinci-
dences.

Secondly s can cut the curve a; then too it coincides with two
rays s. So the second group consists of 7 double coincidences.

Finally a single coincidence is formed when « coincides with @’.
The number of these coincidences evidently amounts to 27 (Bn —4) —
2n (n — 1) — 2n = dn (n — 2). From this ensues :

The parabolic points form a twisted curve (spinodal line) of order

An (n — 2).

1) In Satmon-I'tepter we find on page 638 by mistake 7?-+-32-+ 2 instead of
ning.

On page 643 we find the derivation of the number of fourfold tangents and of
the numbers of tangents 45, ¢s,9,9 and {5,5

( 721 )

Mathematics. “The equation of order nine representing the locus
of the principal axes of a pencil of quadratic surfaces. By
Mr. K. Bers. (Communicated by Prof. J. CARDINAAL).

1. In These Proceedings of Jan. 28» 1905 appears a communication
by Prof. CARDINAAL: “On the equations by which the locus of the
principal axes of a pencil of quadratic surfaces is determined.”

2. Prof. CARDINAAL deduces three non-homogeneous equations of
order two between two variable parameters 4 and /, and tries to
arrive at the equation of the demanded surface by elimination of
these parameters. The result obtained by him (8) seems to be an
equation of order 12. This is incongruent with the result arrived at
geometrically, which made an equation of order nine to be expected.
This incongrueney is attributed to factors, which the equation arrived
at may contain, but these factors are not indicated.

3. The method of elimination described in my paper ‘Théorie
générale de Vélimination” (Verhandelingen, Vol. VI, n°. 7) gives the
means to set aside this incongruency and to determine in reality the
equation sought for by Prof. Carpinaan.

To this end we can start from his equations (5) after having made
them homogeneous with respect to the variable parameters, which
may be done by assuming the equation (1) of the pencil of surfaces
in the form:

pA +AB=0.

If now we develop the equations (5), they assume the following form :
(a,,4,+4,,4,+0,,A,)u’"+(a,,3,+0,,B,+a,,B,+b,,A,+6,,A,+6,,4,)Aut\

4(b,,B,+b,,B,+b,,B,)2+A,pktA ak=0,
(a,,A,+a,,4,+0,,A4,)u’+(a,,B,+0,,B,+a,,B,4b,,A,+b,,A,+),,A,)at

+b,,B,+b,,B,+b,,B,)2+Auk+B,dk=0 , (a)
(a,,4,+4,,A,+4,,4,)u°4(a,,B,+4,,B,+0,,B,4),,A,+b,,A,+0,,A,)aet+

+(b,,B,+-b,,B,.+b,,B,)+A,uk+Bak=—0.

The coefficients of these equations are linear functions of the

variable coordinates 7, y and z. To simplify we can introduce the
following notations :
ERO Ne ne ARL
Pe = Taal, a. Cre S Ta Ae :
Pr dn Ait ass + a,,A. ,
Q, = 4,,B, + 4,,B, + 4,,B, + b,A, + beds + b,,A,,
Q, =4,,B, + 4,,B, + 4,,B, + b,,A, + bad, + bad,
Q, Url Sn Belg ar Op dap a ba, dt brada 6,4, ,
R, = },,B, + 6,,B, + 6,,B, ,
i Dees Gielen SS ban
R, = 6,,B, + 6,,B, + 4,,B, ,

( 722 )

by which the equations («) pass into the following :
Py + Qa + RA + Ak + Bak =0,
Po? + Q,ap +R, + Auk + Bak = 0.) EN
Py? + Q,4u + R,2 + Ayu + Bak =0, |

4. Which condition now must exist between the coefficients of
these equations if they are to allow of a mutual system of roots?
The answer is that no condition is demanded for this. These equations
are namely satisfied independent of the value of the coefficients by
the system of roots:

A= 05 wi 0 e arbitrary.

The result arrived at by applying the method indicated in § 118
of my paper. “Théorie générale de l’élimination” agrees with this.
According to this method we should have to find for the resultant
the quotient of two determinants successively of order 15 and of
order 3. In the case under consideration where we have
a0) AO ande RAN
we always obtain, in whatener way we choose the determinants,
as quotient a quantity which is identically zero.

So the above-mentioned equation (8) can be nothing else but an
identity.

5. This result having been fixed it is no longer difficult to answer
the question how, to obtain the equation of the demanded locus.
To this end we must express the condition that the equations (4) are
satisfied by a second system of roots.

The condition in demand is, that all determinants are equal to zero
contained in the assemblant (85) appearing in § 118 of the already
mentioned paper. Applied to the equations (4) it gives but one
equation, namely

P, P P,

| 2 8
| Q, Fi Q, LE Q; P,

RP aR eRaRe oh Ps a
A, Q, A, Q, A, Q, |
B, R, Q, B, R, Q, B, R, Q, |—=0,
| R, R, Ri}
| BS A, il |
Bel Ba Be
B, B, B,

this being the equation of the demanded locus. It is of order nine
agreeing to the geometrical researches of Prof, CARDINAAL.

Physics. — “A formula for the osmotic pressure in concentrated
solutions whose vapour follows the gas-laws”’. By Dr. Pu.
KoHNsTAMM. (Communicated by Prof. J. D. van prr Waats).

§ 1. The formula for the osmotic pressure may be derived in two
different ways: by a thermodynamic and by a kinetic method. When
putting these two in opposition I mean by no means an absolute contrast,
on the contrary I believe an opinion which I hope soon to treat
more fully elsewhere — that without an equation of state based on
kinetic considerations thermodynamics has nothing to start from
and that therefore we can only oppose “purely kinetic” and “thermo-
dynamie-kinetie”” considerations.

Not numerous are those who have tried to find formulae for the
osmotic pressure of more concentrated solutions by a thermodynamic
method. Only Honpivs Bonpincu') and after him Van Laar ?) have
pointed out that it appears from the theory of the thermodynamic
potential that the concentration of the solution should not be taken
into account in the form w, but as log (1—2) and that for further
approximation a correction term of the form «c* must be applied,
and lately the latter has again come forward to advocate with great
zeal the validity of this result.

More numerous are the attempts to determine the osmotic pressure
in concentrated solutions by direct, molecular-theoretic methods; I
may mention those of Brenig *), Noyes ©), BARMWATER *), WIND ®).

This fact is surprising because Van “r Horr himself, though he
has a definite conception of the nature of the osmotic pressure, has

never dared to base his equations on it, but has clearly indicated
as basis of his theory of the osmotic pressure the thermodynamic
considerations, by means of which he derives the osmotic pressure
from the gas-laws. And; it is the more surprising because all
these attempts wish to follow the train of thought which led
Van per Waats to his equation of state, though VAN per Waars
himself has clearly shown, that in his opinion the osmotic pressure
must not be sought in this way, but by the thermodynamic method, in
connection with the equation of state given by him. That notwithstanding
this so often the other way has been followed, seems noteworthy to

1) Diss. Amsterdam 1893.

8) Zsch. phys. Ch. 15, 466 (1894).
8) Zsch. phys. Ch. 4, 444.

4) Zsch. phys. Ch. 5, 53.

5) Zsch. phys. Ch. 28, 115.

6) Arch. Néerl. (2) 6, 714.

me on account of the predilection which it shows for purely kinetic
considerations. The reasons why | do not share this predilection in
this case, will appear from another communication, occurring in
these Proceedings; here L shall confine myself to the thermodynamic
method, and specially to the form given by VAN pur Waars.

§ 2. In § 18 of his Théorie Moléculaire Vay per Waars treats the
ease, that of a binary mixture the first component can expand through
a given space, whereas the other is confined to a part of that space.
He demoristrates that for equilibrium a difference in pressure between
the parts of the space is required which for dilute solutions has the
value indicated by the law of Van r Horr. In this a thesis is used,
which is very plausible (and which moreover may be proved in the
same way as the condition for equilibrium in the general case)
that namely equilibrium is established when the thermodynamic
potential of the first component is the same in the two parts of
the space. I shall here apply this condition to a binary mixture
of arbitrary components and arbitrary concentration, the vapour
of which follows the gas-laws, and which is in equilibrium with
one of the components in pure condition under the pressure of its
own vapour through a semipermeable wall. How such an equili-
brium might be reached in reality in a special case, and whether
this would be possible, need not be discussed.

§ 3. We assume that there are (1—v) molecules passing through
the membrane and x non passing molecules, then the thermodynamic
potential of the first substance in the mixture is

‘Ow ‘Ow
WE fp, ap) 0 el. =d Gal ==

7 7
7 . 0
= | payer MRT a), | Ge) dv + F(T)
sf EIT

v

in which the integrations must be extended from a volume y so
large that all the laws of ideal gases apply there, to the volume in
question, F(T) being a function of the temperature, which occurs
here only as an additive constant. In order to be able to carry out the
integrations, we require — as mentioned above — an equation of
state p = (Oe):

For this purpose I shall adopt Var per Waars’ equation with
constant 4; though in this way we certainly do not get strietly
accurate results, yet we shall be able to decide about the quantities
which must occur in the formula.

§ 4. If in fig. 1 the isotherm of the mixture is indicated and

the horizontal line is drawn according to the well-known law of
Maxwerr *), then the pressure indicated by that line is what vAN DER
Waats calls the pressure of coincidence of the mixture and denotes
by the symbol p.. The volumes at the end of that line we call
Gc, and vr,
mixture in equilibrium in the above mentioned way.

and p, and #, represent pressure and volume of the

Now the integral f var may be split into three parts:
Le

Ye ve 7
fos | pdr + pdr
Ver Vn

Vo

vo

For the middle quantity we may write:
Ve,

Jae = Pe (Ve — Ve)

Ve,

As according to our suppositions the vapour follows the gaslaws,
we have:
Pe ve, = MRT.

For the same reason we may replace p in the third integral by
1 Through a mistake in the plate this line is drawn much too high here, Also

the form of the isotherm is imperfectly represented. But the figure is merely given
as a schemalic representation,

(126 )

MRT/v. Carrying out the integration we get MRT log y/vc,, for
which we may also write MRT log pe/p,. We get then:

Vey

[pe + porto = | pdv + po vo — Pe re, + MRT + MRT I p/p;

Vo Uo

§ 5. Let us consider the first three terms. The first is represented
in the figure by the area C+ D, the second by A + B, the third
by B-+ D. The three terms together are therefore A + C. If now
as we assumed, the vapour is very dilute, and therefore the tempe-
rature far from the critical, hence also the isotherm very steep when
cutting the line of coexistence liquid-vapour (or strictly speaking: at
the pressure p,, which however is very near the line of coexistence
on the liquid side), then we may neglect C by the side of A, and
we are the more justified in this as the pressure p, is higher, so the

mixture in question more concentrated. For C= {plo— D. If we

Uo
introduce

and integrate, we get:

a a MRT a
MRT 1 (v,,— 6) +-—— MRT 1 (v,—b) = ( :) (ve, = Vo)
n

cr Vo Vod Va

A MRT a MIT
—— )%— |t

vo=b Vo ve, —b Va
If we arrive at very high pressures, », — h approaches zero and

numerator and denominator become both infinite, but the denominator
of a higher order than the numerator. It is already apparent from

the form of the isotherm which becomes steeper and steeper, that
when neglecting C by the side of A we make proportionally a smaller
mistake the higher p, is. And that the neglect is allowed for small
osmotic pressures is selfevident. We may therefore put for the three
terms discussed in this §:
A = (po — pe) Vo.
7
‘(9p zie
$ 6. It remains to calculate the term { (2) ae This integral

v

’
Ve,

Ves yt
too we separate into three parts | + { ao f The last integral

vo Vey Veg
is now zero according to the law of Avoerapo. The middle one
we find from the equation already used above:

2
Ver

by differentiating, taking into consideration that the limits of the
integral are functions of wv.

We get:
2 a d a a
pe p ve |? pe Ver Ve,
= de 4- | p— (ve —Ve,) + Pe—— Pe St
| Een | [ = Ora. 5 li dc EE
Vo,
Now at the limits of the integral p is pe; we retain therefore on
the left and the right only the first members.
Ve,
De 9 > (dp
Finally the first part | Er dv. As we were allowed to neglect
Pad
Ve, Vo
free. we might be inelined to think that this term too might be
Uy
omitted.

But as follows form the equation of state:
Op MRT db daa

de — (v—b)? dir Wace per

this integral appears to be of higher order than the other for small

values of » — 6. We therefore retain it. Carrying out the integration
we get:

MRT db dage Wve,
vb dev _{vo

Here we may substitute p + a/v*, for MRT/v—h, so that our
expression for the thermodynamic potential becomes:

| ae
My, = MRT1(1-2) + pavo — pevo + MRT + MRT lp./p,-2 ee (ves =ve) +

i d ( ie db ( a a | dal 1 1 KT

C2 = Wie Ee TEN er at (ea TEER

daz be! dz [v‚,' vo’ | See OE pet)
ie

This value must now be equated to the thermodynamie
potential of the same substance in pure condition. As we suppose

it to be under the pressure of its own vapour, the quantity to be
calculated is the same as the thermodynamic potential of its saturated
vapour, 1. e.
of

foe + Peoex. “coer. a ik (EL)

Vcoex.
where we denote by the index covx. that the quantity must be taken
on the line of coexistence. Now is on account of the assumed
validity of the gaslaws

7

a)

ae =F Peoex. Ucoex. = MRT Upeoex./p, + MRT

Voer.
If we equate the expression obtained here with that of the
preceding §, then #(7), MRT and MRT logp, neutralise each other
on both sides. What is left we may write in this way :

db Bt pe(1—2) Ope
(Po — Pe) vo & — |= — MRT log ————— + a (ve, — Ve, ) +
da Peoex. De

da / 1 1 db ( 1 1
WE Se cian Oe. ae lee ees

Now ve, and v, can never differ much. If the osmotic pressure
of an aqueous solution amounts e.g. to 1000 Atm., these volumes
differ only a few percents. In the two last terms, which themselves
can only be correction terms, we may therefore put 7, =v in any
case, so that those terms vanish. Further we may neglect v, by the
side of vy, and write MRT logp. for vo. Our equation becomes then:

MRT |, Pe (1—2) Mogpe
(SoS SS ba St
OT db vu Poes. da
Vo — & —
i da

The remaining v, may, of course, not be replaced by ve, first
because this expression occurs here in the principal term and then
because the substitution of 7, for v, would of course be more felt
in a term of the order 1/v—é/ than in 1/v. But in any case, when
we have really to do with osmotic pressures, the pressure will never
be so large that we could not compute v, with the aid of the coeffi-
cient of compressibility of the saturated liquid without any difficulty.

§ 8. The quantity pop, which we have found, is not identical
with the osmotic pressure; the latter is rather po—peoer., but the
transition of one quantity to the other is without any difficulty. If

( 729 )

we neglect in our formula the terms, which are multiplied by a by
the side of those in which this is not the case, if we put po=Peoer.
and if we take woor. instead of 7, which is permissible for very
dilute solutions we get:

MET
P = po—Peoex, = — —— log| 1—«a
Veoer.

which gives the well-known formula of vaN r Horr when the og
is developed and the higher powers omitted.

I wish to point out, that also a more accurate treatment yields
the logarithmic form which Borpinen and van Laar have advocated
— and there could not be any doubt but it must be so — but that
it also shows that van Laar’s statement’) was too absolute when
he asserted that a correction term need never be applied in the
numerator voe (Or Vo) in Connection with the size of the molecules.

In the second place I draw attention to the fact that we find the
osmotic pressure exclusively expressed in what VAN DER Waars has
called thermic quantities (in opposition to caloric quantities). It
appears to be unnecessary to take into consideration the heat of
dilution or other quantities of heat, which van ’r Horr *) seems to
deem necessary for concentrated solutions and which Ewan *) has
taken into consideration. Even if we had avoided all the introduced
neglections, so when we had not assumed, that the vapour follows
the gaslaws, nor that v,=v,, may be put in some terms, nor that
the area C’ may be neglected compared to A, nor (the most important)
that 4 is constant, we should evidently not have had to deal with
any quantity of heat. This seems important to me, as both theore-
tically and experimentally the caloric quantities are much less accessible
than the thermic ones.

Physics. — ‘“‘Ainetic derivation of vax ’r Horr’s law for the
osmotic pressure in a dilute solution.” By Dr. Pu. Konnsraum.
(Communicated by Prof. vax per Waats).

§ 1. When we leave out of account the more intricate theories
as that of PorrriNG ©), who tries to explain the osmotic pressure
from an association of solvent and dissolved substance, and that of

My Ibe:

2) K. Svenska Vet. Ak. Hand. 24. Quoted by Ewan Zsch. phys. Ch. 14
409 en 410.

8) Zsch. phys. Ch. 14, 409 en 31, 22.

4) Phil. Mag. 42, 289.

( 730 )

3ACKLUND '), who seems?) to require even ether waves to explain it,
chiefly two theories have been developed about the nature of the
osmotic pressure: the static and the kinetic theory. The first theory
finds warm advocates in Pupin *) and BARMWATER *); it seems however
doubtful to me whether they have closely realised the consequences of
their assertions. At least the latter brings forward as an objection
to the kinetic nature of the osmotie pressure: “Ein molekulares
Bombardement in einer Fliissigkeit ist mir immer etwas sonderbar
vorgekommen’’; notwithstanding he considers the equation of state
of Van per Waars by no means as a “sonderbar” instance of false
ingenuity, but as an example to be followed. However this may be,
he who does not want to break with all our conceptions about
heterogeneous equilibrium, will not be able to explain such an equili-
brium in another way than statistically i.e. as a stationary condition
of a great number of moving particles. This does, of course, not
detract from the fact that the question may be put what forces are
required to bring about that state of equilibrium. This implies that
the adherents of the static theory need not be altogether mistaken
when they assert that the cause of the osmotic pressure is to be found
in forces of attraction. On this point I shall add a few remarks at
the end of this communication.

§ 2. Of much more importance than this static theory of the osmotic
pressure is the kinetic theory. The great majority of its advocates
(I shall speak presently about the few exceptions) take as their basis

the equality, which bas been proved experimentally and by means
of thermodynamics, of the osmotic pressure and the gas pressure (the
pressure which the molecules of the dissolved substance in the same
space would exercise, when they were there alone and in rarefied
gas state) and derives from this that they have both the same cause
in this sense that the dissolved substance is present in the two cases
in the same state and so acts in the same way; this is then
expressed in about this way that the solvent converts the dissolved
substance into the rarefied gas state. This conception seems doubly
remarkable to me; first because it seems to be pretty well generally
prevailing *), secondly because it alone seems to me to be able to

1) Lunds Univ. Aarsskrit 40.

4) I know his paper only from an abstract in the Beibl. 29, 375.

3) Diss. Berlijn 1889.

4) Diss. Kopenhagen 1898 and Zsch. phys. Ch. 28, 115.

8) It is naturally difficult to give a proof of this opinion, therefore I shall only
adduce the following citations as a confirmation.

“If we look a little more closely into the matter, we find that in the case of
dilute solutions, at least, there is far more likelihood of the dissolved substance

(fais)

explain, why the theory of the osmotic pressure has become so quickly
popular, whereas Gisss’ method for the solution of the same problems
was scarcely noticed. In fact the view mentioned possesses all qualities
required for great popularity: it seems to give a very simple, clearly
illustrating explanation for the striking law discovered by van ’r Horr;
it is allied to the universally known gaslaws; it seems to make us
acquainted in the osmotic pressure with a quantity, which is as
characteristic for the dissolved state as the well-known external pressure
for a gas. On the other hand it does not seem to carry weight
that this “explanation” is, properly speaking, no more than an
explanation of words, which leaves undecided exactly that which
had to be explained, viz. how it is, that the solvent acts on the
dissolved substance in this way. It is, however, worse that this
explanation clashes with everything we know of liquids and gases,
and therefore is to be rejected. We need only think of the well-
known experiment with a bell jar, closed at the bottom by a
membrane, filled with a solution of cane sugar and placed in a
vessel with pure water, which forces its way in till equilibrium
has been established. If now the pressure P, exerted on the mem-
brane, was a consequence of the fact, that the dissolved substance
in the bell jar was in a state which more or less resembles the
gasstate, then those molecules of the dissolved substance would have
to exert the same pressure also on the glass wall of the bell jar, in
other words, the water molecules would exert the same pressure

being in a condition comparable with that of a gas.” (Waker, Introduction to
Physical Chemistry, 148).

“Ich glaube dargethan zu haben — im Gegensatz zu der zur Zeit allgemeinen
Auffassung — dass es nicht notwendig ist eine freie Bewegung der gelösten
Moleküle wie für die Gase anzunehmen. Wenn ein fester Körper in einer Fliissig-
keit gelöst, oder eine Flüssigkeit mit einer anderen gemengt wird, so wird eine
neue Fliissigkeit erhalten, von deren Molekülen es nicht gestattet ist, andere
Beweglichkeit anzunehmen, als diejenige, die Fliissigkeiten charakterisiert.”’ (Barm-
WATER l.c. pag. 143). “Aus den klassischen Arbeiten von van “ry Horr und ARRHENIUS
geht nun hervor, dass die Körper bei Gegenwart von Lösungsmittel thatsächlich
mehr oder minder dem Gaszustand näher gerückt werden,” and a little before :

“Andererseits konnte ich mir.... nicht verhehlen, dass gerade diese Gegenwart
und Einwirkung des Lösungsmittels doch die notwendige Vor-
bedingung für den Eintritt des gasiihnlichen Zustandes sei;.... daher ist

aber ein gasähnlicher (also kinetischer) Zustand nur unter dieser Einwirkung
vorhanden und hört sofort auf, sobald diese Einwirkung beseitigt ist. Es sei be-
tont, dass diese Auffussung durchaus nichts Neues bietet, duss sie vielmehr wohl
einem Jeden eigen ist, der den Begriff des osmotischen Druckes kennen gelernt
hat.” Brepie. le. p. 445 and 444). The italics are mine, the spacing the cited authors’.

Finally cf. Van Laar’s address in the “Bataafsch Genootschap”, p. 2 and 3 and
the example cited there.

50
Proceedings Royal Acad. Amsterdam. Vol. VII.

( 732 )

on that wall from the inside and from the outside (of 1 atm.). This
now is a perfectly unacceptable result, as immediately appears from
What follows. Let us imagine the same solution as in the bell jar
inclosed in a eylinder with a piston under the pressure of its satu-
rated vapour p— Ap, where Ap is the decrease of vapourpressure.
The cane sugar molecules contribute nothing to that pressure or hardly
anything *), as appears from the fact that they cannot pass into the
vapour (at least not in a measurable degree); all the pressure is
furnished by the water molecules. Now we compress the liquid, till
it has got a pressure P+ p, it is now in perfectly the same con-
dition as the liquid in the bell jar, when we except the immediate
neighbourhood of the membrane. On the supposition made just now
the water molecules would exert a pressure p against the piston, the
sugar molecules a pressure P, i.e. the pressure of the latter would
have increased by an amount about 1000 times that of the former,
whereas their initial pressure was at least a hundred thousand times
smaller. And the result would be that the, let us say 2, sugar mole-
cules, which are found to every 1000 water molecules would exert a
pressure twice as great as the 1000 particles together. It is beyond
doubt that the pressure P+ p on the piston or the glass wall of
the bell jar is exclusively exerted by the watermolecules, and if he
meant this, Lornar Meyer was certainly right when he asserted *),
that the osmotic pressure was a result of the collisions of the solvent.

Also in this respect the theory of the gaslike character of the dissolved
substance falls short, as it leaves perfectly unexplained why in an
isolated solution, e.g. a cane sugar solution, which in a glass vessel
stands under atmospheric pressure, nothing is perceived of the gaslike
character of the dissolved substance. For that in this case solvent
and dissolved substance are less closely in contact than in the
osmotic experiment, cannot seriously be asserted.

§ 3. If therefore we must not seek the explanation of the
laws of the osmotic pressure in a particular condition of matter,
characteristic of dilute solutions, then the remarkable fact formulated
by Van ’r Horr calls the more peremptorarily for an explanation.

Nobody less than Lorentz and BOLTZMANN have made attemps to
do this *), but even their endeavours do not seem to me to have solved
the problem entirely. In saving this | agree with Prof. Lorenrz’s own
opinion, at the beginning of his paper he terms it a “freilieh nur zum
Teil gelungene Untersuchung”. As to the reasons of this partial failure,
however, L shall most likely differ in opinion with Prof. Lorentz.

1) Perhaps the pressure of these molecules would even prove to be negative,

2) Zsch. Phys. Ch. 5, 23.

( 733 )

For what is the case? The behaviour of liquids is entirely dominated
by the occurrence of the quantities « and 5 in the equation of
state. Only matter in dilute solution seems to emancipate itself
from it, according to the law of Var ‘r Horr, where neither
the « nor the 5 occurs. This fact calls for an explanation.
Now it is not difficult to understand, why the @ can disappear
here; the membrane is bonnded on one side by the solution, on
the other side by the pure solvent. If we now think it thin com-
pared to the extent of the sphere of action, then it is clear that at

the membrane the force = which works towards the solution, is
ve
a
in first approximation neutralized by the force —— towards the other
ty?
side. It is more difficult to see why also the 4 vanishes, i.e. why
the molecules of the dissolved substance seem to move as through a
vacuum, instead of through a space, which is occupied for a very
great part by the molecules of the solvent.

Just on this most important point Prof. LoreNtz’s paper leaves us
in the dark, for so far as I have been able to see. And it seems to
me beyond doubt, that in the first place this is due to an inaccurate
interpretation of the term “kinetic pressure”. According to Prof.
Lorentz it is always equal to */, of the kinetic energy of the centres
of gravity of the molecules which are found in the unity of volume.
It is therefore independent of the volume of those molecules. Now
this would only be a question of nomenclature, if not that kinetic
pressure was also defined as the quantity of motion, carried through
the unity of surface in the unity of time by the motion of the
molecules; and that this quantity is dependent on the number
of collisions and so on the volume of the molecules does not seem
open to doubt to me after KorreweEG’s proof *). In agreement with
this the kinetic pressure is represented in the equation of state by
MRT/v—b. In consequence of his definition Lorwxrz replaces this

*) Zsch. phys. Ch. 7, 37 and Arch. Néerl. 25, 107.

2) Zsch. phys. Ch. 6, 474 and 7, 88.

3) Verslagen Kon. Ak. Amst. (2) 10, 363 and Arch. Néerl. 12, 254. Compare
also the simpler, perhaps even more convincing proof for one dimension in Nature
44, 152. As the attentive reader will notice Prof Lonenrz’s proof (1. e. 59) does
not take into account the collisions and the fact ensuing from them, that a quantity
of motion skips a distance or moves with infinite velocity for a moment. And
the admission of the validity of Korrewec’s reasoning appears, as it seems lo me,
already from the fact, that Prof. Lorentz has to assume for the solid bodies intro-
duced by him, that they are immovable (Il. c. 40) or of infinite mass (l. c. 42)
which comes to the same thing in this case.

50*

( 734 )
quantity by MRT/v, and so his paper cannot give any elucidation on
the point which requires it most. But that notwithstanding we owe
to Lorentz’s labour a considerable widening of our views, will as I
hope, appear from the continuation of this Communication.

Also BonrzMaNn’s paper leaves us in the dark as to the question
why the quantity 6, which in other cases plays such an important
part for liquids, seems to have no influence on the value of the
osmotic pressure. In the equations, which he draws up, he never
takes the size of the molecules into account *) and it does not appear
why he does not do so. Further he stops at the result, that the
osmotic pressure is equal to the sum of the pressures exercised by
the two kinds of molecules, without discussing the part played by
the different kinds. For these reasons I cannot see a satisfactory
solution of our problem in BOLTZMANN's paper either.

§ 4. To arrive at a solution it seems in the first place necessary
to give three definitions.

dst. Given a fluid. Placed in it a body of perfect elastic impermeable
substance, which does not exert any attraction on the molecules
of the fluid. The thickness of this body (or this surface) be infinitely
small; let us suppose it to have an area of 1 em’. The “kinetic
pressure” in that fluid is then the quantity of motion in unity of
time transferred by the molecules of the fluid to this body (or obtained
in the elastic collisions from this body).

2nd, In the second place I imagine a body’), which is distinguished

1) See speciaily 1. c. 475 equation (4), which is evidently meorrect, when part of
the cylindre is not open to the centres of the molecules, because it is occupied by
distance spheres of other molecules.

2) That I assume that the body does not attract the molecules of the fluid, is
for simplicity’s sake, but it is not essential. If we imagine a wall, which does
attract the fluid, more molecules will reach its surface (cf. the footnote p. 739)
and hence will impart a greater quantity of motion to the wall. But on the other
hand the particles of the surface will now be drawn into the fluid with an} equally
greater force. The elastic displacement of the particles of the surface of the solid
wall, and with it (with sufficient elasticity) that of the layers lying under it, in
other words the pressure which propagates in the solid body, and which would
be measured with a manometer of any kind, will be perfectly the same in the
two cases. If we wish to take also negative external pressures into account, we
shall even have to give the definition by means of an attracting body, because

in this case a non-altracting body would not even be reached by the molecules
of the fluid. (Cf. the well-known fact that for the observation of the negative
pressure slrongly adhering walls are required). In this case the impulse of the
attraction of the molecules is simply greater than the quantity of motion which
they impart to the wall (and which may still be very great), the elastic displacement
is therefore not from the fluid, but towards it.

Also in the case that we wish to take capillary layers into account, our definition

(735 )

from the just mentioned body only by its being very thick compared
to the sphere of action of the molecules. The quantity of motion
transferred by this body per unity of time to the molecules, is called
the ‘external pressure” in that fluid.

34. In the third place I place in the fluid (which I now suppose
to be a mixture) a body, which is distinguished from that mentioned
under 2 only by the fact that the molecules of one component (solvent)
pass through it without any change in their velocity. 1 shall leave undis-
cussed here whether such a body can actually occur. The pressure
to which this body is now subjected, and which might be measured
e. g. by the elastic displacement of the particles of its surface, I
call the “osmotic” pressure in that solution.

From these definitions it is already clear that in dilute solutions
the osmotie pressure defined here must be of the order of the kinetic
pressure exerted by the dissolved substance, and not of that of the

a
external pressure. For these two differ, in that — has disappeared
7

for the kinetic pressure, and this will also be the case for the osmotic
pressure defined here, as appears from the reasoning given above
($ 3). I shall further show, that in dilute solutions this osmotic
pressure has the value indicated by the law of Van ’r Horr, and that
in any ease it is as great as the well known experimentally intro-
duced and measurable osmotic pressure, i. e. the difference in external
pressure of solution and pure solvent under the pressure of its own
vapour. in equilibrium through a semipermeable wall.

calls for fuller discussion. First of all this applies to what we have just now
said, for just as for negative pressures so also in the capillary layer, as
Van per Waats has shown in his theory of capillarity, the attraction of the
surrounding layers is a necessary condition for stable equilibrium. But further,
as Hursnorr has shown (These Proc. 8, 432 and Diss. Amsterdam 1900), the
above defined quantity does not obey the law of Pascan any more, because mea-

sured in the direction of the layer and perpendicular to it, it has a different value.
In this case we might perhaps speak of a total external pressure, which might
be split into an external fluid pressure and an external elastic pressure. The
consideration of capillary layers round a free floating sphere, teaches us further,
that the “external” in the name “external pressure” must not be understood
in such a way, as might easily be done, viz. that the reactive force of this
pressure, as it prevails in a certain point, would act in points outside the system
in question, which would always be more or less arbitrary, as we may choose
the limits of our system arbitrarily. The assertion: the external pressure is in a
point of the fluid so great, comes simply to this, that wen I should place a
strange body at that place, without altering the condition more than necessary
for this, this body would experience a pressure of such a value, and would
suffer an elastic modification in form which corresponds to it, so differing in the
capillary layer in different directions.

( 736 )

$ 5. For this proof I must refer to a formula of Cravsmvs used
by me already before’). Imagine a point which can freely move
in a space W. Crausmus®) shows — which is already plausible
beforehand — that the number of collisions of this point per second
against a wall of area S is proportional to S/W (the factor of
proportion depends only on the velocity of the point).

Let us now consider a wall as defined under 2, and draw a
plane parallel to that wall at a distance */, 6 (6 is the diameter of
the molecules, which we think spherical); this plane we call plane
of impact, because the centre of a molecule, which strikes against
the wall, lies in this plane. Now we apply Cravstus’ formula to
this wall. In this we must allow for the fact that the centre of a
molecule cannot move freely throughout the volume of the fluid;
for within the distance spheres (spheres drawn round the centre of
every molecule with a radius 6) it cannot come; instead of 7 we
have therefore to put v—2b, when 2/*) is the volume of the distance
spheres. Now the whole plane of impact, however, is not accessible
to collisions either, part of it also falls
within the distance spheres. In order to
fix this part we draw two planes at
distances A and / + dh parallel to the
plane of impact. We determine how
many centres of molecules are found
between them and what part of the
plane of impact is within their distance
sphere. In order to find what part of
the plane of impact falis at all within
distance spheres, we must integrate with respect to A between O
and */, 5. It appears then, that instead of S we must put S(1—6/r)
in the formula for the number of collisions against the wall, so

Fig. 1.

that the pressure becomes proportional to

or in first approximation

) These Proc. VI. 791.

2) Kinetische Theorie der Gase, 60.

5) For simplicity L confine myself to the first term, even if we have to deal with
liquids; this is permissible here, because the cther terms have no more influence

on our question (the derivation of the law of Van ‘r Horr) than the first.

1
)
nO
$ 6. Now we apply the reasoning of the preceding paragraph to
the collisions of the dissolved substance on a wall defined as under
3. We assume the solution to be so diluted, that the volume of the
molecules of the dissolved substance may be neglected compared

with the whole volume. For simplicity — though it is not essential
to the proof — we assume now also that the molecules are spheres.

Then here too the available space must again be put equal to r— 2;
but the part of the plane of impact, accessible to collisions, is now
different. For as the molecules of the solvent pass through the
wall, their centres may now just as well be on the other side of
the plane of impact. We have therefore not to integrate with respect
to A from O to */, 6, but from —*/, oto + '/, 6, which evidently
vields the double value. The pressure on the wall becomes therefore
proportional to

heal ——?)

v—2b v

so that the influence of the molecules of the solvent vanishes and
vaN Cr Horr’s formula is proved for the quantity defined by us.

§ 7. That this quantity has further always the same value as
the quantity which may be measured experimentally, is proved as
follows. Let us think the action of the membrane in such a way
that it suffers the molecules of the solvent to pass through freely,
but repels those of the dissolved substance perfectly elastically.
Something similar would take place when the membrane worked
as a ‘molecule sieve”, i.e. when the pores were such as to allow
the molecules of the solvent (thought smaller) to pass, the others
not. According to the definition the latter will then exert a
pressure on the membrane equal to our osmotic pressure. The other
molecules passing through the wall unmolested, there is no mutual
action with the wall, and so they co not exert any force on it.

1) If one should object to the train of reasoning followed here, one can find in
Boiraann’s “Gastheorie” a proof for this formula which intrinsically agrees per-
fectly with that given in this paper, but will appear stricter to some. There one
will also find the above given integration carried out.

*) It is clear that we shall get the same result, when we do not take 20,
but f (b/v.) for the voiume of the distance spheres. For as the place of the plane
of impact wilh respect to the molecules of the solvent is quite arbitrary in our
present case, the part of the plane of impact, which lies within the distance spheres
will stand to the whole area in the same proportion as the volume of the distance
spheres to the whole volume,

(738)

The experimentally measurable difference in pressure on either side
of the membrane must therefore have the same value as the quantity
defined by us.

Lorextz *), however, has shown that the assumption made here
concerning the membrane is by no means necessary. On the contrary ;
if we assume that the membrane is thick compared with the sphere
of action, that its substance fills a volume large compared with the
apertures present and that it feebly attracts the molecules of the
dissolved substance, whereas these are strongly attracted by the
solvent — none of which are improbable assumptions — we arrive
at the result, that none of the dissolved particles reaches the membrane,
much less exerts a pressure on it; the membrane is then quite
surrounded by the pure solvent. And that this case is really the
usual one in nature is made probable by the fact, that it is by no
means always the smaller molecules which pass the membrane, as
we assumed above. The membrane seems therefore not to work as
a molecule-sieve. We are then easily led to suppose that the mem-
brane does not exert a positive repulsion at all on the non-passing
substance, but that it only attracts those particles much less strongly
than the solvent, so that the dissolved particles do not pass through
the membrane, because they occur but extremely rarely in its neigh-
bourhood. This view is supported by the fact, that only those
substances seem to be non-passing which are not easily converted
to vapour, and so cannot reach the limits of the liquid in virtue of
their own thermal motion alone.

However this be, also in this case our conclusion holds good.
For when the molecules of the dissolved substance do not (or only
in an intinitely small number) reach the membrane, two planes will
be found not far from the membrane, A where the molecules of the
dissolved substance still have their normal density, B where this
density has diminished to zero. Between B and the membrane we
find then pure solvent. If we wished to discuss such a layer fully,
we should, of course, have to give a theory, as VAN DER WAALS
has given for the transition liquid vapour’), extented to a mixture
in the way van Expik*) has done. But for our purpose this is
fortunately not necessary. We need only observe, that the layer
AB as a whole has now exactly the same influence on the condition
of motion of the dissolved molecules as the mathematical upper surface
of the membrane had just now. The layer AB as a whole will now,

Det

8) Verh. dezer Ak. (2) 1; Arch. Néerl, 28, 121 and Zsch. phys. Ch. 13, 657.

5) Diss. Leiden 1898.

( 739 )

just as the membrane just now, be pressed downward with a force
equal to the osmotic pressure defined by us, and transfer this force
to the underlying layer of the pure solvent, which is pressed outward
with this force. But this pressing force is evidently equal to the
difference in pressure which may be measured experimentally *).

§ 8. Thus it seems to me that van “rT Horr’s law for dilute solu-
tions is kinetically explained in the same way as the law of Borrr-
Gay Lussac-Avocapro for dilute gases and that of vaN per W aars
for liquids and gases, i.e. we have obtained an kinetic insight how
these laws result from the condition of motion in the homogeneous
mass, while we have left out of account what happens in the
eventually (probably always) present unhomogeneous bounding layers.

It appears from the explanation convincingly, that vaN Laar goes
too far, when he states *), that we cannot speak of osmotic pressure
in an isolated solution. Here too this notion has a clear physical
signification, and the laws which govern it, are to be derived.

1) This hydrostatic proof may easily be replaced by a purely kinetic one, though
the latter is somewhat more elaborate. The layer AB, which (in consequence of
course of the neighbourhood of the membrane) behaves as a layer of water, through
which the dissolved substance cannot penetrate (Cf. Nernsr’s well-known osmotic
experiment) imparts to the molecules of the dissolved substance per second a
quantity of motion equal to the osmotic pressure defined by us, and receives itself
an equally large quantity in opposite sense, which it transfers to the underlying
layers, as the kinetic theory teaches. (See e. g. Bourzmans, Zsch. phys. Ch. 6, 480).
Now the whole mass of water, which is in the neighbourhood of the membrane, (on
either side, reckoned on one side from B, on the other from a plane, so far from
the membrane that the latter does not act on it any more), does not move downward,
so it must receive an equally strong but opposed impulse, which, of course, cannot
issue from anything but the membrane. Of what nature the forces acting here are
is quite unknown. It cannot be the ordinary molecular attraction, for then the
denser liquid found above the membrane would probably be drawn more strongly
downward than that found ander it upward. We might think of friction in the pores, but
it would then have to be different in one direction from that in the other; in short I dare
not venture on any conjecture about this. This alone is certain, such forces must
exist, at least if the case put by us ever actually occurs. This appears already
from the fact that the pure solvent aboye the membrane is subjected to a higher
pressure, so has a greater density than under it. Such an equilibrium occurs for
all kinds of kinetic questions (liquid vapour, gas under the influence of gravity),
but the necessary condition is always a force, which at a cursory examination
seems to have the result, that the velocity of the molecules in one part (so the
temperature) would be higher than in the other, but in reality only proves to have
influence on the densiiy. The membrane, which furnishes this impulse, receives an
equally strong one back from the reaction, and so here too, though indirectly, we
see a force equal to the osmotic pressure defined by us, exercised on the membrane
from the inside to the outside.

*) Chem. Weekblad 1905, N°, 9, § 3. Voordracht Bat. Gen. 3.

( 740 )

Whether this renders it desirable for us to give it a prominent place
in the theory of solutions and make all the rest proceed from it, is
a question to which IT wish to revert in a separate paper.

First I must add this observation. The insight obtained in the
nature of the osmotic pressure enables us to examine what quantities
must occur in the formula for more concentrated solutions. In the
first place it will no longer be true for concentrated solutions, that
the term “2 vanishes, both because on the two sides of the mem-
brane the density v differs, and because the concentration and so
the a will differ. Further — as appears from our proof — for
higher concentrations the volume of the molecules will assert its
influence, and not only that of the dissolved substance, but also of
the solvent. For as on the two sides of the membrane the density
differs, the part of the plane of impact that falls within the distance
spheres of these molecules, will no longer be represented by the
above given value. As finally the molecules are of different size,
when the terms 5, and 4, occur, the term 4,, is sure to appear.
The formula found in this way will certainly not agree with the one
found in the preceding communication by a thermodynamic method,
for the latter is derived from the equation of state with constant 4,
whereas the kinetic considerations exclude all doubt that 6 is a
function of the volume. If ihere should be a real diminishing of
the size of molecules when passing beyond the membrane, then this
fact is also to be taken into account.

Far be it from me to make an attempt to draw up such an
equation. To achieve this, it would be required, as appears from
what precedes, that one should be able to surmount at least all the
obstacles which stand in the way of an accurate equation of state.
And if this might be done — the preceding paper proves it — the
final formula could be found in a way, which would not expose
us again to the danger of making errors. [ shall therefore not enter
into the question either, in what way the formula derived in a kinetic
way can satisfy the first requirement that may be put to every formula
for concentrated solutions: that it yields the value «© for the case
that the substance passing the membrane has perfectly vanished from
the solution.

§ 9. I shall just add a single remark on the question whether
our kinetie view implies that the so-called statie theory of the osmotic
pressure, which ascribes the cause of the phenomenon to attractive
forces, is entirely wrong? It seems to me that from what Lornntz
has proved it appears that we must answer this question in the
negative. It is true that we have seen that the attraction of solvent

( 741 )

and dissolved substance begins to play a part only im sensibly con-
centrated solutions, and that we have to explain the osmotic pres-
sure by a “moleculares Bombardement”. But the case treated by
Lorentz shows that the whole osmotic phenomenon might possibly
exclusively be the consequence, not so much of the presence of
attractive. forces, but just of the reverse, of the want of attraction
between the molecules of most solid substances and certain other
solid substances which form membranes. If the adherents of the
statie theory mean no more than this with their assertion: that the
osmotic pressure must be explained from forces of attraction, then
they seem to me for the present secured against every attack.

Physics. — “Osmotic pressure or thermodynamic potential’. By
Dr. Px. Konnstamm. (Communicated by Prof. J. D. van
DER WAALS).

§ 1. The theory of thermodynamic functions, through which
Gipps has enabled us to derive from the equation of state of a system
in homogeneous condition, what heterogeneous equilibria will oecur,
has attracted attention only in a very limited circle during a series
of years. However great the region opened for investigation by Gipps
was, the methods indicated by him seemed so abstract, that only
very few dared to grapple with them. At a stroke this was changed,
when in 1885 Var ’r Horr succeeded in replacing these methods
in appearance so abstract, by another, that of the osmotic pressure,
which strongly appeals to the imagination. The theory of solutions,
which up to that time had only existed for a few, rapidly became
one of the most frequently treated and discussed subjects of physies
and chemistry; sinee then it has continued to enjoy undivided
attention.

It stands to reason, that the attention, which now for twenty years
has been so lavishly granted to the questions of heterogeneous equili-
brium, have also been conducive to making GiBBs’ methods for the
solution of such questions known in a wider circle. But though Gress’
name may be counted among the most famous and widely known
names in the sciences of physics and chemistry, yet even now his
methods cannot be said to have been universally accepted.

The adherent of a mechanical (or, if one prefers, statistical) natural
philosophy has by no means reason exclusively to regret this course
of affairs, for he sees in it a clear indieatien, that the views whose
truth he advocates, are by no means so antiquated, nay even dead,

( 742 )

as they are often declared to be. And if the current opinion —
which certainly greatly contributes to the greater popularity of the
osmotic pressure compared with that of the thermodynamic potential
— were really correct, that we can form a clear idea of the
physical meaning of the first quantity and ot of the second, then
there could not be any doubt for him which method to prefer, if
for the rest the circumstances were quite the same.

But this. current opinion seems to me hardly tenable and on the
other hand I believe that in many respect the thermodynamical
potential is preferable to the osmotic pressure, and that therefore it
will be advisable to put the question whether it would not be
better to return to the older method both for scientific investigation
and for instruction ?

§ 2. This question has lately again been put forward by Mr.
VAN LAAR in an address for the “Bataafsch Genootschap” at Rotter-
dam *), which was followed by an article “Over tastbare en ontast-
bare grootheden” (On palpable and impalpable quantities) *). Though
1 readily admit, that these papers have induced me to consider the
problem of the osmotic pressure specially, there would not be any
reason for me to diseuss Mr. van Laar’s views here, when only
his address had appeared. For IT can fully subseribe to the general
tendency of this paper though of course TL would not be responsible
for every statement, as moreover has already appeared from my

preceding communications in these Proceedings — and L should
therefore only have to consider what in my opinion would have to
be added to his address. His second paper, however — and in

this I have specially in view $ § 6 and 7, pointed out as the gist
of his paper by the author himself — Mr. van LAAR seems to me
to harm rather than to promote the good cause, which he has
espoused with so much ardour, and already for this reason 1 feel
it incumbent upon me to protest against this part of his reasoning.
I think that 1 accurately represent the gist of it as follows: It is
true that we cannot form a clear idea*) of the nature of the ther-
modynamie potential, but we cannot do so for the osmotic pressure
either. Nor is this surprising, for the improved philosophical insight
of the last years gives us the conviction that our natural philosophy
never works with any but fictitious (though sharply defined) ideas,
1) Also published in Chem. Weekblad, 1905, N° 1.

2) Chem. Weekblad, 1905, NO, 9.

3) Mr. van Laar speaks of a “palpable conception” (tastbaar begrip). It would
lead me too far if I would account for the reason why I think that IT may,
nay even ought to substitute the term chosen here for it.

which must not and cannot claim in the least to represent the real
nature of things. It is also owing to this insight, that several voices
have been raised of late in favour of the use of the thermodyna-
mical potential.

§ 3. Now I think that I have convincingly proved the incorrect-
ness of the second thesis in the preceding paper, and as 1 gladly
and with full conviction range myself with the “tastbaarheids-
menschen,” (those who want to form a clear idea of the physical
meaning of each term used), whose opinion Mr. van Laar severely
condemns, his reasoning would lead me to take side against the
thermodynamic potential party when T could subseribe to his first
and his last thesis more than to his second. This however, is by no
means the case.

The last philosophical-historical thesis I can, naturally, not discuss
here and | confine myself therefore to that concerning the physical
meaning of the thermodynamical potential. [t seems to me that we
can form an idea of this quantity which need not be inferior
to that of any other statistical quantity. That Mr. Van Laar has
overlooked this fact seems chiefly owing to two circumstances of
which it may appear that one can have hardly any influence, for it is
simply a question of nomenclature. Following a common way of
speaking, which does not seem to me the less reprehensible for the
fact that it is of frequent occurrence, Mr. Van Laar does not give
the name of “thermodynamical potential” to the quantity introduced
by GaBBs into science by that name, but to one of the other functions
introduced by Gipps, the &-funetion. There are more reasons than
only a feeling of deference, which make this undesirable. The real
(Gras) potential zs really a potential, i. e. it is constant in a space
where equilibrium prevails, and its not being constant means, that
there is no equilibrium. At least when there act no capillary or
external forces; and in this case the resemblance of the thermody-
namic potential with the potentials of other energies stands out per-
haps the more clearly. For in this case we need only add to the
(Gipps’) thermodynamic potential the other potentials, which exist in
that space in order to get a quantity, the total potential, which now
also is constant throughout the space in case of equilibrium. The
S-funetion has neither the one property, nor the other, except when
we have to deal with a simple substance without capillary layers,
in which case it becomes identical with the thermodynamic potential.

If now also in $$6 and 7 Mr. Van Laar had directed his atten-
tion instead of to the $-function, to the real potential, as he has done
in $ 4, where he carries out his calculations by means of it, it would
probably not have escaped his notice that he wrongly represents the

( 744 )

thermodynamic potential (whether it be in one sense or in the other)
as the last, most fundamental quantity, which determines the internal
condition of a body. As such we cannot take others than v and 7
(if necessary of course w, y, etc.); that this is not only a subjective
“point of view” appears perhaps most clearly from the study of the
theory of capillarity, as van per Waats has given it.

§ 4. From this follows naturally, that we must try to form
an idea on the relation between the thermodynamic functions and
these fundamental quantities, and this does not seem so very difficult
to me just with regard to the thermodynamic potential. Let us only
consider the following. Thermodynamics teach, that however composite
the equilibrium may be, the total potential of every component must
be the same in two phases which are in equilibrium; the kinetie
theory, or in plain language, common sense that in all those cases
equilibrium is only possible when an equal number of particles of
each substance passes from the first phase into the second phase and
vice versa. Now Van per Waats has shown *) that in the case of
equilibrium of vapour and liquid, whether in a simple substance or
a binary mixture, the two conditions are simply the same fact stated
in different terms. It does not seem hazardous to me nor jumping
to conclusions to conclude from this that these two conditions, which
are always at the same time fulfilled or not fulfilled, also in other
cases will agree in signification and that therefore the physical meaning
of the thermodynamic potential *) of an homogeneous phase, on which
no external forces act, is nothing but the number of particles which
per second reach a wall as defined in the preceding communication
§ 4 under 2, if this wall is thought in the midst of that homo-
geneous phase.

1) Verslagen Kon. Akad. Amsterdam (4) 3, 205 and Arch. Néerl. 30, 137.

2) I choose purposely the words “that the physical meaning of etc.” and not
“that the thermodynamic potential is equal to etc.” For the equality of the
two quantities would require an “absolute” scale of thermodynamic potential.
For from the equality of the conditions mentioned follows only:

Mia SEN) EC en ee RE
where Fis such a function, that Mg is a one-valued function of V and reversely
N of Mp. This however, is not of material influence, for formula (1) expresses
only, that we begin to count the thermodynamic potential from another point
than the number of particles (which agrees with the fact that our thermodynamic
potentials always include an undetermined constant) and that we make use of
another unity when measuring one quantity than when measuring the other.
There is therefore perfect concordance of our case ard that of the temperature
measured e.g. according to Celsius and certainly nobody will object to the statement,
also when he thinks of this temperature scale, that the physical meaning of the
temperature is the mean vis viva of the centres of gravily of the molecules.

( 745 )

Yet this definition requires some further elucidation, because the
number of molecules under consideration reaches a bounding plane
of the phase, which does not exercise any attraction on those par-
ticles, whereas on the particles discussed above and whose number has
been calculated by Var per WAALS, viz. those which pass from the one
phase into the other, a force does work directed to the other phase. But
this difference is in my opinion, only apparent. Also in the equations
arrived at by van per Waars, one member refers exclusively to one
phase, the other to the second phase; there are no terms in them
consisting of factors, one of which refers to the first phase, another
to the second. That we had to arrive at that result, may be easily
understood, for the thermodynamic potentials themselves refer either
to the one or to the other phase and are quite determined by the
condition of that phase.

That at least in the definition of the thermodynamical potential
one number may be put instead of the other, appears as follows.
Let us consider a liquid in equilibrium with its vapour. The number
of particles that now passes, per unit of area, through the bounding
laver is that which Var per Waars treats of; let us now place
on this liquid a layer of a substance which does not attract the
molecules; let this layer be thick with respect to the spheres of
action and provided with narrow channels. The number of particles
that penetrates into these channels on either side is the number,
which we used in our definition. Now I assert that the introduction
of this layer cannot disturb the equilibrium of the homogeneous
phases *), i.e. their pressure and concentrations will not change. For
if this had been the case we should have been able to construct
with the aid of such a layer a so-called perpetuum mobile of the
second kind, and should have come in conflict with the second law
of the theory of heat. From this follows that equality of the number

1) The equilibrium in the non-homvgeneous, capillary layer és disturbed by
introducing such a wall. For, as vay per Waars has shown (cf. the footnote p. 735)
the equilibrium in a plane of such a layer is only stable in consequence of the
attractive forces exercised by the surroundings. When introducing the solid layer
in question the condjlion in the transition layers will be considerably modified,
which might also be anticipated. This does not affect our reasoning, for by the
word “homogeneous” we have positively excluded these transition layers in our
definition. That this was necessary in any case appears already from the fact, to
which we have already called attention above, that the thermodynamic potential
for such layers is no lenger the quantity which determines the equilibrium, but
that it is replaced by the total potential. We must therefore certainly not have
recourse to such layers, in order to get acquainted with the thermodynamic poten-
tial in its simplest signification.

( 746 )

meant by Van per Waars implies equality of that used by us in
the definition, and that we may therefore substitute the latter for
the former in the definition of the thermodynamic potential.

§ 5. In this way we have obtained a clear idea of the nature
of the thermodynamic potential, which so far as I can see is in no
respect second to that of temperature, external pressure, kinetic
pressure, number of collisions, mean length of path ete. That for
all this it is not always easy to derive in a special case the value
of the thermodynamic potential from this kinetic meaning is self-
evident, as well as that it will possibly always be more desirable
to derive the thermodynamic potential by means of thermodynamic
functions than from direct kinetic considerations. It is true that we
do not avoid the latter in this way either, but we make use of the
result of these considerations, as it is given in the equation of state.
In these two respects, however, the thermodynamic potential is in
no way inferior to the osmotic pressure, as appears from my two
preceding Communications, specially from $ 8 of the second.

§ 6. Mr. van Laar informs us, that in connection with his address
he had been asked “to supply something as a substitute” for the
osmotic pressure and the kinetic conception of it, something that
“conveys some meaning”’.') This request seems by no means so
unreasonable to me as it seemed to Mr. van Laar and I think that
I have complied with it in the preceding pages. Now I may be
allowed to prove that this “something else” is at the same time
“something better”.

First of all it seems not very appropriate to me to give a quantity
of pressure such a prominent place in the theory of mixtures. As
soon as we deal with this theory in general, i. e., include also
external forces and capillary phenomena (which are very often of
great importance, I need only mention critical points) it appears,
that the pressure is a quantity we may only handle with great
caution and which may certainly not be treated as fundamental
variable.*) In a much higher degree this objection holds for the
osmotic pressure. For this is, as we have seen, nota quantity which
is characteristic for the state in which the solution is; the peculiar
laws of the osmotic pressure are not due to the fact that matter
in dilute solution is in a particular, peculiar condition, they originate
— in their generality — only from our arbitrariness, which by means
of fictitious ideas, calls peculiar conditions into existence on paper,
which never exist in reality. For let us not close our eyes to this

1) Chem. Weekblad 1905 No. 9, § 3. The inverted commas are Mr. van Laar’s.

2) Cf. the footnote on p. 735.

undeniable fact — and least of all should they do so who are so
averse to “hypotheses” — that though all those semipermeable walls

may be realised in a few cases, vet we have on the other hand thou-
sands and thousands of eases, where we have not the slightest
foundation for belief in their existence. What reason can there be
for assuming, that there will ever be found a wall permeable to
toluol, but not to benzol, and another wall, permeable to benzol
and not to toluol, and what else is it but a fiction to speak of a
wall, permeable to cane-sugar and not to water. (For also this is
necessary, see Van ’r Horr, Vorlesungen II, 24). And let us even
put the most favourable case: that such walls existed really, does
it not remain perfect fiction then to try and treat the theory of
concentrated solutions with them? We need only bear in mind that
steel, our strongest material, however thick it is taken, can hardly
bear pressures above 5000 atms, what to think then of a semiper-
meable wall for which such a pressure is but a trifle. And now I
do not in the least object to such fictitious ideas when they are quite
unavoidable — this is sufficiently proved by what precedes — but
what is the use of using them, when we have another quantity of
simple signification, which 7s characteristic of the condition in which
the mixture is, which can be defined solely from the properties of
the substance with which we have to deal?

To this comes another difficulty. He who works with the osmotie
pressure — history teaches it — is but too apt to consider a mixture
not as an individual, which must be examined in itself and must be
known from itself, but as another substance (solvent), more or less
modified by the presence of the ‘dissolved substance”. In this way
we lose quite sight of the fact, that the two components in a mixture
are present in exactly the same condition (the singular theory of the
“oaslike nature” of the dissolved substance proves it); we begin to
overlook, that ‘dissolved substance” and “solvent” are perfectly arbitrary
names, which have only a right to existence when we confine
ourselves to one of the two extreme cases; we are led to try and
explain the properties of a substance from those of another, which
is often in quite different circumstances; we begin to apply all
kinds of hazardous approximations and compromises; we get to the
most extraordinary association and dissociation theories. How fruitful
on the contrary the opposite method is, the whole work of Van Dur
Waats, the experimental and theoretic material (inter alia on the
behaviour of mixtures with respect to the law of corresponding
states) gathered specially at Leiden may prove.

§ 7. Now one may object to this, that all these are theoretical

DL

Proceedings Royal Acad. Amsterdam. Vol. VII.

( 748 )

objections of more or less value, but that they are outweighed by
the practical advantage that calculations with the osmotie pressure
are so much simpler than with the thermodynamic potential, but
this objection lacks all foundation. For kinetic calculation cannot be
meant in this, and for the thermodynamic calculation it holds on
the contrary, that when making use of the thermodynamic potential
we need not take one step, which we are not obliged to take in exactly
the same way when making use of the osmotic pressure. In order to
prove this, L should like to reprint and follow step by step the
proof given by Van ’r Horr in his Vorlesungen, but as this proof —
carefully selected by Van ’r Horr from considerations partly from
himself, partly from Lord Ray.eien, partly from Dr. Donnan, so
undoubtedly the finest and simplest to be found — covers two pages
in print, I shall only indicate the principal operations and put in
juxtaposition the operations, which are required for the thermody-
namic potential with the same neglections.

1. Remove from a solution of 1. The thermodynamic poten-
osmotic pressure P a quantity of | tial is:
solvent, occupying a volume wv. 7
The substance yields an amount Mu=pe+ | piv MRT ee

of work — Pv.
ee
Day i il (2) de
Òr/or

pe becomes here pr,
2. Negleet the change in vapour 2. Neglect the variability of p
tension and the eontraction of the | with « and the compressibility of

solution. (This is not expressly | the liquid, then

stated, but is evidently necessary |, ve,
for the proof). Op \ ; f
proot) | —]dv=0 en pdv = 0.
Oa:
. . %o ro

3. Let the quantity of dissolved 3.
substance, dissolved in v, evaporate Vey
diosmotically ; let its volume be es

: ‘ | pdo = pe (ve, — ve)
V, the work done is:

Ur
p V cy

(when we neglect v by the side

of WA):

( 749 )

+. Let the vapour expand to 4.
infinite volume, the work done is: 5
7 7 | pee = ee
{pac = MRT I= = e Vey
a J Veg ir
ze 2
5. Now press the vapour again 5. The thermodynamic poten-
into the solution, then a work is | tial of pure water is:
done by the substance : /
ce) y Mu = pv + | pdv -+ VAHL)
— {pte a #
J g
V uy
fow — MRTI— ; pv = Pe Ves
Vey —_——
Vg
6. The total quantity of work 6. The two potentials are the
must be zero, so: | same, so:
Bor | (pope) %, = — MREU(I—e)

which in spite of the different notation is the same, when log (1 — x)
is replaced by — z.

So it is seen that to every integration on the right corresponds
an operation on the left of exactly the same nature, though it does
not always refer to the same substance. The only difference is that
on the right the integration is carried out directly and that on the
left pistons and membranes are worked with. Now I do not think
that any one can easily set greater store by a clear physical meaning
of operations than I do, but that we should not be able to carry
out an integration along an isotherm without bringing in two pistons
and three membranes, seems rather too much of a good thing.

§ 8. And now we have considered the most favourable case :
dilute solutions; how is it with more concentrated ones ? It will
certainly be possible to devise also for them cycles so that the
calculations introduced in my first paper may be carried out without
mentioning the name: “thermodynamic potential’, but it will not be
found possible by a thermodynamic method to draw up a formula for the
osmotic pressure without determining the integrals occurring in it.

In this way it would seem as if the two methods were essentially
the same; it is not so, the osmotic pressure method has drawbacks,
of which the other is free. For what is it that we really wish to
learn by the cwo different methods ? Not the osmotie pressure itself,
and the properties of the solutions under that pressure, that is
for concentrated solutions; in sensibly compressed state. What we

SL

( 750 )

aim at are the properties of solutions under the pressure of their
saturated vapour: lowering of the freezing point and the vapour
tension, rise of the boiling point and coefficient of division or more
accurately (ef § 6 above) freezing point, boiling point, vapour tension
of the mixture and the concentration of coexisting phases. And this
does not only apply to physicists and chemists, who rarely if ever
work with membranes, but also to biologists, to whom they are of
the greatest importance. For differences in pressure of about ten
atmospheres will probably hardly ever occur in biologic experiments
and a fortiori not in the living organism either. The equilibrium
between two solutions will therefore never be established by diffe-
rence in pressure, but by the difference in concentration required
to make the pressure equal. So we have not to deal here either
with compressed solutions. *)

For the calculation this implies that when making use of the
thermodynamic potential we need extend the integration along the
isotherm only to the pressure pe and the thermodynamic potential
may then be determined with sufficient approximation from the
well-kwown formula for the vapour pressure :

log = a == 1)
Pk ú

thoneh it be with the factor 7 determined experimentally instead
of the value 4 found theoretically. But if the osmotic pressure is
used we can naturally not do without the integration up to p,

(in the term | 2 ze) and it is exactly this part of the isotherm
\. ec v
which is known the least, where e. g. the variability of b is felt
strongest, even the only term, on which it has influence when the
just mentioned formula for the vapour pressure is used. Quite
unnecessarily therefore the result is made less reliable by the intro-
duction of the osmotie pressure.

And supposed even that we had found the desired expression,
of what use could it be to us? It is true that the quantities, which
we really wish to know and which I mentioned before, are con-

1) For this reason hardly anything would be lost when in the discussion of
really osmotic questions it was made a rule to treat them without “osmotic pres-
sure” and simply to introduce the concentrations on either side of the wall; whereas
in this way there would be a great gain in lucidity of expression, witness the
example cited by Mr. van Laar (lc. § 5). For the interpretation given there may
be correct or incorrect, it can hardly be denied that the cited phrases may be
easily misunderstood in the sense of the well-known question of Puris, which has
so repeatedly been seriously discussed, how e.g. a Call, solution of no less than
53 atms. osmotic pressure could be kept in a thin glass vessel without making it
burst asunder!

nected in a simple way with the osmotic pressure in dilute solutions,
but we have not a single reason to assume this also for concentrated
solutions, or rather we may state with almost perfect certainty that
this will not be the case. How on the other hand those quantities
may be determined with the aid of the thermodynamic potential,
Van per Waars taught us already fifteen years ago.

§ 9. And let us finally not forget that though solutions of non-
volatile substances at low temperatures do play an important part
in nature, yet they are not the only substances which exist, nor the
only ones which deserve scientific consideration. And yet, the theory
of the osmotic pressure must necessarily be confined to them. One
is so customed to derive the laws of the rise of the boiling point and
the decrease of the vapour tension from the osmotic pressure, but
it is generally forgotten, that many mixtures have on the contrary lowe-
ring of the boiling point and rise of the vapour tension *), and that at
any rate if the dissolved subsiance is but in the least volatile, the
changes in boiling point and vapour tension cannot be derived
any more from the osmotic pressure. And it is obvious why. It is
inherent in every definition of the osmotic pressure, that it can only
be applied to those cases, in which one component may be separated
from the mixture in pure condition, as Nernst has clearly stated
for the first time. Hence this does not only exclude the whole region
of higher temperatures, at which all substances become more or less
volatile, but also all cases of not perfect separation in the liquid
or solid state. Also the lowering of the freezing point is touched by
this objection. It is true that the lowering of the freezing point may
be computed from the osmotic pressure, but only when, as in water
and sugar, the solid substance, which deposits, is not of variable
composition. Solid solutions and mixed crystals, which attract at
present so much attention in chemistry, cannot be treated in this way.

Physical chemistry in its present state reminds us strongly with
regard to its quantitative part, of the navigation of a people, which
does not yet know the compass. The coasting-trade is carried on
with great vigour, the same limited region is traversed again and
again; but they do not dare to venture on the main sea far from
the coast, and with reason, for great is the danger of ruin in the
towering waves of random hypotheses. This can only be remedied
by a trustworthy compass. Physical chemistry may obtain it if it
will abandon the method of the osmotic pressure and adopt that of
the thermodynamic potential in connection with a well-grounded
equation of state.

1) Cf. Théorie Moléculaire § 17.

Astronomy. — “Approximate formulae of a high degree of ac-
curacy for the relations of the triangles in the determination
of an elliptic orbit from three observations.” By J. Werper.
(Communicated by Prof. H. G. vAN DE SANDE BAKHUYZEN.)

The places in space occupied by the observed planet or comet at
the instants ¢,, ¢, and ¢, are indicated by P,, P, and P,, the posi-
tion of the sun is indicated by Z.

For the determination of an elliptic orbit we mainly proceed as
follows: first by means of successive approximation we derive the
distances P,Z=r,, P,Z=r,, P,Z=r, from the data of theobser-
vations, from which distances the elements of the orbit are directly

computed without using the intervals of time. From the obtained
ellipse we can again derive the intervals of time in order to test the
accuracy of the results and compare them with the real ones. In
case they perfectly agree, the ellipse found satisfies all the conditions
of the problem, but as a rule this is not so. The cause of it is
that, in order to calculate the distances 7,, 7,, and 7,, we use
: : ; triangle P ZP,
approximate formulae to express the relations — —
triangle P,ZP,
triangle P,ZP,
triangle P.ZP,
three distances to be found, while neglecting the terms of the 294, 3*4
or 4» order with respect to the intervals. Indeed, different expressions
have been proposed for n, and #,, some recommending themselves by
greater simplicity, others by greater accuracy, but, so far as I know,
in the general case of unequal intervals none of them contain the
quantities of the fourth order with respect to the intervals.

De

and == n, in terms of the intervals of time and of the

The errors in the calculated distances 7,,7, and 7, and those in the
elements of the orbit derived from them are generally of the same
order as that of the terms omitted in the expressions for 7, and 7,.

Accurate and at the same time simple expressions for 7, and 7,
have been given by J. W. Grpss’).

The purpose of this paper is to develop, according to Gress’ method,
expressions for n, and 7, which include the terms of the 4 order;
at the same time a new derivation of Gripes’ relations is given.

In the ellipse sought let P be the position of the heavenly body
at the time ¢, # and y its heliocentric rectangular coordinates in the

DJ. W. Gises: On the determination of elliptic orbits from three complete
observations. Memoirs of the national academy of sciences. Vol. IV, 2; p. 81.
Washington 1889.

( 753 )

plane of the orbit, and 7— ZP, then « and y satisfy the following
differential equations

a v dy y

as ei ——_=ij,

rae r Gun r
wherein we have put rk (t—t,) as independent variable instead
of the time 4; rt is therefore the time reckoned from the epoch of
the first observation and expressed in the unit for which, in the
solar system, the acceleration —1 at a distance from the sun
which is adopted as unit of length; # is the constant of Gauss
[log k: = 8.235 581 4414 — 10).

While designating the rectangular coordinates of PPP, by corre-
x $ : EEEN ae

: ant EY UT triangle P, ZP
sponding indices I remark that n= ————- > ——————_—_
LY 3—Y 125 triangle P,ZP,
satisfies a similar differential equation as a and y, namely:
dn n
—_- == — — =?
dt? Ps
At the times (v= Oi (ei 0,);- & CS)
the values of 7 are 0 +n, + 1 and
Sar Ns 1
the values of 7 0 = ——
ia ihe

Consequently in the development of # in a series of ascending
powers of t after Mac Laurin, the terms of the power zero and 2
will be wanting. If in this expansion we do not go farther than the
4h power of t, we require only 3 indefinite coefficients which may
be eliminated from the following 4 relations:

ns a eee eee L
l= Kr, zie K,r," sie Kr oR IA

a min Oan weet fe

es IOA Iers

The remaining relation yields an expression for », in 7,, T,, 7,7
and the remainders /,, F, f, and F,.

The indices which I have used for the remainders, indicate the
order of these terms with respect to +; /,, for instance, which
begins with Kr,’ is of the 4' order of tr, which is evident when
we express the coefficients A in terms of the derivatives of for
«=O and develop the latter by means of the differential equation
for 2 as products of 7
development :

3

» For clearness I shall here give this

( 754 )

dn

==> ni — zt, where z is put for —
at’

dn A 1 og

Fie — Zn — din

dn Ma

Te (4zi —#)n + (2? — 32) ú

(

d
From the differential equation rs (r*# + r) = 0, satisfied by 7, we
at

can derive the following differential equation for <= —:
5

which may serve to eliminate £ from the higher derivatives.

2

d° z 8 2
rset eee

For t= 0, nm is equal to zero and #= K,, hence

C : < 5 7 7 ie 2;
Kak en EK K, = —— KK.

sed 120 hee
If we substitute the expressions for the coefficients A in the second
of the 4 relations, this becomes :

2 | 1 1 2,°—82
LS te ic -= — 2,1,” — — 2,c,° He Tee

and from this it clearly appears that A, and the other coefficients

a ‘ ; 1
K, in so far as they depend on the intervals, are of the order —.
©,

2

From the 4 relations with the indefinite coefficients AG, A,, A, we
find by eliminating the latter :

Tae Atte
„ 2 1 2°3 3 3 ,
Bil ¢ 7 Siva
12 Pe

From this equation I solve :

n= iS Es ee! 1 bie F. En
2 rt | 3

2 2
T, +-T,T.—T,

i
Dn 3
__t, 127, R
DI 5 inn
Tv, 1 T, TT, —Ts
127,°

where

( 755 )
Re Seater oe T; \ F “Eats Te)
Jennie 12 as 5 | 4 in 2 12
R = ciel
d ‘le Pie CN Pie

[rs
This remainder is apparently of the 4° order with respect to the
intervals. If we neglect the terms of higher order than the fourth
we can replace in R,: f, by Art”, F, by Kr”, f, by 20K,r,* and
F, by 20K,r,° ; and we obtain as supplementary term, accurate to the
fourth order

1
== a IE (es 5e T;) (tr, a T;) (27, Fri T;) (tr, a 2r,) ’
J

which expression vanishes on account of the last factor, in the case
of equal intervals.

The corresponding approximate formula for 7, can be derived by

: … triangle PZP. ke ; :
developing the relation ——~——___., depending on the time, in ascend-
triangle P,ZP,

ing powers of &(t, — tf) and further by proceeding in the same manner
as we have done for 7,. The result for », is derived from the pre-
ceding result by interchanging the indices 1 and 3, in which case
t, stands for & (¢, — ¢,), hence :

pee ipa Cre:
Es
Ga 12r, x
I= Ea = R,
T, 1 T, ar An T,
Dn 3
12r,

The remainder of 7, is not only of the same order as that of 7,,
but even in the 4% order it has the same absolute value, with a diffe-
rent sign however. This appears clearly when, using the relation
TT Ht, we express the correction of the 4% order for 7, in
terms of t, and r,; this correction takes the following form, which is
symmetrical with respect to rt, and r,:

|E
3 Kier, (ar vole, Hat) (EE)

In the remainder of n, the coefficient L, may be assumed equal
to A. Therefore these approximate formulae always give for
n, FN, an accurate value (comp. p. 758), including the terms of
the 4% order of the interval.

The denominators of these expressions for 7, and 7,, although here
different in form, are indeed identical; the expressions themselves
agree with those derived from the fundamental equation adopted by
GisBs between the 3 vectors ZP,, ZP, and ZP, which can be easily

3

reduced to the form :

rat Ts HT Ta
= (: Tae ZE

This equation is satisfied by the real places of the objeet when
we neglect a residual of the 5% order with respect to the intervals
of time. This signifies little, however, when compared with the
accuracy of the places calculated after Gipps’ method, which rigo-
rously satisfy them; for each set of vector corrections A ZP,, 4 ZP,
and A ZP does not lessen the agreement below the 5' order with
respect to the intervals of time, provided they satisfy the condition
NZE AZ AZE

1 T 8 2

Ts, 2

and are not below the 38'¢ order with respect to those intervals.

Because in GiBBs’ method the relations 7, and 2, contain errors
of the 4t" order, it would follow from this that the places computed
after this method are inaccurate in the 4" order also. But thanks to
the circumstance that GiBBs’ method includes for n, +, the terms
of the 4 order in all cases, its results are yet correct in terms of
the 4% order.

This special feature of Gipps’ method has been pointed out by
K. Weiss *).

In order to obtain for n, and ”, expressions including in all cases
the 4 order of the intervals of time and containing besides them

1 1
only ——=2,, — =, and a I have used the relation derived
ve
1 2 3
i) 1 =
on p. 754 K,= =O Ker

Starting from the development
n= Krt + Kyr? + Krt‘ + K,t° + remainder of the 5 order
I can make use of the following relations between the coefficients
k, the quantities z,, z,, 2, and n,.
NEE en a ec WGC Nes UA a ie ij
1= K, Ts ir K, Tan =i K, ts Si kK, Ts. zl Hy
+ 6K,t, + 12K, 7,2 + 20K, t° +f,
GK. Ie OK ORE ME
0 = K,z,+ 6K,

ne

1) E. Weiss, Ueber die Bestimmung der Bahn eines Himmelskörpers aus drei
Beobachtungen. Denkschriften der Mathem. Naturw. Classe der Wiener Akademie.
Bd. LX (1893).

Ci

By eliminating A, Aj, K, and K, we derive from them the
following equation:

1 (ar, 9)
Ce ee 60r,

Bay pia 2r, 2 +2r, Tart, —3r,° 2. t,’t,(4t, —3r,) NS
ie 60r, 720 DN NE
*(2r, = bt.)
Eg Eek eho /\ ea.
( ‘a zin Gor, 1
can (Bette tiant ain! nnn),
er 60r, 720 ey)

For shortness I replace the expressions which only depend on the
intervals of time by single letters, putting

i __ ty (2r, or) des _ Ts (2t,—9r,)

Alas Nee Ser Ag

a 60r, 601,

ore ~2r,” oT —2r, st, +37," —2r,° —2r,* *7,—27,7,°+3t

Bog pn Ce = berek Se B39 nn ia
60r, 607,

(Gi TT, (ár, —ôr,) bing = TT, (ár, = 3r,)

Soe 720 i 720

then the equation, solved with respect to ”,, yields for this relation
the following expression :
7, 14 Aso 21 + Bao 23 + C32 21 23

— x OAN RE Jf
a eee en z9 ie 4 00)

The ees R, contains the quantities #,, 7;, #, and /,;
these I set, in order to form the value of R, in the 5 order
with respect to the intervals of time, 7, = K,t,°, /, = Kr
Va 30K, rj andersO 59 then find:

eh KON (ee a Tee AU Cae UN

As the root of the 42 power equation 1— #— «? —2*+a*=0
lies between zero and 1, viz. « =0,5806, ae terms of the 5‘ order
will vanish from the residual, if rt, = 0.5806 r,.

We obtain the corresponding approximation for 7, when we derive
an expression from that for #, by interchanging everywhere the
indices 1 and 3, hence

wt, ,1+ Aiozs + Buoer + Cis 2s 21

<= bed tock (VAL
k T, s 1 + Aoi zs + Boi z2 4+ Cai 23 22 ey

The meaning of the new letters agrees with the rules tor the
interchange of the indices Ll and 3.

jj cme 8) Ve
i 607, 607,
tente! pent tate Be geese
es 60 1, ae 607,
Gees NEK)
oe 720 ZO)

In the remainder which belongs to this expression for 7,, the term
of the 5 order :
4 LZ, (t,{—1,° tt —t, Tr), (t,—T,)
will vanish if 7, = 0.5806 ae ees the em can never vanish
at the same time for 7, and for 7,

jb fficient of ihe developnaty nn
[, occurs as coefficient of 7, in the development of ———————

2 triangle P, VTA
in ascending powers of T—=k(t‚—t), while K, indicates the coefficient
ae . ees TAI ,
of t° in the development of ———_——\. where the variable rt

triangle P,ZP,’
means Á (t—1t,).

If the first of these developments were performed in powers of
k (t—t,) = — t, there would exist between each pair of corresponding
coefficients a relation implying that its sum with regard to +, would
be of one order higher than the coefficients themselves. Therefore,
neglecting terms of higher order than the 5'", we may assume that
the coefficients A, and £, are identical in absolute value, yet differ
in sign.

Of a similar relation I have made use on p. 755, where in the
remainders of the 4" order I assumed the coefficients identical. In
the new expressions for 7, and m, we can now, by putting 1,=—A,,
derive the following value for the remainder of the 5" order of n,--n,:

4K, 1,1,T, (t,—7,) (2t,? + 7,7,).

Therefore when the intervals of time are equal, the error in

n, +n, is of the 6% order.

If according to the indicated method we include the terms of the
4th order, we find for the 3'¢ relation nst =
triangle P, ZP, ny
mee ite As i22 + Banes + Car2228
ny (Aes eB ee Asen

and with it as Sue of the 5 order

> aa)

NT
+ 4 K, = aa (Gan aE t,t = Tet + CeCe -++ Te)

From one of the ae from Gauss’ Theoria Motus (Libr. II,
Sect. 1 ce. 156—158) 1 have computed the 3 relations according to
the formulae 1, II and III. The rigorously correct values of those

(759)

relations and the results of GriBBs’ expressions for this example I
borrow from P. Harzer’s Bestimmung und Verbesserung der Bahnen von
Himmelskirpern nach drei Beobachtungen p. 8. *)
The heliocentric motion of the planet Pallas was from the 1st to
the 3¢ observation 22°33’.
logt, = 9.8362703 logt, = 0.0854631 logt, = 9.7255594
log 7, = 03630906 logr, = 0.5507163 log r, = 0.3369508
These values for log r are also taken from Harzer and differ a
little from those according to Gauss.

Results for log n, and for log n,
GIBBS 9.7572961 GrBBs 9.6480108
formula IT 9.7572928 formula I 9.6480167
rigorous 9.7572923 rigorous 9.6480201.

Formula III yields: log 2298907237:
n,

With the given logarithms agree the following values:

n, Ns eesti:
rigorous = 0.5718654 0.4446518 07775491
Gak 0.5718641 f.l 04446484 f. II] 0.7775418

differences _ — 0.0000007 + 0.00000384 + 0.0000075

From the expressions given for the remainders of the 5 order
I calculated that they are in the ratio of — 9, + 72 and + 140. If we
compare these numbers. with the residuals, it appears that for our
example they would vanish to the 7" decimal if we succeeded
in including also the terms of the 5 order in the expressions.

As to the calculation of the quantities A and 6 dependent on
Tt, and t,, I remark that it may be performed quickly if we modify
these forms in the following way :

ve cae G Eaves =] aes hes Se Fg (stat
ONT raze I) (EA ea i
mn BEATE) maa EE ret)
Az = — 56 = ES =) 136} == 45 = eee ee)
Moe BEE) syns Bll ated
dn) meme ere
mmh) mri oA)

1) Publication der Sternwarte in Kiel, XI.

(760 )

Chemistry. — “A few observations on autocatalysis and the trans-
formation of y-hydroxy-acids, with and without addition of
other acids, conceived as an ion-reaction.” By Dr. A. W. Visser.

(Communicated by Prof. H. J. HAMBURGER).

1. The general equation for catalytic processes as proposed by
OstTWALD ') is:

== hk FSE NAS DIB oen

in which Z4f(w) indicates the changeable catalytic influences. He
remarks here that he cannot give a general method for drawing
a conclusion from the observed progress of a definite reaction, as to
the form of the function f(x).

Whilst studying the transformations of cane-sugar by invertase and
of salicin by emulsin?), I have shown, that, by introducing a correct
measure for the intensity of a catalyzer, the changeable catalytic
influences which occur here, could be indicated during the whole of
the progress of the reaction and it appears to me, that the method
given in my dissertation for determining that change in intensity
during the progress of the reaction may be called a general method
for determining the above mentioned function.

The most simple case imaginable is this, that the change in intensity
of the catalyzer with the change in concentration of the substance
acted upon is constant. In that case :

dt
"Ene oe
therefore :
Ikk,

In this equation is J—=k, when C—0; therefore 4, is the
intensity of the catalyzer when all has been converted, whilst 4,
is the increase in intensity, when the concentration diminishes with
the unity.

Schematically this may be represented as follows :

1) Osrwarp, Lehrbuch der Allgem. Chem. IL, 2, p. 270.
2) Visser, Dissertatie, Amsterdam 1904. A translation will shortly appear in the

Zeitschr. f. phys. Chem.

( 761 )

C, C
Fig. 1.
The reaction-equation for an unimolecular reaction, where the inten-
sity of a catalyzer plays a role, may be represented by :

dC ;
——=kCT.
dt
In the supposed case, therefore, by:
BS SEE
En t (k, — ‚CC.

This is the same equation as the one proposed by Osrwarp *) for
positive autocatalytie processes, but it seems to me that by introducing
the intensity-idea the constants, occurring in the formulae obtain a
more definite significance.

Then we have for the negative autocatalysis :

ai
HO hake
krikt

and here /, is again the intensity when all has been converted and
k, the decrease in intensity when the concentration diminishes with
the unity.
Schematically, this may be represented as follows :
J

ee ees Fig. 2.
1) Lehrb. der Allgem. Chem. II, 2 p. 263.

( 762 )

By positive autocatalysis must then be understood an increase in
the intensity of the catalyzer during the progress of the reaction ; by
negative autocatalysis a decrease in that intensity.

2. As examples of reactions in which autocatalysis occurs, may
be mentioned the spontaneous transformations of solutions of y-hydroxy-
acids into their lactones. The acid is split up into ions and according
to Paut Henry’), who studied the transformation of y-hydroxybutyric and
hydroxyvalerie acids, these would convert the unaltered portion
of the hydroxyacid and therefore act only catalytically. I have put
to myself the question whether these transformations may be consi-
dered as being ion-reactions, as it seems to me that it is more rational
to view them in this way and as such to bring them within the
laws of the mass-action, as we are ignorant as to the true nature
of catalysis. Profiting by the researches of Pau, Henry, I have
arrived at a very satisfactory result.

According to this view, the hydroxy-acid is at any moment in
equilibrium with its ions and these are converted into the lactone
according to the scheme:

y-hydroxy-acid = positive ions + negative ions I
positive ions + negative ions = lactone + water I

As has been stated, Pau Henry thinks that in this transformation
the non-split portion of the hydroxy-acid only changes into lactone
and that the ions exercise only a catalytic action, for he observes,
that on addition of HCl or H,SO, the dissociation-degree of the
hydroxy-acid is diminished, that is to say, the equilibrium is shifted
towards the side of the hydroxy-acid and the concentration of the
negative ions of the hydroxy-acid diminishes and if now the dissociated
portion of the acid were converted, the reaction-velocity would not
rise quite so much under these circumstances. It seems to me that
this argument is not correct; the concentration of the negative ions
of the hydroxy-acid diminishes by addition of H-ions in the form of
HCl, but on the other hand the concentration of the positive H-ions
rises enormously and in order to construct the reaction-equation we
must multiply these two concentrations by each other if we view

the reaction as an ion-reaction.

3. In the first place the spontaneous transformations of the y-hydroxy-
acids must be considered. In constructing reaction-equations it must
be borne in mind that the transformations of y-hydroxy-acids into

lactones is not complete.

LZ phys. Chem. X, p. 111.

( 763 )

Suppose the initial concentration of the y-hydroxy-acid
Suppose the concentration after a time ¢

C,
&
Suppose that the portion p of the hydroxy-acid is then split into
ions, the concentration of the hydroxy-acid will then be (1 — p) C
and the concentration of the ions pC.
According to equation (1)

oC
P =S

lI ll

1—p

Jk is the dissociation-constant of the hydroxy-acid.

Suppose the concentration of the ions after a time ¢= y, then is:

y PCE ARC HEK
The scheme:
positive ions + negative ions = lactone + water,

gives as reaction-equation, when the concentration-change of the
water is neglected

dy ; : :
aor rg Ot ee lt):
In this
kC,—y?—ky
——

k
is the concentration of the formed lactone.
The above differential-equation gives, after introduction of this value
for C,— C and after integration
yt =1(y — a) —l(y — B) + const.
For t=0 is y=y,, that is the initial concentration of the ions
and the equation becomes:

we WEE) oa)
Tt ye)
This equation is the same as the one deduced in my dissertation
for all bimolecular reactions where an equilibrium is formed.

In this equilibrium :

Yo = 3 {V4LC, + 2? — A} the initial concentration of the ions,
PIVO ER the concentration of the ions after

a time f,
B= y,= 3 V4kC, +2 — the end-concentration of the ions,

n= DE (see p. 14 of my dissertation).

The C, and C were determined by titration with barium hydroxide.
k is the dissociation-constant of the hydroxy-acid and could be
obtained by conductivity-determinations.

yt
Nm

Proceedings Royal Acad. Amsterdam. Vol. VIL.

( 764 )

4. In the spontaneous transformation of the hydroxy-acids the C,,
has not been determined and this is necessary in order to calculate
B and «. The question therefore arises how these may be determined.

es wed dy :
In the condition of equilibrium — — — 0 therefore:
dt

ky y= k, (C, ni, C)
y= 3IWARC + — Kl, so
EO EO

PRT, OEE)

nD

bo

x
we
|

|

=~

C',, the initial concentration is known, also & the dissociation-con-
stant. If now we know = the reciprocal value of the equilibrium-
a
constant of the transformation :
positive ions + negative ions = lactone,
C,, may be calculated.

This equilibrium-constant may be found from a series of deter-
minations where the end-condition has been determined of a definite
quantity of hydroxybutyrie-acid when HCI was added, therefore having
H-ions as active constituents. On pg. 112 of his treatise the author
states that to 20 ce. of the hydroxy-acid was added 5 ce. of the acid
to be investigated (HCI or H,5O,); according to table 64 on pg. 116
5 ec. of N HCl are added, the solution therefore becomes N/, HCI
and in case of complete splitting of the HCI in that concentration also
quadri-normal in H-ions.

In the condition of equilibrium the positive H-ions of the y-hydroxy-
acid and those of the HCl and the negative ions of the hydroxy-
acid are therefore in equilibrium with the lactone.

From the data of the above mentioned table 64 we may calculate that
hy .

HOP alll Ome

1
The transformation of hydroxyvaleric-acid also leads to an equi-
librium, but, whereas in the condition of equilibrium the hydroxy-
butyric-acid was converted to the extent of about 65°/,, the hydroxy-
valeric-acid had been converted to the extent of 95 °/
In this transformation was found for :

fie

k, A
== 15 x 107,
ky,

Paur Henry disregards in this transformation the occurring equili-
brium as it is shifted so much towards the side of the decomposition-
products. This should not happen if this transformation is regarded

( 765

as an ion-reaction as shown from the data on pg. 766 ; then the end
concentration of the ions is in the one case 0.000701 and the
initial-concentration 0.002690 and in the second case the end-concen-
tration of the ions is 0.000501 and the initial-eoncentration 0.001885.

The following tables derived from a series of determinations by
Paunt Henry show that the above reaction-equation is a correct one

For the spontaneous transformation of y-hydroxybutyric-acid we
find: (see table p. 766).

5. It has been shown by the writer, that, if so much of a foreign
acid (HCI, H,SO,) is added that the concentration of the H-ions during
the whole progress of the reaction may be taken as constant, the
usual reaction-equation applies; therefore in this case that of the
unimolecular reaction in which an equilibrium is formed.

If we consider these reactions as ion-reactions it may be easily
proved, that in the supposed case that equation appears.

If the concentration of the hydroxy-acid at a definite period is C, that
of the added H-ions C and p the part of the hydroxy-acid which has
split up into ions, then Chydroxy-acid — (1 — p) C; Coios =p C+ C';
Oner: ions — P C.

This acid is in equilibrium with its ions, consequently,

a) all
p(pC+C) _
1l—p
If so many H-ions have been added that this concentration may
be regarded as constant, pC + C' is a constant and the above
equation becomes :
D
BEES i,
1--p
that is to say, whatever value the concentration of the hydroxy-acid (C)
attains, p remains constant and during the whole of the progress of
the reaction, the concentration of the negative ions will amount to the
same part of the hydroxy-acid present. In this case we therefore, have:

negative ions + positive ions = lactone + water.

Suppose the concentration of the negative ions = y, and that of
the positive ions = Cy, then if the concentration of the lactone
= Cr and that of the water = Cy we have:

CRE k Ale (Ô! ' ! ' al
— =k, GHY —k,CwCr= Ty ke Chr=k,y—k,(C,—C);
then Cy and Cyr may be taken as constant.

It has been shown above that during the progress of the action

Tab. 92 and 93. PAUL HENRY p. 128. | Tab. 94 and 95 PAUL HENRY p. 128

0, =0.17166 0, =0.3390
£ —0.000741 a = 0.001313
2 =— 0.000703 2 =—0.001318
EEE NE
tin hours. | y 7 tin hours. y je
0 | 0.001810 — 0 0.002549 | —
28 | 0.001742 | 0.00067 24 | 0.002476 | 0.00085
66 0.001634 71 18 | _0.002372 97
120 | 0.001520 7 72 0.002274 | 107
1481] | 0.001476 68 123 | 0.002118 | 110
172 0.001448 66 151/, | 0.002049 | 109
219 | 0.001377 66 1741/, | 0.002006 106
=| 0.000741 Ee 192 0.001925 119
| x 0.001313 =

The following tables may serve for the spontaneous transformation
of y-hydroxyvaleric-acid.

Tab. 66. PAUL HENRY. p. 118. Tab. 72 and 73, PAUL HENRY. p. 121,

C, =0.3580 C, = 0.1769

& =yo = 0.000701 a =0.000501

« =—0.000703 a =— 0.000502
0 0 002690 — 0 0.001885 a
240 0.002607 | 0.000032 390 | 0.001830 | 0.0000193
450 0.002541 29 1170 0.001705 296
1170 | 0.002350 30 1860 | 0.001615 | 226
1500 0..002° 66 st 2640 | 0.001520 230
1890 0.002188 30 3300 0.001460 225
2810 0.002053 28 4080 0.001885 227
3530 0.001942 28 4710 0 001335 225
4310 0.001895 27 5550 0.001275 224
4940 0.001773 27 6900 0.001185 | 225
6170 0.001580 | 26 0 0.000501 | —
7740 0.001557 25
ee) 0.000701 —

Jijid
dC 1
dy je Pp
From this and the previous equation follows :
dC ak Y k ry. id al " Y ¥
— —=— C— — (¢, — C) =k," € — k," (€C, — C)
die P p

and this is the differential-equation for a unimolecular reaction in
which an equilibrium is formed.

6. If we add instead of HCI or H,SO, an acid like acetic-acid
which is partly split into ions, then, although H-ions are being added,
the concentration of the H-ions during the whole progress of the
reaction must no longer be considered as constant.

Suppose the initial-concentration of the hydroxy-acid = (,, that
of the acetic-acid C’ and that of the hydroxy-acid after a time ¢ (
and let us suppose that a portion a of the hydroxy-acid is split up
into ions and a portion 5 of the acetic-acid then :

Chydroxy-acid = (d WEE CHions— al ZIT bc"
Cacetic-acid il b) C; Cheg. ions hydroxy-acid — ac’

lil

At each period the hydroxy-acid is in equilibrium with the

( neg. ions acetic acid

H-ions and its negative ions and the same applies to the acetic
acid, therefore the two following equilibrium-equations apply :
a (aC +bC’) b(aC+bC’)

TE —— k, and — PE == hee

The dissociation-constants 4, and /, of the hydroxy-acid and the
acetic acid are nearly alike, consequently @ — 6 and

Gaza)
Stes — ON 0020
En
_ VAR th(CC) +P —
a — a
2 (cs (")
C tons hyd aa CVE (WIE SUE
neg. tons hydroxy-acid — EW ial Se fs
NEEN

= = ee GG
CH-ions = a (C HC) — 4 (V4k(C 7+’) =F El TRE js

From this follows:

C= or f(y +h A V(y + hb)? + 4k C}.

We again have:
negative ions + positive ions = lactone, therefore :

dy CHC 4
EL EE Oi
dt 1 ( (0! ) y 2 ( 0 )
kk, +k ;
ae ‘ley EHV FD AEC |» kyk, 0
Suppose gd kt Vy ER AO = 2,
De 0
then 5 U — SRO cam = k
dy 2 +4kC'
and = a
2k 7
and when we call EEE ==
1 dt 2? + Ak C!

ON de 2\@=2khz— Ae Oye Wk.) INC
et Ak C! oF
2° (Nkk, + 2k) z? +2 (NER, — UC — Nk, C,) 2 LANE EO

Therefore :
atpty=Nkk, + 2k (a); al Hay + By =2(NFE,—2kC'—NE,C,) (0);
= apy = 4 NI KG on a RER)
From (/) and (c) follows:
28 (NIE, — 2k0’ — Nk?C,) + ANEL,C!

aty= = 32 a = Sloane
‘
ANRC’
a= — ET (e)
vP

From (a) and (d) follows:
a? — (Nkk, + 2h)3? — 28(Nk%, — 2kC' — NIC) —ANEI,O . (fF)

dz
If there is equilibrium — ae becomes O and this happens when the
(

denominator of the above differential-equation becomes nought. The
equation which we then obtain in z is the same as equation (/) in B,
consequently @= 2, (the value of 2 in case of an equilibrium being

x

established).
Again introducing the value for .V in equation (f) we obtain :
f k. 5
a3 — as 97.4 A2 ) sr yall “il 9: ee :
p= lt INN ie I (9)
ee Heb hee Ron
(aar 7 me in i a 7

(769%)
For the constants p, g, 7 and s we find:
AEC! aS ALG: gr ALC y AKC!
p= LL Ir Nr ae MONI
apy a(a—y)(a—B) Bla BB) y(a—y) (8-y)

‘The following tables have been constructed from the observations
of Pavurt Henry on the aetion of acetie-acid on hydroxyvaleric-acid.

a k, 5 -
For this, #=="0.0000207 and Ps — 15 X 10-‘ (p. 764). Given the
1

values of C,, (initial concentration of the hydroxy-acid) and C’ (con-
centration of the acetic-acid) 8 may be caleulated from equation (4)
a and y may then be calculated from (d) and (e).

For this transformation the reaction-equation becomes,

Á ls = an a ply Sn
en en

Tab. 74 and 75, PAUL HENRY. p. 123. | Tab 76 and 77. PAUL HENRY. p. 123.

C= 0.1708 p= 26976,89 (== ADE) p= 25905,42
C'= 0.2058 g=— 232,46 C'= 0.01977 q=— 497,543
z= 0.004195 r=— 933,84 2= 0.001653 r=— 499,052
z= —0.004153 s = —27005,45 4——0.001597 s—=—925906,92
y= 0.000035 7= 0.000024
t in hours z | a ae t in hours z a: we: =
0 0.005495 — 0 0.004083 | —
210 | 0.005439 | 0.014 390 _ | 0.003947 | 0.0126
390, | 0.005349 21 4170 | 0.003706 128
4170 | 0.005296 | 21 1860 | 0.003521 128
1860 | 0004953 | 21 2640 | 0.003330 127
2640 | 0.004806. | 21 3300 | 0.003206 128
3300 | 0.001692 | 22 4080 0.003082 126
4080 | 0.004591 | 23 A710 | 0.002587 125
4710 | 0.004529 23 | 5550 | 0.009859 127
5550 0.004435 | 25 | |

|

These tables also give satisfactory values for the reaction-constants.

(770 )

Physics. — “Application of the Baroscope to the Determination of
the Densities of Gases and Vapors.” By Artnur W.~Gray.
(Preliminary Notice.) (Communication No. 94a from the
Physical Laboratory at Leyden by Prof. H. KAMERLINGH ONNES).

For determining the densities of gases, especially while flowing
continuously, the principle of the baroscope has been variously
applied by Firzerraip*), LOMMEL *), SmeerT and Dirr*), Musians*),
Precut’), and others. In the apparatus here described the aim has
been great sensitiveness combined with simplicity, ease of operation
and small volume.

reg a
0 5 10.

The accompanying figure illustrates the essential features. A capillary
glass tube carries at one end a closed bulb, and at the other a
henuspherical shell of the same diameter, weight, and kind of glass.
This is fastened to a horizontal quartz fiber stretched on a glass
frame, and carries a small mirror M, so that rotations about the
quartz fiber®) as axis can be measured with telescope and scale.
The whole is placed within a glass tube containing a sensitive ther-
mometer of some sort, and communicating with a manometer.

1) G. WP, Firzeeratp. Worlschritte der Physik 41, 102, 1885.

2) E. Lommer. Wied. Ann. 27, 144, 1886.

3) A. Siegert and W. Dürr. Zs. f. Instr.k. 8, 258, 1888.

5) M. Mesrans. Comptes Rend. 117, 386, 1893.

5) H. Precut. Zs. f. Instr.k. 13, 36, 1893.

6) The use of the quartz fiber was suggested by the delicate chemical balance
of Nernst and Rirsenrerp, Beibl. 28, 380, 1904, to which Prof. KAMERLINGH ONNES
had drawn my attention. Much more delicate instruments are, however, the quartz
thread gravity balance of Turetirat, and Potnock R. 8. Trans. 193, A, 215, 1900,
and the magnetograph of Warson, Proc. Phys. Soc. London, 19, 102, 1904.

Cune)

If the instrument has onee been calibrated, the scale reading gives
immediately the density of the gas within; while the thermometer
and the manometer permit the calculation of the density under
standard conditions, if the compressibility of the gas is known. The
calibration may be made either with a single gas whose density at
various pressures is known with sufficient accuracy for any one
temperature, or by employing in turn several different gases under
known conditions of pressure, temperature and density, or with a
rider. Counterpoising the closed sphere with the hemispherical shell
of equal surface tends to eliminate errors that would be introduced
if the apparatus contained a vapor which condensed on the glass.
The instruments should, of course, be protected from changes of
temperature by proper jacketing or by immersion in a liquid bath.
A fixed reference mirror (not shown in the figure) is desirable to
indicate any change in the leveling of the apparatus.

In order to get an idea of the sensitiveness that could be expected
from such an instrument, some rough preliminary measurements
were made.

The dimensions were as follows:

Diameter of bulb 1.0 em.

5 3 capillary beam Oni
Length ze u A TAO! sop:
Mass of entire suspended system 0.67 gms.
Length of quartz fiber 4 em:

The apparatus was filled with dry air, and the scale readings
noted for various pressures ranging from 0.3 cm. to nearly 90 em.
of mercury. With a fiber about 0,005 em. in diameter and the scale
255 cm. from the mirror, 0.1 mm. change in the deflection was
gm.
liter
density; and this was the same for all densities tried; that is to say,
a change of 0.1 mm. in the scale reading indicated a change of
about one part in 6000 in the density of air under ordinary con-
ditions. The scale might easily have been placed much farther from
the mirror and the sensitiveness could have been greatly increased
by using a larger bulb, a longer beam, and a longer and thinner
fiber. And since the change in deflection is, in the first approximation
at least, directly proportional to the change in density, an accurate
knowledge of the deflections for a few densities is sufficient for the
calibration of the instrument. Certain corrections, as, for instance,

found to indicate a change of about 0,0002 change in the

for the effects of changes of temperature on the quartz fiber, must,
of course, be applied when the greatest accuracy is desired.

This instrument was devised in order to follow the course of a
separation of atmospheric gases by fractional distillation at low tem-
perature, which Prof. KAMERLINGH ONNes wished to be made and to
be eontrolled by density measurements; but it is evident that its use
is not confined to this field. It might be used for determining the
densities of gases or vapors under various conditions, and therefore,
their compressibilities ; but it is especially useful as an indicator of
minute changes of density. Professor KAMeRLINGH Onnus has already
suggested its use to determine the composition of coexisting vapor
and liquid phases in cases where a chemical analysis would be
difficult or impossible, for example, in a mixture of two of the
inert gases of the atmosphere.

Constructional details and refinements, together with the results
of more careful and more varied tests will be communicated in a
later paper.

(May 25, 1905).

CONTENTS.

ABSORPTION. LINES (Double refraction near the components of) magnetically split into
several components. 435.

ABSORPTION SPECTRA (Dispersion bands in). 134.

ACETONE (On Px-curves of mixtures of) and ethylether and of carbontetrachloride and
acetone at 0°C, 162.

ACID (Sulphoisobutyric) and some of its derivatives. 275.

ADMIXTURES (The influence of) on the critical phenomena of simple substances and
the explanation of TEIcHNER’s experiments. 474.

ALGEBRAIC PLANE CURVE (On an expression for the class of an) with higher singu-
larities. 42.

— (On an expression for the genus of an) with higher singularities. 107.

— (On the curves of a pencil touching an) with higher singularities, 112,
ALGEBRAIC PLANE CURVES (On nets of). 631.

— (On linear systems of). 711.

ALGEBRAIC SURFACE (Some characteristic numbers of an). 716.
Anatomy. J. W. LANGELAAN: “On the Form of the Trunk-myotome”. 34.

— A. J. P. van DEN Broek: “On the genital cords of Phalangista vulpina”. 87.

— E. pre Vries: “Note on the Ganglion vomeronasale”. 704.

— J. W. van Bissentck: “Note on the innervation of the Trunkmyotome”. 708,
ANGLES (On the equation determining the) of two polydimensional spaces. 340.
ARBITRARY HIGH RANK (On moments of inertia and moments of an arbitrary order in

spaces of). 596.
Astronomy. C. Eason: “On the apparent distribution of the nebulae”. 117.
— C. Easton: “The nebulae considered in relation to the galactic system”, 125.
— J. Weeper: “A new method of interpolation with compensation applied to the
reduction of the corrections and the rates of the standard clock of the Observatory
at Leyden, Hohwii 17, determined by the observations with the transitcircle in
1903”, 241.

— J. A. C. OUpEMANs: “A short account of the determination of the longitude
of St. Denis (Island of Réunion) executed in 1874”. 602.

— J. Werprr: “Approximate formulae of a high degree of accuracy for the

relations of the triangles in the determination of an elliptic orbit from three

observations”. 752.
Proceedings Royal Acad, Amsterdam. Vol. VIL. 53

1 ClO NTE Nees

ASYMMETRIC syNTHPSIs (On W. Marckwatp’s) of optically active valerie acid. 465.

ATEN (A. H. W.). On the system pyridine and methyliodide. 468.

AUTOCATALYsIs (A few observations on) and the transformation of y-hydroxy-acids,
with and without addition of other acids, conceived as an ion-reaction. 760.

AXES (The locus of the principal) of a pencil of quadratic surfaces. 341.

— (The equations by which the locus of the principal) of a pencil of quadratic
surfaces is determined. 532.

— (The equation of order nine representing the locus of the principal) of a pencil
of quadratic surfaces. 721.

AZIMUTH (Determinations of latitude and) made in 1896 —99. 482.
BAKHUIS ROOZEBOOM (H. W.) presents a paper of Prof. Eve. DuBois: “On the
origin of the fresh-water in the subsoil of a few shallow polders”. 53.

— presents a paper of J. J. van Laar: “On the latent heat of mixing for asso-
ciating solvents”, 174.

— presents a paper of Dr. A. Smits: “On the phenomena appearing when in a
binary system the plaitpointcurve meets the solubility curve” (8rd communica-
tion). 177.

— presents a paper of Dr. J. J. Buanksma: “On trinitroveratrol”. 462.

— presents a paper of Dr. S. Tymstra Bz: “On W. Marckwat.p’s asymmetric
synthesis of optically active valerie acid”. 465.

— presents a paper of Dr. A. H. W. Aven: “On the system pyridine and methyl
iodide”. 468.

— presents a paper of J. J. van Laar: “On the different forms and transformations
of the boundary curves in the case of partial miscibility of two liquids”. 636.

— presents a paper of Dr. F. M. Jancer: “On miscibility in the solid aggregate
condition and isomorphy with carbon compounds”. 658.

— and i, H. Bicuner. Critical terminating points in three-phase lines with solid
phases in binary systems which present two liquid layers. 556,

BAKHUIJZEN (H. G. VAN DE SANDE). v. SANDE BAKHUIJZEN (H. G. VAN DE).

BARENDRECHT (H. P.). Enzyme-action. 2.

BAROMETRIC HEIGHT (On a twenty-six-day period in daily means of the). 18.

BAROSCOPE (Application of the) to the determination of the densities of gases and
vapors. 770.

BEMMELEN (J. M. VAN) presents a paper of Dr. H. P. Barenprecut: “Enzyme-
action”, 2,

— presents a communication of Prof. Evs. DuBois: “On the direction and the
starting point of the diluvial ice motion over the Netherlands”, 40.

— On the composition of the silicates in the soil which have been formed from
the disintegration of the minerals in the rocks, 329.

BENZENE SERIES (On the preservation of the crystallographical symmetry in the sub-
stitution of position isomeric derivatives of the). 191.

BENZENES (The nitration of disubstituted). 266.

BENZOL (On the intramolecular oxydation of a SH-group bound to) by an orthostanding
NOg-group. 63.

CONTENTS. II

BENZPINACONES (On intramolecular atomic rearrangements in). 271.

BENZYLPHTALIMIDE (On) and Benzylphtal-iso-imide, 77.

BES (K.). The equation of order nine representing the locus of the principal axes of a
pencil of quadratic surfaces. 721.

BESSEL FUNCTIONS (The values of some definite integrals connected with). 375.

— (On a series of). 494,

BEIJERINCK (M. w.). An obligative anaerobic fermentation sarcina. 580.

BINARY MIXTURES (The derivation of the formula which gives the relation between
the concentration of coexisting phases for). 156.

— (The conditions of coexistence of) of normal substances according to the law
of corresponding states. 222.

BINARY SYSTEM (On the phenomena appearing when in a) the plaitpointcurve meets
the solubilitycurve. 177.

BINARY SYSTEMS (Critical terminating points in three-phase lines with solid phases
in) which present two liquid layers. 556.

BISSELICK (J, W. VAN). Note on the innervation of the Trunkmyotome. 708.

BLANKSMA (J. J.). On the intramolecular oxydation of a SH-group bound to benzol
by an orthostanding NO,-group. 63.

— On trinitroveratrol. 462.

BLOK (s.). The connection between the primary triangulation of South-Sumatra and
that of the West-Coast of Sumatra. 453.

BLOOD (On the osmotic pressure of the) and urine of fishes. 537.

BOESEKEN (J.). The reaction of FriepeL and Crarts. 470.

BOLK (L.) presents a paper of A. J. P. van DEN BROEK: “On the genital cords of
Phalangista vulpina’. 87.

Botany. H. P. Kuyper: “On the development of the perithecium of Monascus purpu-
reus Went and Monascus Barkeri Dang”. 83.

— C. A. J. A. OupEmans: “On Leptostroma austriacum Oud., a hitherto unknown
Leptostromacea living on the needles of Pinus austriaca, and on Hymenopsis
Typhae (Fuck.) Sacc., a hitherto insufficiently described Tuberculariacea, occurring
on the withered leafsheaths of Typha latifolia”. 206.

— C. A. J. A. Oupemans: “On Sclerotiopsis pityophila (Corda) Oud., a Sphae-
ropsidea occurring on the needles of Pinus silvestris”, 211.

— Miss Tine Tammes: “On the influence of nutrition on the fluctuating varia-
bility of some plants”. 398.

— B. Syrkens: “On the nuclear division of Fritillaria imperialis L”. 412,

— J, M. Janse: “An investigation on polarity and organ-formation with Caulerpa
prolifera”. 420.

BOULDERS (Contributions to the knowledge of the sedimentary) in the Netherlands.
I. The Hondsrug in the province of Groningen. 2 Upper silurian boulders.
1st Communication. Boulders of the age of the eastern Baltic Zone G. 500.
2nd Communication, Boulders of the age of the eastern Baltic Zones Zand J, 692.
BOUNDARY CURVES (On the different forms and transformations of the) in the case of
partial miscibility of two liquids, 636.
53*

IV CeO N LE TNS Tes:

BRAIN in Tarsius spectrum (On the development of the). 331.
BRANCH PLAIT (The transformation of a) into a main plait and vice versa. 621.
BRANCHINGS (On the) of the nerve-cells in repose and after fatigue. 599.
BROEK (A. J. P. VAN DEN). On the genital cords of Phalangista vulpina. 87.
BRUYN (C. A. LOBRY DE). v. LOBRY DE Bruyn (C. A).
BRUYN (H. E. DE). Some considerations on the conclusions arrived at in the com-
munication made by Prof. Eve. Dugors, entitled: “Some facts leading to trace
out the motion and the origin of the underground water of our sea-provinces”’ 45.
BUCHNER (B. H.) and H. W. Bakuuvis RoozeBoom. Critical terminating points in
three-phase lines with solid phases in binary systems which present two liquid
layers. 556,
CARBON COMPOUNDS (On miscibility in the solid aggregate condition and isomorphy
with). 658.
CARBON DIOXIDE (The validity of the law of corresponding states for mixtures of
methylchloride and). 285. 377.
CARBONTETRACHLORIDE (On Px-curves of mixtures of acetone and ethylether and of)
and acetone at O°C, 162.
CARDINAAL (J). The locus of the principal axes of a pencil of quadratic sur-
faces. 341.
— The equations by which the locus of the principal axes of a pencil of quadratic
surfaces is determined. 532.
— presents a paper of K, Brs: “The equation of order nine representing the locus
of the principal axes of a pencil of quadratic surfaces”. 721.
CARVACROL (The inversion of caryon and eucarvon in) and its velocity. 63.
CARVON (The inversion of) and eucarvon in carvacrol and its velocity. 63,
CAULERPA PROLIFERA (An investigation on polarity and organ-formation with). 420.
Chemistry. H. P. BARENDRECHT: ‘“Enzyme-action”. 2.
— C. A. Lory pr Bruyn and S. Tymsrra Bz: “The mechanism of the salicylacid
synthese’. 63,
— J. J. BLANKSMA : “On the intramolecular oxydation of a SH-group bound to benzol
by an orthostanding NO,-group”. 63.
— J. M. M. Dormaar: “The inversion of carvon and eucarvon in carvacrol and
its velocity”. 63.
— J. J. van Laar: “On the latent heat of mixing for associating solvents”. 174.
— A. F, Houteman: “The preparation of silicon and its chloride”, 189,
— A. F. HOLLEMAN: “The nitration of disubstituted benzenes”. 266.
— A. P. N. Francuimont and H. FRIEDMANN : “On za'-tetramethylpiperidine”, 270.
— J. Mont van CHARANTE. “Sulphoisobutyric acid and some of its derivatives”. 275,
— P.J. Monracne: “On intramolecular atomic rearrangements in benzpinacones”’. 271.
— J. Our Jr.: “The transformation of the phenylpotassium sulphate into p-phenol-
sulphonate of potassium”. 328.
— J. F. Suyver: “The intramolecular transformation in the stereoisomeric g-and
B-trithioacet and g-and #-trithiobenzaldehydes”. 829.
— J. W. Dito: “The viscosity of the system hydrazine and water”. 329,

CONTENTS. Vv

Uhemistry. J. M. vaN BemMe ten: “On the composition of the silicates in the soil which
have been formed from the disintegration of the minerals in the rocks”. 329.

— A. F. HorLEMAN: “On the preparation of pure o-toluidine and a method for
ascertaining its purity”. 395.

— J. J. BLANKsMA: “On trinitroveratrol”’. 462.

— 8. Tymsrra Bz.: “On W. Marckwa.p’s asymmetrie synthesis of optically active
valerie acid”, 465.

— A. H. W. Aven: “On the system pyridine and methyliodide”. 468.

— J. BöeseKEN: “The reaction of FrIEDeL and Crarrs’”’, 470,

— J. J. van Laar: “On some phenomena, which can occur in the case of partial
miscibility of two liquids, one of them being anomalous, specially water”. 517.

— H. W. Baknuis RoozeBoom and E‚ H. Bicuner: “Critical terminating points
in three-phase lines with solid phases in binary systems which present two
liquid layers”. 556.

— J. J. van Laar: “On the different forms and transformations of the boundary-
curves in the case of partial miscibility of two liquids”, 636.

— J. J. van Laar: “An exact expression for the course of the spinodal curves
and of their plaitpoints for all temperatures, in the case of mixtures of normal
substances”. 646.

— F. M. Jarerr: “On miscibility in the solid aggregate condition and isomorphy
with carbon compounds”, 658.

— F. M. JarGer: “On orthonitrobenzyltoluidine”. (66.

— F, M. Jararr: “On position-isomerie Dichloronitrobenzenes”. 668.

— A. W. Visser: “A few observations on autocatalysis and the transformation of
y-hydroxy-acids, with and without addition of other acids, conceived as an ion=
reaction”. 760.

CLIMATE (Oscillations of the solar activity and the). 368.
COMPLEX (On a special tetraedal). 572,
COMPLEXES (On a group of) with rational cones of the complex. 577.

— of rays (A group of algebraic). 627.

COMPONENTS (Double refraction near the components of absorption lines magnetically
split into several). 485.

CONDITIONS of coexistence (The) of binary mixtures of normal substances according
to the law of corresponding states. 222,

— (The determination of the) of vapour and liquid phases of mixtures of gases at

low temperatures. 233.

CONGRUENCE (a) of order two and class two formed by conics. 311.

conics (The congruence of the) situated on the cubic surfaces of a pencil. 264,

— (A congruence of order two and class two formed by). 311.

CORRIGENDA et addenda, 382.

CRAFTS (The reaction of FrrepeL and). 470.

CRITICAL PHENOMENA (The influence of admixtures on the) of simple substances and
the explanation of TrtcHNER’s experiments. 474,

CROMER FOREST-BED (On an equivalent of the) in the Netherlands, 214,

VI CONTENTS,

Crystallography. I’, M. Jarcer: “On Benzylphtalimide and Benzylphtal-iso-imide”. 77.
— F. M. Jarcer: “On the preservation of the crystallographical symmetry in the
substitution of position isomeric derivatives of the benzene series’. 191.
cuBic surfaces of a pencil (The congruence of the conics situated on the). 264.
curve (On an expression for the class of an algebraic plane) with higher singu-
larities. 42.
— (On an expression for the genus of an algebraic plane) with higher singularities. 107.
— (On the curves of a pencil touching an algebraic plane) with higher singu-
larities. 112.
— (The relation between the radius of curvature of a twisted) in a point P of the
curve and the radius of curvature in P of the section of its developable with
its osculating plane in point P. 277.
curves (On Px-) of mixtures of acetone and ethylether and of carbontetrachlo-
ride and acetone at 0°C, 162.
— (On nets of algebraic plane). 631.
— (An exact expression for the course of the spinodal) and of their plaitpoints
for all temperatures, in the case of mixtures of normal substances. 646.
— (On linear systems of algebraic plane). 711.

a
DALFSEN (B. M. VAN), On the function 7 for multiple mixtures. 94,

DEDUCTION (Simplified) of the field and the forces of an electron, moving in any given
way. 346.
DEKHUIJZEN (M. €). On the osmotic pressure of the blood and urine of fishes. 537.
DENSITIES (Application of the Baroscope to the determination of the) of gases and
vapors. 770.
DEVENTER (cH. M. VAN). On the melting of floating ice. 459.
DICHLORONITROBENZENES (On position-isomeric). 668.
DILUTE SOLUTION (Kinetic derivation of van ’t Hoff’s law for the osmotic pressure
in a). 729. ,
DILUVIAL 1CE MOTION over the Netherlands (On the direction and the starting point
of the). 40.
DISPERSION (Spectroheliographic results explained by anomalous). 140.
DISPERSION BANDS in absorption spectra. 134.
— in the spectra of d Orionis and Nova Persei. 323.
prro (J w.). The viscosity of the system hydrazine and water. 329.
DORMAAR (J. M. M.). The inversion of carvon and eucarvon in carvacrol and its
velocity. 63.
bDUBOIS (EUG.). On the direction and the starting point of the diluvial ice motion
over the Netherlands. 40.
— (Some considerations on the conclusions arrived at in the communication made
by Prof.), entitled: “Some facts leading to trace out the motion and the origin
of the underground water of our sea-provinces”’. 45.
— On the origin of the fresh-water in the subsoil of a few shallow polders. 53,
— Gn an equivalent of the Cromer Forest-Bed in the Netherlands. 214,

en a

CONTENTS: VII

FAR (On the relative sensitiveness cf the human) for tones of different pitch, measured
by means of organ pipes. 549.
EASTON (c.). On the apparent distribution of the nebulae. 117.

— The nebulae considered in relation to the galactic system. 125.

— Oscillations of the solar activity and the climate. 368,

EINTHOVEN (w.). On a new method of damping oscillatory deflections of a gal-
vanometer, 315.

ELECTRON (Simplified deduction of the field and the forces of an), moving in any
given way. 346.

ELECTRONS (The motion of) in metallic bodies. I. 438. II. 585. III. 684.

ELLIPTIC ORBIT (Approximate formulae of a high degree of accuracy for the relations
of the triangles in the determination of an) from three observations. 752.

ENERGY (On artificial and natural nerve-stimulation and the quantity of) involved, 147.

ENZYME-ACTION. 2,

EQUATION (On the) determining the angles of two polydimensional spaces. 340,

— of order nine (The) representing the locus of the principal axes of a pencil of
quadratic surfaces. 721,

Equations (The) by which the locus of the principal axes of a pencil of quadratic
surfaces is determined, 532.

ERRATUM. 329, 485. 633.

ETHYLETHER (On Px-curves of mixtures of acetone and) and of carbontetrachloride
and acetone at 0°C. 162.

EUCARVON (The inversion of carvon and) in carvaerol and its velocity. 63,

EXPANSION COEFFICIENT (The) of Jena- and Thiiringer glass between + 16° and
—182°C, 674.

FERMENTATION SARCINA (An obligative anaerobic), 580.

FIBRINGLOBULIN (On the presence of) in fibrinogen solutions. 610.

FISHES (On the osmotic pressure of the blood and urine of). 537.

FLOCCULUS CEREBELLI (Degenerations in the central nervous system after removal of
the). 282.

FORMULA (The derivation of the) which gives the relation between the concentration
of coexisting phases for binary mixtures. 156,

FORMULAE of GULDIN (The) in polydimensional space. 487.

FRANCHIMONT (a. P, N.) presents a paper of Dr. I’. M, Jancer: “On Benzyl-
phtalimide and Benzylphtal-iso-imide”. 77.

— presents a paper of Dr. F. M. Jaraer: “On the preservation of the erystallo=
graphical symmetry in the substitution of position isomeric derivatives of the
benzene series”, 191,

— presents a paper of P. J. Montagne: “On intramolecular atomic rearrangements
in benzpinacones”, 271.

— presents the dissertation of Dr. J. Moun van CHARANTE: “Sulphoisobutyric
acid and some of its derivatives”. 275.

— and H. FrIEDMANN. On gz'-tetramethylpiperidine. 270.

PRIEDEL and Crarts (The reaction of). 470,

VIII CONTENTS,

FRIEDMANN (u.) and A. P. N. FRANCHIMONT. On zz'-tetramethylpiperidine. 270.

FRITILLARIA IMPERIALIS L. (On the nuclear division of). 412.
a . .
FUNCTION 7- (On the) for multiple mixtures. 94.

GALACTIC sysTEM (The nebulae considered in relation to the). 125.
GALVANOMETER (On a new method of damping oscillatory deflections of a). 315.
GANGLION VOMERONASALE (Note on the). 704.

GAS LAWs (A formula for the osmotic pressure in concentrated solutions whose vapour
follows the). 728.

casrs (The determination of the conditions of coexistence of vapour and liquid phases
of mixtures of) at low temperatures. 233.

— and Vapors (Application of the Baroscope to the determination of the den-
sities of). 770.

GEEST (J.) and P, Zeeman. Double refraction near the components of absorption
lines magnetically split into several components. 435.

GENITAL CORDS (On the) of Phalangista vulpina. 87.

Geodesy. 8. Brok: “The connection between the primary triangulation of South
Sumatra and that of the West Coast of Sumatra”. 453.

— J. A. U. OupEmaxs: “Determinations of latitude and azimuth made in 1896—
99”. 482,

Geology. Eve. Dugors: “On the direction and the starting point of the diluvial ice
motion over the Netherlands’, 40.

— H. B, pr Bruyn: “Some considerations on the conclusions arrived at in the
communication made Ly Prof. Eve. Dusors, entitled: “Some facts leading to
trace out the motion and the origin of the underground water of our sea-
provinces”. 45.

— Eve. Dugors: “On the origin of the fresh-water in the subsoil of a few shallow
polders”. 53.

— Eve. Dunors: “On an equivalent of the Cromer-Forest-Bed, in the Nether-
lands”. 214,

— H. G. Jonker: “Contributions to the knowledge of the sedimentary boulders
in the Netherlands. I. The Hondsrug in the province of Groningen. 2 Upper
Silurian boulders. 1st Communication: Boulders of the age of the eastern Baltic
Zone G°. 500. 2nd Communication: Boulders of the age of the eastern Baltic
Zones H and J”, 692.

GERRITS (G. C,). On Px-curves of mixtures of acetone and ethylether and of car-
bontetrachloride and acetone at 0°C. 162,

Grass (The expansion-coeflicient of Jena- and Thüringer) between + 16° and —
182°C. 674.

GOLD WIRE (Comparison of the resistance of) with that of platinum wire. 300.

GRAY (ARTHUR w.). Application of the Baroscope to the determination of the
densities of gases and vapors. 770.

GULDIN (The formulae of) in polydimensional space. 487.

ey ee

CONTENTS. IX

HAMBURGER (tH. J.) presents a paper of Dr. A. W. Visser: “A few observations
on autocatalysis and the transformation of y-hydroxy-acids, with and without
addition of other acids, conceived as an ion-reaction”’. 760.

HEAT OF MIXING (On the latent) for associating solvents. 174.

HEUSE (w.) and H. KAMERLINGH ONNEs. On the measurement of very low tem-
peratures. V. The expansioncoéfficient of Jena- and Thüringer glass between
+ 16° and — 182°C. 674.

HOEK (P. P. c.). An interesting case of reversion. 90.

HOFF’s LAW (Van ’t) (Kinetic derivation of) for the osmotic pressure in a dilute
solution. 729.

HOLLEMAN (a. F.). The preparation of silicon and its chloride. 189.

— The nitration of disubstituted benzenes. 266.

— On the preparation of pure o-toluidine and a method for ascertaining its
purity. 395.

— presents a paper of Dr. J. Borsrxen: ‘The reaction of Frreprn and Crarrts”. 470.

— presents a paper of Dr. F. M. Jarcer: “On Orthonitrobenzyltoluidine”. 666.

— presents a paper of Dr, I. M. Jazcer: “On position-isomeric Dichloronitro-
benzenes”’. 668.

HONDSRUG (The) in the province of Groningen. 500. 692,

HUBRECHT (A. A. W.) presents a paper of Prof. Tu. ZinuneN : “On the development
of the brain in Tarsius spectrum’. 331.

HUISKAMP (w.). On the presence of fibringlobulin in fibrinogen solutions. 610.

NYDRAZINE and water (The viscosity of the system). 329.

HYDROXY-AcIDs (A few observations on autocatalysis and the transformation of y-),
with and without addition of other acids, conceived as an ion-reaction. 760.
HYMENOPSIS TYPHAE (Fuck.) Sace. (On) a hitherto insufficiently described Tubercula-

riacea, occurring on the withered leafsheaths of Typha latifolia, 206.

ICE (On the melting of floating). 459.

ICE MOTION over the Netherlands (On the direetion and the starting point of the
diluvial). 40.

INERTIA (On moments of) and moments of an arbitrary order in spaces of arbitrary
high rank. 596.

INNERVATION (Note on the) of the Trunkmyotome. 708,

INTEGRALS (Evaluation of two definite), 201.

— (The values of some definite) connected with Bessel functions. 375.

INTERPOLATION (A new method of) with compensation applied to the reduction of the
corrections and the rates of the standardeloek of the Observatory at Leyden,
Hohwü 17, determined by the observations with the transitcircle in 1903, 241,

INTRAMOLECULAR atomic rearrangements in benzpinacones, 271,

INTRAMOLECULAR OXYDATION (On the) of a SH-group bound to benzol by an ortho-
standing NO,-group. 63.

INTRAMOLECULAR REARRANGEMENTS (On), 329.

INTRAMOLECULAR TRANSFORMATION (The) in the stereoisomerie z+ and (-trithioacet and
a- and #-trithiobenzaldehydes. 329,

x CONTENTS.

INVERSION (The) of carvon and eucarvon in earvacrol and its velocity. 63.

ION-REACTION (A few observations on autocatalysis and the transformation of y-
hydroxy-acids, with and without addition of other acids, conceived as an). 760.

ISOMORPHY (On miscibility in the solid aggregate condition and) with carbon com-
pounds. 658.

JAEGER (f, M.). On Benzylphtalimide and Benzylphtal-iso-imide.- 77.

— On the preservation of the crystallographical symmetry in the substitution of
position isomeric derivatives of the benzene series. 191.

— On miscibility in the solid aggregate condition and isomorphy with carbon
compounds. 658.

— On Orthonitrobenzyltoluidine. 666.

— On position-isomeric Dichloronitrobenzenes. 668.

JANSE(J. M.). An investigation on polarity and organ-formation with Caulerpaprolifera.420.
JONKER (H. G.). Contributions to the knowledge of the sedimentary boulders in
the Netherlands. 1. The Hondsrug in the province of Groningen. 2. Upper
silurian boulders. Ist Communication: Boulders of the age of the eastern Baltic
zone G. 500, 2nd Communication: Boulders of the age of the eastern Baltic zones
H and Z. 692.
JULIUS (w. H.). Dispersion bands in absorption spectra. 134,
— Spectroheliographic results explained by anomalous dispersion. 140.
— Dispersion bands in the spectra of 3 Orionis and Nova Persei. 323.
KAMERLINGH ONNES (H.) presents a paper of B. Mritin«: “On the measure-
ment of very low temperatures. VII. Comparison of the platinum thermometer
with the hydregen thermometer. 290, VIII. Comparison of the resistance of gold
wire with that of platinum wire.” 300.

— presents a paper of Dr. J. E‚ Verscuarrett; “The influence of admixtures on
the critical phenomena of simple substances and the explanation of TercuNer’s
experiments”, 474.

— presents a paper of Arraur W. Gray: “Application of the Baroscope to the
determination of the densities of gases and vapors”. 770.

— and W. Hevse. On the measurement of very low temperatures. V. The expan-
sion-coeflicient of Jena- and Thüringer glass between + 16° and — 182°C. 674.

— and C, Zakrzewski, Contributions to the knowledge of van per Waals y surface,
IX. The conditions of coexistence of binary mixtures of normal substances
according to the law of corresponding states. 222.

— The determination of the conditions of coexistence of vapour and liquid phases
of mixtures of gases at low temperatures. 233.

— The validity of the law of corresponding states for mixtures of methylchloride
and carbon dioxide, 285, 377.

KAP TEYN (w.). The values of some definite integrals connected with Bessel functions. 375.

— On a series of Bessel functions. 494.

KLUYVER (J. ©.) presents a paper of Prof. Epmunp LANDAU: “Remarks on the

tel)

paper of mr. KLuyver : “Series derived from the series a 66.

CONTENTS. xr

KLUYVER (3. C.). Evaluation of two definite integrals. 201.
KOHNSTAMM (Pi). A formula for the osmotic pressure in concentrated solutions
whose vapour follows the gas laws. 723.
— Kinetic derivation of van *£ Horr’s law for the osmotic pressure in a dilute
solution. 729.
— Osmotic pressure and thermodynamic potential. 741.
KORTEWEG (D. J.) presents a paper of Mr. Frep. Scuus: “On an expression for
the class of an algebraic plane curve with higher singularities”. 42.
— presents a paper of Mr. Frep. Scuuu: “On an expression for the genus of an
algebraic plane curve with higher singularities”, 107.
— presents a paper of Mr. Frep. Scuun: “On the curves of a pencil touching an
algebraic plane curve with higher singularities”, 112.
— and D. pe Lance. Multiple umbilics as singularities of the first order of except-
ion on point-general surfaces, 386.
KUYPER (H. P.). On the development of the perithecium of Monaseus purpureus
Went and Monascus Barkeri Dang. 83.
LAAR (J. J. VAN). On the latent heat of mixing for associating solvents. 174.
— On some phenomena which can occur in the case of partial miscibility of two
liquids, one of them being anomalous. 517.
— On the different forms and transformations of the boundary-curves in the ease
of partial miscibility of two liquids, 636.
— An exact expression for the course of the spinodal curves and of their plait-
points for all temperatures, in the case of mixtures of normal substances. 646.
LANDAU (EDMUND). Remarks on the paper of Mr. Kiuyver: “Series derived

from the series s———”.
ii

ACORN
il

LANGE (D. DE) and D. J. Korrewee. Multiple umbilics as singularities of the first
order of exception on point-general surfaces. 386.

LANGE (s. J. DE). On the branchings of the nerve-cells in repose and after
fatigue. 599. ;

LANGELAAN (J. w.). On the Form of the Trunk-myotome, 34,

LATITUDE (Determinations of) and azimuth made in 1896—99. 482.

Law of corresponding states (The conditions of coexistence of binary mixtures of
normal substances according to the). 222.

— of corresponding states (The validity of the) for mixtures of methyl chloride

and carbon dioxide. 285, 377.

LEPTOSTROMA AUSTRIACUM OUD., (On), a hitherto unknown Leptostromacea living on
the needles of Pinus austriaca. 206.

LIQUID LAYERS (Critical terminating points in three-phase lines with solid phases in
binary systems which present two). 556,

LIQUID PHasES (The determination of the conditions of coexistence of vapour and) of
mixtures of gases at low temperatures, 233,

Liguips (On some phenomena which can occur in the case of partial miscibility of
two), one of them being anomalous, specially water. 517.

XII OPO Ne DEE N tas.

Liquips (On the different forms and transformations of the boundary-curves in the
case of partial miscibility of two). 636.

LOBRY DE BRUYN (C. A.) presents a paper of Dr. J. J. BLANKsMA: “On the
intramolecular oxydation of a SH-group bound to benzol by an orthostanding
NO»-group”. 63.

— presents a paper of J. M. M. Dormaar: “The inversion of carvon and eucarvon
in carvacrol and its velocity”. 63.

— presents a paper of J. Om Jr: “The transformation of the phenylpotassium
sulphate into p. phenylsulphonate of potassium”: 328.

— presents a paper of J. F. Suyver: “The intramolecular transformation in the
stereoisomeric 2- and Z-trithioacet and z- and £-trithiobenzaldehydes”. 329.

— presents a paper of J. W. Dito: “The viscosity of the system hydrazine and
water”. 329.

— and S. Tymstra Bz. The mechanism of the salicylacid synthese. 63.

LONGITUDE of St. Denis (Island of Réunion) (A short account of the determination
of the) executed in 1874, 602.

LORENTZ (H. A.) presents a paper of Prof. A. SOMMERFELD: “Simplified deduction
of the field and the forces of an electron moving in any given way”. 346.

— The motion of electrons in metallic bodies. I. 438. If. 585. IIL. 684.

— presents a paper of J. J. van Laar: “On some phenomena, which can occur
in the case of partial miscibility of two liquids, one of them being anomalous
specially water”. 517.

— presents a paper of J. J. van Laar: “An exact expression for the course of
the spinodal curves and of their plaitpoints for all temperatures, in the case of
mixtures of normal substances”. 646.

MAIN PLAIT (The transformation of a branch plait into a) and vice versa. 621.

MARCKWALD’s (On W) asymmetric synthesis of optically active valeric acid. 465,

MARTIN (K.) presents a paper of Prof. Eva, Dupors: “On an equivalent of the
Cromer-Forest-Bed in the Netherlands”. 214,

— presents a paper of Dr. H. G. Jonker: “Contributions to the knowledge of
the sedimentary boulders in the Netherlands. 1. The Hondsrug in the province
of Groningen. 2. Upper Silurian boulders, 1st Communication: Boulders of the
age of, the eastern Baltic Zone G@”. 560. 2nd Communication: “Boulders of the
age of the eastern Baltic Zones H and J”, 692.

Mathematics. Frep. Scuuu. “On an expression for the class of an algebraic plane
curve with higher singularities”. 42,

— Epm. Lanpau: Remarks on the paper of Mr. Kiuyver: “Series derived from

mm
the series tlm)» 66.
mm

— Frep. Scuun: “On an expression for the genus of an algebraic plane curve
with higher singularities’. 107.

— Frep. Scuun: “On the curves of a pencil touching an algebraic plane curve
with higher singularities’. 112.

— J. C. Kruxrver: “Evaluation of two definite integrals”. 201.

bee

da

ee ee

CON DEN 1D 8: XII

Mathematics. Jan pe Vries; “The congruence of the conics situated on the cubic

surfaces of a pencil”. 264,

— W. A. Versruys: “The relation between the radius of curvature of a twisted
curve in a point P of the curve and the radius of curvature in P of the section
of its developable with its osculating plane in point P”. 277.

— JAN pe Vries: “A congruence of order two and class two formed by conics”. 311.

— P. H. Scrovre: “On the equation determining the angles of two poydimensional
spaces”. 340.

— J. CARDINAAL: “The locus of the principal axes of a pencil of quadratic
surfaces”. 341.

—- W. Kapteyn: “The values of some definite integrals connected with Bessel
functions”. 375.

— D. J. Kortewec and D. pr Lance: “Multiple umbilies as singularities of the
first order of exception on point-general surfaces”. 386.

— P. H. Scuoute: “The formulae of GuLpiN in polydimensional space”. 487.

— W. Kapreyn: “On a series of BrsserL functions”, 494.

— J. CARDINAAL: “The equations by which the locus of the principal axes of a
pencil of quadratic surfaces is determined”. 532.

— P. H. Scuoure: “On non-linear systems of spherical spaces touching one
another’’. 562.

— Jan DE VRIES: “On a special tetraedal complex”. 572.

— Jan DE Vries: “On a group of complexes with rational cones of the com-
plex”. 577.

— R. Meumxe: “On moments of inertia and moments of an arbitrary order in
spaces of arbitrary high rank”. 596.

— Jan DE VRIES: “A group of algebraic complexes of rays’, 627.

— Jan DE Vries: “On nets of algebraic plane curves’. 631.

— Jan pe Veres: “On linear systems of algebraic plane curves”. 711.

— JAN DE Vries: “Some characteristic numbers of an algebraic surface”. 716.

— K. Bes: “The equation of order nine representing the locus of the principal
axes of a pencil of quadratic surfaces”, 721.

MEASUREMENT (On the) of very low temperatures, V. The expansion coefficient of
Jena- and Thüringer glass between + 16° and — 182°C. 674. VII. Comparison
of the platinum thermometer with the hydrogen thermometer. 290, VIII, Camparison
of the resistance of gold wire with that of platinum wire. 300.

MECHANIsM (The) of the salicylacid synthese. 63.

MEHMKE (k.). On moments of inertia and moments of an arbitrary order in spaces
of arbitrary high rank. 596.

MEILINK (B.). On the measurement of very low temperatures. VII. Comparison
of the platinum thermometer with the hydrogen thermometer. 290. VIII. Compa-
rison of the resistance of gold wire with that of platinum wire. 300,

MELTING (On the) of floating ice. 459.

METALLIC BODIES (The motion of electrons in). I, 438. II, 585. LIL, 684.

XIV CHOPNET ENNE DASS

Meteorology. J. P. van per Stok: “On a twenty-six-day period in daily means of

the barometric height”. 18.
— C. Easton: “Oscillations of the solar activity and the climate”. 368.

METHOD (On a new) of damping oscillatory deflections of a galvanometer. 315.

METHYL CHLORIDE and carbon dioxide (The validity of the law of corresponding states
for mixtures of). 285. 377.

METHYL 1opIpE (On the system pyridine and). 468.

Microbiology. M. W. Bersertnck: “An obligative anaerobic fermentation sar-
cina”’, 580. ;

MINERALS in the rocks (On the composition of the silicates in the soil which have

been formed from the disintegration of the). 329.
MISCIBILITY (On some phenomena which can occur in the case of partial) of two

liquids, one of them being anomalous, specially water. 517.

— (On the different forms and transformations of the boundary-curves in the case
of partial) of two liquids. 636.

— (On) in the solid aggregate condition and isomorphy with carbon compounds, 658,

MIXTUREs (On the function 7 for multiple). 94.

— of normal substances (An exact expression for the course of the spinodal curves
and of their plaitpoints for all temperatures, in the case of). 646. :
MOLL (J. w.) presents a paper of Miss Tine Tammes: “On the influence of nutrition
on the fluctuating variability of some plants”. 398.
— presents the dissertation of B. SrPKeNs: “On the nuclear division of Fritillaria
imperialis L”. 412,
MOLL VAN CHARANTE (J.). Sulphoisobutyric acid and some of its dee 275.
MONASCUS purpureus Went and Monascus Barkeri Dang. (On the development of the
perithecium. of). 83. 4
MONTAGNE (Pe. J.). On intramolecular atomic rearrangements in benzpinacones, 271.
MOTION of electrons (The) in metallic bodies. I, 438, IL. 585. III. 684.
MUSKENS (L. J. J.). Degenerations in the central nervous system after removal of
the Floceulus cerebelli. 282.
NEBULAE (On the apparent distribution of the). 117.
— (The) considered in relation to the galactic system. 125.
NERVE-CELLS (On the branchings of the) in repose and after fatigue. 599.
NERVE-STIMULATION (On artificial and natural) and the quantity of energy involved. 147.
NERVOUS system (Degenerations in the central) after removal of the Flocculus
cerebelli. 282.
NITRATION (The) of disubstituted benzenes. 266.
NOVA PERSEI (Dispersion bands in the spectra of 3 Orionis and). 323.
NUCLEAR DIVISION (On the) of Fritillaria imperialis L. 412.
NUMBERS (Some characteristic) of an algebraic surface. 716.
NUTRITION (On the influence of) on the fluctuating variability of some plants. 398.
OLIE JR. (9). The transformation of the phenylpotassium sulphate into p. phenol-
sulphonate of potassium, 328.

GEOPNEDSEPNE DISS XV

ONNES (H. KAMERLINGH.). v. KAMERLINGH Onnzs (H.).

ORDER of exception (Multiple umbilics as singularities of the first) on point-general
surfaces, 386,

ORGAN-FORMATION (An investigation on polarity and) with Caulerpa prolifera, 420.

ORGAN-PIPES (On the relative sensitiveness of the human ear for tones of different
pitch, measured by means of), 549.

ORIONIS (Dispersion bands in the spectra of 5) and Nova Persei. 323.

ORTHONITROBENZYLTOLUIDINE (On). 666.

OSCILLATIONS of the solar activity and the climate. 368.

OSCILLATORY DEFLECTIONS (On a new method of damping) of a galvanometer. 315.

OSMOTIC PRESSURE (On the) of the blood and urine of fishes. 537.

— (A formula for the) in concentrated solutions whose vapour follows the gas laws. 723.
— (Kinetic derivation of vaN ’T HoFF’s law for the) in a dilute solution. 729.
— and thermodynamic potential. 741.

OUDEMANS (ce. A. J. A). On Leptostroma austriacum Oud., a hitherto unknown
Leptostromacea living on the needles of Pinus austriaca; and on Hymenopsis
Typhae (Fuck.) Sacc., a hitherto insufficiently described Tuberculariacea, occurring
on the withered leafsheaths of Typha latifolia, 206.

— On Sclerotiopsis pityophila (Corda) Oud., a Sphaeropsidea occurring on the
needles of Pinus silvestris. 211.

OUDEMANS (J. A. C.) presents a paper of S. Brok: “The connection between the
primary triangulation of South Sumatra and that of the West Coast of
Sumatra’. 453,

— Determinations of latitude and azimuth, made in 1396—99, 482.
— A short account of the determination of the longitude of St. Denis (Island of
Réunion) executed in 1874. 602.

OXYDATION (On the intramolecular) of a SH-group bound to benzol by an ortho-
standing NO,-group. 63.

PEKELHARING (c. A.) presents a paper of Dr. M. C, DEKHUIJZEN ; “On the
osmotic pressure of the blood and urine of fishes”. 537.

— presents a paper of Dr, W. Hurskamp: “On the presence of fibringlobulin in
fibrinogen solutions”. 610.
PENCIL (The locus of the principal axes of a) of quadratic surfaces, 341.
— (The equations by which the locus of the principal axes of a) of quadratic
surfaces is determined. 532.
— (The equation of order nine representing the locus of the principal axes of a)
of quadratic surfaces, 721.
PERIOD (On a twenty-six-day) in daily means of the barometric height. 18.

PERITHECIUM (On the development of the) of Monascus purpureus Went and Monascus
Barkeri Dang. 83.

PHALANGISTA VULPINA (On the genital cords of). 87.
PHASES (The derivation of the formuia which gives the relation between the concen-
tration of coexisting) for binary mixtures. 156,

XVI ClO! N TRAN Ts:

PHENOMENA (On the) appearing when in a binary system the plaitpointeurve meets the
solubilityeurve. (8rd communication). 177.
— (On some) which can occur in the case of partial miscibility of two liquids,
one of them being anomalous, specially water. 517.
PHENYLPOTASSIUM SULPHATE (The transformation of the) into p. phenolsulphonate
of potassium. 328.

Physics. B. M. van DALFSEN: “On the function = for multiple mixtures”. 94,

-- W. H. Junius: “Dispersion bands in absorptionspectra”. 134.

— W. H. Junius: “Spectroheliographic results explained by anomalous dispersion”. 140.

— J. D. van per Waats: “The derivation of the formula which gives the relation
between the concentration of coexisting phases for binary mixtures”. 156.

— G. C. Gerrits: “On Px-curves of mixtures of acetone and ethylether and of
carbontetrachloride and acetone at 0°C”. 162.

— A. Smits: “On the phenomena appearing when in a binary system the plait-
pointeurve meets the solubilitycurve”. (8rd communication). 177.

— H. KAMERLINGH Onnes and C. ZAKRZEWSKI: “Contributions to the knowledge
of VAN DER Waals’ p-surface. IX. The conditions of coexistence of binary mixtures
of normal substances according to the law of corresponding states”. 222.

— H. KAMERLINGH Onnes and C. Zaxrzewsk1: ‘The determination of the conditions
of coexistence of vapour and liquid phases of mixtures of gases at low tempera-
tures”, 233.

— H. KAMERLINGH Onnes and C. Zaxrzewsx1: “The validity of the law of cor-
responding states for mixtures of methyl chloride and carbon dioxide”. 285. 377.

— B. Merrink:’“On the measurement of very low temperatures. VII. Com-
parison of the platinum thermometer with the hydrogen thermometer”. 290.
VIII. “Comparison of the resistance of gold wire with that of platinum wire.” 300.

— W. H, Jvrrus: “Dispersion bands in the spectra of $ Orionis and Nova Persei’’. 323.

— A, SoMMERFELD: “Simplified deduction of the field and the forces of an electron,
moving in any given way”. 346.

— P. Zeeman and J. Geest: “Double refraction near the components of absorption
lines magnetically split into several components.” 435,

— H. A. Lorentz: “The motion of electrons in metallic bodies”. L. 438. IL.
585. IIL. 684. 3

— Cu. M. van Deventer: “On the melting of floating ice’. 459.

— J. E. Verscuarretr: “The influence of admixtures on the critical phenomena
of simple substances and the explanation of TercHNeER’s experiments”, 474, _

— J. D. van per Waats: “The transformation of a branch plait into a main plait
and vice versa”. 621.

— H. KAMERLINGH Onnes and W, Ileuse: “On the measurement of very low
temperatures. V. The expansion coefficient of Jena- and Thüringer glass between
+ 16° and- 182° ©”. 674.

— Pu. Kounstamm: * A formula for the osmotic pressure in concentrated solutions

whose vapour follows the gas laws”. 723.

CONTENTS, XVII

Physiology. Pu. Konnstamm: “Kinetic derivation of van ‘r Horr’s law for the osmotic
pressure in a dilute solution”. 729.
— Pu. Konyxstamm: “Osmotic pressure and thermodynamic potential”. 741.
— ArtHuur W. Gray: “Application of the Baroscope to the determination of the
densities of gases and vapors”. 770.
— H. ZWAARDEMAKER: “On artificial and natural nerve-stimulation and the
quantity of energy involved”. 147,
— L. J. J. Muskexs: ‘Degenerations in the central nervous system after removal
of the Flocculus cerebelli”. 282.
— W. ErtHoven: “On a new method of damping oscillatory deflections of a
galvanometer”. 315.
— M. C. DEKHUDZEN : “On the osmotic pressure of tle blood and urine of fishes”. 537.
— H. ZWAARDEMAKER Cz.: “On the relative sensitiveness of the human ear for
tones of different pitch, measured by means of organpipes”. 549,
— S, J. pr LANGE: “On the branchings of the nerve-cells in repose and after
fatigue”. 599.
— W. Huisxame: “On the presence of fibringlobulin in fibrinogen solutions’. 610.
PLACE (T.) presents a paper of Prof. J. W. LANGELAAN: “On the form of the
Trunk-myotome”, 34,
— presents a paper of HK. pe Vries: “Note on the Ganglion vomeronasale”. 704.
— presents a paper of J. W. van BrssericK: “Note on the innervation of the
Trunk-myotome”’. 708,
PLAITPOINTCURVE (On the phenomena appearing when in a binary system the) meets
the solubilitycurve. 177.
PLAITPOINTS (An exact expression for the course of the spinodal curves and of their)
for all temperatures, in the case of mixtures of normal substances. 646.
PLANTS (On the influence of nutrition on the fluctuating variability of some). 398.
PLATINUM WIRE (Comparison of the resistance of gold wire with that of). 300.
POLARITY (An investigation on) and organ-formation with Caulerpa prolifera. 420.
POLDERS (On the origin of the fresh-water in the subsoil of a few shallow). 53.
POLYDIMENSIONAL SPACE (The formulae of GULDIN in). 487.
POLYDIMENSIONAL SPACES (On the equation determining the angles of two). 340.
PotassiuM (The transformation of the phenylpotassiumsulphate into p-phenolsulphonate
of). 328.
POTENTIAL (Osmotic pressure and thermodynamic). 741.
PYRIDINE and methyl iodide (On the system). 468.
QUADRATIC SURFACES (Lhe locus of the principal axes of a pencil of). 341.
— (The equations by which the locus of the principal axes of a pencil of) is
determined. 532.
— (The equation of order nine representing the locus of the principal axes of a
pencil of). 721.
RADIUS of curvature (The relation between the) of a twisted curve in a point P o
the curve and the radius of curvature in P of the section of its developable

with its osculating plane in point P. 277.
Proceedings Royal Acad. Amsterdam. Vol. VII. 54

XVIIL CONTENTS:

RATIONAL CONES of the complex (On a group of complexes with), 577.

rays (A group of algebraic complexes of). 627.

REACTION (The) of FriepEL and Crarts. 470.

REFRACTION (Double) near the components of absorption-lines magnetically split into
several components, 435.

REVERSION (An interesting case of). 90.

ROOZEBOOM (i. W. BAK HUIS). v. BaKmurs RoozeBoom (H. W.).

SALICYLACID synthese (The mechanism of the). 63.

SANDE BAKHUYZEN (H. G. VAN DE) presents a paper of C, Easton: “On
the apparent distribution of the nebulae”. 117.

— presents a paper of C, Easton: “The nebulae considered in relation to the
galactic system”. 125.

— presents a paper of J. Weeper: “A new method of interpolation with com-
pensation applied to the reduction of the corrections and the rates of the
standardoloek of the Observatory at Leyden, Hohwii 17, determined by the
observations with the transitcircle in 1903”. 241.

— presents a paper of J. WeEDER: “Approximate formulae of a high degree of
accuracy for the relations of the triangles in the determination of an elliptic
orbit from three observations”. 752.

SARCINA (An obligative anaerobic fermentation). 580.
SCHOUTE (Pe. H.). On the equation determining the angles of two polydimensional
spaces. 340.

— The formulae of GULDIN in polydimensional space. 487.

— On non-linear systems of spherical spaces touching one another. 562.

— presents a paper of Prof. R. MeumKE: “On moments of inertia and moments of
an arbitrary order in spaces of arbitrary high rank”. 596.

SCHUH (FRED.). On an expression for the class of an algebraic plane curve with
higher singularities. 42.

— On an expression for the genus of an algebraic plane curve with higher singu-
larities. 107.

— On the curves of a pencil touching an algebraic plane curve with higher sin-
gularities. 112.

SCLEROTIOPSIS PITYOPHILA (Corda) Oud. (On), a Sphaeropsidea occurring on the
needles of Pinus silvestris. 211.

: ; ge(1)
SERIES derived from the series Dar 66.

sILICATES (On the composition of the) in the soil which have been formed from the
disintegration of the minerals in the rocks. 329.

SILICON (The preparation of) and its chloride. 189.

SMITS (A). On the phenomena appearing when in a binary system the plaitpoint-
curve meets the solubilityeurve. (3rd communication). 177.

SOLAR ACTIVITY (Oscillations of the) and the climate. 368.

SOLUBILITY curve (On the phenomena appearing when in a binary system the plait-
pointcurve meets the). 177.

KCTOPNS TEEN WEISS XIX

SOLVENTS (On the latent heat of mixing for associating). 174.

SOMMERFELD (A). Simplified deduction of the field and the forces of an electron,
moving in any given way. 346.

SPECTRA (Dispersion bands in the) of 3 Orionis and Nova Persei. 323.

SPECTROHELIOGRAPHIC results explained by anomalous dispersion. 140.

SPHAEROPSIDEA (On Sclerotiopsis pityophila (Corda) Oud., A) occurring on the needles
of Pinus silvestris. 211.

SPHERICAL SPACES (On non-linear systems of) touching one another. 562.

STANDARDCLOCK (A new method of interpolation with compensation applied to the
reduction of the corrections and the rates of the) of the Observatory at Leyden,
Hohwii 17, determined by the observations with the transitcircle in 1903. 241.

ST. DENIS (Island of Réunion) (A short account of the determination of the longitude
of) executed in 1874. 602.

STEREOISOMERIC g-and @-trithioacet (The intramolecular transformation in the) and
a-and #-trithiobenzaldehydes. 329.

STOK (J, P. VAN DER). On a twenty-six-day period in daily means of the barometric
height. 18.

SULPHOISOBUTYRIC ACID and some of its derivatives. 275.

suMATRA (The connection between the primary triangulation of South-Sumatra and that
of the West-Coast of). 453.

SURFACE (Contributions of the knowledge of van per Waats’y-). IX. The conditions
of coexistence of binary mixtures of normal substances according to the law of
corresponding states. 222.

SURFACES (Multiple umbilics as singularities of the first order of exception on point-

general). 336,
SUYVvER (J. F.). The intramolecular transformation in the stereoisomeric g- and

B-trithioacet and z- and @-trithiobenzaldehydes. 329.
SYPKENS (B.). On the nuclear division of Fritillaria imperialis L, 412,
SYSTEMS (On non-linear) of spherical spaces touching one another. 562.

— (On linear) of algebraic plane curves. 711.

TAMMES (TINE). On the influence of nutrition on the fluctuating variability of
some plants. 398.

TARSIUS SPECTRUM (On the development of the brain in), 331.

TEICHNER’s experiments (The influence of admixtures on the critical phenomena
of simple substances and the explanation of), 474,

TEMPERATURES (The determination of the conditions of coexistence of vapour and
liquid phases of mixtures of gases at low). 233,

— (On the measurement of very low). V. The expansioncoefficient of Jena- and
Thüringer glass between + 16° and — 182° U, 674, VII. Comparison of the
platinum thermometer with the hydrogen thermometer, 290. VILI. Comparison of
the resistance of gold wire with that of platinum wire. 300.

— (An exact expression for the course of the spinodal curves and of their pluit-
points for all), in the case of mixtures of normal substances. 646.

TETRAEDAL COMPLEX (On a special). 572.

XX CONTENTS.

TETRAMETHYLPIPERIDINE (On zz'-). 270.

THERMODYNAMIC POTENTIAL (Osmotic pressure and). 741.

THERMOMETER (Comparison of the platinum thermometer with the hydrogen). 290.

THREE-PIIASE LINES (Critical terminating points in) with solid phases in binary systems
which present two liquid layers. 556.

TOLUIDINE (On the preparation of pure o-) and a method for ascertaining its purity. 395,

TONES of different pitch (On the relative sensitiveness of the human ear for), measured
by means of organ pipes. 549.

TRIANGLES (Approximate formulae of a high degree of accuracy for the) in the deter-
mination of an elliptic orbit from three observations. 752.

TRIANGULATION (The connection between the primary) of South-Sumatra and that of
the West-Coast of Sumatra. 453.

TRINITROVERATROL (On). 462.

TRITHIOACET (The intramolecular transformation in the stereoisomeric g-and @-) and
g-and -thrithiobenzaldehydes. 329.

TRUNK-MYOTOME (On the form of the). 34.

— (Note on the innervation of the). 708.

TUBERCULARIACEA (On Hymenopsis ‘Typhae (Fuck.) Sace.. a hitherto insufficiently
described), occurring on the withered leafsheaths of Typha latifolia. 206.

TYMSTRA BZ. (s.). On W. Marckwatp’s asymmetrie synthesis of optically active
valeric acid. 465.

— and C. A. Losry pe Bruyn. The mechanism of the salicylacid synthese. 63.
umMBILIcs (Multiple) as singularities of the first order of exception on point-general
surfaces. 386.

URINE of fishes (On the osmotic pressure of the blood and). 537.

VALERIC Aci (On W. Marckwatp’s asymmetric synthesis of optically active). 465.

vapors (Application of the Baroscope to the determination of the densities of gases
and). 770.

vapour (The determination of the conditions of coexistence of) and liquid phases of
mixtures of gases at low temperatures. 233.

— (A formula for the osmotic pressure in concentrated solutions whose) follows
the gas laws. 723.

VARIABILITY of some plants (On the influence of nutrition on the fluctuating). 398.

VERSCHAFFELT (J, E.). The influence of admixtures on the critical phenomena
of simple substances and the explanation of TErcHNER’s experiments. 474,

VERSLUYS (w. A.). The relation between the radius of curvature of a twisted curve
in a point P of the curve and the radius of curvature in P of the section of
its developable with its osculating plane in point P. 277.

viscosity (The) of the system hydrazine and water. 329.

VISSER (a. w.). A few observations on autocatalysis and the transformation of
y-hydroxy-acids, with and without addition of other acids, conceived as an
ion-reaction. 760.

VRIES (e. DE). Note on the Ganglion vomeronasale. 704.

CONTENTS. XXI

VRIES (HUGO DE) presents a paper of Prof. J. M. Janse: “An investigation on

polarity and organ-formation with Caulerpa prolifera”. 420.

VRIES (JAN DE). The congruence of the conics situated on the cubic surfaces of
a pencil. 264.

— A congruence of order two and class two formed by conics. 311.
— On a special tetraedal complex. 572.

— On a group of complexes with rational cones of the complex. 577.
— A group of algebraic complexes of rays. 627.

— On nets of algebraic plane curves. 631,

— On linear systems of algebraic plane curves. 711.

— Some characteristic numbers of an algebraic surface. 716.

WAALS (VAN DER) y-surface (Contributions to the knowledge of). IX, The conditions
of coexistence of binary mixtures of normal substances according to the law of
corresponding states. 222,

WAALS (J. D. VAN DER) presents a paper of B. M. van DAurseN: “On the

function - for multiple mixtures”. 94,
D

— The derivation of the formula which gives the relation between the concentra-
tion of coexisting phases for binary mixtures. 156.

— presents a paper of G. C, Grerrars: “On Px-curves of mixtures of acetone and
ethylether and of carbontetrachloride and acetone at O°C’. 162.

— presents a paper of Dr. Ca. M. van Deventer: “On ihe melting of floating
ice”. 459.

— The transformation of a branch plait into a main plait and vice versa. 621.

— presents a paper of Dr. Pu. Kounstamm: “A formula for the osmotic pressure
in concentrated solutions whose vapour follows the gas laws.” 723.

-— presents a paper of Dr. Pa, KounstamM: “Kinetic derivation of van ’r Horr’s
law for the osmotic pressure in a dilute solution”. 729.

— presents a paper of Dr. Pu, KounstamMm: ‘Osmotic pressure and thermodynamic
potential”. 741.

WATER (On the origin of the fresh-) in the subsoil of a few shallow polders. 53.

— (The viscosity of the system hydrazine and), 329.

— of our sea-provinces (Some considerations en the conclusions arrived at in the
communication made by Prof, Eve. Dugors, entitled: Some facts leading to trace
out the motion and the origin of the underground). 45.

WEEDER (J.). A new method of interpolation with compensation applied to the
reduction of the corrections and the rates of the standardclock of the Observa-
tory at Leyden, Hohwii 17, determined by the observations with the transitcircle
in 1903, 241. .

— Approximate formulae of a high degree of accuracy for the relations of the
triangles in the determination of an elliptic orbit from three observations, 752.

WENT (F.A. F.C.) presents a paper of H. P Kuyper: “On the development of
the perithecium of Monascus purpureus Went and Monascus Barkeri Dang.” 83.

XXII CONTENTS.

WIND (c. H.) presents a paper of Dr. C. Easton: “Oscillations of the solar activity
and the climate.” 368.
WINKLER (C.) presents a paper of Dr. L. J. J. Muskens: “Degenerations in the
central nervous system after removal of the Flocculus cerebelli.” 282,
— presents a paper of Dr. 8. J. pr LANGE: “On the branchings of the nerve-cells
in repose and after fatigue.” 599.
ZAKRZEWSKI (c.) and H. KaMERLINGH ONNEs. Contributions to the knowledge of
VAN DER WaaLs’ yp surface. IX. The conditions of coexistence of binary mixtures
of normal substances according to the law of corresponding states. 222.
— The determination of the conditions of coexistence of vapour and liquid phases
of mixtures of gases at low temperatures, 233.
— The validity of the law of corresponding states for mixtures of methylchloride
and carbon dioxide. 285. 377.
ZEEMAN (P.) and J. Gees. Double refraction near the components of absorption
lines magnetically split into several components. 435.
ZIEWEN (TH.). On the development of the brain in Tarsius spectrum. 331,
Zoology. P. P. C. Hoek: “An interesting case of reversion.” 90.
— Tu. Zrenen: “On the development of the brain in Tarsius spectrum.” 331.
ZWAARDEMAKER (H.). On artificial and natural nerye-stimulation and the
quantity of energy involved. 147.
— On the relative sensitiveness of the human ear for tones of different pitch,
measured by means of organ pipes. 549.

Koninklijke Akademie van Wetenschappen j
te Amsterdam,

PROCEEDINGS

OF THE

SECTION OF SCIENCES

eN

NO Or ME Er VEE

(2nd PART)

AMSTERDAM,
JOHANNES MÜLLER.

July 1905.

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prin
BL ad
W cy

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