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Koninklijke Akademie van Wetenschappen
te Amsterdam.

PROCEEDINGS

OF THE

eee ON O PosS.c 1 EN CES.

«94

AVE (S)ALA UPN ast) WV/S EEE

AMSTERDAM,
JOHANNES MULLER.
June 1906,

(Translated from: Verslagen van de Gewone Vergaderingen der Wis- en Natuurkundige
Afdeeling van 27 Mei 1905 tot 27 April 1906. DI. XIV.)

Koninklijke Akademie van Wetenschappen
te Amsterdam.

PROCEEDINGS

OF THE

ree olrON- OK SCLlEN CES

——e>-_4—

WA (SYA Ey SF ANE Bay Wa EE

(ist =PA Ra)

AMSTERDAM,
JOHANNES MULLER.
December 1905. 47 se

(Translated from: Verslagen van de Gewone Vergaderingen der Wis- en Natuurkundige
Afdeeling van 27 Mei 1905 tot 25 November 1905. Dl. XIV.)

CrOeN Ti: NTS:

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Proceedings of the Meeting of May 27 IO O Same cum: Vices 5: Lacy tones 1
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KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM.

PROCEEDINGS OF THE MEETING
of Saturday May 27, 1905.

SSS

(Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige
Afdeeling van Zaterdag 27 Mei 1905, DI. XIV).

Gi) INP GEIS Bes

L. Bork: “On the development of the cerebellum in man”. (First part), p. 1. (With one plate).

E. Verscuarrerr: “Some observations on the longitudinal growth of stems and flower-stalks”.
(Communicated by Prof. Huco pr Vrigs), p. 8.

G. C. J. Vosmarr and H. P. Wrsman: “On the structure of some siliceous spicules of Sponges.
I. The styli of Tethya lyncurium”, p. 15.

Jan pE Vries: “On pencils of algebraic surfaces”, p. 29.

J. J. van Laar: “On the shape of the plaitpoint curves for mixtures of normal substances”.
(2nd Communication) (Communicated by Prof. H. A. Lorentz), p. 33. (With one plate).

J. J. van Laan: “Some remarks on Dr. Pu. Konnsramm’s last papers”. (Communicated by
Prof. H. A. Lorentz), p. 49.

W. A. Versiuys: “On the rank of the section of two algebraic surfaces”. (Communicated by
Prof. P. H. Scnourer), p. 52.

The following papers were read :

Anatomy. — “On the development of the cerebellum in man.”
(First part). By Prof. L. Bonk.

(Communicated in the meeting of March 25, 1905.)

On account of the fact that the lobulisation of the adult cerebellum
of Primates generally and of man in particular, deviates in various
respects from that of the remaining mammals, so that a homologisation
of the lobes of the cerebellum of Primates occasionally presents
difficulties, I undertook an investigation concerning the development
of the grooves and lobes of the cerebellum in man, in order to try
to elucidate certain obscure points in the anatomy of the Primate-
cerebellum in this way. This investigation comprises some forty
cerebella of human embryos, varying in length from crown to sole
from 5 to 30 em.

Proceedings Royal Acad. Amsterdam. Vol. VIII.

(2)

All the objects had been hardened in formol in situ; alcoholic
material cannot be used for the study of this developmental process.

A complete systematic investigation of the formation of the lobes
in the cerebellum of man has until now not been carried out. Rerzius
gives in his well-known standard work — Das Menschenhirn —
a great number of pictures of developmental stages, also of the
cerebellum, but a reasoned explanation to them is lacking. Of an
earlier period we moreover mention the communications of KOLLIKER
and Scuwa.sn, of more recent date those of KurrHan, Eniior Siri
and Crarnock Brapiry. In general, however, these investigations
have been made with material which for this purpose was insufficient
and as a consequence of this, opinions have become current which
[ have found to be wrong. This is more especially the case with the
view concerning the way in which the suleus horizontalis develops.
Particularly with a view to the individual variations which arise
especially in the later period of the lobulisation, it is essential to
carry out the investigation with an extensive material, if we want
to form a clear and continuous idea of the process and if we desire
to distinguish well what is norm here and what exception.

In the morphogenetic process of the human cerebellum three
periods may be distinguished, the first period is that of the development
of the “cerebellar lamella” until the appearance of the first cortical
groove; the second and third periods are those of the formation of
grooves and lobes, during which, in the second period, those grooves
appear which are generally characteristic for the cerebellum of mam-
mals, in the third, the specific grooves and lobes of the Primate
cerebellum. In this first communication only the first and second
periods will be deseribed.

Fig. 1, 2, 3 and 4 are sufficient to give an idea of the develop-
ment of the ‘cerebellar lamella’ until the time of appearance of
the first groove. Fig. 1 has been taken from a foetus of 5 cm.
length from crown to sole. The curvatures of the pons and neck
have reached their maximum. The cerebellum appears as the already
fairly thickened “cerebellar lamella” of Mitancovics. It is remarkable
that the thickening is turned intraventricularly in man whereas in
the rabbit and pig (CHarNock Brapiey) and the sheep (KurrHan) it
is exactly the extraventricular face which is most prominent.

From figs. 2a and 3a it appears that the convexity of the intra-
ventricular plane becomes greater and greater, while the outer plane
is only slightly vaulted. As a consequence of this, the cerebellum
has in Fig. 3 acquired a triangular shape in the section with one
extra- and two intraventricular planes; of these latter one faces

(3)

basalwards, the other caudalwards. The original caudal edge of the
“Lamina cerebellaris’” has consequently been shoved upward together
with the insertion of the epithelial ventricular roof. Since in the
mean time the plica-chorioidea has been formed, the peculiar condition
has arisen, described by Hts for the human cerebellum, of which
KurrHan, however, denies the existence, since he could discover no
trace of it in the sheep, although a complete developmental series
was at his disposal. The anterior piait of the plica chorioidea, the
so-called lamina chorioidea, is now stretched parallel to the intra-
ventricular plane of the cerebellum which is turned backwards and
this gives origin to a slit-shaped space between this plane and the
lamella mentioned. His described this in this way that the lamina
chorioidea partially encloses the cerebellum like a sac and how far
this is the case appears from fig. 36 where the cerebellum is seen
laterally. Now in fig. 4 nothing is found any longer of this condition,
the plica chorioidea is now inserted at the edge of the cerebellum
which now is turned to the back. Also in this respect I can confirm
the observations of His against those of KurrHan, that namely the
lamina chorioidea lays itself upon the intraventricular plane and
coalesces with this. Through this the latter has become an extra-
ventricular plane and the plica chorioidea has obtained a new
secondary line of insertion with the cerebellum. At the same time
the outer plane has in this way become convex, the inner plane
shows as a first indication of the ‘tent’? in its posterior part a
shallow groove which is to be distinguished as Incisura fastigii.
The primitive line of insertion of the lamina chorioidea has to be
sought in fig. 4 at the top of the extraventricular plane, laterally it
lies more caudally, as follows from fig. 34. This developmental stage
of the cerebellum in man seems, by the peculiar way in which it
thickens, to differ fundamentally from that of other mammals, where
the cerebellar lamella retains in the section a more flattened lenti-
cular shape.

At first the thickened lamina cerebellaris has the shape of a semi-
ring, standing vertically on the anterior part of the longitudinal axis
of the rhombencephalon and laterally passing into the still smooth
regio pontis without a sharp border (fig. 24). Soon the lateral parts
of the lamina cerebellaris show a fairly strong clublike swelling
(fig. 3b) by which a clear demarcation between cerebral base and
cerebellum is formed. These lateral swellings remind us of the bilateral
origin of the cerebellum in lower vertebrates (observed e.g. by SCHAPER
in Teleosteans). Yet this lateral demarcation is only temporary ; as
soon as the pons begins to differentiate, it disappears again and arises

1*

Cz.)

anew only at a much later stage when the cortex has already become
amply lamellised.

In the mean time, during the thickening of the lamina cerebellaris,
developmental phenomena have taken place in the bordering region
between Mesencephalon and Rhombencephalon, giving rise to the
formation of Plica encephali dorsalis (Kuprer), Isthmus rhombence-
phali and Velum medullare anterius. In the youngest stage represented
(fig. 1) the anterior edge of the Lamina cerebellaris passes directly
into the mesencephalic roof, only the posterior edge of this latter
is a little inwardly invaginated. An Isthmus rhombencephali or Plicé
encephali dorsalis do not yet exist. In the next stage of development
(fig. 2a) the mesencephalon has obtained a clearly defined posterior
wall, vertical to the roof; the inward invagination of the posterior
edge of the roof is still in existence and is partly visible in fig. 3,
but has disappeared in fig. 4 on account of the thickening of the
posterior wall of the mesencephalon. In fig. 2 the plica encephali
dorsalis has developed, bordered in front by the posterior wall of the
mesencephalon, at the back by the lamina cerebellaris. The formation
of the Pliea is accompanied by a_ rotation of the Lamina cere-
bellaris, the anterior edge of which is now no longer situated at
the front but below and as a consequence of this the Isthmus
rhombencephali is now also indicated in principle. Next the bottom
of the plica encephali dorsalis becomes broader, there arises between
the thickened lamina cerebellaris and the mesencephalon a thin middle
plate (figs. 3a and 4), the first origin of the velum medullare anterius.
The further details of this stage and the following stages will be
extensively described elsewhere.

The lobulisation of the cerebellum in the second stage is characte-
rised by the fact that the grooves which divide the surface of the
cerebellum into several regions originate with a single exception in
the median plane and from there extend laterally. These interlobular
erooves are consequently unpaired with one exception and divide
the foetal cerebellum of man into a number of lobes, which can be
homologised without difficulty with those which tf learnt to be typical
for the adulf mammalian cerebellum.

The median section of a cerebellum with indications of the grooves
that appear first, is given in fig. 5. The ineisura fastigii has been
shifted more to the front compared with tig. 4. On the extraventricular
plane two grooves can be clearly distinguished, one, a little rostrally
from the top of the cerebellum, another at a short distance from the
margo myelencephalicus. Which of these two arises first I have not
been able to make out, evidently they both arise about simultaneously,

(5)

since in three cerebella of this stage I found both of them already
present in each (total length of the foetus 8 to 10em.). The anterior
groove is the suleus primarius (1), the typical principal groove, easily
recognised in every mammalian cerebellum, separating the two lobes
of the cerebellum, the lobus anterior and lobus posterior. The posterior
eroove is the sulcus uvulo-nodularis (7/7) (sulcus postnodularis of
Exnuiorr Smirx, sulcus praeuvularis of Ziman, Fissure 1V of Coarnock
Brapuey). It borders the nodulus in front, i.e. in the direction of
the mesencephalon. Between these grooves a still shallow depression
is visible on the upper part of the posterior plane, the first indication
of the fissura secunda (ELniorr Smiri, mzh/, suleus inferior anterior
of Zimuen, fissure 7 of Cuarnock Brapiry). The cerebellum, seen at
this stage from behind, is somewhat biscuitshaped (fig. 6) and lies with
the front planes of its lateral parts against the occipital poles of the
hemispherical vesicles. Besides the two mentioned grooves, proceeding
from the median line, the cerebellum possesses at this stage already
a suleus which is bilaterally symmetrical and lies at a short distance
of the margo myelencephalicus. This groove (p), which develops in
a latero-medial direction is the homologon of the groove which |
have distinguished in the mammalian cerebellum as fissura para-
floceularis. It borders in front the already slightly prominent so-called
recessus lateralis. The anterior wall of this recessus lateralis has been
distinguished by K6LiIKER as gyrus chorioideus. It seems to me that
the name “Gyrus floccularis” is more characteristic, since from this
narrow cerebellar seam which is already marked out at so early a
stage, the flocewli are later formed.

In a successive stage (Figs. 7 and 8) the suleus primarius (1) has
become deeper and the fissura secunda (2) has become a distinct
groove; moreover a first secondary groove has arisen in the lobus
anterior. Later this Jobus is separated into four small lobes by three
grooves; I have not been able, however, to make out whieh of
these three is the first to appear. That the sulcus in the lobus anterior,
seen in figs. 6 and 7, which is first in appearing, is really the
groove, distinguished by Ennion Smirn as suleus praeculminatus, |
have not been able to confirm, while also from a comparison of my
human cerebella of this stage with corresponding figures given by
Cuarnock Brapiey for the rabbit and pig, it appears that this first
groove lies in the lobus anterior of man farther away from the sulcus
primarius than with the two animals mentioned. So I cannot decide
whether this first groove in the lobus anterior in man is homologous
with the “Fissure 1° of CuHarnock Brapiry. Also in the Jobus

posterior a new groove has appeared in the median line, between

(6)

the suleus primarius (1) and the fissura secunda (2). This groove,
indicated in the following figures by 4, is the sulcus praepyramidalis
(mihi) (sulcus inferior posterior of Zinnen, fissure suprapyramidalis
of Exsior Sr, fissure II] of Crarnock Brapiuy). This fissura
praepyramidalis borders in front the pyramis and soon reaches the
length of the fissura secunda. This latter can in its further develop-
ment lengthen itself regularly in a lateral direction, or otherwise
there independently arises (fig. 84 2') in the hemisphere at a short
distance above the fissura parafloccularis (jp) a groove which soon
becomes confluent with the fissura secunda. While at the same time
the recessus laterales bend out further, the fissura parafloccularis
(fig. 7, 8b, p) becomes confluent with the suleus uvulo-nodularis,
by which the gyri flocculares form with the nodulus a part which
is marked off from the remainig cerebellum. ELiLior?r Smita mentions
that the fissura parafloccularis can also flow together with the fissura
secunda (2). This observation 1 can confirm for other mammals on
account of the structure of the adult cerebellum; with the embryonic
material of man I have not observed such a case. At a later stage
the fissura secunda does terminate, above in the fissura parafloccularis.

In the cerebellum of a foetus of 18 em. the hemispheres are no
longer swollen balloonlike, but have, when seen from behind, obtained
the more angular form which now is characteristic for them during
a longer period of development (fig. 9). The median zone is still
a little depressed, even in the posterior part of the lobus anterior.
The sulcus primarius (1) lies still relatively far at the back, the sulens
praepyramidalis (4) has already pretty far advanced into the hemis-
pheres, but in such a way that the lateral parts with the transversally
proceeding middle part form an obtuse angle, with the opening
downwards. This peculiar shape forms during the successive stages
of development, in which the interpretation of the grooves is not
always easy, an excellent diagnostic for the suleus praepyramidalis.
The fissura secunda (2) has advanced as far as the lateral wall of
the cerebellum so that the regio tonsillaris (tig. 9a 7) is now bordered
on all sides. This region is always more or less swollen in the shape

of an egg. The gyrus floccularis is divided by a longitudinal groove
into two small lobes. At this stage consequently the uvula with the
appertaining lateral parts and so also the nodulus with its lateral
regions are already differentiated in principle. This rapid developmental
process contrasts strongly with the still very simple condition found
in the remaining part of the lobus posterior and supports to some
extent the opinion of Eniiorr Swiru who looks upon the complex
of uvula with tonsils, nodulus with floceuli, as a more independent

CE)

lobe of the cerebellum. Then the peculiar surface division in the
median line between suleus primarius (1) and fissura secunda (2)
deserves notice. The sulcus praepyramidalis (4), namely, at first
always divides this region into two unequal parts; the lower half,
the greater, situated between suleus praepyramidalis (4) and fissura
secunda is the origin of the pyramis, while from the very narrow
upper half, situated between sulcus primarius (1) and suleus praepy-
ramidalis, must originate: declive, folium vermis and tuber vermis.
In this respect a parallelism can be noticed between the phylogenetic
and ontogenetic development of the cerebellum. For the narrow
region between sulcus primarius and = suleus praepyramidalis is
homologous with that lobulus which in the median section of the
mammalian cerebellum I have distinguished as lobulus C, and which
only in the Primates attains a very strong development.

The frontal plane and median section of the cerebellum of a foetus
of 15 em. are given in fig. 10. This stage of development is important
because now the foetal human cerebellum shows the same lobulisation
which I Jearnt to be the fundamental type of the mammalian cere-
bellum generally, a stage which only lasts a short time, since now
soon the grooves appear that characterise the Primates generally or
the Anthropoids and man more particularly and the homologa of
which are missed with other mammals. For as will be seen from
fie. 106, now in the median section, as well the lobus anterior as
the lobus posterior, is divided by three grooves into four lobuli.
With the mammalian cerebellum | have distinguished the four lobuli
of the lobulus anterior as lobulus 1, 2, 3 and 4, the latter being
situated immediately before the suleus primarius, the fowr lobuli of
the lobus posterior I distinguished as lobulus @ (homologous with
the nodulus), lobulus 4 (homologous with the uvula), lobulus C,
(homologous with the pyramis) and lobulus C, (homologous with the
complex of declive, folium vermis and tuber vermis). It will be seen
by a comparison with the investigations of CHarnock Brapiey, that
the stage with man, sketched in figs. 9 and 10, has a strong resem-
blance with a developmental stage which the cerebellum of other
mammals (pig and rabbit) traverse before the final lobulisation of the
cerebellum. However much the Primate cerebellum may in its final
form differ from that of other mammals the groundplan of its lobuli-
sation is, as is evident from the now sketched period of development,
not different from: that of other mammals. In the next following stage,
however, it follows a development of its own, grooves occur, intro-
ductory to the lamellisation of the cortex, which are specific for the
Primates and which will be described in the second communication.

(8)
Botany. — “Some observations on the longitudinal growth of stems

and flower-stalks”. By Prof. E. Verscuarrenr. (Communicated
by Prof. Hugo pe Vrtrs).

(Communicated in the meeting of March 25, 1905).

Superficial observation already shows that in many cases the
erowth of stems, leaf- and flower-stalks is greatly dependent on the
organs which they bear: buds, leaf-lamina, flowers. When these
latter are removed the growth of the axial parts is generally arrested
and they even die after a shorter or longer time.

In literature I have not found any investigations mentioned,
attempting to analyse this phenomenon more closely; e.g. in the
‘ase of flower-stalks, to find out whether excision of certain parts of
the flower had as much influence on the growth of the stalk as the
removal of the entire flower. I have now been able to make this
out for some vernal plants by measurements of growth and T shall
in what follows give a short account of the results.

I chose preferably flowers for this purpose, since here at the top
of the same spindle organs of different physiological functions occur
together and so the experiments admitted of greater variety. In one
ease, that of Mranthis hiemales Salisb., 1 shall describe the course
of the investigation and its results a litthe more in extenso; the
other examples will be more briefly dealt with.

The stem of Lranthis, as will be known, bears at its top a single
flower and close under it, united to a sort of broad collar, a whorl
of three green sitting parted leaves. As long as the stem is still
under the ground, its top is sharply bent downward and the still
perteetly closed flower hangs down, protected by the three leaves,
still yellow then, which envelop it. As soon as the top of
the stem has come above the ground and also the flower has
come free, this latter raises and soon unfolds itself; then the basal
collar spreads out and turns green. The measurements of growth
were made in the stage between the period when the stem is not
yet visible above the ground and that, in which, after the petals
and stamens have fallen off, only the fertilised pistils remain. About
this time the longitudinal growth stops. Whether afterwards, during
the ripening of the fruits, a new period of growth begins, as in
other plants, | have not investigated.

The plants, serving for the investigation, were placed in a hothouse
of the Botanical Garden at Amsterdam, in which the mean tempe-
rature was 20°? C. and in which the specimens developed very rapidly
and entirely normally.

(9 )

I will first show by a few examples that the presence of the
organs on the top is necessary in order to cause the stem to grow
normally in length.

The stem of an Hranthis was on February 4. 1905, 40 mm. lone,
measured from the base near the rhizoma to the junction of the
leaf-whorl. Placed in the hothouse the plant was at first measured
daily, afterwards every other day; for briefness’ sake I shall here
only give the length reached by the stem after every four or five days.

Date 4.2.05 8.2 3.2 17.2 22.2 26.2
Length in mm. 40 89 135 154 162 162

In the same time the development of a stem on which leaves and
flower had been cut away, was:

Date 4.2 8.2 13.2 ALE
Length 49 52 54 5

i)

Oo

Another example of growth with a normal stem:

Date 5.2 9.2 13.2 17.2 22.2 26.2
Length 44 98 128 145 150 150

and of a stem, bereft of leaves and flowers:

Date 5.2 9.2 13.2 17.2
Length 97 103 104 104

Whereas with normal /ranthis-stems the top with the flower on
it, had in the hothouse after a couple of days, entirely erected itself,
on the other hand the hook-shaped curvature of the stem without
flower or leaf-whorl, partially remained and it was only very slowly
that its extremity raised itself to some extent. This need cause no
wonder, if it is remembered that the disappearance of this curvature
is caused by asymmetrical growth of the top of the stem.

Now in a series of Lranthis plants the organs on the top of the
stem were only partly removed; e.g. the three green leaves, the
petals, the stamens, the pistils. The length of the stems was measured
from day to day. The result of these experiments has been very
clear. As long as the green leaves remained undamaged, the growth
of the stem might be called normal. At the utmost the stem remained
a little below its normal length if the whole flower or certain parts

( 10°)

of it were cut away. On the other hand the growth was very con-
siderably impeded by removing the whorl of green leaves. This will
be seen from the following measurements.

Eranthis-stem, on which only the three leaves under the flower
have been preserved, the flower itself having been removed :

Date 7.2 11.2 15.2 19.2 22.2 26.2

4

Length in mm. 51 107 134 141 141 141

Another example of the same case:

Date 6.; 10.2 13.2 17.2 22.2 26.2
Length in mm. 58 104 129 135 135 135

Eranthis-stem of which the basal whorl has been cut away, the

flower remaining intact:

Date 6.2 10.2 13.2 17.2 19.2
Length in mm. 86 96 97 100 100

Another example of the same case:

Date 7.2 11.2 15.2 Ao

Leneth in mm. 59 72 74 74

Henee a stem which had been bereft of its flower grew in length
in a period of twelve days 176 °/, in the first and 183 °/, in the
second experiment, this increase in length being only 16 °/, and 25 °/,
respectively in the same time with a stem on which the flower had
been preserved but the whorl of leaves removed.

The influence which the presence of the leaf-whorl has on growth
follows clearly enough from this. Also in the other eases which I
investigated, the growth of stems that bore flowers only, may have
been a little greater than of stems from which the leaf-whorl as
well as the flower had been removed, it is certain, however, that
the longitudinal growth is chiefly regulated by the presence of the
green leaves.

A related facet is that after removal of the leaf-whorl the flower
raises itself only very slowly and often only partly.

Although the supposition is not very probable, it might be presumed
that the observed effect of the three leaves is caused by the cireum-
stance that they have to provide the stem with food. That tliis

els)

is not the ease follows from the fact that the same results are
obtained in the dark and that consequently the presence also of
the non-assimilating leaves renders a strong longitudinal growth of
the stem possible, which does not occur if only the flower is preserved
on the top. It will be superfluous to mention figures in this respect.

No more does it appear necessary to give in extenso the measu-
rements proving that removal of the pistils, the stamens or the
petals has with Hranthis little or no influence on the longitudinal
growth of the stem. On the other hand it is not superfluous to
remark that the leaf-whorl must be pretty completely cut away if
we want soon to arrest growth. The three green leaves namely
show basal growth themselves and if their foot is not damaged, this
latter may appreciably grow in size in the course of a few days;
at the same time the stem continues growing in length.

Example: foot of the three green leaves kept; also the flower
intact.

Date 8.2 41.2 A582 20.2 26.2
Length in mm. 54 81 113 145 145

Already on the 13 the leaf-whorl had considerably grown out;
at the edge nothing of the nature of a wound could be seen any more.
In the same time a stem of 102 m.m. length on which the leaf-
whorl had been completely cut away, the flower remaining intact,
had only reached a length of 117 m.m.

If one should be inclined to think that not the presence of the
whorl of green leaves but the intact condition of the junction of
the leaves on the stalk is the principal point here, | must remark
that of this junction zone a layer of tissue may be removed all
round without the longitudinal growth being materially affected.
Alzo from the somewhat vaulted receptacle a part may be removed
or the middle part may be hollowed without any other consequences
than would ensue on the plucking eff of the floral parts situated on it.

Finally we remark that Hranthis-stems, cut off near the junction
on the rhizoma can continue growing for days when they are put
with their feet in water and then show the same behaviour as whole
plantlets. Besides, the presence of one out of the three green leaves
is sufficient to render a considerable growth in length of the stem
possible; e.g. lengthening from 53 to 89 mm. in two weeks. That
also with /ranthis-leaves the growth of the leaf-stalk depends on
the presence of the leaf-disk will now be obvious; I have ascertained
myself of it by measurements, however.

Galanthus nivalis l. enables us to observe phenomena of a diffe-
rent kind in this same respect. With this plant also, the stem termi-
nates in a single flower which, however, when it is fully developed
and unfolded, hangs on a thin, limp, flower-stalk. This is implanted
on the top of the stem, where also two coalescent bracts are found
which enveloped the flower-bud before its unfolding. Henee we must
here investigate the influence of the terminal organs On the growth
of the stem as well as on that of the flower-stalk.

Concerning the longitudinal growth of the stem, we find that it
is completely independent of the presence of the flower. A single
example will suffice to show this. The stem was measured from the
point where it appears from the bulb to the impiantation of the
bracts; these latter still surrounded the flower-bud: in a the plant
remained undamaged; in / bracts and flower were cut away to
the foot.

Date 13.2 16.2 20.2 23.2 26.2

Length in mm. a. 90 133 1957 161 162
b. 46 60 90 105 108

On the other hand, the growth of the flower-stalk stops as soon
as the flower is removed. The influence of the flower on this organ
is even so great that already after a couple of days the Stalk of
cut flowers turns yellow at the top and soon dies from above down-
ward. The measurements show that the Ovary plays if nota prepon-
derant, yet a considerable part here. So the flower-stalks of flowers
Which already Opened, grew from 28.2.05 to 6.3.05, in two cases
from 16 and 44 mm. to 23 and 24 mm.; a flower of which the
perianth was removed, in the same time from 17 to 21 mm., while
two flower-stalks Without their flowers Measuring 20 and 14 mm.
had reached 22 and 16 Inm. the next day, but after that died off.
Cutting the stamens has no great influence on growth; yet growth
remains very small if stamens as well as perianth are ‘removed, so
that with Galanthus the ovary reeulates the growth of the flower-
stalk to a ereat extent but not exclusively. On the other hand the
flower-stall remains alive as long as the Ovary is still present on
tS top.

Exactly the same behaviour jis shown by Narcissus Pscudo-
Narcissus L., where the stem continues srowing when the flower
is eut, but the flower-stalk stops growing and dies, if the Ovary
is wanting, | may add here that for the growth of the stem it

makes no difference Whether its top is cut above or below the

(13)

swelling occurring at the point where the bracts and flower-stalk
are implanted, so that this zone also has no importance for the
erowth of the parts under it. Also stems of Galanthus and Nar-
cissus, cut in the basal part and hence separated from the bulb,
or even parts of them, if they were taken from plants with their
flower-buds still closed, continue to grow vigorously whether the
flower-bud be present or not.

Tulipa Gesneriana 1. shows something different again. Here the
flower is born by a leafed stem; the internodes which are placed
near the base stop growing sensibly towards the time that the flower
becomes visible from the outside and is about to open. At this
stage, however, the upper internode with the flower at the top,
still grows considerably in length. For this the presence of the
flower is absolutely necessary. The upper portion of the stem is
arrested in growth and gradually dies off as soon as the flower is
cut off.

Example: a. flower present; $. flower removed. Only the upper
internode measured.

Data 6.3 8.3 13.¢
Length in mm. a, 42 50 83
b. 44 42 44

From the following measurements the significance of the various
floral parts may be seen :

a. perianth removed.
6. stamens removed.

ce. pistil removed:

Date 6.3 8.3 13.3
Length in mm. a. 36 41 45
b. 46 63 70
peal 51 68

Although removal of each of the individual whorls of organs, partly
suppresses the growth of the upper internode, yet it is seen that
the petals have the greatest influence here. The above is only an
example chosen from several concordant measurements.

Finally some observations were made with Crocus vernus All.
Since the ovary lies fairly deep here, hidden in the tube formed by

( 14 )

the green leaves and the bracts round them, plants that had been
cut open had to be used for the measurements, in which the flower
was laid bare over its full length. For this purpose flowers were
chosen which were still surrounded by bracts and entirely closed
and the top of which became just visible above the ground. It
appeared, however, that at this stage the stem on which the flower
is situated, had reached about its full Jength and only grew a few
millimetres more. The further longitudinal growth which is very
considerable and brings the flower above the ground is nearly wholly
caused by the corolline tube between the ovary and the loose slips
of the perianth. Only to this stage I paid attention. Some measure-
ments of the corolline tube may follow :

a. flower undamaged.
4. corolline lobes removed.
c. corolline lobes, stamens and pistil cut away at the upper

end of the coalescent corolline tube.

Datum 8.3 9.3 AES

Length in mm. a. 46 LOL 108
b. 55 84 84
Cy roll 72 72

So removal of the terminal organs has not remained without
influence on the growth of the corolline tube, but has not been able
to check it to the same extent as in the preceding cases.

It deserves notice that removing the anthers and stigmas did not
prevent the stamens and styles to reach about their normal length.

Summary. The investigation has shown that the normal longi-
tudinal growth of the stem with Eranthis hiemalis is only possible
when the whorl of leaves at the top is present, while the flower
exercises no influence on it. This latter is also the case with
the stem of Galanthus nivalis and Narcissus Pseudo-Narcissus ;
the flower-stalk however, in these two plants, is checked in growth
as soon as the flower is cut, the ovary proving to be of especial
importance. With Tulipa Gesneriana it is chiefly the perianth that
rules the longitudinal growth of the upper internode; with Crocus
vernus, finally, the growth of corolline tube, stamens and style is
in a high degree independent of the presence of petal lobes as well
as of anthers and stigmas.

(15)

Zoology. — “On the Structure of some Siliceous Spicules of Sponges.
I. The styli of Tethya lyncurium, by Dr. G. C. J. Vosmarr
and Dr. H. P. Wiisman, Professors at the Leiden University.

(Communicated in the meeting of April 22, 1905).

After Scuweicerr (1819) had demonstrated that the spicules of
sponges in some cases do not consist of calcium carbonate, Grant
(1826) found them to contain silica, and Bownreank (1841 &) showed
that, in addition to the silica some organic matter is present. He
reached this conclusion through the fact that the spicula when heated,
were partly carbonised. K6niiker (1864) remarked that the brown
or black colour, produced by heating, is certainly not only due to
carbonised organic matter; examined in reflected light the heated
spicula appear white, and the dark spots seen in transmitted light are,
therefore, partly due to inclosed air. THoutnr (1884) found no
organic matter and concluded: “les spicules sont done constitués par
de la silice pure’, which he compares with opal. SoLnLas (1885)
likewise finds that the silica resembles opal. It is now generally accepted
that the spicules of siliceous sponges consist of some kind of opal;
but that in some way or other, organic matter is also present. So far
here is a general agreement of opinion but the chemical analyses which
have been carried out show considerable differences as to the quantity
of water, combined with the silica as a gel. The formulae, given
for the composition vary from 2 (S’O,) + H,O to 5 (Si0,) + H,O,
but must be considered as mere failures. F. E. Scuutze (1904) comes
to the result: ‘dass, entweder die Siphone keinen bestimmten kon-
stanten Wassergehalt haben, oder dass die organisehen Zwischenlamel-
len.... einen je nach der vorgiingigen starkeren oder geringeren
Austrocknung wechselnden Gehalt an Wasser haben”.

It is certain that even in the best cases, the quantity of organic
Matter is so little that it cannot well be ascertained. Its presence
‘an, however, be proved by treating the spicules with hydrofluoric
acid. But there is also some disagreement as to the nature of this
matter and the exact place in the spicules where it is met with.
We shall see that different kinds of spicules vary in this point.

In addition to the silica, which behaves like some kind of opal,
and which we propose to call spécopal, and the organic matter which

F. E. Scuvunzn called spiculine, — modifying the original term used
by HarckeL — in some spicules there have been found traces of

Na, K, Cl, Fe, Mg and Ca, but in such slight quantities that they
ean be left out of consideration for the moment.

( 16 )

As to the structure of the spieules, Gray (1835) had found them
in Hyalonema to consist of layers, which became conspicuous by
heating. These layers concentrically surround a “central canal”, which
is filled out, as Koniuxker (1864) has shown, by an organic mass,
the axial rod. Cuavs (1868) found that the silica which directly
surrounds this central rod, is homogeneous; he called this homogeneous
cylinder the axial cylinder. According to Max Scnunrzn (1860) the
longitudinal striae, which become conspicuous especially after heating,
are due to the fact that layers of silica alternate with very thin
layers of organic matter; the first are, after ScHuLrzn, isotropic,
the second anisotropic. The outer layer is generally found to be of
organic nature.

In all these cases, the investigators described some special kind
of spicule; naturally they have chosen very large spicwes. Gray,
Cpaus and Max Scuvunrzr studied the large rods of Hexactinellida,
such as Hyalonema and Euplectella. It may be asked, how far
their results hold good for spicules of other sponges.

K6otukeR had already found that not in every case the axial
thread is conspicuous. Also it has not been possible to demonstrate
in every case alternating layers of spicopal and of organic matter,
not even where longitudinal lines are evident. There is a great
confusion with regard to the presence of a so-called spicular sheath.
Any accurate determination of the refractive power of different
spicules is likewise wanting.

It seems, therefore, desirable to get some more information about
these subjects. Since F. E. Scuutzn (1904) studied the enormous
spicula of Monorhaphis, it appears useful to investigate, whether
spicules from other groups agree with them as to their structure,
and their chemical and physical properties.

We began our examination by the large styli of Tethya lyncu-
rium. After bringing this point to some certainty, we have com-
pared the results with those obtained from other species.

In the first place we tried to answer the question: Do the styl
of Tethya contain other organic elements than the central rod, either
as a sheath, or as layers between the spicopal, or as both of them.

Of course, the method which at first presents itself for the detec-
tion of organic matter, is the dissolution of the silica by means of
hydrofluoric acid. Former investigators who applied this reagent, have
omitted to give an accurate description of their experiments. SoLLAS
(1888 p. XLIX) put the isolated spicules into a drop of water and
added a drop of hydrofluoric acid. On doing this, one generally
sees that the silica is dissolved and that the central rod remains.

C17)

Adding less hydrofluoric acid, the process does not go to the end. As
it is necessary to cover the slide by means of Canada balsam (SonLas)
in order to preserve the object glass from the disastrous influence
of the vapours of hydrofluoric acid, it is difficult to vary the acidity
of the fluid, in which the spicules are mounted. Also it is impossible
in this way, to exclude the influence of the glass. We had therefore,
to construct an apparatus allowing the concentration of the hydrofluoric
acid to be varied without danger to the lenses of the microscope.
At first we tried ebonite, in combination with glass, covered by a
layer of celluloid, which is sold in solution under the name of zapon.
But as this method did not satisfy our purposes, we tried another
and we think that we have found a good and rather simple device.
The adjoined figure needs little explanation. Out of a sheet of trans-

parent celluloid, 1 mm. thick, is constructed a case, abcd"); the
bottom measures 6X 10 em., the height is 1.5 em. In the midst
of the case a circular rim of celluloid, high 5 mm., is joined to
the bottom. Another case of celluloid «'b'c'd' measures 4.5 < 6 em.
bottom and 8 mm. height. In the middle of the bottom a square of
2 2.5 em. is cut out. This opening is covered by a sheet of thin cel-
luloid, no more than 0.5 mm. thick, measuring 2.5 « 3.5¢em. This
thin sheet A is joined with O'c' air- and watertight by means of soft
paraffin. The spicules under examination are placed on the bottom
of the case ab'c'd', e. g. in a drop of water. To flatten this drop
it is covered by a very thin film of celluloid. In the exterior part
of the case abcd is poured out some liquid paraffin p, in the interior
commercial hydrofluoric acid (//), diluted with 3 or 4 parts of water.
Now the case db'c'd' is reversed and put into abcd. As we have
found that hydrofluoric acid in the gaseous state after some time

1) The carefully cut and cleaned sheets of celluloid are easily united by means
of acetone. The parts that should be united are pressed together, and by means
of a small brush a_ little drop of acetone is applied. The celluloid itamediately
sticks together. The corners are afterwards, to diminish ihe chance of leakage,
cemented another time with zapon. Celluloid is got at the D. Celluloid Fabrik
Leipzig.

1) Very thin films of celluloid are got by pouring out zapon on a glassplate,
in the way that collodion plates are made, and tearing it off the glass after drying.

2

Proceedings Royal Acad, Amsterdam. Vol. VIII.

( 18 )

passes through thin strata of celluloid, it might be expected that
the very thin layer of celluloid used in the above manner would
be no obstacle for the acid to attain the spicules. Even the retaining
of the reaction in this way is an advantage, as it may now be
observed without the use of very dilute solutions.

In this way we have constructed a littke apparatus which has
answered to owr purposes in several respects, viz: absence of glass,
slow reaction, rather great security for the lenses of the microscope,
and the possibility of interrupting the reaction at any moment.

As we found it unmaterial whether the spicules (obtained from
sponges preserved in alcohol) were isolated by means of artificial
gastric juice (after some days at 35° C) or by boiling with hydro-
chlorie acid, we preferred the latter method. Such spicules, having
been boiled with hydrochloric acid for some minutes, washed out
repeatedly with water, either with the aid of a centrifuge or not,
and dried afterwards at the ordinary temperature, are the objects
investigated by us, if another treatment is not expressly mentioned.
By placing some spicules in a drop of water, and covering them
with the thin film of celluloid, one can first study under the
microscope whether they are normal in their aspect, uninjured ete.
Also, and Vethya is a proper object for this, which spicules are open,
which closed. Then we expose the preparation to the vapours of the
hydrofluoric acid. The commencement of the reaction is more or less
retarded, depending upon the concentration of the acid, the quantity
of water and the thickness of the film of celluloid; but at any rate
the spicules begin to be dissolved after some 30 or 50 minutes.

It is best to give attention to the fractures, as the reaction is here
to be seen at first.

Biirscunt (AYOL) already remarked that the dissolution of the silice
may occur in different ways. We can confirm this observation for
the styli of Tethya. Observing what happens at the broken end of
a spiculum, the opal surrounding the axial thread is seen to be
hollowed out in the shape of a cone. The top of the cone is very
sharp, and becomes still sharper if the reaction proceeds. BivrscHni
says (I. ¢. 258—259): “Man kénnte wegen dieser so haufigen
Bildung einer trichterformigen Auflésungshéhle an den Enden auf
die Vermuthung kommen, dass die Angreifbarkeit und Léslichkkeit
der Schichten von aussen nach imnen, gegen den Achsenfaden sue-
cessive zunechme. Kine solehe Annahme scheint jedoch zur Erklarung
der Erscheinung nicht néthig, vielmehr diirfte sie sich schon daraus
hinreichend erlautern, dass die Flusssiure almahlich in den gedff-
neten Achsenkanal eindringt und gleichzeitig auch in dem Masse

(19 )

starker wirkt, als der Achsenkanal dureh Auflésung erweitert wird,
indem dann eine grdssere Menge der Siure zur Verfiigung steht.”
We believe, on the contrary, that it really follows from the obser-
vation that the inner parts of the spicopal are more easily dissolved
than the outer ones. For we see the sharp conical funnel long
before any trace of reaction is to be seen on the rest of the fracture.
The borders remain intact for a considerable time. And we get the
same view by boiling in a solution of caustic potash. Only when
the conical hole has attained a certain depth, the dissolution of the
spiculum from the exterior commences. Finally there remains a
kind of tube, formed of silica in which the axial thread is laying
isolated, until all the spicopal has entirely passed into dissolution.

If we study the way in which the hydrofluoric acid acts on com-
pletely intact spicules, some difference may be seen, according as we
have to do with sharply pointed needles or with those of which the
apex is rounded off. Blunt styli, i.e. transitions to strongyli, resist
the acid for some time; but once the dissolution has begun from
the exterior, the process proceeds regularly, the spiculum becomes
thinner and thinner.

It seems that in pointed styli, the apices are first attacked ; in
such eases we see the axial thread gradually coming free by external
dissolution of the spicopal. Sometimes it may be observed that, in
addition to the dissolving process as described above, a hollowing
out along the axial thread takes place. In other cases, however,
this is not seen, and we get half dissolved spicules in which the
axial thread is partly freed, partly enclosed in a coat of spicopal,
thus strongly resembling whips; the more so as the thread is gene-
rally flexible, whereas the rest is still straight and rigid.

Biirscuit has already remarked that there are sometimes seen “durch
lokale starkere Auflésung der Kieselsubstanz zellenartige Vertiefungen
der Nadeloberflache’. “Indem diese Vertiefungen schliesslich zu Léchern
werden, die bis zum Achsenkanal reichen, wird dieser der Flusssiiure
zuganeglich und nun beginnt von diesen Lochern des Kanals aus
die innere AuflOsung der Kieselsubstanz unter Entwicklung zweier
trichterformiger Hohlen....” This observation we can confirm; we
consider it another proof that the spicopal in the neighbourhood of
the central thread is more easily dissolved than the peripheric mass ;
the observation can hardly be explained in another way.

Spicules, dissolved in the described way, show, that there remained
not only after the dissolution of the spicopal an organic central thread,
but also a very thin coat, which covered the exterior layer of the
spicopal. This coat, which represents the true spicule sheath, is

9%

( 20 )

extraordinarily delicate; consequently it is easily torn or shrunk.
Still, we are convinced that it exists, but the examination must be
carried out with the utmost care. In some preparations we found it
in the greater part of the objects. That it is not always observed
may partly be due to the treatment of the spicules with hydrochloric-
acid, partly by its being destroyed already during the life of the
sponge. The axial thread is likewise not always visible, or at least
not over the whole length of the spiculum. On carefully dissolving the
spiculum it may be observed that, while the cylinder of silica grad-
ually diminishes its diameter, a very thin line shows the dimensions
it originally possessed.

When the spicules are observed in water, the limits show them-
selves as rather broad black bands, as the refractive index of the
spicopal is considerably higher than that of water. When the dissol-
ving process goes on, the black bands gradually approach each other,
and the thin line, the optical section of the spicule sheath, becomes
conspicuous. When the object is now studied in acid fuchsine, the
central thread stains intensely red as it is set free, and the sheath
becomes faintly reddish in the mean time. Organic layers, so called
layers of spiculine, such as can easily be demonstrated in the large
needles of Hexactinellida, are nowhere met with in Tethya. In some
cases we saw something which resembled them, but in every case
we could explain the phenomenon by a folding of the sheath. Conse-
quently we conclude that layers of spiculine are absent in Tethya.
And we cannot agree with Mincnin (2900), who says about spicula
in general (l.¢. p.40): “the mineral matter is deposited round it
(viz. the axial thread) in concentric lamellae of colloid silica, alter-
nating with lamellae of organic nature’.

As to the spicule sheath, writers do not agree. What F. E. Scuutzn
calls “Spiculascheide” in his last publication (1904) is not homologous
with what we indicate with the name of sheath. That we notwith-
standing use this term has historical reasons, as it seems to us that
the word is originally used for formations belonging to the spicule
itself, homologous to the product which is found in caleareous spicules,
where its existence had been first demonstrated. In this sense
MINcHIN applied the term, and he is the author of the newest and
best general treatise on Porifera.

KOLiKER (1864) may be regarded as the discoverer of the spicule
sheath. It seems to us beyond doubt what Kéniker meant by it,
uthough we acknowledge that his opinion is not always expressed
with the utmost clearness, and that from the beginnine there had
existed some confusion of ideas about this organ.

( 21 )

KOLLIKER says (I. ¢. p. 64—65), speaking about “Nardoa spongiosa”:
“Ausserdem finden sich dann noch nach der Auflésung der Spicula
durch Essigsiure, zahlreiche Liicken, welche diese Bildungen enthalten,
die allen von einer scharfen Linie begrenst sind, wie bei Dunstervilia.
Bei Nardoa glaube ich mich davon iiberzeugt zu haben, dass diese
scharfe Linie der optische Ausdruck einer selbstandigen Scheide der
Spicula ist .. 2’ What KOLLiknr means by the word “selbstandig”’
becomes clear when we read that in every canal spicules project,
in which, after treatment with acetic acid, ‘an der Stelle des in die
Flimmereandle hineinragenden Strahles der genannten Spicula zarte
Scheiden leer zuriick (bleiben). These sheaths are, according to
KOLikER, perhaps a “Rest von Bildungzellen.” However, he adds :
“freie Spicula zeigen, der Einwirking der Kssigséure ausgesetzt,
keine solehe Scheide....”. What is meant here with “freie’ spicula is
not evident. It can hardly mean anything else than isolated spicules.
If they be isolated mechanically, the sheath is as obvious as in
spicules in situ; if they be isolated chemically, then of course the
absence of a sheath is no proof at all.

It is easy enough to repeat K6niikrrs experiments, especially in
using specimens with thin walls, as e.g. Leucosolenia. If a fragment
of a wide tube of L. rariabilis which has been cut open, is spread
out in water under the microscope, and carefully treated with acetic
acid, the carbonate of lime is seen to be gradually dissolved, and
soon the sheaths, with sharply defined outlines, exactly as deseribed
by Koénniker, are visible. The sharp outlines are especially clear
on spots where the spicules are wholly enclosed by parenchyma; in
projecting spicules the conical sheath of spiculine is seen to remain
as a homogeneous, extremely thin film; still more striking, perhaps,
is the phenomenon if the sponge is stained on the object glass, e. g.
with carmalumn (Grisinr). The carmalumn, which is generally some-
what acid, causes the calcite to dissolve, and stains the sheath purple;
the sharply defined outlines (optical sections) appear dark-purple,
while the projecting spicules are faintly purplish. If such prepara-
tions are examined in glycerine or in Canada balsam, the spots
where the carbonate of lime has been dissolved, or where it is still
present, are hardly to be discerned. When we use the polarising micros-
cope, however, the presence of calcite becomes immediately visible.
Consequently, there really exists a special layer of organic substance
whieh tightly covers the spicule, which can be isolated with the spiculum,
but which cannot be separated from it otherwise than by dissolving
the carbonate of lime. Doubtless it is this organic layer which
Kouuiker called “Scheide”. In this sense also Mrvcuin uses the word,

( 22 )

The question now arises how far siliceous spicules are likewise
enclosed by such organic coats, homologous to the sheaths of the
caleareous spicules. We are of opinion that this is actually the case;
the delicate organic film which we found covering the spicules of
Tethya we consider as the homologon of the spicule sheath of
‘aleareous spicules. Such products have been already observed. Nout
(1888 p. 16-17) says: “Noch ist fiir die Spicula von Desmacidon
Bosei eines Ueberzugs von organischer Substanz Erwahnung zu thun,

. . . Hauptsichlich nach Behandlung der Praparate mit eimer
Hollensteinlédsung .... weniger deutlich mit Acidum pyrophos-

phoricum, manchmal auch mit Picrocarmin wurde derselbe sichtbar.
Stifte, die isoliert, ohne Ueberzug von verkittendem Spongin, iiber
die Hialfte frei aus dem Schwammegewebe hervorstanden oder auch
solche, die ganz frei lagen, waren besonders nach der Silberfarbung
gleichmassig mit einem lichtbraunen Ueberzuge versehen, der trotz
seiner geringen Dicke doppelte Konturen erkennen liess und die
Stifte gleichmassig tiberdeckte .... Die Spicula von Desmacidon
Bosei besitzen also einen homogenen hautartigen Ueberzug von
organischer Substanz, der verschiedene Farbstoffe aufnimmt. Wir
wollen ihn als Spicula-Oberhaut bezeichnen ... .” Although Not
sees in this coat something else than what K6LikKer found in calea-
reous spicules, we suppose them to be equivalent. Just as K6LLIKER
indicates that his “Scheide’, is perhaps a “Rest von Bildungszellen”’,
so Noni writes that his “Oberhaut’? may be “der Rest der die
Nadeln bildenden Zellen’’.

Sonias deseribed in the same year (1888) such a sheath, which
beeame perceptible after treatment with hydrofluoric acid. Description
and drawing (l. ¢. p. XLIX, Pl. XLIII, fig. 18), regarding the
spicule of ‘Dorypleres Dendyi’, leave nothing to be desired as to
clearness. “Although at first sight the acid appears to remove all
the substance of the spicule except the axial rod, careful observation
will show that this is not the case, for a delicate film of organic
matter also remains behind; it has the form of a hollow sheath,
corresponding in form and position with the outermost boundary of
the original spicule; between it and the axial rod the whole of the
spicule is completely removed. The spicule thus consists of a central
organic axis, surrounded by coneentrie layers of opal, the outermost
of which is invested in a spicule sheath of organie matter or rather
of organic matter in intimate association (chemical union ?) with
silica”. Our results regarding the presence of such a sheath in
spicules of other species we hope to give in a next publication ; for
the present we deal only with the spicules of Tethya.

( 23 )

It is evident, that if we are right in our conception of the spicule
sheath, other coats which sometimes are found surrounding spicules,
may not be called sheaths.

F. KE. Scnuntze describes in his last paper with his well known
accuracy such surroundings from the enormous needles of J/ono-
rhaphis. We vegret not to agree with him in calling this formation
“Scheide”. As Scnutze demonstrated it to be not only an investment
of each spiculum for itself, but also a means of joining different
spicules together, and as this is trne for other spicules also, we propose
to eall it periapt"). Sottas (1880 p. 401%) described a similar coat
surrounding spicules from /sops phlegraed. Bérscutt found it in
Tethya lyncurium, a fact which we can confirm. The periapt is
composed of connective tissue with conspicuous fibrils and cells;
it has therefore, nothing to do with the spicules as such and may
be left out of consideration bere. Mutatis mutandis, the periapt
behaves to the spicule sheath as a perimysium to the sarcolemma.

We have seen already that when the spicopal is dissolved first of all
the axial rod appears. The presence of such an organic thread in
macroscleres is no more doubted. Although we have no more
doubt that the axial thread is normally present, it cannot be denied
that in some cases it is wholly or partly absent. We consider
such cases, however, to be pathological.

With regard to the shape of the rod Bérscui1 (1901 p. 253)
writes: “eigenthiimlich ist das Querschnittsbild des Fadens, das....
stets deutlich dreieckig erscheint, gleichgiiltig ob der aiussere Umriss
des Nadelquerschnitts selbst etwas dreiseitig oder ganz kreisrund ist.
Vielfach ist jedoch auf den Nadelquerschnitten zu erkennen, dass
der Querschnitt des Achsenfadens sechseckig erscheint, indem die
Eeken des dreiseitigen Umrisses regelmassig abgestumpft sind”,
Consequently, the central thread in such cases, e. g. in Tethya,
would be a triangular rod and not a cylinder. Whereas already
Bowerbank (1864) apparently had seen something like this in Geodia,
PF. E. Scautzn (1904) arrived at the conclusion that the axial threads
in Hexactinellida are cylindric. In view of this contradiction we
thought it necessary to submit the axial thread of Tethya to :
careful investigation. In order to judge about the shape of a thread
in transverse section, we followed the method of Bivscuii by grinding
spicules in an agate morter. In the powder, procured in this way,
there is found always a sufficient quantity of particles of approximately

1) =. p1za7w 1 bind together.

2) Not 1890; — apparently this is a misprint in Scautze (1904 p. 204), — also
not p. 41:0—441 but 400—401.

( 24 )

cylindrical shape, although that the fracture is irregular. If these
pieces are mounted in glycerine,’) it is possible, as the spicopal almost
entirely disappears from the eye, to judge with certainty in which
position the thread is seen, whether oblique or not. Paying attention
only to those which are undoubtedly seen in transverse optical section,
we found by careful focussing that they were really triangular with
the angles cut off. In the second place we studied isolated axial
threads. By moving somewhat the coverglass, a quantity of little
pieces break off, generally almost transversely. At the same time
it can be observed that the little fragments turn over, owing to the
movement of the fluid. In this way we saw plenty of them from
all sides. With high power (Zeiss, homog. imm.) we found that in
this ease also the transverse section is triangular. In spite of our
astonishment that the axial rod in Vethya is triangular we cannot
but agree with Birscuri’s observations. We did not see varicosities
nor sharp restrictions ; normally the isolated rods are perfectly smooth.
The diameter remains the same, with exception of the extremities ;
these are, in uninjured threads, either sharply pointed or rounded
off, according to the well known shape of the styli themselves.
Deviations of this rule seem to us to be pathological.

Not less strange than the shape appears to be, is the consistency.
This may be the reason that previous authors so little agree. Whereas
we find, after treatment with hydrofluoric acid, that the free axial
threads on one hand are very flexible, so that they can form elews,
we see on the other hand that in many cases they break offat once
if touched with a needle. We remarked already that they generally
break at right angles to the axis. In a certain sense BirscHL is
right, therefore, if he calls them ‘“spréde’, but it is by no means the
sort of brittleness of for example a thread of glass. The consistency of
the axial thread can be best compared with agar-agar. Here a certain
flexibility is likewise combined with the property of suddenly break-
ing. Similar phenomena are known of gels at a certain point of
dehydration.

The axial rod in Tethya is, taken as a whole, not homogeneous.
First of all we observe, especially in threads stained with iodine, a
double contour. This is easily demonstrated on uninjured, isolated
threads as well as in transverse sections; in the triangular figure,
mentioned above, the wall is both inside and outside triangular. This
wall is comparatively thick — about '/, or ‘/, of the total diameter.
We may consider the axial thread as a tube, filled with something ;

1) Refractive index n = 1.4508. Cf, infra.

(25)

whereas the wall seems to be homogeneous, the contents are homo-
geneous or rather granular, apparently of a softer consistency than
the move rigid wall. This we conclude from the faet that curved
or bent axial threads under the microscope resemble curved or
bent indiarubber tubes, filled with a fluid or semi-fluid substance.
Solid, flexible cylinders never show such abruptly bent figures.

A remarkable phenomenon is to be seen in broken spicules under
the influence of hydrofluoric acid. As stated above, the spicopal is
dissolved in a_ peculiar way, the central canal being hollowed out
in the shape of a funnel. If we now only take into consideration
the eases where the thread is broken at the same place as the
spicopal, we see the thread gradually shrinking somewhat under the
influence of the hydrofluoric acid. However, the wall and its contents
do not shrink equally. The result is that the contents somewhat
pour out beyond the wall. It is not improbable that Biirscunt has
seen this; at least his illustration (fig. 24 on pl. XX1) strongly
resembles what we observed. But Biscuit explains it in another
way; he believes the thread to be restricted ‘‘manschettenformie”’.

According to birscut: the axial thread consists of a proteid
substance. With F. E. Scnunzm we can confirm this in the main
points. Boiled in Minnon’s fluid the threads in ground spicules turn
yellow. This staining is especially distinct in pieces where a part of
the thread is lost, and where, consequently, the axis of the spicule
is partly uncoloured, partly filled with a yellowish thread. Isolated
axial threads or threads partly freed by solution of the spicopal are
easily stained with iodine. Treated with nitric acid (25°/,) they
swell somewhat and acquire a faint yellow colour, which becomes
darker by subsequent addition of ammonia. Heated with nitric acid
the threads dissolve ; likewise in caustic potash. We may conclude,
therefore, that the central rod if not wholly, at least partly consists
of some proteid. Observed under the polarisation microscope no
trace of anisotropy could be seen.

With regard to the styles of Vethya lyncurium we thus arrived
at the conclusion, that they are composed of an organie axial thread
and an organic spicule-sheath, between which elements the spicopal
is deposited. We failed in demonstrating any trace of organic (spiculine)
lamellae. But still we found, that under special circumstances
longitudinal, resp. concentric striae were distinctly seen. We have
to look for an explanation of this fact.

In order to avoid a_ possible misunderstanding or confusion we
wish at once to draw attention to the fact, hitherto rather neglected,
that one has to distinguish the various /ayers of spicopal from their

(26 )

limiting plains. We hope to show that the well-known stripes are
nothing but the optic sections of such limiting plains, and that they
ave independent of eventual differences of the layers. It is easy
enough to microscopically demonstrate such limiting plains in layers
in artificial siliceous gels. If a coat of not yet coagulated siliceous
gel is poured out over another one freshly coagulated, and if this
is repeated, it becomes evident that the consecutive layers of gel in
the beginning do not unite. Only in drying the layers become one
mass; still, in section the limiting plains are very conspicuous. This
experiment teaches us that in a siliceous gel a lamellar structure
can appear, wherein the consecutive layers are separated by visible
limits, without interference of another substance e. g. of an organic
lamella. We will come back to this fact later.

Several ways are open to us for the study of the spicular structure,
but they are not easy. Besides dissolving the spicopal by means of
hydrofluoric acid and carefully watching the process, the method of
heating has been applied since Gray (1835) showed that this brings
out more distinctly the lamellar structure. Imma (1901) was the first
to point out the effect of different media. We shall see, that some
spicules mounted either in Canada balsam or in glycerine, so widely
differ in aspect that on first sight one believes that one has to deal with
entirely different sorts of spicules. We thought it necessary, therefore,
to begin by determining the refractive index of spicopal of various
spicules somewhat more accurately than hitherto done. As far as
we know of there exists no other information than given by SoLLas
(1885), who states in general that “the refractive index of sponge-
silica is... . that of opal or colloidal silica, and not of quartz”,
and that the spicules come nearest to invisibility when “mounted
in chloroform, which possesses a refractive index of 1.449”.

In determining the refractive index of the spicules we used the
method, since Sonnas generally used also in mineralogy, viz. to find
in what fluid the spicule can no longer be scen. Perhaps there is a
still better criterion to make out how much a spicule differs from
its medium and in whieh direction, viz. the appearance of coloured
borders. In order to avoid the effect of fluids wich might influ-
ence the amount of water contained in the spicopal, but, on the
other hand to demonstrate just this influence, we used fluids which do
not mix with water as well as such which were diluted with water.
The refractive indices were determined by the refractometer of Abn,
Which has the great advantage of enabling us to work with ordinary
daylight and to determine any number of indices, between the micros-
copical work. In spite of the apparatus of Purrricu being more

C2Y)

accurate we used, therefore, an Abbr, the more so as it turned out
to be sufficiently accurate for our purposes.

Among the fluids, not mixable with water, we took advantage of
the series liquid paraffin (7m = 1.4759), petroleum (mn = 1.4568),
benzin (rn = 1.3994) and petroleum-ether (72 = 1.8780).

We succeeded by using mixtures of petroleum and benzin in fixing
the refractive index of the spicules at 1.4508—-1.4510. In order to
give an idea of the degree of accuracy that can be attained in this
way, we may state that undoubtedly a difference is to be seen between
spicules, mounted in a mixture of 20 cc. of petroleum with 3 ce.
of benzin (7 = 1.4500) and mounted in a mixture of 20 ce. of petro-
leum with 2.5 ec. of benzin (7 = 1.4510). For a aqueous watery solution
we used dilute glycerine, and found with this medium also a com-
plete disappearance at m= 1.4508. In addition to these media we
studied the influence of air, methylic alcohol, water, potassium acetate,
creosote, oil of bergamot, venetian turpentine, oil of red cedar wood,
oil of lemon, oil of thyme, firoil, oil of peppermint, oil of cloves
(pure or mixed with alcohol), Canada balsam and monobrom-
naphtaline. Practically however, we used more especially the two
fluids mentioned above.

In glycerine with = 1.4508 indeed the spicopal of some spicules
disappears completely, and only the axial rod remains visible as a
light bluish thread. In -all styli, the central thread has a higher
refractive index than the spicopal. By careful examination (Zeiss
Apochr. 8.0c.4) in most spicules a light, sharp line may be seen
as border, and a system of longitudinal striae between the axial rod
and the border. With low power the lamellar structure does not
become conspicuous, though the borderline is still visible. Most
probably this thin line, whieh exhibits double contours with high
power, represents the organic spicule sheath.

By these experiments it becomes at the same time very evident,
that the axial thread can be partly absent; on these spots the
light bluish band (the central rod) abruptly ceased. If the glycerine
has entered into the central canal, only an indication of the spicopal
is visible; if air has entered, of course this is directly visible
by the lower refraction. The aspect of styli isolated by means of
boiling with hydrochloric acid, either with addition of potassium
chlorate or without, or by digestion by means of artificial gastric
juice, or by heating with sulphuric acid and potassium-bichromate,
fundamentally agree.

Quite another aspect is shown by spicules which have been dried
for some days in the presence of anhydrous phosphoric acid. We have

( 28 )

studied spicules which had been dried in this way for some days
at an ordinary temperature, and also in Vieror Mnyer’s toluol bath
at 104°. The heating, however, had no influence on the aspect.
The determination of the refractive index is a little less accurate than
with ordinary spicules, because on drying the lamellar structure had
become somewhat more conspicuous, and therefore the disappearance
in glycerine is a little less perfect. Still we could determine in
petroleum-benzine the refractive index m= 1.4052—1.4055; with
diluted glycerine the same results were obtained.

If the dried spicules, mounted in glycerine with = 1.4055, and
with the border of the coverglass well shut by means of vaseline,
are left to themselves, gradually from the outside the refraction is
seen to increase, and afier one day the spicules become again highly
refractive. If they are now as much as possible separated from
adhering glycerine and transferred into glycerine of 2 = 1.4508, it
is seen that they disappear in this medium, and consequently have
absorbed again their original quantity of water. Spicules, which have
been dried by P,O, and are exposed to the air afterwards, behave
in the same way. On the other hand we have examined the behavy-
iour of spicules which, after being isolated and washed, were not
dried in the air, but immediately after removing the adjacent water,
were mounted in glycerine with = 1.4508. We could indeed see
a very slight difference with spicules which had been dried in the
air at an ordinary temperature, more than corresponded with possibly
adherent water, but too smail to be measured. However, it appears
that, even at the ordinary temperature, the spicopal gives off
some water.

Consequently we have demonstrated that the spicopal is a form of
hydrated siliceous acid, which may give off water in an atmosphere,
dried by P,O;, which diminishes hereby in refractive index, and
Which may again absorb the original quantity of water by immersion
in a watery solution or by exposure to moist air. The spicopal also
behaves with regard to the absorptive power for water just as a
gel, as has been shown by the well known and intricate researches
of VAN BEMMELEN.

(To be continued).

(29 )

Mathematics. — “On pencils of algebraic surfaces.” By Prof.
JAN DE Vriks.

1. Let a pencil (/") of surfaces /” of order » be given, inter-
secting in the base-curve o,

The principal tangents in a point S of o to the surfaces of (2)
form a cubic cone having the tangent s to 6 for edge.

For, if (f”) is indicated by

n

Cae KO A) ek oy oe Seen aC)

and if y, are the coordinates of .S, the substitution a, = y, + e2,
/ 5 : 6 A n ; 2 F
furnishes in connection with «@,—=0O and 6) = 0 for a point Z on
a principal tangent the conditions

n—1 n—2 2

— 2 2 n—2
ay ras aCe Ozi—— ION CymieGz =|A Os Oz. 0,

so that the locus of the principal tangents touching in JS has as
equation

=) 9 Ye

n—1 ]

Lr 2 Qn — lee

dy Dy a@zb,—ay by azbz=0 . ss . (2)

If Z is a fixed point, Y a variable one, this equation represents
a surface of order (2n— 8) determining on 6 the points SS) which
are points of contact of principal tangents through Z.

The principal tangents in pots of the base-curve form a con-
gruence of order n? (2n —3) and of class 3n?.

The inflexional tangents of a pencil of plane curves c" enveloping
a curve of class 3n(n— 2), the complex of rays of the principal
tangents of #” is of order 38n (mn — 2).

2. <A principal tangent f, in S becomes four-pointed tangent ¢,,
“7 ¢ : r ‘ ; : —3 3 —373
if for a point Z lying on it the relation @, “a:+2b, “b:=0 holds
good. So the tangents ¢, touching in S belong to the biquadratic
cone

no: 3 N— GS!) nl Vid 6
ay by az bs — ay b, Gz0s 0 Oo pre Matis eet (8

As the cones (2) and (3) have the tangent s represented by

n—1 1

Cigna —10) 7, By C—O hee e rateyis 2 (4)
in common, the point S will be the point of contact of eleven four-
pointed tangents.

For a,=0 and 6;=0 the equations (2) and (3) represent the
figure formed by the surface of tangents (s) of o and the scroll 7,
of the right lines ¢, having their points of contact on 6.

To determine the order of 1, I search for the number of points

( 30 )

of intersection of the indicated figure with the right line z, = 0,
2,—0. Substitution in (2) and in (3) and elimination of z, and z,
gives an equation containing the coefficients of (2) and (3) succes-
sively in the orders 4 and 8. Hence the resultant in the coordinates
y is of order 4(2n — 3) + 3(2n— 4) or 14n — 24.
So the number of points of intersection is 2/7 (7m — 12).
Applying the same treatment to (4) I find for the order of (s)

the well-known number 27? (2 — 1).

The four-pointed tangents having their points of contact on the
base-curve 6 form a scroll of order 2n? (6n— 11), on which o is
eleven fold.

For 28 this seroll passes into the locus of the right lines on
the surfaces of a pencil (7), thus into the scroll of the trisecants
of o? which is of order 42. For 2n? (62— 11) we now find 126,
corresponding to the fact, that each trisecant does duty as right line
t, for three points JS.

On each surface #" the points of contact P, of four-pointed
tangents ¢, form a curve of order 7 (11m — 24). As 6 is evidently
an elevenfold curve of the locus of the points of contact P, belonging
to the /” of the pencil this locus has in common with every /” a
locus of order n (112 — 24) + 11n?.

The points of contact of four-pointed tangents form a surface of
order 2 (11n — 12).

For n=838 we find the scroll of trisecants of order 42.

3. As the tangents ¢, passing through a point Z form a cone of
order 8n(2—2) and two principal tangents have thei point of
contact in Z the locus of the points of contact P, of the right lines
f, containing Z is a curve of order 38(n—2)-+ 2 with double point
in Z.

Each of those right lines ¢, cuts the surface #” osculated by it in
(n— 8) points Q more. The locus of the points Q is a curve of
order 37 (n— 2)(n — 8) + (n? + 2) (rn —3) or 2(n—3)(n—1)(2n—1).
For, through Z pass (2? + 2) (1m — 3) tangents ¢,, osculating the surface
indicated by Z in an other point *).

To find the number « of the coincidences of P, with Q, I make
use of the well-known formula

e=ptq—-%
which appears when the pairs of points ?, Q are projected by a
pencil of planes. Each point P belonging to (n—8) pairs we have

1) See Gremona—Curtze, Oberflichen, p. 66.

( 31)

p = (8n* — 6n + 2) (n — 3). Farther more ¢ = 2 (n — 3) (n—1) (2n—1),
whilst the number of right lines PQ resting on an axis is of course
equal to 3n(m— 2)(m— 3). So ¢«= 2 (nm — 3) (2n? — 3n + 2); this
is also the number of four-pointed tangents through a given point «.

The number of right lines ¢, in a given plane is equal to the
number of points of undulation on the curves ec” of a pencil; this
number I have determined in a preceding paper ').

The four-pointed tangents form a congruence of order

i)
2 (n — 3) (2n? — 38n 4 2) and of class > (n—3)(n?+n?—8n-+4).

a

4. If we wish to apply the above-mentioned formula of coincidence
to the pairs of points of intersection Q,Q' on the right lines ¢, through 7
we have to substitute p = q¢ = 2 (nm — 3) (n — 1) (Qn — 1) (n — 4) and
g = 3n (n — 2) (n — 3) (n — 4). For each point Q belongs to (n—4)
pairs and each right line ¢, bears (2 — 3) (n — 4) pairs. We then
find «= (72 — 3) (n — 4) (5n* — 6n + 4), i.e. the number of tangents
fg2 through the point Z.

In the above-mentioned paper I have determined the number of
right lines having with a curve of a pencil (e") a three-pointed and
at the same time a two-pointed contact.

The two-three-pointed tangents form a congruence of order

(n — 3) (n — 4) (5n? — 6n + 4) and of class

(n— 4) (n — 3)? 10n* 4 35n* — 21n? — 80n + 20).

2

5. Each principal tangent ¢, having its point of oseculation in a
point S of the base-curve bears still (2 — 3) points of intersection Q
with the surface £” osculated by it. As JS is point of contact of 11
four-pointed tangents the locus of the points @ will have an eleven-
fold point in JS. As an arbitrary plane through S evidently contains
3 (n — 3) points Q (§1) the order of the curve (Q) is equal to (82 -- 2).

When applying the formula ¢=p-—+q—yg to the pairs @, Q’
which the cubic cone with vertex |S bears, we have to put
p= = (8n + 2) (n— 4) and g = 3 (n — 3) (n — 4). Then we get
& = (n — 4)(8n +13). So this is the number of tangents é32 for
which the point of oseulation lies in |S.

In other words, 6 is an (1% — 4) (82 + 138)-fold curve on the locus
[R,| of the points of osculation of tangents tz. to surfaces of (2).
Now the points of osculation of the right lines ¢32 of an #” forma

1) *On linear systems of algebraic plane curves”, (Proceedings, April 22, 1905.)

(32 )

curve of order 1(m—4)(8n?-+5n—24)'). So [R,| has in common

with an J of the pencil a curve of order

n (n—4) (8n?-+5 n—24)+-n? (n—4) (8n-+13)=—n (n—4) (6n?-+18n—24).
The points of osculation of the three-two-pointed tangents of (E)

form a surface of order 6 (n—1) (n— 4) (n+ 4).

6. To determine the order of the cone formed by the double
tangents of (4) of which a point of contact in S lies on the base-
curve 6, we notice that the tangent s in S to 6 is intersected by a
3) double points

pencil in an involution of order (1 — 2). Its 2(n
are points of contact of double tangents touching in |S too. So sisa
2(n—8)-fold edge of the indicated cone.

In each plane 9 through s we can draw out of S n(n—1)—6
tangents to the curve of intersection of ¢ with the surface /” touching
¢ in S. From this ensues that the indicated cone is of order
(n — 3) (nm + 4).

The locus of the second points of contact AR, of the edges of this
cone has evidently in S an elevenfold point, where it is touched by
the eleven right lines ¢,. So the curve (/,) is of order

(n— 3) (n + 4) 4+ 112 =r? + n —1.

Every edge of the cone intersects the surface doubly touched by
it in (7—4) points V more. The locus of these points passes
(1 —4) (8n +18) times through S, where it is touched by the right
lines ¢39 oseulating in S. As each plane through S bears moreover
(n?--n—1 2)}(n—4) points V, the curve (V) is of order

(1 — 4) (n? + 4n + 1).

Now the number of coincidences of /, with J can be determined

again by means of the formula «= p-—+q—y. We find
&= (n?-+n—1) (n—4) + (n—4) (vw? + 4n +1) — (n— 8) (n +4) (n—4) =
= (n— 4) (nv? + 4n + 12).

This is the nnmber of tangents és2, of which the point of contact
lies in S, thus at the same time the multiplicity of the base-curve
on the surface [2,| of the points of contact of surfaces of pencils
with right lines ¢;0. Taking into consideration, that the points of
contact R, form on the surface /” a curve of order

n(n—2) (n—A) (n? + 2n + 12)
we find that | #,| has with /#™ an intersection of order
n (n — 2) (n— 4) (n? + Qn +12) + n?(n—4) (nr? + 4n 4-12).

1) See inter alia my paper: “Some characteristic numbers of an algebraic surface.”

‘Proceedings, April 22nd, 1905).

(33 )

The points of contact of the three-pointed tangents of (F") form
a surface of order 2(n— 4) (n* + 2n? + 10n — 12).

7. Through the tangent s in S to 6 we can make to pass four
tangent planes to the cubic cone of the principal tangents (§1). So
S is a parabolic point on four surfaces of the pencil. Therefore o is
a fourfold curve on the locus of the parabolic points.

As the parabolic points of an #” lie on a curve of order 4n (n—2)
the locus under consideration is cut by each of the surfaces 2” in
a curve of order 4n(n—2)+ 4n? = 8n(n—1).

The locus of the parabolic points of the surfaces of a pencil (F")
ws a surface of order 8(n—).

\

Chemistry. — “On the shape of the plaitpoint curve for mixtures
of normal substances.” (Second communication). By J. J. van
Laar. (Communicated by Prof. H. A. Lorentz).

1. In a previous paper’), starting from van pprR WAALS’ equation
of state, in which 4 is assumed to be independent of v and 7, I have
found for the equation of the spinodal curves at successive tem-
peratures (I.c. p. 690):

Rr = =| «(02 & + o(-—0) |, eeu ea aaKGL)

and for that of the plaitpoint curve in its v, 2 projection (Le. p. 695):

(1-264 (0 2) n— Be (Lo | + alo—0y| 82(1—2) (0-8 V0) +

Fao) (r=3) [= 0, OAs dee ee MO)

In this G=2+4 a(v—b), r=}, Ye, —},Ve,, c= Y'o,—Ya,,
and B=), — 6,.

The equations (1) and (2) hold for the so-called symmetrical case,
where not only 6,, = '/, (6, + 6,) is assumed, but also a,, = Ya,a,.
These hypotheses lead to:

b=(1—a)b, + 0b, ; a=[A—a) Va, +2ya,).
The equation (1) had been given already before by vAN DER WAALS

in implicit form’), for after some reduction his general equation

1) These Proc. April 22, 1905, p. 646—657.
*) Cont. Il, p. 45; Arch. Néerl. 24, p. 52 (1891).

wo

Proceedings Royal Acad. Amsterdam. Vol. VIII.

( 34 )

: : Pes ; ih aly aka
passes really into (1) after substitution of the values of - Daeg

2
and = in accordance with the above hypotheses.

3ut the equation (2) may be said to have been derived here for
the first time in the above simple form. It is true that van DER WAALS
gave a differential equation of this curve’), and derived an approz-
imate rule for its shape’), but he did not arrive at a general final
expression. Nor has Kortewee arrived at it in his very important
papers: ‘Sur les points de plissement’” and “La théorie générale
des plis, ete.’ *) In his final equation (73) (le. p. 361) there occur,
besides 7, still several functions g (v), $(v), w(v) and x (v), which
have been given respectively by the equations (37), (38), (40) and
(74) (l.e. p. 850 and 361). Kortrwre’s equation is one of the 9
degree with respect to v, but it is easy to see that it may be reduced
to one of the 8 degree (l.c. p. 361). It appears from our deriva-
tion that this degree may be reduced to the 4. In a later paper ‘*)
KortrwreG confines himself to a full discussion of the plaitpoints in
the neighbourhood of the borders of the w-surface.

I think that one of the reasons for failure in this direction is due
to the intricate form of the differential equation of the plaitpoint
curve, when we use the y-function. The $-function on the other
hand leads to simpler expressions. Already the differential equation

2
or == (0) (oye

for the spinodal line at given temperature, viz. &
@ pT
Ps

Ou ‘ :
& = 0, is much simpler than the corresponding expression in wp.
& Th

P>

And to get the plaitpoint curve, we have only to combine

Ou, vee Oates
— |} =0 with sO
Oa )»,7 0x? )T

2. We shall now examine the shape of the curves given by (1)
and (2) more closely, and specially for the case that 8 = 0, i.e.
b, =6,=b. The calculations are rendered very simple in this way,
and it is obvious from the adjoined figs. 1—4, that when 6, is not
=4,, so ? not =9O, the results will be modified only quantitatively,
but by no means qualitatively. We shall come back to this in a
following paper.

1) Verslagen Kon. Akad. Amsterdam, 4, p. 20—30 en 82—93 (1896).
2) Id. 6, p. 279—303 (1898),

3) Arch. Néerl. 24, p. 57—98 en 295—368 (1891).

4) These Proc. Jan. 31, 1903, p. 445,

(35)

As 6=2+a(v— b)=av—BYyéa passes into av for B=0, we
may write for (1):
2 9

is =| (1 — 2) a?vo? + a (v — 6)? | {os as, 4 ile)

and (2) is reduced to:
a(1— a)a*v* [(1 —22)v] + Vile] BoC e2e+a(e-t) (0-88) |=. (2a)

Let us put these equations into a more homogeneous form.

Asa=[Va,+2(Va,—V a,)]? = (Va, + va)’, we may write for (1a):
9 2
RT =— E (1 -—«) @ + (Ya, 4+ 2 @)? (: — =) |
v

v

ees E (Geary a (e a ) (2 = =|
Vv a Uv

If we now put:

this last equation becomes :
2a?
Tight ARS E (—w2)+(¢g+2?0— oy |
Let us now introduce the “third” critical temperature 7,. This
temperature is the plaitpoint temperature at v= 4, i.e. that at which
the limiting curve lying in the limiting plane v = 6 (see fig. 1 of my
previous paper cited above) reaches its maximum, and is represented
by (o = 1):
20°
RT, = x, (1 — ae) aie

But as in the case 6,=6, for a, the value 1/, is found (the
maximum of the now parabolic curve), we get:
Sie a’

b
Our equation for RZ’ becomes therefore :

Tigh hy ee OCT E d—a+(g4+2)(1 — oy |

And if henceforth all temperatures are expressed in multiples of
T,, we have finally, putting

RT, =

LS oe
mes

t= 4wW E (1 — «) + (g + 2)? (1 — o)? | . (16)

(36 )

In this simple form the equation is very suitable for caleulating
successive spinodal curves. It is of the second degree with respect
to «x, of the third degree with respect to w. For a given value of
rt we have therefore only to put successively w —1, 0,9, 0,8 ete.
down to 0, and then we find the corresponding values of w by
solution of ordinary quadratic equations.

The equation (2a) becomes after division by a (1 — 2) a’v':

a Lia's a/,2(1 — 4/,,) (1 — 3b/,
a—29+"8(1—<) [s 2 a BO ee Ben Pe |=,

Va Ya,

i.e. as —=—+e=9g+24
a a
(1 — 2x) + (g + 2) (1 —o)? Ee ae aM “(1—«)(1 30) | = 0... 20

This equation of the plaitpoint curve is of the third degree with
respect to xv, of the fourth degree with respect to o.

3. Before discussing the equations (10) and (26) more fully, we

shall first derive a few relations between 7, 7, and 7’.
1 2

i /,@
As RT, =

8
(see above) and RT, = 7 + we find immediately :

From this follows that for values of g << */, V3 (= 1,380) 7’, will
be < 7T,; i.e. the lower critical temperature of the two components
will then be lower than the critical temperature of mixing of the
two liquid phases at v = 0d.

Va,

1
Ned ps) et ee
VYa,—Va, 7

For g =0 is T, =o xX 7,3 forg=—ois 7, = 7, borg — aie
(see above) 72/7, = (1 + ‘/, 3)? = 1/,, (48 + 24 3) = 3,138.

It will also prove important to know the amount of the pressure
for all points of the spinodal curves. For this purpose we reduce
the equation :

to

( 37 )
RY _—Vatea)? RT

a
— matt OES a ay
£ v(1—4/,) v ~~ e(1—w) 2? ne
This becomes on account of «? = 2bRT, (see above) and 7/7,=r:

nee

p= REE = Bengal

o
b 1—w

Let us express p in the critical pressure p,. (As, namely, the
pressure p, corresponding to 7, (v=) is evidently =o, p cannot

é RT Te 16 2 RT.
be expressed in p,). Asp, ='/, ; * and r, = 57 PI = ° 9’,
hence — when we put
iY
SS S158
ign
__ 27 T 4 ! s
nice obet ae ee) sevretis. a Mie aS)

This equation may be used, when t is already known from (18).
If this value is, however, substituted, we get:

Roa at é
Poel a (l-—a) 4+ 2 (¢+2)? (L—w)? — (g+2) (—o) |
i.e.
age o 9 1 5 Mt 2 1 a)
C= , = wv (1—a) + (g4+-2)? (1—o) ( —20) | eg Loe Gi)

4. Better than descriptions and calculations the adjoined figures
1—4 represent the different relations which may present themselves
in the discussion of (16) and (2), combined with 3 or (3a). We
shall therefore confine ourselves in the following to what is strictly
indispensable.

Two principal types occur, according as g < 1,43 or > 1,48.
Fig.1 with g=1 is a representative of the one type, fig. 2 with
y = 2 of the other. The transition case y = 1,48 is represented in
fig. 4.

a. Description of the case gy=1 (fig. 1 and 1a).

There are ¢wo plaitpoint curves, one of which extending from
C, to C,, the other from C, to A. The latter, however, may only
be realized down to a point between C, and &,, where it is touched
by the spinodal line + = 0,63 °).

1) See Korrewes, |. c. p. 305 (fig. 12) and plate F, to F,. (The plaitpoint - has
already disappeared in the limiting line v = in our case). R, is a so-called point

de plissement double hétérogéne. Cf. also van per Waats, These Proce. V, 310,
Oct. 25, 1902.

(38 )

Beyond the point R, the temperature, and with it also the
pressure, decreases, as is to be seen from the succession of the
different spinodal curves, so that in the p, T-diagram (fig. 1a) the
plaitpoint curve C,R,A shows a cusp at R,, and begins to run back.

It is known that this ease is realized with mixtures of C,H, and
CH,OH, ether and water (Kurnrn), ete. It is the principal type I,
as I have fully described it in one of my two preceding papers *).

Remarkable and quite unexpected is the fact that this type may
be realized for mixtures of normal substances. It was formerly
believed that such deviating plaitpoint curves were only possible
when at least one of the two substances is anomalous. This, however,
seems not to be the case; more and more the conviction gains ground
with me that the anomaly of one or of both components only
accentuates the phenomena sharper or brings them into attaimable
regions of temperature.

It is also striking in fig. 1a, that the curve C, C, has the same
appearance, viz. with an inflection in the middle part, as the typical
curve as observed by Kunnun for C,H, + CH,OH (see fig. 1 of my
just cited paper). Only in our case there is not yet a pronounced
maximum and minimum, as with the mixtures of C,H, with the
strongly anomalous substance CH,OH.

The type of fig. 1 oceurs for comparatively small values of g.
According to the equation given in § 3 the proportion 4/7, = 4
corresponds with g~=1. The eritical temperatures of the two
components must, therefore, lie comparatively far apart.
eT
Pans ane
T,=1, as has been done in the figure, then 7’, —0,59 and 7, = 2,37.

1

As fi/7, = is considerably higher than 7. If we put

b. Some mathematical and numerical details.

The plaitpoint curve C,C, touches the line «=*/, in C,, the
curve AC, touches the line «=O in A. Moreover the curve C,C,
touches the line wa='*/, once more in D, and it does so at
w = ?/,(v= 1,55). In C, and C, no contact takes place.

When g becomes <1, and approaches to 0 (42/7, then becomes
larger and larger and approaches to oc), then the curve CA approaches
the straight line «=O more and more and the curve C,C, the
dotted curve in the figure, which continues to present a clearly

1) These Proc. VII p. 636—638, April 22, 1905.

(39 )

pronounced inflection point up to the last’). For values of gy > 1,
the curve C,C, lies partially on the left of the curve «='/,, and
the point of contact at D passes into two points of intersection.

By an approximate solution of (26) and substitution in (1°) and
(3) of the values found, the following points of the two plaitpoint
curves are calculated. (The other values of @ or a are either imaginary
or do not satisfy).

Pall
Curve C,C, Curve C,A
z=05 06 0,7 08 09 4 w=0,330,4 051) 06 07 O08 09 4
a 0,49 0,43 0,39 0,36 0,33 | ¢=0 0,021 0,041 0,042 0,023 0,010 0,0017 0
z=41 4,78 1,98 213 2.96 2.37 | = =059 063 062 051 033 016 0042 0
w=o 645,75 5,05 4514 |<=1 41,15 108 0 —309 —864—169 - 97

It is seen that the pressure begins to be negative for points in the
neighbourhood of A. This is not remarkable; also for a simple
substance the points of inflection in the ideal isotherms reach to
within the region of the negative pressures. Though the pressures
in some points on the spimodal curve are negative, this is no reason
why those on the connodal curves should be so.

The limits of the region of negative pressures on the spinodal
curves may be easily fixed (see the dotted curves in fig. 1) by solution
of the equation (see (3a))

22 (1 — «) = (p + «)? (1 — w) (2w — 1).
If we put here (1 — w) (20 —1)=4@, we find:
pra eae Oy
2+6
In this way we calculate for g=1:
OO See Or Ole 106) 90:5
OF O04 O07 20:07 0104570
i O;84s 20 for Ona" O84, Ak
That 2 approaches to —27 for e=0, w=1, r=O follows

—

: ‘ : T
immediately from (3). For as "ELE approaches to 0, as we shall
82)

27 1
prove presently, 2 —=— a (2 ) =, independent of the

value of g.

1) For this plaitpoint curve g=0 the following points are easily calculated:
o— 09) 08 OF 016 O15 O14
x —0,507 0,528 0,567 0,623 0,712 0,853
The equation (2b), viz., passes then into the following quadratic equation in xw:
x7w? (9 — 100 + 3w?) — 3xe (2 — w) +1—0
The other value for ~ is always > 1.
?) The maximum lies at #=0,54; x is then about = 0,043.

(40)

In this we must notice that in the immediate neighbourhood of
the point A, a inereases with the utmost rapidity from — 27 to
+o, when we pass the above considered border curve; im the
point A itself this transition takes of course place suddenly, For
27 2a(1—«)

yg’ ee

when w = 1, a approaches to = o, according to (3a), except

; : 2a(1—z)
in the case that w is exactly = 0, when (see further) aS == 0)

the following term yielding then the finite value —27. This follows
also from the figure, because the border curve, which separates positive
from negative pressures, passes through the point A.

T
and ——
—@ 1—w

That on the plaitpoit curve the expressions
approach to 0 for 2=0, w=1, r=—0 at A, follows from (26).
For putting «=A and 1—w=d, we get:

1+ go? goto Pi ele
A

ay ‘ :
or as 3gd* may be neutralized by 1, 1 — 2g? A = 0, from which

A : EMR yA
follows, that at the point A Fe = 2y*, so remains finite. So A is of

Lb A
the order d*, so that —— = 5 really approaches to 0 at A. From
=e CLD. C

this follows also the contact. And as according to (16) t approaches to
4 (A + g’d’) = 4970? (A being of the order 6°) for z=0,0=1,

T
approaches to O at A.
l—w

In the same way the plaitpoint curve C\C, touches the line
2—="/, for z= ‘/,,@=1. For, for 2 V0 -- A), 0 — ie

equation (26) becomes:
-2+@ + ioe [38 +)" |=0,

; A
which appoaches to — A + 3 (g + ‘/,) 0? = O, yielding i = 3(g+7/,),

Bias A
so again finite. So A is now of the order d?, and so 5 again = 0,

which proves the contact at C;.

I call attention to the fact, that on account of the small values
of A a large portion of the curve C,C, from C, as far as beyond the
point D may be calculated very accurately, by writing for (26) (g = 1):

(41)
—4+7/,0— 0/890 —0) G0 — 9],

so that A=/,(1 -—)? E Sule 30) (80), — 1)

From this follows eg. for o = 0,9, 0,8, 0,7, 0,6 resp., for A
0,022, 0,029, 0,004, 0,029.

The contact at D. If we put in (2b) c= 7*/,, then 1— 227=—0,
and hence:

(+ 4) — 0) E sited AE 1/9" (bes Se) ip.

This yields besides @ —1 (the point C,), also:
a. 3
Rep=)

9
ee eo ———————
Peet ly (2p-++1)"

For g=1 this yields two equal roots w» = ?/,, which proves the
contact at D. For g< 1 the roots become imaginary, so that then
C,C, no longer cuts the line «= '/,, but keeps continually on its
right, whereas for g >1 two points of intersection are always
found. So is e.g. for p=2 w="%/,, (close to C,) and o=7/,
(lying on the other branch between C, and C;, (see fig. 2)).

In order to facilitate the tracing of the different spinodal lines, it
is to be recommended to fix the limiting values of + for «=O,
ey == ay ——"/..). Also: for 7——=s/,. it is; easy to. calculate: x:
From (10) follows e.g. for z=0, p=1:

t = 4w (1 — a)’.

(1 — w) (8H — 1)

hence:

This yields:
o=—109 08 O07 O6 O65 04 033303 0,2 Oj,
+=0 0,036 0,128 0,252 0,384 0,50 0,576 0,593 0,588 0,512 0,324

For «—1 these values become simply + times larger, (g + 2)?
then being = 4.

For z= '/, we get,

tr=—of{1+9(1—wo)},

yielding :
le 0-95) 9 1OL9 Ose O57) (0,602,055 0,4 0,33 0,3
7 =—1. 0,971 0,981 1,09 1,27 1,46 1,62° 1,70 1,67 1,62.

For w=1 we get simply:

t= 4a (1 — 2),

from which follows:

( 42)

z=0 O, 0,2 0,3 ORL Os OG RON OS O19 1
r=0 0,36 0,64 0,84 0,96 1 0,96 0,84 0,64 0,36 ©

Le /oe
fis"

ca, E (1 — 2) + 4/, (@ + 0» |

Finally we get for wo =

yielding :

z= 0 O1 O02 08 04 O5 O6 O77 O08 ,0;0 ae
— 0,593 0,837 1,07 1,28 1,48 1,67 1,84 1,99 2,13 2,26 2,37.
It appears from the diagram (see also above for # = 1/.), that the
temperature from C, to C, is not continually ascending, but that it
shows a minimum very near C,. This causes the spinodal line r= 1
not to pass through C,, but to remain under it. The point C,, where
t is also —1, is an isolated point belonging to that line. Just beyond
C, the two branches of one and the same spinodal line intersect in
a double point; beyond that place the course is normal; between
C, and this intersection the spinodal line has two separate branches,
one of which encloses the point C,. Now the question arises,
whether this will be the case for every value of g. If we solve a
from (10), we get:

x (20 — w’?) — @ (: + 2y (1 — oy) ++ i= —g’(1— oy) 08

This gives for « two roots of the same value for given values of
rt and w, when
4g(g+ 1) —ow)’? 4+ 1—1(2—o0)=—0. a
The value of a is then:

at elo)

a

w(2—o)
Now it follows from the value of the above given discriminant,
that it becomes =O for two values of w. So two branches of a

spinodal line intersect, when those values of w become the same. From

49 (p +1) «? — © (80 (p +1) — ") fe (4 (p+1)+ 121) =0

follows, that w has two roots of the same value, when
= = I6pigtl) 7 oe
ee ee SPE),

And t+ being 1 at C,, the minimum disappears only, when +
becomes =1 in the above expression. And this is evidently only
the case for g=o, i.e. when 7 and 7, should have the same
value. Hence in general there will always be found a minimum in
the neighbourhood of C,. For g=1 we find r= 0,970, w = 0,94,
e— 0,506; for p=2 we fnd += 0990; o=0:398) c= 0508.
etc. etc.

( 43 )

It may be easily demonstrated that in the neighbourhood of C,
such a minimum never appears in our case. For from (416) follows
16

with «= ae Oe

16
nv tole by +d oy],

After substitution of «=A, w='!/,(1+ 4), we get, neglecting
A’, which is justified by the sae

a+ a] eS (04+ S)0— yo |r.

1
or as Eats =e eee ee 5 and so Eerie, ce 20) eal eos
ee at as yy J) =) J?
4g? Pp 2 4 ’
yielding :

ele ee eke 6% 8
SS 51 | =) || == ats I 5
|

The spinodal line 7’=— 7, touches, therefore, the axis «=O for
every value of g, and, at least on the assumptions made by us
concerning a and 6, a minimum can therefore never appear in the
neighbourhood of C,, in consequence of which the spinodal lines in
the immediate neighbourhood of C, would enclose this point.

Finally some corresponding values of 2 and w are subjoined, which
determine the shape of the spinodal liner = 1(7’= 77). By solution
of the quadratic equation

co) E (1 — «) + (1 + 2)? 1 — | = |

follows immediately :
ae Oee 0p a 0662) 05, 0,4 = 0.38 03° 0:2: 0,1
za=0,5 0,403 0,292 0,227 0,184 0,164 0,182 0,182 0,806 0,679
0,743 1,004
So this line cuts the axis «= 1 for w = 0,7, and henceforth only
one solution satisfies. 2 becomes evidently 1 for w (1— w)? =‘*/,,,
yielding about w = 0,07.

From the above derived equation 4g(g+-1)(1—)?-+-1—1(2—w)=0,
which was the condition for two equal values of x, we find g=1,7r=1:
8H? — lbo + 7=0,
from which, besides w=1,w='/, follows. To this belongs then
a—="'/,,— 0,524. Between wo=1 and w=0,875 we find only

imaginary values for z in the above table.

( 44 )

As to the spinodal line 7 = 7, (x = 0,59), we calculate « = 0,0019
for w — 0,30, whereas 2 = 0,006 corresponds to w = 0,40.

As to the shape of the spinodal lines for great values of v (vapour
branch) i.e. when t and @ approach to 0, follows immediately

from (10):
t= 4w E (l1—2#)+ (9+ | = 4w G + (29 + 1) | ‘

a Cae b
If we substitute 7/7, = T: a for r, and 5 for w, we get:
2a?
RT = — G +(2g9+1) «| :
Vv

a
After substitution of @ = as this becomes :

pee
a RE E Hy —aye|,

From this follows that the vapour branches of the spinodal lines
in their v, x-projection will approach more and more to straight lines,
which will cut the axes «=O and «—1 at distances proportional
to the quantities a, and a,.

5. Let us now consider the second type, which occurs for g = 2.

a. Description of the case y = 2 (fig. 2 and fig. 2a).

The two plaitpoint curves of fig. 1, viz. C\C, and C,A have met
for g about 1,43 (see fig. 4), after which two new ones have been
formed, now CC, and C, A. This case, which is found for compa-

ih

ratively large values of gy, for which the proportion i approaches
1

more and more to unity, is the usual one or the normal one. It is

the principal type ITT, as described in one of my two preceding papers *).

The region of negative pressures on the spinodal lines extends
now all over the v, z-diagram, from «=O to «= 1, and is bounded
by the two dotted curves (see fig. 2) above and below.

The spinodal line belonging to r= 1,35 touches now the curve
C, A in the point #,. Again the plaitpoints are not realisable from
a point between A, and C, to A (see the footnote in §4 at a.)

Beyond &, the temperature and with it the pressure decreases,
so that in the p,7’ diagram (see fig. 2a) the curve C,R,A runs back

1 Le. p. 642—644.

(45 )

again from f,. In Rk, the pressure is already negative, and it
becomes again = — 27 p, in A. (See §4 at b).

When g=—2, we find easily from the equations derived in $8,
that then 72/7 = 2'/, and 1/7,= “‘/,,. So if 7, is again =1, then

27

st, and, f= 5,33. Now r. is higher than 7’.

b. Some mathematical and numerical details.

Much having already been derived in § 4, it will suffice to give
some few values.
Of the two plaitpoint curves the following points were calculated

g=2
r=0 Cio Ose cos (05 06 OF 08 09 -1
o = 0333 0395 0403 041+ o41— 040 og9t os 0359 0347 039° 1 Cupp
7= 297, 287 304 338 370 400 428 460 487 510 593 | C,C,
x=1 146 162 1,90 211 2% 292 237 296 298 225
z— 0. - 001 “od Oe) 08: of of
c= 09 | O8l O78 060°) 0,65) 09a3ana?

Curve C,A.

ll

0 Obs, O81. 128), 195 (1,26 1,04 anal
r=-—27 —173 —790 516 —462 —398 49 and w |
The separation between the negative and positive pressures on the
spinodal curves is given by
o= 1 09 0,894 0,606 06 0,5
0 031 040 0,40 0,31 O
1, 70}50" O}40 0,40 0,50 1
The places where w has here two equal values, are easily found
from the value of x given in § 4. Evidently we must have then
ae = (1 — ow) (20-—1)=7/,,. This gives w —0,894 and 0,606, =
1,(8+7/,/3). For g=1 O@ would have to be '/,, and there are
no values of w which satisfy this condition.
For the calculation of the different spinodal curves it is convenient
to know the limiting values of t again. We find for «= 0:
Hg Wa ee os eoOY 1575 2s 2,25, 2,50) 2575) 8
7 = 00351, 94,19 74568 2, 2,20) 2,30, -2,36 2:37

For «=1 these values are all 2'/, times greater.

—

For z= '/, we find with the same values of w:

t~=1: 1,60 2,53 3,20 3,63 3,88 3,99 4,04 4,04
w =1 yields the same values as in § 4 for p=1.
w ="/, yields:

( 46 )

2=0. 0, 0,2: 03 0,490 520607 70'S, 10:95:
7 = 2,37 2,73 3,08 3,41 3,73 4,03 4,33 4,59 4,86 5,10 5,33

6. We may now determine, where the transition represented in
fig. 4, takes place. (The place of the point P is also drawn in
figs. 1 and 2 ')).

If we put 1—w=~y in the equation (24) of the plaitpoint curve,
then

y (dy—2
(=) +e+9r (9+ et « gy i) =0. (a)

’ of of
Now in the double point sought ae must be O and ae must be 0,
a y

when f denotes the first member of (a). This gives:

— 20 (Le) + (1-22)? + 8y* | 20) (@ +9) + « L—0)} +

+8@+9)yGy-29=0,..... 06
and after division by 6y (w+ g):
wx (1—a) + (@+gyyQy—1l)=0..... 0
Substitution of the value of «(1—vx) from (c) in (a) gives:
(120) +(e + 9)y (8 ZS vo) =10,
or
by
2y
So we have to solve y, « and @ from (a’), (>) and (c). Substitution
of 1— 22 from (a), and «(1 —2) from (ce) in (6) gives, after
division by (# + )*y:

1— 37\°
~ sit igen ean
— oy

+ 3y7 (8y — 2)= 0,

(1 — 22) + (e + 9) 9? — tl scutes ee sé. (7)

+ 8y*

“1—2
i.e. after multiplication by (1 — 2y)?:

mm Cl) eae PM i ye
+ 3y* (L — 2y) }— y (L — 3y) + (1 — 2y)?| + 8y? (8y—2) (1—2y)? = 0,

from which y may be solved. The above equation gives:

1) This point must be thought more to the left. In fig. 4 no contact but inter-
section takes place in the double point P.

(A)

— 2 (1 — 2y)* + 99 (1 — 8y)* + By? (1 — 2y)(y? + 2y — I) = 0,
or dy — Ldy* 4- 29y° — 27y? + 12y —2=—0,
i.e. after division by (y—1)’:
dy” — by + 2,

1 ae a A V3.
As it is obvious that y cannot be larger than 1, only:
y=1—1/, ¥3 = 0,4226

yielding :

satisfies here.

If we substitute the value #-+ g from (a’) into (c), we get:
(1—2y)°
y* (I—3y)?

In this the last fraction passes into */, (1 + 3), after substitution
of y= 1 —"'/, V3, so that we get for a:

«(1 —#)—*/,0 + Y8)

xv (1 — x) — (1 — 22)?

1 — 4¢ (1 — | == 0}
hence:

a1 —2)="/,(—1+4 3),
giving:

— sf +1/, (V6 — //2) ,2412 or 0,7588.

It is obvious from the figure, that only the first value satisfies,
V1Z.:

«=’/, }i EA (VAb = v2 — 0,2412.

The value of is finally found from (c):

state #(l—a) =#) ESeeS
+ OF ay = ee + V8)
giving «+ 9=1/,(8V2+/6), hence g='/,(—1+/24 //6)=1,482.
As y=1—%*/,V3, w='/,//3, i.e. the intersection takes place
ab ub /73 = 1,732 b.
As mentioned before T,= 7, for g = 1,30 (see § 3). For py = 1,43
T, is already < T,. For 4/7,="/,,q? we find easily the value
1,215, while 2,887 is found for %/7 = (1 + 1/¢)?.

7. Besides the cases, given in figs. 1 and 2, representing the
principal types I and III, there is another important type, viz. II,
of which I also gave a full description in my previous paper, which
I have already cited several times’). The p,7-diagram of this case

1) 1. c. p. 663—667.

( 48 )

is given in fig. 4a. Kupnen met with it, among others, in the case
of mixtures of C,H, with ethyl- and some higher alcohols. Also
triethylamine with water is a well-known instance.

This case is evidently found, when the plaitpoint curve C,C, of fig. 2
assumes the shape drawn in fig.3. We may namely imagine that
when the two curves C,C,and C,A approach each other, a deviation
from the straight course may be found on the left side of C,C,,
specially if 6, should not be =6,, by which the point C, would
therefore be shifted to the left, to the side of the small volumes. At
all events the anomaly of one of the two components can give
rise to the occurrence of this second principal type, as I showed in
a preceding paper.

From the shape of the different spinodal curves it is obvious that
from C, the temperatures first increase, as far as the point of contact
at R,. The temperature is then 7” (see fig. 3a). But between R, and
R,', where the plaitpoint curve is again touched by one of the
spinodal curves, the temperature decreases, and so also the pressure,
so that in the p, 7-diagram of fig.3a the line R, R,' runs back
again, as in fig. 1a the line &, A and in fig. 2a the line R, A, having
in this case two cusps in R, and L,’.

Here the points between A, and R,', and also those on C,R, and
C,R,' in the neighbourhood of R, and R,' can again not be realized,
and the consequence will be the occurrence of a three phase
equilibrium *).

As I already observed in one of my previous papers (l.c. p. 646),
after the two liquid phases 1 and 2 have coincided in the neigh-
bourhood of the point &’,, here too, separation of the two liquid
phases must take place again — provided the temperature be sufficiently
lowered — and this will take place in the neighbourhood of the
point, where one of the spinodal curves in FR, touches the plaitpoint
curve (, A. This is also represented in the p, 7-diagram of fig. 3a.

When comparing figs. 1, 2 and 8, we see clearly the connection
between the three principal types and their transition into each other.
The connection is given by the different course of the two plaitpoint
curves in figs. 1 and 2, which (see fig. 3) may pass continuously
into each other with changed circumstances of critical data of the
two components.

1) Cf. van per Waats, Continuitiit Il, p. 187, and These Proceedings V,
p. 307—11 Oct. 25, 1902.

( 49 )

Physics. — “Some remarks on Dr. Pu. Kounstamm’s last papers.”
By J. J. van Laar. (Communicated by Prof. H. A. LORENTZ).

1. With interest and full approval I read Dr. Kounstamm’s three
papers on the osmotic pressure *). From them it appeared to me that,
practically, he perfectly agreed with me. Only with regard to a few
points there are differences of opinion — only in appearance, however,
as I shall show in what follows.

On pages 723—729 |. c., namely, Kounstamm gives also a thermo-
dynamic derivation of the osmotic pressure, which seems to lead to
a somewhat different result from mine. He finds, namely, in the
numerator finally the quantity ae instead of v,. [I use here
my notation; v, is the molecular volume of the pure solvent (Konn-
STAMM'S v,), vz that of the solution, in which the dissolved substance
is present with a concentration x (K.’s v,)|. But here he overlooks
that according to his approximations v, may be written for the
latter. For on page 726 an integral is neglected, among others
on the strength of the fact that v,— 6 approaches to 0. He puts

db db

therefore v,—6, in consequence of which Og re nee pe =
Av x

= b—2(b, — b,) =6,. This however, is the value of 6 or v, when

i— 02 S0. 0), 3

So KonnstamM finds exactly the same thing as I found already in
1894 in a much simpler way. In my method no integral need be
split into three parts, and we need not neglect anything but the
compressibility of the liquid (which is of course also done by
KounstamM), so that my result (the compressibility excepted) is per-
feetly accurate, which cannot be said of that of Konnstamm.

2. The above mentioned method has been repeatedly published
by me. [Z. f. Ph. Ch. XV, 1894; Arch. Teyler (Théorie générale),
1898; Lehrbuch der math. Chemie, 1901; Arch. Teyler (Quelques
remarques sur la théorie des solutions non-diluées), 1903; and recently
in the “Chemisch Weekblad’, 1905, N°.9]. The derivation may
follow here once more.

If there is namely, equilibrium between the solution with the
concentration « under a pressure p, with the pure solvent with a
concentration O under the arbitrary pressure p, (e.g. that of the
saturated vapour, or of the atmosphere etc.), the molecular potentials

1) These Proc. VII, 723—751.

Proceedings Royal Acad. Amsterdam. Vol. VIII.

(50>)
of the solvent in the two liquid phases (separated by a semiperme-
able membrane only passable by the solvent) are the same. Hence:
BOSCH M)o a a oe Pee (t)
But evidently we bave the identity

P
Of,
(9, po) = (0; p) — Op Ip «
Po
Of,
Here a =v, (for meaning of v,, see §1). So we have also:
P
1 (020) = HOP) = f evap.
Po
If we now assume v, to be mdependent of the pressure — which
Konnstamm thinks perfectly permissible — we get:

U (0, Po) =H (0, p) — % (Pp — Po) -
Substituting this in (1), we get at once:

i
Dy p00 . . . . ° (2)
0

by which the osmotic pressure is immediately brought into connection
with the difference of the molecular potentials of the pure solvent
and of that in the solution, both under the same pressure p.

Now we can in the usual way replace uw, — u, by its value. We
find then, as has been frequently derived:

2
ae ERT Iogee

(1+re)? v,—),
in which the latter terms is often neglected, and @¢ and 7 have the
known meaning.

In this way the apparent deviation with regard to v, has been
disproved. My statement, therefore, that in the numerator for », no
correction term need be applied (see Konnstamm, p. 729), was by
no means “too absolute’.

ete.

— RT log (1 — 2)

3. When reading through Konnstamm’s paper, I was further
struck by the following in my opinion inaccurate assertions.

On p. 739 it says: “It appears from the explanation convincingly,
that van Laar goes too far, when he states, that we cannot speak
of osmotic pressure in an ?so/ated solution.”

I fully maintain this view. For in the kinetic explanation of

( 51 )

KounstamMm the osmotic pressure in an isolated solution is established,
only when he places semi-permeable walls or planes in it. But then
it is of course no isolated solution any more! What I demonstrate
is no more than this: Without semipermeable membrane no osmotic
pressure. And to this Komnstamm will certainly not have any objection,
witness the cited question of Pupin how it is possible, that e.g. a
CaCl,-solution of no less than 58 atm. could be held in a thin glass
vessel without bursting it! I do not see very well, what objection
KonnstamMM can have to my assertion. For this is the core of the
question, with regard to which he proves to be quite of my opinion
in another place (cf. p. 742).

4. What Konnstamm further observes on pages 742—4 with
regard to the idea “thermodynamic potential”, and what he says on
“palpable conceptions’ may be very well left undiscussed here. For
this is only a question of words, which does not affect the real
nature of the affair at all. Every one who works with the thermo-
dynamic potential, means with it the ¢-funetion of Grpps, which
perfectly determines the condition of equilibrium, as it must be
minimum in this case.

Finally I may only be allowed to point out that Dr. Kounstamm
has evidently misunderstood me, where he says that he thinks the
request to supply something “as a substitute’ for the osmotic pressure
and the kinetic conception of it less unreasonable than it seems
to me (p. 746).

I, namely, spoke of the osmotic pressure in an isolated solution.
And I very distinetly added: nothing can be put in the place for
what does not exist. And I wrote further, that the wsual (faulty)
kinetic conception of the osmotic pressure (i.e. where there are semi-
permeable membranes) must be replaced by a perfectly new kinetic
explanation, in which inter alia, the process of diffusion at the mem-
brane is put more into the foreground (Ch. Weekbl., 1905, N°. 9).

And where Konnstamm himself has made a very laudable attempt
in this direction (I. c. p. 729—741) to explain the osmotic pressure,
I have after all reasons for satisfaction, though he has wisely aban-
doned the idea of drawing up an equation for non-diluted solutions
in this way.

And as to the thermodynamic derivation, in this KoHNsTAMM has
been less fortunate in my opinion; where he has tried to substitute
for my perfectly exact, and yet so simple derivation an indirect,
elaborate derivation, the result of which on account of some neglections
cannot even lay claim to perfect accuracy.

( 52.)

Mathematics. — “On the rank of the section of two algebraic
surfaces.” By Dr. W. A. Vurstuys. (Communicated by Prof.

P. H. ScuHovure.

1. In this paper I intend to prove the relation new to me
r—=m,n, + m,n, — 2d — 3y,. . . - - - (A)

where 7 is the rank of the curve of intersection s of two algebraic
surfaces S, and S,, respectively of the degree m, and n, and of the
class m, and m, and possessing in d points an ordinary contact
and in y points a stationary contact. Some applications of this
formula are given too.

Formerly I proved’) the following extension of a well-known
formula *)

rn, n, (n, +n, — 2) — 2 (m,§,+-2, §,+d) — 3 (n,»,+7,r,+4), - (B)
where §,,&,,7,,%, represent the degrees of the nodal and cuspidal
curves. of the two surfaces S$, and §,. Formula (A) shall first be
proved for the case that S, and S, are developables. If we wish
to apply formula (4) to developables the numbers of double gener-
ating lines w, and w, must be added to the orders §, and &, of the
nodal curves and the numbers of stationary generating lines v, and
v, to the orders », and », of the cuspidal curves.

Formula (4) becomes

rn, n, (rn, + n, — 2) — 2 fn, (§, + @,) + 2, (§, + w,) + J} —
= {ny (v, ala Us) =F Ne (v, ote v,) ai x} oy Soe oes, (C)

2. Let A*S be the second polar surface of the degenerated surface
S,-+ 8, with respect to the arbitrary point P. This surface A*S
is of the degree (n,-+,—2) and meets the curve of intersection s
of S, and S,, this curve being of the degree n,n,, in nn, (n,+n,—2)
points.

These points of intersection are 1st the triple points of S,-+-S,
through which the curve s passes and 2°¢ the points of s for which
the tangent plane to one of the two surfaces passes through P.
The triple points of S,-+ 8, through which the curve s passes
are the points in which a double line of one of the two surfaces
meets the other surface. So these triple points are:

1st. The (x,», + »,»,) points in which a cuspidal curve of one of
the surfaces meets the other one. These points are cusps on the
curve of intersection s, they are indicated by Cremona as points 4

1) Verstuys, Mémoires de Liége, 3me serie, t. VI. Sur les nombres Pliickériens ete.
2) K. Pascat, Rep. di Mat. Sup. I, p- 325.

(53

and must count according to him for three points of intersection of
the nodal curve, thus here of the curve s with AS‘).

and, The (2,v,47,7,) points in which a stationary generating line
of one of the surfaces meets the other one. These points, also cusps
on the curve s, are indicated by Crrmona as points 7 which must
count according to him for three points of intersection of the nodal
curve s with A?S”*).

3, The (n,§, + 7,§,) intersections of S, or S, with the nodal
curve of the other surface. According to Cremona each of the branches
of the nodal curve meets A*S*) one time ina triple point t. Through
each of these bane t pass two branches of s, which is a nodal
curve on S,+5,; so each of these triple points counts for two
points of intersection of s with A*S.

4th, The (n,o, + 7”,@,) nodes of s in which a double generator
of one of the surfaces S, or S, meets the other one. According to
Cremona the nodal curve s meets A?S*) two times in such a
triple point *.

The surface S, +S, possesses still more triple points, among others
the cusps 6 of the cuspidal curves; these points do lie on A?S, but on
the curve of intersection s they do not; so they do not belong to
the points of intersection of s with A?’S,

3. Through P pass m, tangent planes of the surface S,. A gene-
rator of jS,, along which one of the m, tangent planes through P
touches /S,, meets 7, times the surface S,. Each of these points of
intersection is a point on s also situated on A’S. Such a point of
s and of A’*S counts for one point of intersection, according to
Cremona °). So A*S is met by the curve s in (mn, —-- m,n,) points
for which one of the tangent planes passes through P.

This gives the relation:

myn, (n, + n, — 2) = mn, + m,n, + 2 fn, (§, + @,) + 2, §, + wo} +
Gre ay Emmet ey) - . 2 « ()
Comparing the equations (C’) and (D) we get immediately
Pits, ap Wein, === Oy Oe ao ea ((4))
The degree of a developable being the rank of its cuspidal curve,
we can write for this formula:
r—— mr, + mr, — 2d — dy.
1) CremonaA—Courtze, Oberfliichen § 108. :
*) CremonA—Courrze, loc. cit. § 100.
8) Gremona—Currze, loc. cit. § 109.

4) Cremona—Currze, loc. cit. § 101.
°) Cremona—Curtze, loc. cit. § 99.

( 54 )

4. The formula (7) and hence also the formula (A), which is now
proved for the case that the two surfaces are developables holds still
good when S, and 8, are arbitrary algebraic surfaces. Let §, and »,
represent the degree of the total nodal curve and total cuspidal curve
of JS,, likewise & and », for S,. One of the formulae of Prickmr
applied to an arbitrary plane section of S, gives

mm

SS ee}
>= n? —n, —26,—387,,

or

== 2 ae Re fe peas
0=n,? —n, —m, — 26, — 37,.

In like manner an arbitrary plane section of S, gives

0=n,?—n, —m, —2§,—3r,

hence

O=n, (n,? —n, —m, — 2§, —3,) + n, (n,? — n, — m, — 2§, —3p,)
or

N,N, (2, + 2, — 2)=m,n, + min, + 2(n,§, +7, §,)+3(n,v, + n,v,)....(D)
combining the formulae ()') and (5) we get the formula (A).

If S, is a plane, n, becomes equal to unity and m, equal to nought,
whilst the curve s becomes a plane section and the rank 7 of s
passes into the class of the plane section. So formula (A) gives for
that class

r=m,—2d—3y,
which is indeed the class of a section of S, with a plane, having
with .S, in d points an ordinary contact and in x points a stationary
contact.

5. If S, is of the second degree and JS, of the degree » and
of the class m, the formula (A) gives for the rank of the curve
of intersection

r= 2(m+n)—2d—3y

if S, is a quadratic cone A? this formula will be proved directly
once more as follows for the sake of verification.

The rank of the curve of intersection s is the number of its
tangents meeting an arbitrary right line, e.g. a generator / of K?.
Each tangent of s, meeting the generator / has three points in
common with the cone /*, in fact the two consecutive pouits it
has in common with s and its point of intersection with /, unless
the latter comceides with the point of contact to s. Each right line
having three pomts in common with A? lies entirely on A?. The
only tangents of s meeting / are thus the generating lines of A’? which
are at the same time tangents of s and the tangents to s at its
points of intersection with /. The generator / of A? meets .S, and
therefore s too 7 times; through each of these points of intersection

( 55 )

pass two consecutive tangents of s. Whence already 2 tangents
of s meeting /.

Tangents of s, being at the same time generating lines of A?, pass
through the vertex 7’ of A* and, being tangents of s, are also tangents
of S,, and therefore situated on the tangent cone A of S,, having 7
for its vertex. Conversely every common generator of the two cones
Kk? and XK is a generator of A’ having with S,, thus also with s,
two coinciding points in common. A right line having with s two
coinciding points im common is either a tangent of s or it passes
through a double point of s. So the common gererators of the
cones A? and A are either tangents of s or they pass through double
points of s. The order of the tangent cone A’, being equal to the
class m of »S,, the number of common generators is 2m. The num-
ber of tangents of s meeting / in the vertex 7’ will be 2m, diminished
by a number still to be determined for the common generators
passing through a double point of s.

If A? has in a point d an ordinary contact with S, the common
tangent plane a in d is a tangent plane of S, passing through 7.
So a is also a tangent plane to the cone A along the line 7’'d.
So the two cones A* and A’ have along the common generator
7S a common tangent plane. The line 7’d must therefore count
for two common generators of the cones A* and A. A point d is
a node of s and with the exception of very particular cases the two
tangents of s in d will not coincide with 7’. So for every point
Jd the number of tangents of s passing through 7’ must be diminished
by two.

The following example proves that for every point x in which
S, and A’ have a stationary contact, the number of generators of
k? touching s must be diminished by three. Let S, also be a
quadratic surface and let the curve of intersection s be a not degene-
rated biquadratic curve F* with a cusp x. Then the line 7% counts
already at least for two common generators of the cones A? and
and is again not a tangent in y to sor R*. If now Ty were to
count only for two common generators the cones A? and AK would
have two more generators in common. These latter two cannot be
two consecutive generators, for in that case * would have two
double points and so it would have to break up. Now it is easy to
see that these two remaining generators are tangents to #* or s at
points for which the osculating plane is a stationary plane. So L'
would have to possess two stationary planes « whilst a £* with cusp
possesses but one stationary plane a‘). The right line 7% must
a » E PascaL, loc, cil. p. 363.

(56 )

therefore count for three common generators of A? and K. The
number of tangents of s meeting the line /, thus the rank of s is
consequently

r= 2n + 2m — 2S — By.

6. The reciprocal polar figure s' of the curve of intersection s of
kK? and S, is a developable circumscribed to a conic c* and to a
surface S' of order m and of class n, whilst the conic c? touches
J times the surface SS’ and osculates it ~ times. If we take for the
conic c? the imaginary circle at infinity the developable s' becomes
the developable focal surface of S'. The rank of s' is the same as
that of s. So we find the theorem:

The rank of the focal developable of a surface of order m and of
class n touching the imaginary circle at infinity d times and osculating
at y times is

r= 2m + 2n — 2d — By.

If S, is a developable the point of contact of a common tangent
plane that is an ordinary plane of S, is always a node of s’).
The developables A? and S, will only then have a_ stationary
contact in a point x, when the common tangent plane is a stationary
plane @ of S,. The line 7y counts thus for four lines of intersection
of the cone A? with the tangent cone A which breaks up into
m planes. It is easy to see that now the line 7y is at the same
time tangent to s at the special cusp x which is a singularity of
order two, of rank unity and of class three *). So a stationary contact
y% gives rise to four lines of intersection of A? with K of which
only one is an ordinary tangent of s lying on A’*. Each point x now
also diminishes the rank of s by three. The reciprocal polar figure
of S, is a curve S' of order m and of class”. Each common tangent
plane of A? and SS, is transformed in a common point of c? and
S'. If the common plane is a stationary plane @ of S, the common
point is a cusp on the curve S'. So we find the theorem:

The rank of the focal developable of a plane curve or a twisted
curve of the degree m and of the class n and of which 6 ordinary
points and y cusps lie on the imaginary circle at mfinity ts

r= 2m + 2n — 2S — 34.
If S’, and S', are the reciprocal polar figures of the surfaces
and JS', are respectively of the degree m, and

fod

(fo
and 3S,,- then |S’

S

2 |
m, and of the class n, and n,.

1

') Verstuys, Mémoires de Liége, 3me série t. VI. loc. cit.
?) HatpHen, Bull. de la Soe. Mat. de France, t. VI, p. 10.

(57)

If the surfaces S, and S, have an ordinary contact in d points,
the common tangent planes in these d points are ordinary double
tangent planes of the developable D circumscribing S, and S,'). The
surfaces S', and JS’, will also have in d points an ordinary contact.

If the surfaces S, and S, have in x points a stationary contact
the tangent planes in these y points are stationary tangent planes
of the developable D’). The surfaces S', and S', have thus also
in x points a stationary contact.

So the rank of the curve of intersection d' of the surfaces S', and
S', is according to formula (A), just as the rank of the curve s,

r= m,n, + mn, — 2d — 3y.

The curve d' being the reciprocal polar figure of the cireum-
scribing developable D, the rank of D is equal to the rank of d.
Whence the theorem:

For two arbitrary algebraic surfaces the rank of the curve of
mtersection is equal to the rank of the circumscribing developable.

Here we have supposed that the points of contact d and y are
ordinary points on both surfaces and the tangent planes ordinary
tangent planes in these points*).

1) Verstuys, Mém. de Liege. 3™¢ série t. VI. De l’influence d’un contact ete.
*) Verstuys, loc. cit.
8) Verstuys, loc. cit.

(June 21, 1905).

KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM.
PROCEEDINGS OF THE MEETING
of Saturday June 24, 1905.

DCG

(Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige
Afdeeling van Zaterdag 24 Juni 1905, Dl. XIV).

ClOeN Bates Ny sa Ss

H. Zwaarpemaker: “On the pressure of sound in Corti’s organ”, p. 60.

H. W. Baxkuvis Roozesoom and J. Ou Jr.: “The solubilities of the isomeric chromic
chlorides”, p. 66.

J. J. Buayksma: “Nitration of symmetric nitrometaxylene”, (Communicated by Prof. A. F.
HoLurMman), p. 70.

H. Kamerzinent Onnes: J. “Improvement to the open mercury manometer of reduced height
with transference of pressure by means of compressed gas”, p.75. IT. “Improvement in the trans—
ference of pressure by compressed gas especially for the determination of isothermals”, p. 76.
(With one plate).

H. Kameruincn Onnes: “Methods and apparatus used in the cryogenic laboratory. VII. A
modified cryostat, p. 77. (With one plate). VIIL Cryostat with liquid oxygen for temperatures
below — 210°C. p. 79. (With one plate). IX. The purifying of gases by cooling combined
with compression, especially the preparing of pure hydrogen”, p. 82.

L. Bork: “On the development of the Cerebellum in Man” (2nd part), p. 85. (With one plate).

A. J. P. van pe Broek: “On the sympathetic nervous system in Monotremes”. (Communicated
by Prof L. Bork), p. 91. (With one plate).

H. G. Jonker: “Some observations on the geological structure and origin of the Hondsrug”.
(Communicated by Prof. K. Martin), p. 96.

J. Weeper: “Approximate formulae of a high degree of accuracy for the ratio of the triangles
in the determination of an elliptic orbit from three observations”. If. (Communicated by Prof.
Hi. G. van pe Sanpe Bakuuyzen), p. 164.

J. A. C, OupEMANS: “Supplement to the account of the determination of the longitude of
St. Denis (Island of Réunion), executed in 1874, containing also a general account of the
observation of the transit of Venus”, p. 110.

F. M. Jarcer: “On some derivatives of Phenylearbamic acid”. (Communicated by Prof.
Pp. van Rompurcu), p. 127.

P. van Rompurcu: “On the presence of lupeol in some kinds of gutta-percha’, p. 137.

P. vay Rompurcu: “On the action of «ammonia and amines on allyl formate”, p, 138.

A. J. Urren: “On the action of hydrocyanie acid on ketones”. (Communicated by Prof.
P. van Romevren), p. 141.

J. J. van Laary “The molecular rise of the lower critical temperature of a binary mixture
of normal components”. (Communicated by Prof. H. A. Lorenrz), p. 144.

F. M. Jarcer and J. J. Buanksma: “On the six isomeric tribromoxylenes”. (Communicated
by Prof A. F. Horzeman), p. 153.

C. Easton: “Oscillations of the solar activity and the climate”. (2nd Communication). (Com-
municated by Prof. C. H. Winp), p. 155.

F. H. Evpman Jr.: “On colorimetry and a colorimetric method for determining the dissociation
constant of acids”. (Communicated by Prof. S. HooGrewrrrr), p. 166.

W. A. Verstuys: “On the number of common tangents of a curve and a surface”. (Com—
municated by Prof. D. J. Korrewra), p. 176.

J. D. van pER Waats: “The shape of the sections of the surface of saturation normal to the
x-axis, in case of a three phase pressure between two temperatures”, p, 184. (With 4 plates).

J. D. van per Waars: “The Y\x-equilibria of solid and fluid phases for variable values of the
pressure”, p. 193. (With one plate).

A. Smits: “On the hidden equilibria in the p-«-diagram of a binary system in consequence
of the appearance of solid substances”. (Communicated by Prof J. D. van per Waats), p. 196.
(With one plate).

A. Suits: “Contribution to the knowledge of the px- and the p7-lines for the case that two
substances enter into a combination which is dissociated in the liquid and the gasphase”.
(Communicated by Prof. J. D. van per Waats), p. 200. (With 2 plates).

The following papers were read :

Proceedings Royal Acad. Amsterdam. Vol. VIIL.

( 60 )

Physiology. — “On the pressure of sound in Corti’s organ”. By
Prof. H. ZWAARDEMAKER.

(Communicated in the meeting of May 27, 1905).

According to the hypothesis of Hetmuonrz-Hensren the vibrations
of sound, penetrating into the inner ear by way of the stapes,
evoke a resonance in the transversely stretched fibres of the mem-
brana basilaris. Strong vibrations are imparted to different fibres
according to the pitch; these vibrations being communicated to the
sensory epithelia of Corti’s organ and then becoming the stimulus
for definite nerve-fibres. We recognise the tone by the nerve-tibres
which are affected in this way.

That such short fibres as the transverse fibres of the membrana
basilaris can resound to the relatively deep tones of the human
seale is explained by Hetmnontz 1st by the resistance in the fluid
and in the soft cell-masses (Ciausius’ cells); 2°¢, by their being loaded
with Corti’s arches on which again a whole system of cells rests.

At first it was imagined that the fibre vibrates in its entire length
as a freely stretched string. Later attention has been drawn to the
fact that the pars arcuata (the part over which the Corti’s arches
vault themselves) remains largely at rest while the pars pectinata
(the remaining part of the string, not covered by the arches) makes
the greatest excursions. But then the difference in length of the
fibres is no longer sufficient to explain the difference in the pitch
for which they are tuned, so that also a difference in tension and in
load must be assumed. ') Examining the proportions of microscopical
preparations and bearing in mind that the arches are more or less
rigid formations, one is soon convinced that the pars arcuata cannot
possibly resound to the deep tones of the audible scale. It is not
the bottom cells on their upper face that are an impediment to this,
but the large vein on their lower surface. Moreover the transverse
fibrous structure, which is so distinct in the pars pectinata is entirely
absent in the pars arcuata. The property of resounding may on
sufficient grounds be attributed to the stretched and loaded fibres of
the pars pectinata only.

I have tried, as far as this is possible, to reproduce in a model
the conditions prevailing in Corti’s organ. A horizontal steel string,
‘/, millimetre thick and somewhat longer than a metre, represents
a transversely stretched fibre of the membrana basilaris. On this
rests, at one of the ends, a wooden imitation of Corti’s arches. The

1) A. A. Gray, Journal of anat. and physiol. 1900, Vol. 34. p. 324.

( 61 )

other end is fastened, transversely to the direction of vibration, to
the vibrating prong of an electrically driven tuning fork. Now, when
the Corti’s arches are sufficiently loaded (with sponges, or, for
demonstrations, with a hollowed little board on which a drawing
has been stuck), the tension of the string being at the same time
regulated by means of a micrometer screw, it is possible to cause
the system to resound to the tuning fork so that with small deflee-
tions of the fork the deflections of the pars pectinata become very
large. —

In the experiments for study proper, if deserves recommendation
to attach pins to the wooden Corti’s arches, on which smaller and
larger sponges can be stuck in various positions. As long as the spon-
ges are dry the whole system partakes in the vibrations. But when
water is dropped on them, which is sucked in by the sponges and
makes them heavier, the damping system is brought to rest and a
node is formed at the base of the outer pillar. The string can be
prevented from sinking down too much, by attaching the fixed extre-
mity of the Corti’s arch to a spring, which keeps it up. The free
extremity of the arch is placed loose on the string. Sometimes it is
a little difficult to obtain only vertical movements of the string,
but by moving the fixed point of support of the string forward or
backward, one always succeeds in this.

We find then:

1. broad deflections of the pars pectinata.
2. immovability of the pars arcuata.

3. immovability of the Corti’s arches.
4. immovability of the loading mass.

This immovability is not absolute, of course; on the contrary, the
floor, the table, everything in the room vibrates under the influence
of the tuning fork, but the movements are infinitely small compared
with the excursions of the pars pectinata and are so insignificant,
moreover, that a photograph of the parts, called immovable, shows
absolutely sharp definition. On the same photograph the pars pectinata
is seen in the extreme positions, which it reaches with broad amplitude.

The conditions of the model have purposely been so chosen that
they correspond in general outlines to the conditions actually found
in Corti’s organ. A complete imitation is impossible, but within the
limits of technical practicability we have reached here, without any
preconceived opinion, what can be achieved with the ordinary means
of the laboratory. Now if we may see in the described model a
more or less happy imitation of reality and to this assumption we
are especially entitled by the manner of loading, then it follows that

5*

( 62 )

also in the organ itself as well the Corti’s arches as the loading cells
remain at rest. But then we must drop all the ideas, which have been
broadly developed during a long time, about the impact of the ciliae
of the hair-cells on the membrana tectoria, on the bending of the
ciliae, ete. Rest prevails in the system of arches and a covibration
of them is necessarily excluded.

Yet the imparting of stimuli, which was supposed to be explained
by the co-vibration of the hairs, need not remain a mystery, if
attention be paid to the effect of sound-pressure.

In a paper, entitled “the pressure of vibrations”, Lord RayinicH *)
has treated a simple case, which is nearly identical with ours. It is
the case of a string, itself infinite, but vibrating between two rings,
one fixed, the other sliding. When the string vibrates the sliding ring
is pressed outward, towards the extremity, with an average force

E : ‘ : F ;
f=: E being the energy of the vibration, / the length of

the string.

The base of the outer pillar is in the case of the sliding ring.
According to Rerzius the pillar is one with the semi-solid cell-mass
of the bottom-cell; from this cell it would originate and form a
whole with it. In this way at the same time an attachment and a
small movability in the cell-mass have been obtained.

But the pillar is not only in juxtaposition with the fibre, but also
presses on it by the inertia of the large cell-masses with which it
is connected, as soon as the fibre begins to execute movements. Hence
the vibrating fibre will in this place present a node and the load
itself will necessarily have a great influence on the conditions of
tension during the vibration.

So the pillar has a double function: 1. that of the movable ring
of Rayneiu, 2. that of carrier of the inertia of a damping and
loading mass. In its first quality it receives a pressure in the direc-
tion of the modiolus, a pressure which can be perfectly measured
by Rayieren’s formula.

In the model this pressure can even be demonstrated. For this
purpose the pillars were removed and the base of the outer pillar,
which imparts a node to the string, was replaced by a brass lamella,
provided with a slit. The split lamella grips the string like a miniature
fork. In this way the node is preserved. As the lamella is 19,5 em.
long and 0,1 em. thick, it possesses a certain mass, which does not
press on the string since the lamella is placed normally to it, but
gives a distinct damping as soon as the string vibrates.

1) Lord Rayueien, Philosoph. Magazine (6) UL 1902, p. 339.

(63 )

Besides, at its place of attachment the lamella has been made
considerably thinner (thickness 0,02 em.) and consequently flexible
over a length of 6 centimetres. The result is that the lamella, although
accurately placed in the node of the vibrating string, will slightly
deviate outward as soon as the excursions have become large enough.
The force by which the little fork is driven outward is undoubtedly
extremely small. Accordingly the deviation did not exceed 3 mm.
with a semi-amplitude of the string of 0,4 em. The new position
can easily be fixed photographically and be compared with the
position of rest which is assumed as soon as the string stops vibrating.
This renders it possible to measure the force. But from a physiolo-
gical point of view it has no meaning to perform the actual measu-
rement on the model although it would be important if if could be
performed under the actual conditions, for this pressure must be the
immediate cause of hearing. This will be easily perceived with
respect to the sensory elements at the modiolus side of the pillars.

The pressure of sound acting at the base of these outer pillars is
in the direction of the string and hence of the modiolus. It has a
component in the direction of the pillar itself. Through this the outer
pillar, the upper end of which presses loosely against the capitulum
of the inner pillar, is displaced parallel to itself and the cells at the
modiolus side of the system must necessarily be compressed, although
slightly. The pressure which they experience is either entirely con-
tinuous or periodically feebly variable. Beginning at the foot of the
pillar the pressure varies from a maximum at the extreme deflection
of the string to zero in the position of equilibrium. Higher up in
the system these differences will probably for the greater part have
disappeared, though they may remain to some extent. The pressure,
however, is at all times positive; it never becomes negative, as would
be the case if the Corti’s arches and the loading cells followed the
vibrations of the string. Since they are at rest, the pressure met with
in the sensory cells at the modiolus side of the inner pillar must
always act in the same sense, which is in the direction of the modiolus.
It is quite possible that also the hairs of the hair-cells experience
its influence, the effect of which will also be in one direction.

The matter is somewhat less simple for the sensory elements situated
at the inner side of the outer pillar. These appear to me to experience
no pressure at all from the outer pillar, which is retained in the soft
cell-mass of the bottom cell. On the other hand such a pressure is
present from the side of Hunsrn’s cells and also to some extent from
the side of the supporting cells.

We are at liberty to consider this cell-group, situated at the exterior

( 64 )

of the directly sensory elements, also as a Rayneien ring. We shall
have to try this the sooner, since in birds the pillars are absent and
so we cannot regard these formations as essential. If we try again
to find in Corti’s organ an analogon of Ray.eien’s movable ring,
and in abstracto it is always admissible to seek such an analogy,
we may never restrict ourselves to the arches alone. For by doing
this we should deny the essential meaning of analogy for the physiology
of hearing.

So Hensen’s cells may also be regarded as a movable RayY.eicu
ring. They also rest with a relatively narrow foot on the fibres of
the membrana basilaris, near the foot of the pillar, when the human
organ of hearing is studied. They will also exert a damping and
loading influence on the vibrating fibres by their inertia. They will
also cause a relative node and be shifted laterally, in the direction
of the modiolus, by the vibration. But if this is the case they also
squeeze the sensory elements situated between them and the pillar *).

Beside this lateral pressure, experienced by the cells themselves,
it is not entirely impossible that also the hairs experience a pressure
which they now receive through the agency of the lamina reticularis,
which forms a whole with the capitula of the pillars. This pressure
will then press them against the membrana tectoria with a some-
what varying force, but which is always in the positive direction.

All these reasonings can be simpler for the ear of birds than for
that of man. The pillars are there absent and only the sensory elements
and the supporting cells are found. Also this whole lies laterally on
the fibres of the membrana basilaris and must experience a lateral
pressure of sound.

The here developed conception, which deviates from the current
one, has the important advantage that it reduces hearing to the
perception of a pressure. The mechanical action of the vibration,
which in the old form of the theory of HriMnorrz-HEnsen is vibra-
tory, intermittently positive and negative, now becomes a permanent
pressure of somewhat varying strength, to be sure, but at all times
in the same direction, always positive. Hearing becomes the exact
analogon of touching and all experience gathered for this latter sense
we may try to find again mutatis mutandis, in the physiology of
hearing.

Also small secondary advantages are gained by the new conception.

In the first place the simple juxtaposition of the heads of the

1) For points inward of the node it can be shown in an elementary way that
the masses there present and situated unilaterally, continually experience impulses
having a permanent component in the direction of the node.

( 65 )

pillars (showing no joint like the auditory bones) finds an explana-
tion. For a pressure which is always positive this is sufficient, not
for a vibration. In the second place it explains the varying shapes
and aspects presented by the membrana basilaris in the preparations.
These are very obscure when they concern an integrating part of the
organ, but are explained very easily if what we see in the prepa-
rations, is only a coagulated colloid or elastic mass.

Finally our conception is by no means bound to the theory of
Het_mnortz-Hensen. It is also acceptable to those who would exchange
this theory for that of Ewanp. For Lord Rayieten treats in his paper
also the case of a vibrating membrane: “but a membrane with a
flexible and extensible boundary capable of slipping along the surface,
provides for two dimensions. If the vibrations be equally distributed
in the plane, the foree outward per unit length of contour will be
measured by one-half of the superficial density of the total energy”.

So the theory of the pressure of sound might also be applied
to a membrane such as is imagined by J. R. Ewanp. but his mem-
brane does not answer the conditions mentioned by Ray.eicn, so
that the quantitative relations are not so easily perceived as in the
above developed case.

Finally, concerning the modern theories of bearing which I would
call the pulsatory ones, since they only take into account the bul-
gings of the membrana basilaris, caused hy the piston movement of
the stapes, the hypothesis of the pressure of sound cannot be applied.
For these theories purposely neglect the vibratory movements of the
smallest parts and only take imto account the mass-result. If however
we lose sight of what is the essential thing in a vibration, we also
lose the right of applying the properties of a vibration. In my
opinion there can then be no question of pressure of sound.

The reader will have perceived that the starting-point of our reasoning
was the probability of the fact that the arcuate zone and the arches
vaulting over it remain perfectly at rest. On anatomical grounds this
is very probable. Should it appear later that this rest is not absolute
but only relative, the preceding reasoning is none the less valid.

Only one objection could then be raised, namely the small amount
of the pressure of sound. This would then have to be placed
against another small value, that of the possible movement of the
hair-cells. Hence the question would be a quantitative one. But also
in this ease the two forees, the pressing foree and the thrusting
force, would by no means preclude each other. They would both have
to be present. For the present we prefer, by assuming immovability,
to neglect the thrusting force and only to retain the pressing force,

( 66 )

Chemistry. «The solubilities of the isomeric chromic chlorides’.
By Prof. H. W. Bakutis Roozesoom and J. O1m Jr.

(Communicated in the meeting of May 27, 1905),

At the December meeting 1908, a communication was made by
Bakuuts Rooznpoom and ATEN as to the changes in form which may
occur in the solubility-lines of binary mixtures in dependence on the
quantities of the molecules of a compound which may be formed
from the components in the liquid mixtures. This subject is only
a part of a more extensive problem embracing the equilibria of
phases in systems composed of three kinds of bodies between
which a transformation is possible in liquid (or vapour). If that
transformation takes place with greater velocity than the setting
in of the equilibria of phases, the system will appear externally as
a binary one, although if is in reality ternary, and in order to
explain the course of the equilibria of phases we must take into
account that ternary nature.

In those cases where the third kind of molecules consists of a
combination of the two others no instanee has, as yet, been noticed
where a correct view could be formed with certainty as to the inner
composition of the liquid phase.

We, however, came across an example where this is quite possible,
namely in a case where two isomeric substances may be converted
into each other by dissolving in a third substance. Similar cases may
frequently oceur with all kinds of orgame isomers; but apart from
the fact that their behaviour has been little investigated from the
point of view of the equilibria of phases we often lack the means
to determine the two kinds in solution. That possibility, however,
presented itself with the isomeric chromic chlorides, which not
only may be determined in each others’ presence, but also require
when in solution much more time to reach an equilibrium than is
necessary to reach the equilibrium between solid matter and solution.
This rendered it also possible to study the change of the solubility
as a function of the progressive transformation in the solution.
Finally, this research could also serve to elucidate the cause of the
stability or unstability of the isomers, and the most rational method
of preparing the same from the solution.

It has lone been known that all kinds of salts of trivalent chromium
when in solution undergo molecular transformations depending on
temperature and concentration which are shown by the change in
colour of the solutions, which may vary from green to violet.

Only of late this matter has been better understood when various

( 67 )

modifications of the same salt were successfully isolated in a solid
condition.

In the case of chromic chloride two compounds were found to
exist at the ordinary temperature with 6 H,O. In connection with
his theory on complex compounds, Werner proposed the following
structural formulae :

Cl,
(HEO}

2

(Cr (H,O),} Cl, and Jor i Cl. 2 HO:

The first salt is violet, the second one green. In the first salt the
three chlorine atoms should be capable of ionisation; in the latter
only one. If only these are precipitable by silver solutions ') the
amount of each salt in a mixed solution may thus be quantitatively
determined.

First of all measurements were made at 25° as to the velocity of
the transformation of solutions with different contents of chromic
chloride and as to the final condition which they attain.

A

The result of these last investigations is indicated in the Figure
by the line AGH.
In this figure A stands for the solvent H,O, B for the green

1) We found this not to be absolutely correct but the precipitable chlorine could
n any case be used as a measure for the two salts,

( 68 )

and C for the violet chromic chloride. Both are taken in the
calewlation as hydrates with 6 H,O so that the sum of H,O and the
two hydrates is always taken as 100 (percentage by weight).

The line AGH first runs close to the axis AC. This means that
in weak solutions a final condition is reached in which the chromic
chloride occurs nearly exclusively in the violet modification.

Strietly speaking this means in the condition in which the violet
chromie chloride finds itself the moment it has dissolved. Briefly,
we will eall this the violet condition. If, therefore, we make a
solution of the green chloride of the same concentration the green
chloride will be almost completely changed into the violet salt. This
process proceeds slowly enough to admit of its course being studied,
also to show that both green and violet lead to the same final condition.

If the amount of hydrated chromic chloride exceeds 20 °/, the line
AlG begins to run perceptibly upwards and consequently the final
condition in the solution shifts more and more towards green.

In the point / the final equilibrium is situated near an equal
amount of green and violet. This corresponds with a total of 65 °/, of
chloride’) of which 32.5°/, is green and 32.5 °/, violet.

It will be noticed that we cannot go further than G because the
solution there reaches its saturation. If crystallisation did not take
place the prolongation of the line A/G could be determined. If
this may be represented by GH, the terminal point AH would
indicate the amount of green and of violet chloride in liquid hydrate
of chromic chloride (without excess of water); this point would
therefore lie at about 15°/, violet and 85°/, green. Its determination
is however impossible as the green hydrate melts at 83° and the
violet one at 92°. Although the melted hydrates crystallise very
slowly still it is difficult to keep them liquid down to 25°.

The final condition of solutions of different concentrations thus
being known, the solubility of the two-hydrates at 25° was studied.
The saturation was very soon accomplished, 0) and / represent
the concentrations of freshly prepared saturated solutions of green
and violet chloride.

These, however, soon undergo a modification. In the green solution
violet chloride is formed and conversely. This causes a change in
the solubility which runs along the lines Di’ and HF respectively.
These show that the total solubility of both green and violet increases
as the transformation of green into violet or the reverse proceeds in
the solution.

1) This total amount may be read off on AC or AB if we draw from J a line
parallel with BC.

The solutions of the green chloride do not however run further
than G, where the solution saturated with green also attains the
inner composition corresponding with the equilibrium at the total
concentration. Solutions on G# could only be made by rapidly
dissolving a mixture of green and violet in the desired proportion
and then introducing some solid green chloride. These solutions
would then, however, recede towards G as the point of final equili-
brium of the liquid saturated with green chloride.

The solutions saturated with violet chlorid run along the line LF.
The solution / might be at the same time in equilibrium with green
chloride, but as soon as this occurred the violet would be completely
converted into the green and then the solution containing the green
would again shift to G as a terminal point.

As the line of equilibrium A G H intersects the solubility line for the
green but not that for the violet chlorid, the latter cannot be definitely
in equilibrium at 25° with any solution, consequently at this tempe-
rature the green chloride is the only stable one. Even outside the
solution the violet changes, therefore, after a lapse of time, into the
green; in contact with the solution this takes place more rapidly.
This is the reason why the line //’ cannot always be followed up.

The question now arises how it is possible to separate violet
chloride in the solid condition. This is done by leading gaseous HC]
into solutions containing at most 30°/, of green chloride and which
have been recently heated to 100°.

Addition of HCl at 25° diminishes in a high degree the solubility
of both chlorides.

The two lines DF and EF are shifted towards the left about
parallel to their original positions and about to the same extent. It
will be easily seen that the point of intersection G will also move
towards the left and might finally arrive in the liquid region to the left
of the equilibrium line. In that case this line would no longer
intersect Di but HL; a saturated solution of violet chloride would
then be in inner equilibrium and the violet chloride could be separated
in a stable condition.

This, however, is not the case, because the line AG also moves
strongly towards the left on addition of HCl and consequently the
equilibrium in the solution shifts towards the green side. The investi-
gation showed that the violet chloride is still metastable in contact
with the solutions rich in HCl; the point of intersection /’ therefore
remains, obviously, to the right of A/G even on additon of HCl.

If, however, we heat to 100° before leading HCl, the line A/G
moves very considerably towards the violet side so that it now

( 70 )

intersects the solubility isotherm of the violet chloride at 25°. The
receding of the solutions towards the green, on cooling, now proceeds
with sufficient slowness to enable us to precipitate the violet chloride
at 25° by means of a current of HCl, which diminishes its solubility.

ae

Chemistry. — ‘“Nitration of symmetric nitrometarylene.” By Dr.
J. J. Buanksma. (Communicated by Prof. A. F. HoLiemay).

(Communicated in the meeiing of May 27, 1905).

If symmetric dinitrophenol cr symmetric dinitromethylaniline is
treated with mixed nitric and sulphuric acids, pentanitrophenol or
pentanitrophenyl-methylnitramine is formed *). Consequently the pre-
sence of the two nitro-groups, which are in the meta-position with
regard to the OH or NHCH, group does not prevent the introduction
of another three nitro-groups in the para-position and ortho-positions
in the benzene core. Symmetric dinitroanisol and phenetol yield,
however, on nitration tetranitroanisol. or tetranitrophenetol *); the
hydrogen atom in the para-position with regard to the oxyalkyl group
is not replaced by NO, here. As the methyl group on substitution
in the benzene core behaves in some respects analogous to the
OH and NH, (or NH CH,) groups it seemed of importance to inves-
tigate the conduct of symmetric dinitrotoluene on nitration in order
to ascertain what influence is exercised here by the NO,-groups
in the meta position.

The symmetric dinitrotoluene was, therefore, heated with mixed
nitric and sulphuric acid for two hours on the waterbath; the sub-
stance had not, however, undergone any change. The presence of
the nitro-groups in the mefa-position with regard to the CH, group
consequently prevents the further introduction of nitvo-groups in the
positions 2, 4 and 6. If, however, one of the NO,-groups in sym-
metric dinitrotoluene is replaced by bromine, this substance may be
successfully nitrated. Symmetric bromonitrotoluene yields on treat-
ment with mixed nitric and sulphuric acids three isomeric trinitrobro-
motoluenes which it is, however, difficult to isolate.

The question now arose what result is obtained when one of the
NO,-groups of symmetrie dinitrotoluene is replaced by CH,, in other
words what is the behaviour of symmetric nitro-m-xylene on nitra-
tion? For it is known that m-xylene readily yields 2-4-6-trinitro-

1) Recueil 214, 254.

)

*) 7 23, 111; 24, 40.

Cu)

m-xylene. As in the symmetric nitro-m-xylene the NO,-group is
placed in the meta-position with regard to the CH,-groups it did
not seem impossible that this NO,-group (like the NO,-groups in
symmetric dinitrophenol and symmetric dinitromethylaniline) would
not prevent the further introduction of the NO,-groups in the positions
2, 4 and 6, so that we ought to arrive at tetranitro-m-xylene.
On the other hand the nitration of m-nitrotoluene’) and symmetric
dinitrotoluene gave reason for believing that not four but at most
three nitro-groups would be introduced.

The symmetric nitrometaxylene was prepared according to Wro-
BLEWSKI's directions *).

Two grams of this substance were treated for twenty minutes on
the waterbath with mixed nitric and sulphuric acids; on cooling,
long colorless needles were deposited. These were collected at the
pump on glasswool and recrystallised from alcohol when long, color-
less needles or rods were obtained, m. p. 125°.

The motherliquor of the acid solution was poured into water
which caused a white flocculent precipitate. By reerystallising the
product from alcohol fine four-sided crystals mixed with a few
needles were obtained; the crystals melted at 90° and the needles
at 125°. These crystals could be separated by recrystallisation from
alcohol. From 2 grams of 5-nitro-m-xylene were obtained about 2
grams of the product melting at 125° and 0.5 gram of the produet
melting at 90°. The analysis showed that both substanees had the
composition of trinitroxylene.

In the trinitro-m-xylene (m. p. 182°) prepared by nitration of
m-xylene the nitro-groups occupy the positions 2, 4and 6. We had,
therefore obtained the two as yet unknown trinitro-m-xylenes.

CH; CH, CH,
VEN = NO, va NO, PF NO;
HNO x
| | HesOmamed [125° | and | 90° |
NOz \ 7 CHs se NO, \_// Cis NO, \ fous
NO,

The constitution was determined according to the subjoined scheme.

CH, CH, CH, CH,

MORAN with -NOs/ \ with NOs 7 \(BEOlwith NO; 7 \ Br
125° | NE, py | 85° | Br—> y |146°| HNO; no, |103° |

NO, cu, “He teu, \ / ols cHN\ CK cH, NO

NO, NO, NO, NO,

1) Hepp. Ann. 215. 366.

2) Recently, Witteeropr (Ber. 38. 1473) has described more fully the preparation
of symmetric nitro-m-xylene. I had then already made this preparation according
to Wrostewski's directions. (Ann. der Chem. 207. 94).

(a2)

CH, CH, CU,
NO.“ \NO, NO, 7 XRO; NO, \.NO,
| 90° | ae —> i ~ 197° | Br— =| 175°
NO:\ // CHs ios CIs CH; CH, S ola —S
ey wy \Z
Cy Cis CU, CH, CH,
wa ee, SON _, HINO NO, \.NO. NOz 7 \NOs NO, Z \NO,
| | meyer | 56°| uso, [183° Sy AOC eee NOs a [152° |
\ fella NOs\ 7 CHa NO Ae Cis” \/ CHa
Nil. Br Br
CH, CH, CH, CH,
YER Va NO.,/ \NO, NO, \NO, Zi!
[482 NANO me 832) | nee ioe
Br CH, Br CH, Br CH, CH,‘ / CMs
Ne NZ \Z SY

We therefore see that the substance melting at 90° is 2-5-6-trinitro-
m-xylene; the compound melting at 125° must therefore be 4-5-6-
trinitro-m-xy lene.

Both trinitroxylenes contain a moveable NO,-group ; in the compound
m.p. 90° the NO,-group 5 is under the influence of an ortho- and
a para-placed NO,-group, whilst in the compound melting at 125°,
the NO,-group 5) stands between two ortho-placed groups. By the
action of alcoholic ammonia or methylamine these groups are readily
substituted by NH, or NHCH,.

Of special interest is also the formation of 4-5-6-trinitro-m-xylene
from 4-iodo-5-nitro-m-xylene.

While 4-bromo-5-nitro-m-xylene (see above scheme) and 4-chloro-
5-nitro-m-xylene readily yield trinitrochloro(bromo)metaxylene on
treatment with a mixture of nitric acid (sp. gr. 1.52) and sulphuric
acid, 4-iodo-5-nitro-m-xylene yields 4-5-6-trinitroxylene with elimina-
tion of the iodine atom from the benzene core :

CH, CH,
VeN NO./ \NO>
| 51° | = [165° |
ls Z NO //CH; NOX //Clh
os a or
|
NOX /Cll; CH, CH,
‘ia, Sea w0,/\
| 405° | = 258
BO Ae NON ors
NO,

After this had been observed, the 4-iodo-6-nitro-m-xylene was also
nitrated. Here the iodine atom was also replaced by NO, and 2-4-6-
trinitro-m-xylene was formed m.p. 182°.

(73) )

CH; CH, CH,
NO,/ ROY AS NO: \NO,
128° | — | 86° | = 182° |
SoU \ fous Sods

: I

NU NOs
4-chloro(bromo)6-nitro-m.xylene yields, however, on subsequent
nitration 4-chloro(bromo)2-6-dinitro-m.xylene '). This made two cases
in which an iodine atom was replaced by a NO,-group whilst it
was shown at the same time that Br and Cl were not displaced by
the NO,group under the same circumstances ’).
If we compare the melting points of the three trinitro-m-xylenes
CHs CH; CH;

NO, / NO, NOLAN NOs SX NO,
182° | 1250 | 90° |
Ne is NO, Sf ous NO3\. 7 Cis

NO, NO,

we notice that these are situated the higher the more symmetrical
the position of the nitro-groups in the core is; this consequently
agrees with the rule that the melting point of isomeric substances
is generally the more elevated the more symmetrically constructed
the molecule is.

It was also found that the higher the melting point of the three
isomers is situated the less their solubility in aleohol becomes *).

Dr. Jancer was kind enough to examine the crystals of the three
trinitro-compounds and communicated to me his results.

2-5-6-Trinitro-1-3-vylene t = 90° erystallised from aleohoi.

Triclino pinacoidal. a: 6 : c—=2.8359:1:0,8510 with A= 117°.2'/,'.
eles) Oe)! oe 1002 a4 B=a1067.09) y= 1A. 14/5

Forms: a={100}, 5={010}, p={110} all very lustrous; c={001},
striped parallel (c:a); r= {101}, s= {801}, t= {201}; finally
o = {111} very narrow and dull.

4-5-6- Trinitro-1-3-vylene t = 125° erystallised from alcohol.

Monoklino prismatic. a:6 : ¢=0.5950:1:0.2706 with 6 = 88°.11'.
Large long-prismatie crystals. Forms 6 = {010} broad; m = {110}
also; a = {100} narrow; c= {001} large; r= {101} well developed
o = $110} very narrow. Cleaves well towards r.

2-4-6- Trinitro-1-3-aylene. t= 182° crystallised from benzene --
alcohol. Large thick prismatic very lustrous crystals; well built.

Rhombic-bipyramidal. a: 6:¢ = 0.6587 : 1: 0.5045. Forms a= {100},

1) Ber. 24. 2012.

2) Many cases where COOH, SO;H, I etc. is replaced by Br and this in turn by
Cl or NO; have been described or referred to previously. Ree. 21, 283, 336. 23, 207.

3) Compare Carnettey and Txomson, Journ. Chem. Soc. 53, 782. Lopry pe Bruyn,
Rec. 13. 116.

74 )

>= {110}, b=f010} broad and lustrous; o—= {122} large; r= {102}
narrower; g—= {012} small very completely cleavable towards 6,
fairly so towards a. Optical axial plane is {OOL}; First diagonal is
the a-axis. Faint dispersion, apparent axial angle in @ monobromo-
naphtaline about 70°.

Dr. Janaer intends publishing a more detailed examination later on.

If symmetric nitro-m-xylene is allowed to remain in contact for
some time at the ordinary temperature with nitric acid of sp. gr. 1.52,
the 4-5-dinitro-product is formed'). If the solution is poured into
water a white flocculent precipitate is formed, which when recrys-
tallised from alcohol yields fine, colorless needles m.p. 132°. As this
substance on subsequent nitration with HNO, and H,S0, yields chiefly
trinitroxylene m.p. 125° the NO,-group must have been introduced
into the position 4, for, if it had been introduced into position 2
the subsequent nitration would have formed exclusively trinitroxylene
m.p. 90°.

CH; Ci, CH,
a ILNO VN INO; NOs
aay | 132° | USO, | 195° |

NOK om NOON —= NO3\ /CUs
NO, NOs

An effort to prepare tetranitro-m-xylene from 4-5-6-trinitro-in-xylene
ended in failure. The trinitro-xylene was treated at 150° with HNO,
(sp. gr. 1.52) and H,SO,. The substance was to a large extent
destroyed but a small crop of colorless erystals was obtained m. p.
190°. These crystals were readily soluble in alcohol or warm water ;
the solutions had a strongly acid reaction so that probably one of
the CH, groups was oxidised to COOH. Tetranitro-m-xylene which
ought to have readily yielded trinitro-s.xylidine *) on treatment with
alcoholic ammonia was not found.

As the compound described by Drossspacn *) as trinitro-o-xylene
has been found by Noéxtine*) to be an impure trinitro-m-xylene, four
of the possible six trinitro-xylenes (0. m. and p.) are now known,
namely, the three trinitro-m-xylenes and also trinitro-p-xylene.

Summary. Symmetric nitro-m-xylene yields on treatment with
nitric acid 4-5-dinitro-m-xylene; on nitration with HNO, (sp. gr. 1.52)
and H,SO, two isomeric trinitro-m-xylenes are formed being chiefly
me 4-5-6-trinitro-1-3-xylene m. p. 125° besides a smaller quantity
of 2-5-6-trinitro-1-3-xylene m. p. 90°.

Amsterdam, May 1905.

1) A perceptible amount of 2-5-dinitro-m-xylene was not found.
2) Ber. 28. 2047. Rec. 21. 399,

3) Ber. 19. 2519.

4) Ber. 35. 634.

( 75 )

Physics. — Communication N’ 94° [ and II from the physical
laboratory at Leiden by Prof. H. KamernincH Onnus.

(Communicated in the meeting of May 27, 1905).

1. Improvement to the open mercury manometer of reduced height

with transference of pressure by means of compressed gas.

In Comm. N° 44, Oct. ’98, § 6, is described how the steel capillary
tubes gq were cemented on to the glass tubes im (see figs. 1 and 4
of the plate belonging to Comm. N° 44, of which the part that has
been modified is reproduced here as fig. 2 of the annexed plate). It
was also remarked that from time to time these connections became
defective (they slipped off when soft cement was used, or leaked
when hard cement was used), and that therefore we were trying to
find a better method of connecting. Since that time we have succeeded
in finding a way to make perfectly trustworthy joints. For the useful-
ness of the manometer this is of great importance.

Fig. 1 shows the connection as made now. It is based on the
method of Cainteter to solder glass on te metal, which method has
been mentioned in Comm. N° 27, I, June °96, for similar purposes
and as appears from Comms. N° 85, June ’03, and N°’ 89, Nov.’03,
has stood the proof in several cases where it was applied. The steel
capillary q,, reaches as high as qg,, in the glass tube 6,, which has
a wider part 6,,, containing mercury which is separated from the
junctures of 6,, and p, by a layer of marine glue 4,, in 6,,. Fol-
lowing the above mentioned method, the end of 6,, has been plati-
nised, then coppered galvanically and soldered on to /,.

This connection has quite answered the expectations. With the
former method when the manometer was put under pressure it some-
times appeared that minute cracks had come in the sealing wax,
through which the compressed gas which transfers the pressure from
one manometer column to the other escaped. In case it escaped
rapidly, the mercury of one manometer tube flew into the other
before the pressure could be removed and within a few moments
the apparatus was defect and wanted a thorough rearrangement to
become again fit for use. This has never occurred with the new
connections, and yet they have been used for a long time. If a tiny
opening should have remained in the marine glue, the mereury has
first to pass through if and a mercury drop at p gives warning. Then
the manometer can be freed in time from pressure and when the leak
of the single tube is repaired it may be immediately used again.

6

Proceedings Royal Acad. Amsterdam. Vol. VIII.

( 76 )

Formerly the escape of the gas which tiansfeis the pressure on
ihe mercury with the above mentioned consequences was to be feared
at the connecting pieces of the steel capillaries g with the cocks K
and the |-pieces 7’ (ef. fig. 1, Comm. N° 44). This was especially
dangerous as the place where it happened was often detected too
late. As packing between the flat steel surfaces of the joints for
cocks, flanged tubes, and | -pieces, we use now only a single sheet
of parchment paper. Leaks rarely occur. Moreover in the new arrange-
ment care is taken that each place where gas might escape is kept
under vaselin oil, so that even the smallest leak betrays itself immedia-
tely by a gasbulb raising in the oil.

In order to atiain this the mounting of 7’ and XK is modified as
shown on the annexed plate. Fig. 2 is a part taken from the
plate of Comm. N°. 44 (front elevation). Fig. 3 is a top view of this
part from a section 6. Figs. 4, 5 and 6 next to it show the present
arrangement in front elevation, section, and top view. The drawings
do not require much explanation. H/, is a wooden case (tinlined and
protected from action of mercury by parchment paper), in which are
placed the tuke H (vide Comm. N°. 44) with all the cocks K and
the [-pieces 7 connected with it. (A,, is a leose key on the cock-
needle, #7, is a tap; the contents of the case is about 0.8 hectoliter).

IH. Lmprovement in the transference of pressure by compressed
gas especially for the determination of isothermals.

The advantages of the transference of pressure from the experi-
mental apparatus to the measuring apparatus by means of compressed
gas caused this method to be repeatedly used for experiments at
Leiden. The drawback of it is, however, that much care is required
to make connections which are perfectly tight and that it is a very
elaborate work to seek for leaks, especially when there are a great
number of connections. It may happen that a whole series of experi-
ments loses its value when the existence of a leak is not immediately
detected. For some time, therefore, we have arranged those connec-
tions where the escape of gas is to be feared, (as explained in part I
of this paper for the open manometer) so that they can be covered
with vaselin oil and yet are easily reached. It has appeared that
with the arrangements based upon this principle we can work so
much more securely and rapidly that it more than balances the

small complication which sometimes arises when we carry out

the principle.
As an clucidation we have represented in fig. 7 of the annexed

ar)

plate the way in which of late we have made the connections for
the compression tube for piezometer determinations.

If we compare this figure with plate I, fig. 1, of Comm. N°. 84,
March ’03, where the same parts are designated with the same letters,
and also with the description of it given in Comm. N°. 84, then no
further explanation is required. The upper end of the level glass C",
is bent in order to immerse C",, in the oil vessel C’,,. The cocks
and connections which are not kept under oil, as for instance C,,
are those where a leak must show itself by the outflow of mercury.

Physics. Communication N°, 94° from the Physical Laboratory
at Leiden by Prof. H. Kamernina Onnus. “Methods and appa-
ratus used in the cryogente laboratory. VIL. A modified cryostat.”

(Communicated in the Meeting of May 27, 1905)

§ 1. In several Communications I have described cryostats based
on the use of baths of liquefied gas evaporating at ordinary or lower
pressure. For those eryostats where, as described in Comm. N°. 14,
Dee. 94, I succeeded in maintaining during any desired time a bath
of ‘/, to */, liter of liquid oxygen for measurements at a constant
low temperature by means of a circulation, no vacuum glass at all
was used. The whole method had been worked out before Drwar’s
investigations showed that the vacuum glasses were fit above all for
storing liquid gases.

Nor were any vacuum glasses used in the improved eryostats of
large dimensions described in Comms. N°. 51, Sept. ’99, and N°. 83,
Feb. ’03. When we started the measurements for which these
eryostats were used, we could only obtain sufficiently trustworthy
vacuum glasses which were blown to fit exactly when we were
satisfied with small dimensions.

Since, however, excellent vacuum glasses which are also of large
dimensions are made to fit, especially by R. Burerr at Berlin, it
will be possible in many cases to find vacuum glasses of the proper
size and the just mentioned methods of arranging will be especially
reserved for those cases where we want vertical walls of plane
parallel glass, or when the bath must be of excessively large
dimensions *).

In Comm. N°. 83 IIL § 6 we have already described a cryostat
of small dimensions constructed by means of a vacuum glass. The

1) Gomp. the end of VIII of this Series of Communications.
6*

( 78 )

annexed plate shows such a cryostat in a vacuum glass of much
larger dimensions (9 ¢.m. internal diameter) which during some
years has satisfied high requirements.

The apparatus has served for measurements ') with a differential
thermometer of which one reservoir was filled with hydrogen the
other with nitrogen, for a comparison of a thermoelement with the
hydrogen thermometer (cf. Comm. N°. 89) and for measurements
on the isothermals of diatomic gases (cf. Comm. N°. 69, April ’O1
and N°. 98, April ’02). If the plate is compared with Comm. N°. 83
no much further explanation is wanted. The same letters designate the
same parts. The connections of the cryostat with the regulating apparatus
for constant temperature are the same as on Pls. I, V and VI of Comm.
N°. 83. The stirringapparatus to obtaina wnaiform temperature is moved
by an electromotor as is the case with the eryostat represented there.
During the measurements with the differential thermometer the tempera-
ture was regulated according to the indications of a thermoelement @
(which is described in detail in Comm. N°. 89 published lately).
In the comparison of the thermoelement @ with the hydrogen thermo-
meter one of the thermometerreservous on the annexed plate was
replaced by a resistance thermometer (double cylinder according to
Comm. N°. 98, Pl. I, fig. 2, with improvements which will be deseribed
later on). Moreover in the measurements of isothermals the piezometer
(ef. Comm. N°. 69, Pl. I) was put in the place of the second
thermometerreservoir.

In order to secure a symmetrical distribution of the current in
the bath mica sereens, (which also serve for insulation) are used if
necessary (for instance in the resistance. thermometer), and a tube
similar to the thermoelement @ was mounted symmetrically with
the latter.

The agreement between the mean temperatures of the measuring
apparatus and the temperature indicator is further promoted by
making the mean height of the two equal.

As with the cryostats of Comm. N°. 83 we can reach by means
of this one a constancy to within 0.01° C. For everything relating
to this I refer to Comm. N°. 88.

A silvered vacuumeglass being used, there was arranged a float
(not to be seen in the figure) to show the position of the level of
the liquid.

The stopper and the way in which the thermoelement is fixed

1) The completion of the calculations of these measurements, on the subject of
which we shall soon publish a communication, requires some new determinations
and the application of some corrections,

into it is given only schematically in the figure. In the same manner
as the tube a, the thermoelement is easily introduced and moved
up and down through a tube arranged for this end.

The controlling rod for determining the mean temperature of the
capillary of the thermometer or piezometer could be omitted in
these measurements.

§ 2. The working of the cryostats described in the last section
and in former communications is based upon the cooling caused by
evaporation at the surface of the liquid. Although the temperature
in those apparatus is almost everywhere uniform there yet remains
acolder layer at the surface and a warmer one at the bottom.
In some measurements it is very disturbing that the temperature
at the top of the bath is somewhat, though very little, lower than
elsewhere. In a following communication I hope to be able to give
drawings of a cryostat where the bath is surrounded from the bottom
upwards by vapours of a lower temperature than that of the bath,
so that if we regulate the pressure there is a continual heating
instead of a continual cooling at the surface and the normal condition
that the temperature of the upper part of tbe bath is higher, is reached.

Physics. — “Methods and apparatus used in the cryogenic labora-
tory. VIM. Cryostat with liquid oxygen for temperatures below
— 210°C.” Communication N°. 944 from the Physical Labora-
tory at Leiden by Prof. H. Kamertincu Onnzs.

In Comm. N°. 838 IV (March ’03) I have described how in one

of my cryostats constant temperatures between — 195° C. and
— 210°C. (in round numbers) are maintained by means of liquid
nitrogen. Whereas between — 180° C. and — 195° C. (in round
numbers) oxygen is the proper liquid for cryostats, for the range
between —195° C. and the freezing point of nitrogen the latter

substance offers the advantage that its vapour pressure is several
times larger than that of oxygen and that the quantity of it which
evaporates at the same quantity of heat supplied, can be taken up by
a vacuumpump of a much smaller capacity. Moreover if for evapora-
tion purposes we are obliged to use the same vacuumpump which
also serves for the methylchloride or ethylene, the difficulties are
much less with nitrogen than with oxygen. All these reasons made
us formerly prefer nitrogen for temperatures below — 195° C. For
temperatures below the freezing-point of nitrogen, however, we are

(80 )

obliged to return to oxygen. Although in this case the volume of
eas that must be displaced in order to allow the use of the cryostat
described in the previous paper, becomes very large, still it may
be controlled at — 247° C. by a Burcknarpt-Wriss vacuumpump,
arranged as described in Comm. N°’. 83 V, which can displace
360 M’ an hour. Fortunately we could set an additional pamp working
to this end. Not only that in this way the BurckHarpt-WEIss pump
of the methylehloride circulation (Comm. N°. 87, March ’04) remains
free, but now there is also little reason to use nitrogen for tempe-

ratures between — 195° and — 210°C. With the same cryostat
with oxygen we can produce temperatures ranging from — 180° to

217°C. The arrangement of the cryostat also admits of keeping
the temperature constant to within 0°.01 a 0°.02 C.

On the annexed plate figs. 2, 3 represent a cryostat which is
used for measurements at these low temperatures and which differs
only in a few minor details from that of the last communication
(N°. 94c May °05) to which I refer for the rest (the same letters
denote the same parts). In this eryostat the wires with which the
valved stirrer is moved up and down do not pass over two pulleys
y., so that they run parallel with each other outside the apparatus,
but they leave the apparatus over a single pulley zy,’ diverging from
each other (see fig. 2). Wearing and friction are lessened by this.
In the india-rubber tubes surrounding these wires, at distances of
8mm. small spirals, 8 mm. high and 38 mm. in diameter, are intro-
duced, which prevent the compression of the tubes when the pressure
in the cryostat is reduced. Tube 7), is taken wider and had there-
fore (in order that we still might use the existing apparatus) to be
flattened at the end where it passes into V,,’.

Fig. 1 represents the determination of the isothermals of hydrogen
by means of the piezometers of Comms. N°. 69, 78, 84 § 19,
(general letter P) in this cryostat. The temperature is measured
(cf. Comm. N°. 83 III) with a thermoelement (Comm. N°. 89), and
regulated according to the indications of the resistance thermometer
hk (fig. 4 shows this in bottom view). As in the model given in
Comm. N°. 98, Oct. ’O4, (the small letters added to the general letter
R have the same meaning as in Comm. N°. 93 VIII, § 2), the
resistance thermometer consists of naked platinum wires wound upon
two glass cylinders and of one protecting cylinder. The improvements
which are spoken of already in the previous paper, consist in using
instead of the mica sheets 7, 7, @,, 7, (the form of the supporting
ridges being modified) the glass tubes 7,', 7,', 2,', 2,’ (hence in the
figure /?;, ete.), so that short circuiting between the different parts

( 81 )

of the resistance is better prevented. Moreover in the new construe-
tion the pins g,, (see fig. 4), which surrounded by glass tubes g,,,
fit into the grooves g,, (see fig. 5, twice the dimensions of fig. 4),
prevent the cylinders and the supporting ridges during the mounting
from shifting against each other, which would cause the wires to
break at the soldering places. Lastly, to prevent that (in consequence
of being cooled conduction of heat) water vapour condenses at the
place where the wires ¢,, ¢,, ¢,, ¢, leave the apparatus, the upper
part of the supporting rod / is made of glass. Therefore we have
fastened the cap 4,, to k,, by means of hard solder, and fixed
according to Camnnrer’s method (cf. Comm. N°. 944) the glass rod
k, to this cap.

The connection of the cryostat with the auxiliary apparatus agrees
in principle with that of Comm. N° 83 (especially Pls. IV and VI)
to which I refer for further details. The cryostat represented in this
paper replaces Cr on Pl. IV. At the place of Hvh' Pl. IV, the
vacuumpump was exhausted by a smaller one, which displaces
20 M?* an hour. This forces the oxygen at normal pressure through
a solution of caustic soda in order to keep back the oil carried
along from the vacuumpumps. The small vacuumpump also replaces
AC of Pl. IV (the lead Hvh' terminates into it at Y,,, the tube with
caustic soda replaces D, on Pl. IV). The oxygen, after having
bubbled through the caustic soda solution, can without fear of
explosion be compressed by a BroTHERHOOD compressor arranged as
described in Comm. N°. 51, Sept. °99, and lubricated with glycerine
(cf. Comm. N° 83 IV). It thus replaces HgC on Pl. IV of N°. 83.

The admission of liquid oxygen is sometimes effected by directly
syphoning over liquid oxygen from a vacuum glass into the cryostat.
As a rule, however, compressed oxygen from a cylinder is used
and as generally we have a large quantity of cylinders with com-
pressed oxygen in store, it is supplied from another reservoir than
that in which the sucked off oxygen is compressed (in that case the
connection 2N-D, of plate IV, Comm. N°. 83, does not exist). The
oxygen liquefies in a cooling tube immersed in liquid air (the
nitrogen in CS of Pl. IV Comm. N°. 83, is replaced by oxygen,
the oxygen by air) and thence passes through a (see the annexed
plate) to the cryostat.

As to the way to keep the temperature constant, the only alteration
from what has been laid down in Comm. N°. 83 is that with high
vacua the oil manometer is no longer used and we regulate only by
means of the cock V’,, (Comm. N°. 83) being guided only by signals
according to the readings of the resistance thermometer, The mean

(82 )

temperature is determined exactly as described in Comm. N’, 83.

The cryostat described could be relatively simple, because a vacuum-
elass of large dimensions was used. Excellent though a vacuumglass
may be, it is still always to be feared that it bursts unexpectedly and
so damages the measuring apparatus. Indeed, one of the series ot
measurements was put an end to in this way. Hence, when mea-
suring apparatus are used to which we attach great value, because,
for instance, many other measurements have been made with them,
it is advisable when we want to bring them in baths of constant
temperature below — 210° C., to use the cryostat described in
Comm. N°. 83 III, where, though it is much more complicate than
the one described here, no vacuum glasses are required, and in this
the oxygen can be evaporated at a very low pressure, also with the
aid of the above mentioned large vacuumpump.

Physics. — “J/ethods and apparatus used in the cryogenic laboratory.
IX. The purifying of gases by cooling combined with com-

pression, especially the preparing of pure hydrogen.” Com-
munication N° 94e from the Physical Laboratory at Leiden by

Prof. H. Kamerticu Ones.

§ 1. To separate less volatile elements from a gaseous mixture
by cooling with liquid air belongs now to the ordinary operations
in laboratories. At Leiden it is applied on a fairly large scale to
reobtain ethylene in its pure form after it has been mixed with air.
In the experiments it repeatedly occurs that ethylene is contaminated
with air; from time to time when in the ethylene cycle of the cascade
process the condensation pressure of the ethylene increases, the gas
which remains behind after the greatest part of the gas used in the
eycle is liquefied, is blown off and replaced by pure ethylene, in
order to reduce the condensation pressure to its ordinary amount.
All such mixtures and remnants with a larger or smaller proportion
of ethylene are collected in a large gasholder because of the expen-
siveness of this gas. The ethylene is afterwards frozen out from the
collected gas in a vessel cooled by liquid air.

By cooling at a normal pressure we can from a mixture of gaseous
substances which differ very much in volatility, separate a large
portion of the least volatile substance when we go to temperatures
which though lying above the boiling point of the one, reach far
below that of the other substance. How much of the impurity is
still in the remaining gas is then fairly well determined by the

C33")

Vapour pressure of the less volatile element at the temperature of
cooling. The separation will be much more perfect when we can
also avail ourselves of compression, as it is the case, for instance,
when the gas which we want to purify, at the lowest temperature
to which we can cool, is still above its critical temperature.

If we do not take the pressure too high we may assume roughly
that the degree of purity which we can reach with long continued
cooling, is at the same temperature directly proportional to the
pressure to which we compress. In cases where the gas flows through
a cooled tube, other factors come into consideration, but even then
compression offers a great advantage.

I have availed myself of this operation for a last and thorough
purification of the electrolytic hydrogen (prepared as described in
Comm. N°. 27, May ’96), which is used for piezometers and thermo-
meters, when it appeared that notwithstanding it was led through
drying tubes with phosphorus pentoxide, traces of water still occurred
in the gas. This purification was effected by cooling hydrogen under
strong pressure in liquid air.

A similar method may be recommended to free, for instance, helium
from admixtures of neon and hydrogen. The degree of purity of
the helium can be raised considerably by causing the bath (liquid
hydrogen) to evaporate in vacuo; for this purpose an apparatus is
being constructed *).

§ 2. Pure hydrogen for thermometers and piezometers.

Several improvements have been made to the apparatus for the
preparation of pure hydrogen (described in Comm. N°. 27). Some
of them are deseribed in Comm. N°. 60, Sept. ’00. Later the plate
jf of fig. 6, Pl. II, Comm. N°. 27, was riveted to a platinum wire
(instead of being soldered to the copper wire e) and melted in a
glass tube which is bent down under the mercury on the bottom of
the apparatus and is itself filled with mercury. Further the cock d
was sealed to the bell-jar ce, and the sealing place / is kept under
mercury to be cooled by it; finally the shutting of the apparatus was
made easier as the india rubber stoppers in the cover were replaced
by cone-shaped ones which are pressed on to it by means of a
small plate and tightening screws and as six tightening rods ¢
instead of three as in the above mentioned figure have been made.

1) After this had been wrilten and published in the Dutch Proceedings of the
Academy I found that Dewar in his Bakerian Lecture, Proc. Roy. Soc. Vol. 68,
1901, recommended the method of adding compression to cooling for purifying
helium,

( 84)

The electrolytic hydrogen prepared under excess of pressure in
the improved generating apparatus flows off through a fine regulating
cock (see FR, Pl. I, Comm. N°. 27). It is, however, not directly
admitted into the mercury airpump and the measuring apparatus
which is to be filled, but is first led through a. steel capillary to
the piezometer in a pressure cylinder where pressure is exerted by
compressed air, as was used in the experiments on the condensation of
gaseous mixtures (see Comm. N°. 92, Sept. ’04, Pl. I, fig. 1). The
stem of this piezometer carries a three way stopcock (Comm. N°. 84,
March ’03 Pl. I figs. 2 and 3), to which are connected on the one
side the above mentioned capillary, on the other side a copper cooling
tube (a platinum cooling tube with platinum capillaries would still
have been better), which at either extremity ends in steel capillaries
with connections. A high pressure cock, which admits of a fine
regulation, connects the cooling tube with the mercury air pump
and the measuring apparatus. All the packings are made of cork,
the gas itself has no contact with anything but the metal of the cooling
tube and the capillaries, with glass, or with twice distilled mercury.
After all parts between the generating apparatus and the mereury
airpump have been carefully exhausted, the gas is admitted from
the generating apparatus into the piezometer with the cooling tube,
then the latter are shut off from the generating apparatus and the
mercury in the piezometer is forced up until a pressure of 60 atm.
is reached, the cooling tube. being immersed in liquid air up to the
steel capillaries. At the same pressure the gas is then led through
the regulating cock into the measuring apparatus that are to be filled.

§ 3. Hydrogen for the cycle with liquid hydrogen. The commercial
electrolytic hydrogen is as a rule too much contaminated with oxygen
and air to serve for a circulation of hydrogen. In order to separate
these admixtures we may compress it in a cooling tube immersed
in oxygen, which evaporates in vacuo. The following operation is
simpler still. The hydrogen is compressed and led through a cooling
tube immersed in liquid air under normal pressure into the appa-
ratus where liquid hydrogen is prepared by means of a regenerator
spiral, which apparatus together with a gasholder, the compressors and
drying apparatus forms a cycle. The pressure of compression is now
regulated so that the compressed gas flows out without blocking the
delivery cock of the regenerator spiral, at least not as long as this
cock is opened and shut alternately. The pressure is gradually raised
higher and higher, while the temperature of the outflowing gas falls,
and this is continued until the cock is blocked, and the pause during

(8)

which we-are waiting for the cock to be free again, is used to
remove that which is deposited at the place intended for the liquid
hydrogen. Thus it is not difficult to prepare from the commercial
hydrogen Jarge quantities of hydrogen with less than 1 pro mille of
admixture.

Anatomy. — “On the development of the Cerebellum in Man’.
(Second Part). By Prof. L. Bou.

In the first part of this communication the development of the
Cerebellum is described until the stage in which the sulei appear
typical for the mammalian cerebellum. In this stage it is divided
by the suleus primarius into an anterior and posterior lobe. The
first of these lobes is separated by three grooves into four lobules,
corresponding with the lobuli 1, 2, 3 and 4 of the mammalian
cerebellum. The posterior lobe is also separated by three grooves
(suleus praepyramidalis, fissara secunda and suleus uvulo-nodularis)
in four lobules, corresponding with the lobuli A (nodulus), B (uvula),
C,, (pyramis) and ©, (declive +- folium vermis ++ tuber vermis), which,
with a few exceptions, are to be found in the other mammals. In
these exceptions the suleus praepyramidalis, which separates the lobuli
C, and C,, is missing, as in Erinaceus (ArnBack Curistie Linpr),
Notoryetes (Eniior Smirx), Vesperugo (Cuarnock Brapiuy), Chryso-
chloris (Lucu). In this case the posterior lobe is only built up of three
lobules. The missing of the suleus praepyramidalis in these cerebella
of extremely simple construction gives rise to the supposition that
this fissure is phylogenetically the youngest of the primary sulci of
the cerebellum. This supposition is corroborated by the fact that in
man the suleus praepyramidalis is ontogenetically the last that appears.

After the development of these primary sulci, grooves appear
characteristic for the cerebellum of the primates, and whose homologa
are wanting in other classes of mammals.

In embryos of a length from 16 to 22 ¢.M. arises a groove on
the posterior surface of each of the hemispheres, the lateral part of
which is directed to the obtuse angle of the lateral border of the
cerebellum (Fig. 11 y). The mesial ends of these grooves approaching
each other, penetrate into the narrow lobule which is bordered by
the suleus primarius (1) and by the suleus praepyramidalis (4); after-
wards these grooves unite and divide the lobule in an upper and
lower half. This differentiation however not always proceeds sym-
metrically, so that it may happen, that these grooves do not meet,

( 86 )

but that one of them unites with the suleus praepyramidalis; in
other cases they grow along of each other, causing in this way an
asymmetry of the lobule, which influences the further lobulisation of
this region; these cases however are rather exceptional. The appea-
rance of these grooves is known in literature, and it is generally
believed that they form the suleus horizontalis. In the beginning I
inclined to the same opinion, but the study of the abundant material
which was at my disposition instructed me that this notion is wrong,
and that this groove, which appears symmetrically, is the suleus
superior posterior which separates the lobulus lunatus posterior and
the lobulus semilunaris superior. The suleus horizontalis appears
afterwards in a manner as illustrated in Fig. 12 and 13.

The fact that the sulcus superior posterior arises in an earlier period
of human embryonic life than the sulcus horizontalis seems of interest
in connection with other particulars of comparative anatomy. I found
namely in my researches on the cerebellum of Primates the suleus
superior posterior appearing phylogenetically before the suleus hori-
zontalis. All Primates excepted the Arctopithecidae possess a suleus
superior posterior, whereas a sulcus horizontalis is only to be found
in Anthropoids, although an indication is also to be found in Ateles.
After the formation of the sulcus superior posterior, the lobule between
the suleus primarius and sulcus praepyramidalis quickly increases
in size, the cerebellum becoming convex in its median zone. (Fig. 11
and 12. 1, 4 and y).

In the same period in which the lobulus lunatus posterior -
bordered by the suleus primarius and the sulcus superior posterior —
develops its secondary grooves, a short straight groove appears on
the upperlip of the suleus praepyramidalis. This groove is the suleus
horizontalis (Fig. 12 and 13,2), which, contrary to the general concep-
tion appears in the median portion as an unpaired groove. In the
beginning therefore the region between sulcus horizontalis and sulcus
praepyramidalis is extremely narrow in its median portion, the region
however between the first suleus and the suleus superior posterior
being relatively large. The first of these regions becomes the Tuber
vermis, while the second forms the Folium vermis. If one compares
the size of these regions with those of the Tuber and Folium vermis
of the adult cerebellum it is evident that there must be a very
unequal surface-expansion in these adjacent parts of the cerebellum
and that the development of this organ in man is not as simple as
it appears. And it may be concluded from the fact that the surface-
expansion of the lobules takes place with very different intensity that
the signification of the grooves and lobules is not merely a morphos

( 87 )

logical one. The obvious difference in the extension of the cortex of
the different lobules ought to have a physiological base.

In the stage of development in which the sulcus horizontalis forms
a short straight groove in the forelip of the sulcus praepyramidalis,
the Tuber vermis does not yet reach the surface, whereas the folium
vermis, which in a later period is concealed, still appears broadly
on the surface. This relation is modified by a lamella which mounts
to the surface from the bottom of the sulcus praepyramidalis, and
which pushing forwards the sulcus horizontalis, separates the latter
from the suleus praepyramidalis (Fig. 12 and 13. h, 4). This lamella,
arising from the forewall of the suleus praepyramidalis is the first
“Anlage” of the Tuber vermis.

At the same time the suleus horizontalis has lengthened and pene-
trates into the hemispheres, soon being equal in length to the sulcus
superior posterior (Fig. 15c). These sulci include a cuneiform lobule
with its top directed mesially, on the surface of which arise secon-
dary furrows, even before the sulcus horizontalis has reached the
lateral border of the cerebellum (Fig. 15a 6 and c). This wedge-
shaped lobule is the lobulus semilunaris superior.

The suleus praepyramidalis has also extended into the hemispheres
(ef. Fig. 11 till 15. 4) and with that keeps its typical form for a
long time: namely a median horizontal portion of which the lateral
parts bend sharply down and back. This typical form enables us to
recognize easily this groove. By this course of the suleus a second
cuneiform lobule is formed with its top directed mesially, bordered
above by the suleus horizontalis (1) below by the sulcus praepyramidalis
(4). This region becomes the lobulus semilunaris inferior. It is remarkable
that the first groove which subdivides this lobule also rises from the
upperlip of the suleus praepyramidalis from which again appears
that here exists a focus of very intense surface-expansion. This groove
penetrating into the lobulus semilunaris inferior can be seen in
Fig. 13, 14 and 15 in different phases of development, whereas in
Fig. 16a a second groove emerges from the underlip of the suleus
horizontalis, quite near the middleline, which grows out into the
Iobulus semilunaris inferior. By these two intralobular grooves is
initiated the subdivision of the lobulus semilunaris inferior into three
sublobuli, a fact, to which Zrenun has fixed attention.

The region between the suleus praepyramidalis (4) and the fissura
secunda (2) undergoes but fewer changes and takes part ina slighter
degree in the surface-expansion. For a long while this area is broadest
in its median zone (Fig. 11, 12, 13 and 14) and shows there one
or two short grooves which are however limited to the middle

(88)

region and do not penetrate into the hemispheres; this broad middle-
piece is the Pyramis. The parts of the hemispheres corresponding
to the Pyramis are separated relatively late from the rest of the
cerebellum. This separating is connected with other phenomeng
which ave of importance for the topographical relation of the cerebellar
lobules. The fissura secunda namely, which limites the regions of the
Uvula and the Tonsilla on the upper side, extends originally from one
lateral border of the cerebellum to the other (Fig. 11—14). The
area however, situated above the transversal zone formed by Uvula
and ‘Tonsilla, increases more rapidly in transversal direction than
the Tonsilla does and in this way the latter is enclosed. By this
process the fissura secunda ends no longer at the lateral borders of
the cerebellum, but, if observed from behind, at the myelencephalic
border (Fig. 15). Now it is mainly that part of the hemispheres
which gets situated at the side of the Tonsillae, which by a narrow
lamella remains connected with the Pyramis and develops to the
lobulus biventer.

The region of the lobulus biventer and Pyramis shows in its lamel-
lisation a characteristic that indicates that the surface-expansions of
the middle- and side-pieces are more or less independent of each
other. Already I drew attention to the fact that in an early period
of development one or two grooves appear in the Pyramis which
do not extend into the hemispheres; figures 11-—19 show these
erooves at a number of two or three. Now we see, that the furrows
of the lobulus biventer take their origin quite independently of those
of the Pyramis. For according to my preparations the lobulus
biventer is lamellised in two ways. From the underlip of the suleus
praepyramidalis arises a groove on some distance from the middleline.
This groove lengthening itself in a lateral direction reaches the
margin of the cerebellum and divides the lJobulus into an upper
and under part. In Figures 17, 18 and 19 this groove is indicated
by a 6 and is identical with the suleus bipartiens of ZimHEn. The
lamellisation of both parts of the lobnulus biventer takes place in a
different manner. The upper part of this lobule, which is cuneiform
in shape develops new grooves, taking their origin from the underlip
of the suleus praepyramidalis or from the upperlip of the suleus
bipartiens, which grooves lengthen laterally ; the grooves of the under
part, which is a narrow lobule connected with the Pyramis arise
from the margin of the cerebellum and grow out mesially. Especially
figure 18 shows very clearly this difference in the folding of the
cortex of both parts of the lobulus biventer, which difference gets
more important when it is compared with the mode of folding in

( 89 )

the Uvula and Tonsillae. Like the Pyramis the Uvula too very soon
shows one or two transversal grooves, which do not penetrate into
the Tonsillae. The surface of the last called lobules remains unfolded
for a remarkably long time (Fig. 17 from a foetus of 29 eM. and
fig. 18 from one of 32 ¢.M.) and assumes an oval-shaped rounding.
But, the folding of the cortex once commenced, it proceeds in the
same way as in the under part of the lobulus biventer. For as ean
be observed in Fig. 16 and 19, the grooves begin on the margin of
the lobules, i.e. laterally and grow out mesially. In connection with
this last fact it must be recalled to mind that also the fissure which
separates the Flocculus from the remainder of the cerebellum first
appears at the lateral edge of the latter and lengthens afterwards in
a mesial direction.

The Floceuli and Nodulus have undergone but little differentiation
during the described stages of development; the Nodulus has increased
in surface and in the number of its grooves; while the Floceculus
in an foetus of 35 ¢.M. (Fig. 19c) shows only three lamellae.

A distinct differentiation between the cerebellum and pedunculi
pontis is indicated sharply in foetus of 25 to 30 ¢c.M. (Cf. 15c, 164,
17a and 19c). In the foetus of 29 ¢.M. (Fig. 17a) the fossa lateralis
is sharply bordered while the sulcus superior posterior (%) ends in
its fore border and the suleus horizontalis in its top.

In the development of the human cerebellum some interesting
phenomena may be observed, which are worth to be brought in the
fore-ground and which may be summarised in the following way.

1st. In the grooving of the human cerebellum two stages may be
observed ; in the first stage those grooves arise that in general are
characteristic for the mammalian cerebellum. By these primary
grooves the organ is divided into an anterior lobus, which is sub-
divided into four lobules, and into a posterior lobus, which in the
median plane is also subdivided into four lobules. All these grooves
take their origin in the middleline. Besides these another groove
appears, beginning at the lateral border of the cerebellum (Fissura
parafloccularis). In the second stage those grooves become visible,
that are typical for the cerebellum of the Primates.

2nd, After the primary grooves in the first stage having been formed
the further lobulisation and lamellisation takes place in the second
stage in a regular way. In the anterior lobe all the grooves take
origin in the middleline and lengthen laterally ; the same happens
with the maingrooves of the region between suleus primarius and
bipartiens, but in the last region there also arise grooves in the

( 90 )

hemispheres which are confined to the latter. Finally between the
suleus bipartiens and the margo myelencephalicus of the cerebellum
the grooves begin at the border of the hemispheres and grow out
mesially. The system of the grooves belonging to the Pyramis, Uvula
and Nodulus forms an independent system, which has no connection
with the systems of grooves existing in the adjacent parts of the
hemispheres. Consequently from a morphogenetic point of view three
zones may be discerned. Anterior zone: all grooves arise in the
middleline, grow out into the margin of the cerebellum or end at
some distance of it. Zn this zone the system of grooves is an unpaired one.
Middle zone: the grooves take their origin partly in the middleline
and extend to the margin of the cerebellum, partly they arise in
the middle of the hemispheres to which they are confined. This
system of grooves is a paired one. Posterior zone: there arise grooves
in the middleline which are confined to a narrow band, while inde-
pendently of these a second system arises in the hemispheres. Jn
this zone the system of grooves possesses a threefold character. The
cerebellum of the Primates compared to that of the other Mammals
is characterized by a progressive development of the anterior and
middle zones and a regression of the posterior zone.

3, After the first stage of development having been passed there
arise spheres of intense surface-expansion by the side of others, where
this expansion is minimal. This is the case with: the most anterior
part of the Lobus anterior, which develops into the Lingula; further
with the Folium vermis, which at an early period of development
reaches the surface as a relatively large lamelle, and the Flocculus,
the surface of which enlarges very little. Spheres of intense surface-
expansion are: the region in the middle line, immediately surround-
ing the suleus primarius; the forelip of the suleus praepyramidalis,
from which arises the whole Tuber valvulae; the region between
the sulcus horizontalis and sulcus praepyramidalis. Especially the
human cerebellum is distinguished by the mighty development of
this part.

The facts brought to notice in 2 and 3 lead to the conclusion
that the cortex of the cerebellum is not an organ with a homo-
geneous distributed function, but a well organised entirety with
localised functions.

4th, In general the anterior lobe keeps ahead of the posterior
lobe in development, the lamellisation beginning latest in the caudal
part of the cerebellum.

5th, In connection with the difference in the mode of lamellisa-
tion of the zones described in 2, sulci paramediani are wanting in

(91 )

the anterior zone, in the middle zone they exist, but continuity
between the lamellae of the hemispheres and of the vermis remains,
and in the posterior zone they form a complete division between
the lamellae of the hemispheres and of the vernis.

Anatomy. — “On the sympathetic nervous system in Monotremes.”
By A. J. P. v. pb. Brozk. (Communicated by Prof. L. Bork).

The following deseription contains the results of an investigation
on the structure of the sympathetic nervous sysiem in Monotremes.

For this investigation I had at my disposal a female specimen of
Echidna aculeata and of Ornithorhynchus paradoxus. The sympathetic
system of the two specimens resemble each other in many respects,
i.e. in structure and ramification; in other respects they show
important differences from placental mammals.

In the cervical sympathetic chord we find in Echidna one, in
Ornithorhynchus two ganglia.

The ganglion cervicale of Echidna is (Fig. 1. g.c.) a rather large,
oval-shaped body, situated close above the Arteria subclavia. Singular
or double rami viscerales connect this ganglion with the first as far
as the fifth cervical nerves included. The ramus visceralis of the
first cervical nerve does not enter directly into the ganglion cervicale
but is joined to a nerve that appears at the upper end of this
ganglion and can be traced as far as the base of the skull (Fig. 1. a.)
where it enters into a little foramen. Close to the base of the skull
(Fig. 1. b.) two little twigs branch off this nerve, which go through
the M. longus colli to the vertebral column. Anastomotical branches
of this nerve with the Nervus vagus and the ramus descendens
hypoglossi are under the base of the skull. (Fig. 1. ¢.).

In Ornithorhynchus a_ little part of the cervical ganglion, which
should be considered as the fusion of the ganglion cervicale supremum
and medium in placental mammals, is situated on the atlas as a ganglion
cervicale supremum (Fig. 2 g. ¢. s.) and is connected with the first
cervical nerve. A thick branch of the Nervus vagus enters into the
ganglion from the lateral side; at the medial side the Nervus laryn-
geus superior (Fig. 2 |.s.) leaves it. In its course this nerve contains
a little ganglion before dividing into ramus externus (Fig. 2 r. e. 1. s.)
and internus.

The rami viscerales parting from the second to the fifth cervical
nerves included communicate in Ornithorhynehus with the ganglion

7

‘
Proceedings Royal Acad. Amsterdam. Vol. VIII.

( 92 )
cervicale, which, as in Echidna, is situated close above the Arteria
subelavia.

The communication between the ganglion cervicale and_ the
ganglion stellatum is formed in Echidna by two nervestems, which
form an Ansa Vieussenii round the Arteria subeclavia ; in Ornitho-
rhynchus I only found one nervestem passing backwards round this
artery. In both, the nerve that passes at the back of the artery is
connected with the rami viscerales of the sixth and seventh cervical
nerves, while that of the eighth cervical nerve passes directly into the
ganglion stellatum. This ganglion also receives the rami viscerales
of the first and second thoracic nerves. (The latter in Echidna partly).

The ganglion cervicale sends out some nerve branches to the heart,
close. to rami cardiaci of the vagus nerve (Fig. 1 and 2 r. ¢.). In
Echidna these branches are also sent out from the anterior nervestem
of the Ansa Vieussenii.

The cervical sympathetic system in Monotrvemes differs from that
of the placental mammals not only by the composition and arrange-
ment of the cervical ganglia, but also in the nervus vertebralis,
which is wanting in Monotremes; in both Echidna and Ornitho-
rhynchus the rami viscerales of the cervical nerves run extravertebral.
If mechanical influences be the cause of the origin of the vertebral
nerve (cordon apophyso-vertrébrale of Tuépautr'), it is important to
point out the peculiarity of the cervical vertebrae in Monotremes,
in which the rudiments of the ribs only confuse at a very late stage
of development with the transverse processus of the vertebrae.

The visceral branches of the first and second thoracic nerves run
to the ganglion stellatum; the third till twelfth intercostal nerves
included are connected by very short rami viscerales with the top
of triangular ganglia, situated in the spatia intercostalia.

From the seventh unto the eleventh spatium intercostale the sym-
pathetic chord of Echidna is divided into two chords that run parallel,
the lateral of which is much smaller than the medial one.

From the latter there goes a nerve in caudo-medial direction,
which I could follow as far as the aorta. The rami viscerales of the
ninth and tenth thoracic nerves of Ornithorhynchus divide into two
rami and go to two sequent ganglia. With both animals the sym-
pathetic chord, where it receives the ramus communicans of the
thirteenth thoracic nerve, makes a curve in medial direction and
penetrates the diaphragm in front of the vertebral column.

') ‘Treépautr. V. Etude sur les rapports qui existent entre le systeme pneumo-
gaslrique el sympathique chez les Oiseaux.
Annales des Sciences naturelles. Série 8. Tome VI, p. 1.

(93 )

If one will call this part of the chord homologous to the Nervus
splanchnicus major of placental mammals, it must be pointed out
that in this group the Nervus splanchnicus is the entire sympathetic
nerve. So the segmental caudal limit of this nerve falls in the region
of the thirteenth or fourteenth thoracic nerve.

In the abdominal part some differences appear between Echidna
and Ornithorhynehus. In the former the sympathetic nerve divides
under the diaphragm into two branches of very unequal dimension,
The larger of the two passes in medial direction and disappears in
the corpus suprarenale (Fig. 1 ¢@.s.r.) after having sent some small
branches to the kidney. The second, much smaller branch passes in
caudal direction and is the continuation of the sympathetic chord.

Into the latter first passes a branch, springing from the sympathetic
ganglion of the fourteenth thoracic nerve, after that the rami viscerales
of the succeeding thoracolumbar nerves.

In Ornithorhynchus the sympathetic chord passes under the dia-
phragm into an oblong ganglion. From the medial side of this some
fibres arise, that pass to the suprarenal body (Fig. 2 g. s. 1.) and
the kidney (Fig. 2 n.). With this ganglion communicate moreover
the rami viscerales of the sixteenth and partly that of the seventeenth
thoracic nerve.

In Ornithorhynehus I did not find a ramus visceralis of the fifteenth
thoracic nerve.

In KEcumpna there were separated ganglia where the rami viscerales
of the fifteenth and sixteenth thoracic nerves join the sympathetic
nerve. A ganglion splanchnicum (ARNOLD), which occurs in mammals
in the course of the Nervus splanchnicus major, Monotremes do not
possess. Probably this ganglion is still contained in the corpus supra-
renale; this organ we may therefore call in these animals “ganglion
suprarenale’’.

Following the sympathetic nerve further in the abdomen we see
that it receives the rami viscerales of the nerves in regularly arranged
ganglia of which sometimes two are joined to a single one. (e. g. in
Ornithorhynchus those of the seventeenth and eighteenth thoraco-
lumbar nerve). Here and there I found double rami viscerales which
then pass through the M. psoas in a curve.

On account of what | found in the lumbar part of the sympathetic
chord in other mammals I am inclined to ascribe the splicing of
the rami viscerales to mechanical influences, i.e. to the development
of the processus transversi of the vertebrae and the M. psoas.

A division of the rami viscerales in grey and white rami, embracing

fi

( 94 )

the intercostal arteries, is described by Gasket") from the second
thoracie unto the second Jumbar nerve in man. I did not observe
a similar splicing in the Monotremes.

The most caudal branch of the sympathetic chord to the abdominal
viscera leaves this chord in Ornithorhynchus at the level of the
entrance of the ramus visceralis of the eighteenth thoraco-lumbar
nerve, in Echidna at that of the nineteenth thoraco-lumbar nerve.

So the caudal limit of the visceral nerves in Monotremes is much
lower than in man, in whom this limit is described by Bisnop
HArMAN?), in accordance with GaskELL at the level of the fifteenth
thoraco-lumbar nerve.

In the caudal part of the sympathetic chord I found branches of
it united with the pudendic nerve. In Ornithorhynchus these connect-
ine branches arose from the chord at the level of the sacral and
first caudal nerve, in Echidna at the level of the first till third
caudal nerves. The plexus hypogastricus, which in mammals is found
on the side of the caudal rectum end and the uro-genital canal,
1 found only slightly developed in Monotremes.

A few observations about the ramifications of the sympathetic
branches to the abdominal viscera may be added here.

On the medial side of the suprarenal body two groups of nerve
fibres appear (in the figures each of them is represented by one
single line).

The topmost of the two groups is going to the origin of the Arteria
coeliaca ++ mesenterica superior (Fig. 1 and 2 a. ¢. m.) and passes
into the plexus coeliacus, ii which in Ornithorhynchus two separate
ganglia are to be distinguished. The plexus coeliacus also receives
a branch of the Nervus vagus.

The second group of nerve fibres runs to a little quadrangular
ganglion, situated in the peritoneum at the medial side of the kidney,
which I will call ganglion renale (Fig. 1 and 2 g.r.). The ganglion
renale sends off small nerve fibres to the kidney, to the plexus
coeliacus (or in opposite direction) and continues at its caudal end
into a nervestem, which runs parallel with the aorta.

In Ornithorhynehus this nervestem ends in a ganglion (Fig. 2 g.g.)
which also receives the already described caudal limiting branch for
the abdominal viscera.

') W. Gasket. The sympathetic nervous system. Nature 1895.

2) N. Bisuop Harman. The caudal limit of the lumbar visceral efferent nerves
in man,

Journal of Anat. and Physiol. Vol. 32 pg. 403.

From this ganglion (Fig. 2 g.g.) nerve fibres pass to the Arteria
mesenterica inferior and to the peritoneum in which they could be
traced unto the urmary bladder. In Echidna I found two separate
ganglia. In the topmost of the two enters the caudal limiting branch
from the sympathetic chord, from the other the nerve fibres run in
the same direction as I mentioned for Ornithorhynechus.

When we compare Kehidna and Ornithorhynchus, it appears that
Ornithorhynchus approaches a little more to the condition of placental
mammals in so far as in this species we find a little ganglion
eervicale supremum which is missing in Kehidna and the sympathetic
chord does not enter directly into the suprarenal body as is the
ease in Kehidna.

Description of figures.

Fig. 1: Sympathetic chord and principal branches of Echidna
aculeata &.

Fig. 2: Sympathetic chord and principal branches of Ornithorhynehus
paradoxus 9.

@. ¢. s.: ganglion cervicale supremum.

2. ¢.: ganglion cervicale.

n. 1. s.: nervus laryngeus superior

re. n. 1: ramus externus nervi laryngei sup.

r. ¢.: rami cardiaci.

n. r. nervus recurrens vagi.

n. ph. nervus phrenicus.

a. s. d.: arteria subclavia dextra (too large in’ relation to the
other arteries).

a. i. p.: arteria intercostalis prima.

a. ¢. m.: arteria coeliaca +- mesenterica superior.

g. s. r.: corpus suprarenale.

n.: kidney.
o, r: ganglion renale. ;
e.: ganglion at the origin of the art. mesenterica inferior.

ge of

a. m. 1.: Arteria mesenterica inferior.
s. s.: left sympathetic chord.

(96 )

Geology. “Some observations on the geological structure and origin
of the Hondsrug’. By Dr. H. G. Jonker. (Communicated by
Prof. K. Martin).

For some years already I have been occupied in making obser-
vations on the structure of the Hondsrug. These are far from being
complete yet and also for other reasons it will be impossible for me
io examine that material with proper care for some time to come.
Indeed, I should have had no reason for writing about if so soon,
had it not been for an essay on this subject by Prof. E. Dusors at
Haarlem published some years ago, in which a view is taken alto-
gether differing from the opinion generally adopted till now. As his
Opinions seem to me to be wrong and yet have been propagated to
an undue extent by their insertion into a little book for the use of
schools *) — 1 have to thank Dr. J. Lori for this information —
I consider it my duty at once to develop in a somewhat detailed
criticism, why I do not agree with him in his opinions.

Eve. Dupo: “The geological structure of the Hondsrug in’ Drenthe
and the origin of that ridge. (De geologische samen-
stelling en de aize van ontstaan van den Hondsriug
in Drenthe)?

Roy. Acad. of Se., Proceed. of the Sect. of Se., Meeting of June 28, 1902;
Vol. V, p. 93—103. (Versl. v. d. gew. Verg. d.. Wis- en Nat. Afd. van 31
Mei en 28 Juni 1902; DI. XI, 1, p. 48—50, 150—152.)

Eve. Desois: “La structure gcologique et Corigine du Hondsrug dans

la province de Drenthe.”

Arch. néerl. d. Sc. exact. et natur., Sér. 2, T. VII, p. 484—496; 1902.

Further see:
J. Lor: “Beschrijving van eenige niewve grondboringen, Vy? p. 20—21.
Meded. omtr. d. geol. vy. Nederland, verz. d. d. comm. y. hl. geol. onder-
zoek, n° 33.
Verh. d. Kon. Ak. v. Wet., 2e Sectie, dl. X, n® 5; 1904.

EvuG. Dunois: “On the direction and the starting point of the diluvial
ice motion over the Netherlands. (Richting en uityangs-
punt der diluviale ijsheweging over ons land).”

Noy. Acad. of Se., Proceed. of the Sect. of Sc., Meeting of May 28, 1904;
Vol. VII, p. 40—41. (Versl. v. d. gew. Verg. d. Wis- en Nat. Afd. v. 28 Mei
1904; dl. XIII, 1, p. 44—45).

1) Overzicht van de geologic van Nederland (“Outlines of the geology of the
Netherlands”), written by a teacher for his pupils.

(97 )

H. G. Jonker: “Bijdragen tot de kennis der sedimentaire zwerfsteenen
in Nederland.
I. De Hondsrug in de provincie Groningen.
1. Inleiding. Cambrische en ondersilurische zwerfsteenen.”
Acad. Proefschrift, Groningen, 1904; Stelling XV.

First of all this remark. By the Hondsrug is usually meant a ridge
extending from Groningen to Emmen in a nearly N. W.-S. E. direc-
tion. It should however not be thought that a perfectly continuous
ridge even of but a small height is found here. He who goes by
bicycle from Groningen by way of Zuidlaren, Gieten, Gasselte, Borger,
Odoorn, to Emmen, will frequently find much difficulty in reecogniz-
ing the ridge. As then the connection between the elevations
which in many places are distinctly to be seen, is not always per-
ceptible, and the examination of various of those parts has shown
a great distinction in structure, if is advisable to be very careful in
dealing with conclusions drawn from examinations of one part.
Though it may be probable that the origin of the whole Hondsrug
is attributable to a single factor, this cannot be adopted a priori and
must be proved by comparison of the examination of the single parts.
Dusois has examined the southern part of the Hondsrug between
3uinen and Emmen, and has drawn conclusions from the observa-
tions there, which according to him hold good for the whole Honds-
rug and even for the whole region of our Northern provinces. It
might be expected therefore that the author had tested his new hypo-
thesis by previous observations of the region not examined by him.
This, however, has not been done; it seems to me that, if he had
taken previous researches properly into account especially those made
by Van Canker into the Groningen Hondsrug his opinions would no
doubt have partially changed '*).

In the first communication — for particulars the reader is referred
to the English text, to which the cited pages mentioned below also
refer — the author demonstrates that the nucleus of the Hondsrug
in South-Drente consists in diluvium of the Rhine, over which the
glacial diluvium is pretty regularly spread; on the ridye itself in the
shape of a bed of houlder-sand, seldom attaining a thickness of 1 M.:
on the sides frequently as more or less thick banks of bowlder-
clay. These rather local observations suggest to him the hypothesis
that all the land-ice has not reached our country in a direction
nearly at right angles with the Hondsrug, but on the contrary

1) The essay does not mention a single source. It seems to me that this makes
it difficult for the less expert reader to form an opinion.

(98 )

has flowed in the longitudinal direction of the Hondsrug over our
Northern provinces. In accordance with this was the mutual sliding
of the parts of a split boulder of quartzite observed by Dusors.

That is all. To preclude all misunderstanding I shall quote the
following passage (p. 100):

“The situation of the elevated ridge of preglacial sand side by
side with the long and broad western‘) strip of boulder-clay makes
us also suppose that the direction in which the ice moved was not,
as is still generally admitted, from north-east to south-west or from
north to south *), but the same as the extension of the Hondsrug,
from north-west to south-east. Now with this supposition perfectly
agrees the at first sight paradoxical direction of motion as derived
from the shifted boulder of quartzite.”

After stating this hypothesis the author attempts to explain his
observations more in particular by this. IT need not enter into these
explanations. Be it only said that he tries to support his hypothetical
direction of the ice-flow by a second supposition about the possibility
of the forcing back of the Seandinavian glacial flow by’ one coming
from Scotland in the following words (p. 101):

“Now that it is known that the direction of ice-streams which
ended in North-Germany has often been considerably modified by
the form of the basin of the Baltic and also by the meeting with
other ice streams, it is less surprising, that, notwithstanding the
predominating or exclusive occurrence of Swedish, at least Seandi-
navian*) rock species in the bottom-moraine of our north-eastern
provinees, these can nevertheless have arrived there in north-west-
south-eastern direction. Suchlike factors, as supposed to have modified
the direction of the North-German ice streams, may have been the
cause of the deviations of an ice stream, which, coming from Sweden,
jirst took a south-western direction over Denmark *), till it arrived in
the North-Sea. We do not know how far the ice which came down
from southern Seotland and northern England did progress south-
eastward in the North-Sea; if might be possible, at least, that as a

4) The English text is here not perfectly corresponding with the Dutch (p. 49);
see for this the note on p. 99.

*) I do not know who ascribes the diluvium of our Northern provinees to a
elacial flow directed from North to South.

%) This addition again suggests that the author may think of Norway as the place
of origin. Besides the greater part of the boulders in the ground-moraine of the
Hondsrug is less of Swedish than of Baltic origin. More about this question
below.

4) The italics are mine.

(99 )

very powerful stream it has met there with the ice stream coming
from Sweden and has pushed this back south-eastward in the direction
of Friesland, Groninghen and Drenthe.”

I wish to show first that the glacial cover of the Hondsrug in
Dvusots’s sense does not exist. For this it is sufficient to prove that
in various of the highest places of the Hondsrug boulder-clay occurs.

1st. Moreover Dusois himself, in his second communication, in
which numerous observations of the occurrence of boulder-clay in
South-Drente are enumerated, states its presence in various places in
the Hondsrug between Buinen and Exlo, it is true very near the
Eastern border but in the highest points of the Hondsrug (p. 102)').
At a recent examination of the said section of the N. EK. Loeal
Railway this also proved to me to be the case.

2d, Boulder-clay occurs further at the highest point of the Honds-
rug near Gasselte. There the N. E. L. R. cuts the ground to a
depth of 5 M. and at the same place where a bridge has been
constructed over the new railway (about the highest point of the
neighbourhood) a bed of boulder-clay 2 M. thick is found under a
thin sandy layer of vegetable earth.

3". Moreover I wish just to make mention of a clay-pit near
Zuidlaren, about 1*/, K.M. outside the village, about 300 M. north
of the road from Zuidlaren to Vries. Though there the hilly character
of the Hondsrug is less distinctly to be recognized, it is easy to see
that the mentioned place is one of the highest of the surroundings.
The clay-bed is 3 M. thick there.

4h) Finally I wish to remind the reader of the characteristics
in Groningen and south of it. Though the Hondsrug is hardly
noticed there, it is most characteristic as regards the shape. Well
then, there, in numerous places, boulder-clay occurs very often at
the highest places; on the borders there is usually more sand.

[am not of opinion that from these observations, rather regularly
spread over the Hondsrug, it follows that boulder-clay, quite contrary
to Duxots’s opinion, should chiefly or exclusively occur at the highest
points. For a conclusion a most accurate and extensive examination
is required. | only wish, by the way, to call attention to an

) Sull the author sees no reason in this lo drop his hypothesis, though he
says on p. 102:

“The origin of the Hondsrug according to the hypothesis indieated in the former
communication can thas only be applied to that western strip of boulder-clay”,
but for the rest he maintains the opinion once pronounced as also appears from
his answer to Lorué’s criticism cited before,

( 100 )

opinion already pronounced with regard to this question in the
“Report of the Board of the Dutch Society for the Reclaiming of
Heaths to the Provincial Government of Drente about an inquiry
into the character of the waste land in that Province” '), where
may be read on p. 16—17 inter alia: “Red clay especially occurs
on the Hondsrug and chiefly in its highest parts.”

On the other hand I do think I am entitled to say that from
what is said above appears sufficiently that a distinction between
boulder-sand on the ridge itself and boulder-clay only along the
sides is in reality wanting.

Moreover the author has, in my opinion, not satisfactorily proved
that the boulder-sand of the elevations in South-Drente cannot have
been originated by the wash-out of boulder-clay. He mentions the
following reasons for this (p. 98—99) :

Is. “The hard boulder-clay offers great resistance to eroding
agencies. This appears amongst others from its forming steep and
more or less projecting parts at the coast as the Roode Klif, the
Mirdumer Klif and the Voorst, and even islands, as Urk and
Wieringen.”

Of course, there is no denying the truth of this statement, though
something might be said against it as regards the difference in action
between lateral and normal erosive agencies. But moreover may be
argued against this that the boulder-clay of the ground-moraine has
in a much greater number of places partially or altogether disappeared,
no matter how this may have happened. Besides all kinds of
intermediate stages between original boulder-clay (as original as we know
of, at least) and altogether washed-out boulder-clay may be observed.
In Drente e.g. boulder-clay is nearly every where washed-out so much,
that all limestone has vanished from it. If we consider what impor-
fant quantities of rock have been lost in this way, the powerful influ-
ence of such a solution and wash-out cannot be denied. In other
places, on the contrary, the limestone is found preserved, but nearly
al the finer parts of the clay washed away, so that the boulders
lie in more or less clayish sand (which can also vary in many ways
with boulder-clay which has remained. more or less intact). I mention
these examples to prove that a general appeal to the resistance of
boulder-clay to erosive factors in this particular case is of no value.

1!) “Rapport, uitgebracht door het Dageliyksch Bestuur der Nederlandsche Heide-
maatschapplj aan de Provinciale Staten van Drenthe omtrent een onderzoek naar
den aard der woeste gronden im die provincie.” (Tijdschr. d. Ned. Heidemaatsch.,

Jg. XI, 1900.

(101 )

24, From his observations Dusois has calculated that about '/,, of
the volume of the boulder-sand bed has consisted of boulders and as
in that region boulder-clay is very poor in stones, boulder-clay of
enormous thickness must have been washed out.

I have not repeated this computation, but have this objection that
I have often observed that the percentage of stones in the boulder-
clay — which indeed is very different in various places — increases
very much towards the surface. The required thickness would decrease
very much by it and as we know so little about the original thick-
ness of the ground-moraine, this reasoning does not seem to me to
settle the question.

3", “The boulder-sand contains very little flint, the boulder-clay
very much, everywhere. Flint is the kind of rock most frequently
occurring in the clay (Odoorn, Zwinderen, Nieuw-Amsterdam, Mir-
dumer Kklif, Nicolaasga, Steenwijkerwold, Wieringen, ete.).”

First of all the remark that the places outside the Hondsrug,
mentioned here had better not been taken into consideration. As
regards the Hondsrug the decision that flint in boulder-clay is the
prevailing rock is in its generality no doubt wrong. In clay of the
Hondsrug in Groningen flint is very rare indeed. For example: When
a pit, about 2'/, M. deep and a diameter of 3 M., was dug in the
garden of “Klein-Zwitserland” near Harendermolen (the soil consists
there chiefly of clayish sand and sandy clay, but with very much
limestone), there was not a single flint among some thousands of
boulders which were produced! This is, less strictly taken, every-
where the case there. In the Hondsrug in Drente I found some more
flint in various clay-pits, but it was never predominant. Moreover
through the disappearance of limestone the percentage is doubled.
In this respect the Hondsrug differs very much from other parts of
our glacial diluvium and this in my opinion very interesting cha-
racteristi¢ will have to be explained satisfactorily. — To mention
also. some observations outside the Hondsrue which areue the
reverse: the boulder-sand e.g. near Roden is exceedingly rich in
flint, as it is in Steenbergen, ete.

This flint therefore does not prove anything.

4". “Even the deepest and evidently not washed out parts of the
boulder-sand, which rest immediately on the Rhine-sand, are as a
rule poor in clay.”

I do not know what enables Dupois to state that they are not
washed out. I beg to remind the reader of the vanished lime-stone
it contained and the numerous brown veins sometimes as thick as
an arm, which occur mostly under the boulder-sand in the white

( 102 )

river-diluvium and according to the author himself have come from
the upper-stratum (p. 94—95).

5, “Boulder-clay and boulder-sand are found jointly or the
latter alone without this being expressed in the form of the surface.’

I cannot subseribe to this. My examination has not yet led to an
established opinion, but in some places I ean decidedly conclude
from the relief whether we have to do with boulder-sand or with
clay. I take for example the already mentioned hill near Gasselte,
which runs nearly in the longitudinal direction of the Hondsrug. At
the top we have a thick clay-bed there, which towards the sides

passes into a thinner layer, at the deepest places altogether consisting
of boulder-sand. Further observations in this direction are of course
absolutely necessary before formulating this rule in general.

Taking all this into consideration, I am of opinion that, from the
five mentioned reasons, it does not follow that the boulder-sand eannot
be washed out from the clay. In my opinion the author has not
taken either of the two ways in which this problem can possibly
be solved :

1st the comparative mechanical analysis of boulder-sand and boul-
der-clay ;

2 the study of the general petrographical nature and the charac-
teristics of the surface of the enclosed stones. Nothing has been said
about this, whereas it seems to me that only in this way it might
perhaps be proved whether that is to be looked upon as inner-
elacial-moraine and not as washed out ground-moraine.

Taking all this together it gives sufficient proof that the superficial
structure of the Hondsrug does not correspond with Dusots’s opinion.
The direction of the glacial flow derived from this opinion is not
supported by anything, apart from the piece of quartzite about which
I have little to say. In my opinion only one observation like this
has but very litthe value; a greater number of course would be of
great importance.

Moreover as regards the possibility of a deviation of the direction
of the Seandinavian glacial flow owing to that of Seotland, I wish
to make the following remark :

Not without some surprise have | noticed that DuBois represents the
glacial flow from Sweden as moving first in a South-ewestern direction
to reach our country over Denmark, though one of the principal
results of the examination of our boulders by K. Martin, vAN CALKER
and SCHROEDER VAN DER Konk is that the glacial flow which has
produced the glacial diluvium in the North of the Netherlands has

( 103 )

been a Baltic one. This makes the author’s opinion unexplainable, the
more so as my own researches of the last years especially into the
Groningen Hondsrug have completely confirmed this result. Though
the examination of the diluvial boulders of Groningen will at least
take one or two years more, yet | have got so far that I commu-
nicated part of the first results of my study at the 10 Physical
and Medical Congress at Arnhem. I refer for this to the Proceedings
which will no doubt soon be published and only mention here that
the glacial flow which has created the Groningen diluvium originated
somewhere in North-Sweden, passed the Alands-islands through the
Gulf of Bothnia to the South and South-/ivs¢ and further reached
our country passing between Oesel and Gothland in the longitudinal
direction of the Baltic. About the question whether the direction
particularly derived as regards Groningen also applies to all the
other parts of the Hondsrug, | will pronounce no opinion as yet.
This must be examined more closely. Yet there is no denying the
prevailing Baltic chavacter of the glacial flow. Such a glacial flow
then, would have to be supposed to be deviated about 90° through the
one coming from Scotland. Perhaps after this elucidation Dusots, too,
may abandon the supposition. Otherwise | should like to point out
that, — from a parallelogram with the directions of those glacial flows
being the conterminous sides and that of the Hondsrug the resultant, —
it appears that for such a deviation a force must be ascribed to the
Scotch flow much greater than that of the Baltic, a conclusion which
is diametrically opposed to fact. It is the Scandinavian land-ice which
has forced back the Seotch, witness the numerous Norwegian boulders
on England’s east coast, and not the reverse. Nothing has ever been
heard about English erratics in the Netherlands. If I should find them
this year in the Texel, | hope to communicate this at once. Till
that moment this supposition is altogether unsupported and everything
argues against it.

In the preceding pages I have made an attempt at refuting Dupots’s
supposed deviating glacial flow. This seems first of all necessary to
me, because in case this conception is the right one a great number
of researches into our “Scandinavian diluvium” would become doubtful
and it would be advisable at once to begin a revision. Fortunately
there is no reason for this yet.

As regards the remaining contents of the discussed essay, | wish
to remark that I also object to looking upon the Hondsrug as a whole
as being a terminal moraine by itself. It would carry me too far

( 104 )

here to traee how and on what grounds this name came to be
eeneral. | intend to explain this afterwards at some length.

Finally one remark more. The nucleus of the Hondsrug in South
Drente is rightly said to be ofa fluviatile nature. Yet it may be asked
whether the boulder-sand, to keep to this, immediately rests on if
and whether nothing can be observed there of formations known as
stratified mixed and stratified glacial diluvium. This would indeed be
very striking and in fact this is not always the case, though I must
acknowledge that I have found in the discussed part of the Hondsrug
only with great difficulty some profiles in which somewhat acute
bounding lines may be observed. | must however put off this dis-
cussion. | have mentioned here only so much of my own observa-
tions as was strictly necessary ; in a complete treatise of it I hope
to have an opportunity to enter into the question of the origin of
the Hondsrug more in particulars.

Groningen, Min.-Geol. Institute, June 6, 1905.

Astronomy. — “Approvimate formulae of a high degree of aceu-
racy for the ratio of the triangles in the determination of
an elliptic orbit from three observations UL.” By J. Weeper.
(Communicated by Prof. H. G. vAN pr Sanpw Bakuuyzen).

In connection with my paper on the same subject read on
22 April 1905 IT now intend to derive simple approximate formulae
for the ratio of the triangles, which contain the 3 > times of
observation and the heliocentric distances belonging to them, and
include the terms of the 5 order with respect to the intervals
of time, it bemg easy to add, if necessary, those of the sixth order.
The same problem has been treated by P. Harzer, and in the
developments at which he arrived he attained a much higher degree
of precision’). Nevertheless it appears to me that his publication
does not render mine superfluous because of the different methods
of the treatment and the conciseness of my results.

After the method, followed by Gusps, to derive his fundamental
equation, we find with satisfactory approximation a general relation
between the values of a function /’(t) at the three instants, its
second derivatives with regard to the time #(r) at the same instants

1) P. Harzer, Ueber die Bestimmung und Verbesserung der Bahnen von
Himmelskérpern nach drei Beobachtungen. Met einem Anhange unter Mithilfe
von I, Ristenparr und W. Eperr bereclineter Tafeln. Leipzig 1901. Publication
der Sternwarte Kiel XI.

(105 )

and the two intervals of time t, and t,. The value namely, of the
following expression

fo aE UT, eae’ T Staten a A a
iGer Si) — AGe +4, aa +r, (F+é, = )

where tr, =1, + 1,, is of the 6 order with respect to the intervals.
I use the letters C,, C, and C, to designate the multipliers of the
second derivatives in this expression and put

Y

3 3
TTT, +T, C Ati

aa 12 Es OO Say Taam

Neglecting the terms of the 6 order we then have for an arbi-
trary function of the time, the relation

a One eee Oana eH (Cl —=.0)) oo (NZ)

provided this function and its first four derivatives be continuous
and finite within the interval r,.

Applying this formula to the heliocentric distance + and to 7’, 1
obtain approximate expressions for the semi-parameter p, and the semi-
axis major a of the elliptic orbit. By eliminating p, from the two well-

known differential equations *r* + r=p and 7? = — — — — — sil
find a differential equation which may be easily reduced to

a? 1 1 y : . p-—?r
ii = — |... sceording. to: these — relations: == ———
at a

, ,

= a ill 1
belongs to F=r, and F=2 (- — =) Ko) J SS
7 a
If as before I put 2 for '/,3, the substitution of # =r in formula
1V yields the following equation to determine p,
eee Ca atalmeCal sal Cae ma) Cae (Ge ,) — C,2,.(p — F,)\—= 0
whence

Cc

Ge a r, =f =16 = + (ty + Cyes) *
= Pe Se aan )

Through the substitution of “=r? in IV I obtain the equation

: ay oe eee Al 1 a fa 1 es 1
T,7,7—T,P,° + T%, — 20, 2C, 2C, — == 0
Ty rm @ ay

whence

9 C, C, C,
1 Sl Aa ra SUN ae aes
ELS (VI)

a 2G, + 6,40)

( 106 )

The terms neglected in these expressions for p and are of the
a

3¢ order with respect to the intervals of time.
I shall now proceed to show how we can avail ourselves of these
1 : :
values for p and — for the calculation of the ratios of the triangles.
a

In my previous paper I have demonstrated that the area of
the triangle PZP, considered as a function of += / (t—t,) satisfies
the differential equation #’-++- 2/0. The same differential equation
is satisfied by the area of the triangle P?,ZP, considered as a
function of t=A4(t,—t). The two areas may, according to Mac
Laurin be expressed in series of the ascending powers of t. If
the variable t takes the value /: (¢,—/,) = T,, the two triangles become
equal to P,Z2P,; therefore it will be possible to obtain a new expansion
in series for double the area 4 P,ZP,, by putting in the sum.of the
two former series t=t,. From this new series we can_ easily
remove the terms with the even powers of t.

According to this plan I give here first some higher derivatives
of the function /, expressed in F, F, 2 and derivatives of z with
respect to the same variable.

gn ee

EN (22 =) =e

HV = (423 — 2) F + (2? — 32) F

FV = (— 23 4 437 4 728 — ZIV) P42 (822 — 22M)

FY m= (0. Se cis eee 3 See =b 0s be ee

triangle PZP,

== Pikt—e)|=—AG
[t= FO)

and that of its first derivative is known, viz. /, = Oand /,=-+ }.
The above mentioned expansion in series for APZP, is therefore :

For t=0, the value of the funetion

APZP, ae 1) ct 1h Ge ce
Vp BB Br ag el eee
1 es 7 aa yl He

++ > (182,2, + 102,*— 52,1¥ — z,*) 7 1) = Vil (e— a) eee
a (i (i
0
meee. LZ Rise
The function V; = G |k (t,—0| = G@ (r) and its derivative also
/p

take for ‘= 7, or t=O the values G,=O and Gi, =-+ 3, and so

( 107°)

for this function, beeause it satisfies the differential equation G'--z G0,
the same expansion holds as for /'(t), but while in the series for
APZP, the derivatives are taken with regard to increasing time,
those in that for 4P?,7P must be considered with regard to decreas-
ing time. If we make use of the symbols 4, 2, 21 ete. to denote deri-
vatives of 2 with regard to increasing time, the signs of the odd
derivatives of ¢ in the expansion for AP,ZP must be reversed.
Hence we obtain for AP,ZP:

LSP ZE? ile ice pies ae GE i Nae a
mmm mt 37 hae 8 gy Oa A Dee
1 132.2 102.2 ha» IV +3 mo nl Gvyul 1
alate ( D2a%_ + 02, — 02; ar heist) qt Fine (x — u) du
0

and by summation of the two series, for t= 1r,

A P,ZP. iE ie ae oe A Peuace
BS eee
Vp 1 Zeleol! 4!

1 wae
+|56. 83) + 5 G32 fe —fost—2 224!) (82,2, —2 zee

Ea 103,?—5z,".—2,° i 132,2, +103,? | Tal

2) 9

3

Ef aE (eu) + GY (eu du.

0

It appears that in this formula the terms with even powers of
t, can be transformed into series of terms with the higher odd powers
of r,. In order to do this I derive an expansion in series by which
this aim is reached in a general manner for the difference f(y) — f (2), f
being an arbitrary function which between « and y does not show

. "oe . . i
singularities. Let here t be put for y—a, and m for ied then
+7/s
Fy) — Ff (e) =| J '(m + u) du and after integration by parts:
9 ategha

POSE) E LMJ uf om + 9 de
a tf,

Proceedings Royal Acad. Amsterdam. Vol. VIII.

( 108 )

suai ip
As { uf" (m) du = 0 and f" (m + w) — f" (m) =fr (m + v) dv,
=I as als
we may write instead of uf (m+ u)du, the double integral

u ="),
fo du “f 7"! (nm + v)de which by reversing the order of integration
0 +7/, +7/,
is transformed into f (m ++ v) ww fx du.
—7/, v

I now proceed to integrate with respect to w, and so we obtain
the following relation :

ae ie
; ; Tr, “4 1
*(y) — f(#) = [7 @) + Ff (2) — 3 =i MT (m + v) (tx? — 4 0?) de.
—t/,

The operations may be repeated and by doing so we shall find:

Os 2) — 57 UF @) + Fl +
1.7/5
1
-f- fe — 4u?) (5 1? — 4u?) f¥ (m + wu) du.
384 ;
a Gif

The expansion may be easily continued in the indicated manner,
but for the end I have in view that deduced above goes far enough.

If according to this formula we replace +, — ¢, by
T mee
= (2 + 2,)— oil (<,!Y + <,!Y), terms of the fifth order are neglected,

and if we replace (8 z, 2, — 2 2,17) — (8 z, 4, — 2z,™) by

= (32,” 2 ar 3z,2 oa CE ar 32,7 a 32,2, rma 22, )s

we neglect a quantity of the third order. I propose to terminate
NPI
the expansion of 2 ———— with the term in t,’, then the above
Vp
mentioned substitutions will not alter the order of approximation.

So we obtain the following approximate formula:

( 109 )

> APGP, Relea a ze fig i ee . Iiyas 2
= Vp = Oh a +e 2a) 2 2 St (Gea
” , - 3 y 3 oe Paneer s "4 t :
a a8 $54 $ Hel 4 be) + eZ $54 tte Hey

The development is symmetrical with respect to z, and 2, and
their derivatives, and resolves itself into two parts, which have the
same form, and which depend besides on 7,, only on the value of
z and those of the derivatives of z at one point.

if the following series

r 5

oa T 5 = 3 > y 2
Set eigen = 1 + 22,) = — (82.2, + 12,7 + 842," + 2,°)

ri
Tie
where only ihe odd powers of the variable t occur, be denoted by U(r)
and the corresponding series for z, and its derivatives by U(x), we get:
AP,ZP,

Vp
and the ratios of the triangles may be expressed in the following
way in these functions U:

2

= 2 {U,(r,) + U,.(t;)

AP,ZP U(t,) + Use;
ernie ceaiy = 8s i(t5) +E ! G5 (Vila)
AEP: U («,) + U AGa)
and
AP.,ZP. U U
at eae CIS) (VL1b)

5 ? = .
AP,ZP, U,(t,) ar U,(t,)
In the series U(r) only such differential quotients occur as can be
1
rationally expressed in p and. By means of the known differential
a

equations of the 1st and of the 2"¢ order for

: 2 1 P ‘. Dp il
No i nd —
r a 7? ? r?
; pad nals 1
we obtain by differentiating 2 = —
r
pane (a ae en
a i
LOS Gp eee
a 7
oie ae ; ? Cee ANE
while from the differential equation 21% — 5 — — roa 2 by diffe-

rentiation with respect to + and by elimination of 2” the following
expression is found for 2!V.
Q*

52 s a ) 23
Pa Meseep pes 9) (teehee Dy cay Pea
ZN aes 2 (6; 4 = Om,

As —, p and the 3 heliocentric distances 7 are known, 3’, 2
a

and <'Y can be computed for each of the 3 points P,, P, and P,.
For a circular orbit all the derivatives of z are equal to zero

sin
and the function UU becomes 7s According to the preceding

development we obtain for an elliptic orbit the following approxi-
mate formula for UY, which still contains the 6 power of the
interval :
sint Vz oe r p
8 Quid J 5 ad) ee aT
a r
By means of the values which. take U,, U, and U, for the
values 1,, t, and +t, of the argument +t, we obtain for », and
n, values containing the terms of the 5 order with respect to the
intervals; while the approximation may be extended to the 6" order,
if we add to the above mentioned expression for U:
id
— 2° (— A — 120u + 1702? + 340Au — 1020u°).
5040
where 4 and uw denote :

3 r p
=o
2 a r

Astronomy. — ‘Supplement to the account of the determination
of the longitude of St. Denis (Island of Reunion), executed
in 1874, containing also a general account of the observation
of the transit of Venus”. By Prof. J. A. C. Ouprmans.

When I set about to correct the imperfections left in my first
communication, | began by calculating for the times of observation
of the occultations the correction of Newcomp’s parallactic correction,
mentioned on p. 603 of my previous paper; as said there this
correction amounts to

+ 0"67 sin D + 0"05 sin (D — gq) — 0"09 sin (D+ Q'),
where D stands for the mean elongation of the moon from the sun,
g for the moon’s mean anomaly, and g’ for that of the sun.

(Sali.

D and g could be derived from Tables I and If at the end of
Newcomp’s paper’); there, however, in agreement with the method
introduced by Hansen in his tables of the moon, the unit is not a
degree, but a mean day, so that the numbers derived from those
tables must be multiplied by 12°,19 and 13°,065 respectively in
order to be reduced to degrees. gy’ could be derived with a sufficient
accuracy for our purpose from the tables of Larerrrau in the Conn.
des Temps for 1846.

The corrections found are, however, not to be applied to the true, but
to the mean longitude, called by Hansrn n dz, and therefore must be
reduced to corrections of the true longitude by multiplication by
(1+ 2ecosg....), and these must again be reduced to corrections
of right ascension and declination; for the latter reduction I used
the moon’s hourly motions in R. A. and declination from the Nautical
Almanac, whence the direction of the moon’s motion with regard
to the parallel could be directly derived.

The corrections of the moon’s ephemeris in the Nautical Almanac,
given by Newcomp on p. 41 of his Jnvestigation for each day
from 1 Sept. 1874 to 31 January 1875, were corrected for the
two first months by means of the values found; and in the calcula-
tion of the longitude from the occultations we have now applied
these corrected corrections, instead of the corrections furnished by
the meridian observations. It then would become evident which
corrections were to be preferred, and it soon appeared that it was
the former. The large corrections in declination found for 19 Sept.
and 16 Oct. 1874, (— 4.3 and — 4".1) for instance, in consequence
of which the second occultation observed on 19 Sept. was formerly
rejected, were apparently due to the inaccuracy of the meridian
observations.

I now shall give the details of my calculation. (see table p. 112).

These corrections were added to those given on p. 609 of my
first account as being interpolated from Nurwcoms, and then the
required alterations were made in all the calculations of the oecul-
tations.

3efore I passed on to this second communication, I have once more
thoroughly revised all the computations and thus was able to apply
some corrections; some occultations which had been rejected, could
now be retained after the error had been corrected.

1) S. Newcome. Investigation of corrections to Hansen’s Tables of the Moon;
with tables for their application, forming Part IIf of papers published by the Gom-

mission on the Transit of Venus, Washington, Government Printing Office, 1876.

(112 )

4874 MTGr.| D 9 g! eae ana ant pie Gre Md ie Ae
Sept. 19 598 | 105°5 | 956e7 | 257° || +0764 | —or02 | ovo || --ove2 | +o"e0 | Loves = +0°0% | -Loro1
1 7.5 130.5 | 283.5 | 959 |! +0.51 | —0.02 | —0.05 |] +0.44 | +0.45 | 10.48 — 0.03 | 40.14

2 3.6/ 440.9 | 294.6 | 260 || 40.42] —0.02} —0.06 |] +-0.34 | +0.35°
40.34 = +0.02 | 40.11

9 10.3 | 144.4 | 298.4 | 260 || +0.39 | —0.02| —0.06 || +0.31 | +0 395
96 4.84903 | 347.5 | 964 || —0.42| —0.02| —0.09 || —0.93 | —0.26 | —0 92 = —0.015| —0 42
Oct. 2 41.5 | 267.4 | 69.8 | 970 || —0.67 | —0.01%| —0.00* | —0.69 | —0.71 | —0.79 = —0 05 | -40 07
& 41.6 | 991.5 | 95.9 | 979 || —0.62 | —0.01 | 10.04 || —0.59 | —0.58 | —0.60 = —0.04 | +0.18
45 3.8] 61.7 | 235.3 | 983 || 40.50] —o.01 | +0.02 || +-0.60 | +0.56 | +0.63 = +0.04 | —0.09
16 5.0| 74.3 | 948.8 | 984 || +0.64/ —0.005| 0.00 || +-0.64 | +0.61 | +0.70 = +0.05 | —0.02
17 7.2| 87.7 | 263.2 | 985 || +0.67| 0.00| —0.02 || +0.65 | +0 64 | +0.72 = +0.05 | +0.06
18 8.3 | 100.3 | 276.7 | 286 || 40.66] 0.00| —0.04 || +0.62 | +0.62) +0.68 = +0.045| 10.13
49 7.4| 412.5 | 980.7 | 987 || 40.62] 0.00| —0.06 |] +0.56 | +0.58 | +0.60 = 40.04 | 40.17

C18")

I shall briefly record the modifications of my previous account *).

The following are the numbers originally belonging to the obser-
vations mentioned there: 1, 3, 4, 7, 9, 10, 11, 12, 14, 16, 18 20,
22, 23, 24, 25, 27, 28, 29, 30; 31, 32, 34, 35, 36, 37.

Hence were rejected the numbers: 2, 5, 6, 8, 13, 15, 17, 19, 24,
26, 33, 38, 39.

N°. 2, disappearance of Arg. Z.
the ground of the harbour office on 19 Sept., yielded — 215.31 for
the correction of the eastern longitude. It appeared that this large
value was for the greater part due to the large correction (— 4"3)
applied to the moon’s declination as derived from the meridian
observations. The corrected corrections of Nrwcomsp were — 0s.45
and -+- 0.3 (that of the declination even with another sign), and
the correction of the eastern longitude became — 68.85, not larger
than several others.

Ns. 5 and 6, disappearances of Arg. Z. 311, Nes. 72 and 75
observed by me on the ground of the harbour office on 21 Sept.
It appeared that in reducing these two observations the correction
of the chronometer had been taken from the journal with a wrong
sign. After rectification of this error the results were satisfactory.

N°. 8, disappearance of a 9 magnitude star, observed by me on
the ground of the harbour office on 22 Sept. at 7'38™25s,07, hence
349° after that of 383 Capricorni. | have not succeeded in rectifying
this observation. Judging from the map which by means of ARGELANDER’S

233, N°. 77, observed by me on

Zonae had been constructed preliminary to the observations, it seemed
that the star could be no other than N°. 18 of A. Z. 255, but then
the correction of the eastern longitude would have been ++ 58s,24.
Supposing that an error might have occurred in noting down the
minute of the time of observation, I repeated the calculation adopting
the time to be 1 minute later, but now I got: Corr. of the E. longitude
+ 208,07. The time of observation ought therefore to be taken another
half minute later, but I did not hold myself justified to do so.

There still followed two oceultations, which were missed through
clouds, probably one of these two has been A. Z. 255 N°. 18, and
the star observed by me does not occur in A. Z. Neither ScHOnrELp’s
southern atlas, nor GiLu’s catalogue could help me to arrive at a
conclusion.

N°. 18, disappearance of 73 Piscium on 26 September, N°. 15,
disappearance of 538 Geminorum and N°. 17, disappearance of a star

') In the 3rd column on p. 607 a few clerical or printing errors have crept in:
for Cordoba IIL 1589 read Cordoba XVIIL 1589 and for Cordoba XVII 124 read
Cordoba XVIII 1612.

( 114 )

of the 6'/,"" magnitude, both on 2 October, were recorded as uncertain
and besides the results were too discordant. Full moon had occurred
on 25 September; so these disappearances took place at the bright
limb of the moon, and it is well-known how uncertain their observation
is then. In this ease a sudden disappearance can only be observed
with stars of the 1st or 2°¢ magnitude.

N°. 19, disappearance of b. AC. 5800 on 15 October, also yielded
a large negative correction of the eastern longitude (—13s,14), but
there was no reason for rejecting it; the disappearance took place
at the dark limb, the star was of the 6'/,!" magnitude, hence very
bright in the telescope, and in the journal of observation uncertainty
is not mentioned.

N°. 24, disappearance of A. Z. 223 N°. 48, observed by Mr.
E. F. van pe Sanne Bakuuyzen on 16 October, yielded + 335,70;
N°. 26 disappearance of a star of which the place was 20%5™16s
— 25°10'46", gave — 120s,6. Both had therefore to be rejected.

Nor was I more fortunate with N°. 33. The star as determined at
Leyden gave an unsatisfactory result (+ 21™38s) and I could not
find in the catalogues another star which fulfills the requirements.

I succeeded better with N°. 38. N°. 88 had been noted by
Mr. Baknuyzen as disappearance of § Piscium on 28 October; it
appears however that this star was not oceulted and that the oceulted
star could be no other than 24 Piscium; assuming this, I arrived
at a satisfactory conclusion.

In the case of N°. 39, disappearance of 59 Geminorum at the
dark limb on 26 November, we could only obtain a result, that
was not wholly inadmissible, by assuming a combination of errors.
Although each of these was in itself not quite improbable, it was
thought necessary to reject also this observation.

About N°. 27 I remark that on p. 607 I noted as observers $.B.
i.e. that both Mr. Sonrrrs and Mr. Baknuyzun observed the occulta-
tion (disappearance at the dark limb); as the time recorded by
Mr. Baknuyzen was 4 seconds /ater than that noted by Mr. Sorrrrs, |
accepted the result of the former as the more probable one, the more
so as it agreed better with the other results.

Generally, the endeavours to rectify the oceultations, which at
first seemed to have failed, have cost more work than those where
nothing was wrong.

Finally I must remark that Mr. Sorrmrs himself had corrected a
small error of computation in his reduction of the observations, made
to determine the relative position of our different observing places,
but had neglected to enter the corrected value in the final table of

(115 )

his results. In consequence of this error we must read for the longitude
east of Greenwich of the observing place on the ground of our
dwelling house, as given on p. 604 3'41™48s.06 instead of 488.11,

Taking into account the corrections mentioned above, the list of
results communicated in the Proceedings of March (p. 607 of the prece-
ding volume) must be modified as given at the end of this paper ;
see tables Iw and Id.

We then find as correction of Germain’s longitude :

Using disapp. and reapp. indifferently . . —2s.90+0s.64 (m.error)
resin them separately feo 2 9.) (0 811.220) .) )

It is much to be regretted we did not succeed in observing more
reappearances. There is always a greater chance to observe the
disappearances than the reappearances at the dark limb of the moon.
A short time after new moon until a few days after first quarter
we can easily see with a good telescope on the east side of the
moon stars of the 8", 9% or perhaps even the 9*/,'* magnitude, of
which the disappearance may be easily observed; no preparation
is required for this.

For the observation of reappearances at the dark limb, a prepa-
ration by means of star maps is necessary, which takes up much
time. We must calculate from hour to hour the parallax of the
moon in R.A. and declination and hence derive its apparent places,
draw them on the map, and then derive geometrically the instants
at which the stars considered must reappear. For the most southern
declinations the star maps themselves had to be constructed first by
means of ARGELANDER’s southern Zonae. Moreover it is always desirable
finally to derive more accurate results by a calculation according
fo the known formulae.

The operations described here have been executed as well for the
days preceding full moon as for those following it, and it was our
bad luck that in the latter part of the lination the weather was
always unfavourable.

After this revision of the caleulations a small negative correction
of the longitude of St. Denis according to Germain seems probable,
although its exact amount is uncertain. We have however, still to
consider what follows :

When in 1884 Avwnrs wanted to determine a fundamental meridian
for Australia’), for which purpose he chose that of SypNry, he
used 78 occultations observed from 1873 to 1876 in Windsor (N.

1) Astron. Nachr. Vol. 110 p. 289-—346.

( 416 )

S. Wales) by Trssurr and 18 occultations observed by Enipry in
1874 and °75 at Melbourne. He applied to the ephemerides of the
moon of the Nautical Almanac the corrections of Newcomp’s /nvesti-
gation and took for the relation between the moon’s radius and
the horizontal parallax the value 4 = 0.27264 found by me. (Vers?.
en Meded. Akad. Amsterdam Afd. Nat. 18* Reeks, Vol. X p. 25).

But as nevertheless a constant error might oceur in the obtained
results, he calculated, as a test, a large number of occultations,
which had been observed either at Greenwich or at places of
which the longitude had been determined by telegraph, viz: 31
observed at Greenwich, 25 at Washington, 40 at Nikolajef, 44 at
Oxford, 30 at Luxor, 46 at Strassburg, 13 at Leipzig, 7 at Vienna,
2 at Kénigsberg, 2 at Moscow, 2 at Pulkowa and 1 at Kiel.

Thus he was able to derive the correction to be apphed to a
longitude determined by a disappearance at the dark limb, and found
for this after a graphical compensation :

1873,0 aie COS)
1873,5 =A 1633
1874,0 ae HO:
1874,5 TEGO),
1875,0 at ae
1875,5 soa
1876,0 TESOL
1876,5 + 3,38,
1877,0 4 3.59.

I have on purpose given this table in full to show how constant
is the positive sign of the correction. For the reappearances AuwrErs
found a correction which in the mean was larger by + 0%,23. (Although
this value has been found having regard to weights, it yet seems to
me rather uncertain; I find for its mean error + 0%,64). It appears
from this that this correction is due not to an erroneous value of
the moon’s radius, but to a slowly varying error still left in the
tables of the moon.
and I consider this as

Now if we want to apply this correction
quite justified — we must also take for the relation & between the
apparent radius of the moon and the horizontal parallax the same
value as Auwers has used and hence apply the necessary corrections
to our longitude.

The formulae required for this could be easily derived. Let the
difference in right ascension between the moon’s centre and the place

(117

of the star, after the moon’s parallax, calculated for the point where
the occultation took place, has been added to it with a contrary
sign, be denoted by I; let the geocentric radius of the moon (as
it was used for this calculation) be = R, the horizontal parallax = I
and the difference in declination between the reduced place of the
star and the moon’s centre =v, then we have

er ee eee
i is sec SV RE
and hence
eR
== _ oR,
v— Vv

but as R= Ik
we have 0R= Tok

Pirin = LF Ok
hence ol aes Ok = ee 7

The reduction to be added to the stars R. A. to get that of the
apparent moon’s centre is = 1+ II, where the 24 term is indepen-
dent of / and the upper sign of the first is to be used for disap-
pearances, the lower for reappearances.

If the hourly motion of the moon in R. A. is Ae, the correction

1 ; : pias
of the Greenwich mean time of an occultation is — a < 386005
a
and the correction of the eastern longitude derived from it:
+ J — Bi 2 ; o
— >< 3600s. Now as the assumed value of & was 0,272525
La
we have 0/4: = + 0,000115:
ies 0.000115 eR
and Ola == 3600 < - 7 Nase 7
0. 272525 (Le? — wv") Aa
ele
== == [0,1814] ———_____

(R? — v) Aa’

where the value in square brackets is a logarithm, and the loga-
rithms of the other factors may be derived from the former cal-
culation. The + sign is to be used for disappearances, the
for reappearances.

sign

In this way I have found the corrections given in table II and
thus obtained corrected values for the longitude. I think it best to
use indistinctly the results from disappearances and reappearances.

( 118")

We then find as mean correction of

GrrMatn’s longitude: — 28,15 + Os,79 (mean error)
the supplementary correction according

to Auwmrs is for 1874,80 : + 28,71 + 05,50 *)
and the final correction is + 08,56 + 08,93.

Although our calculations were somewhat modified and a systematic
correction was applied, which seems to be required, we arrive at
the same conclusion as in our first paper, viz. that the correction of
the longitude of St. Denis found by Germain, in so far as we may
judge from the occultations observed by us, is very small. If we
pay attention to the mean error of our result, it is not even certain
whether it is negative or positive, though there is a greater proba-
bility in favour of a small positive correction.

In my previous paper I have not mentioned that the reduction
to 1874 of the places of the stars from all the available catalogues
has been very carefully executed by Mr. H. Kress, “amanuensis” at
the Observatory at Utrecht. The derivation of the most probable
places from the whole material I have made myself.

It will be interesting to record that the meridian observations of
the moon, made at Leyden in Sept. and Oct. 1874 by Mr. H. Haga,
then assistant at the observatory (now professor of physics at the
university of Groningen), has yielded the following corrections of the
places in tne Nautical Almanac, previously corrected according to
Nrwcoms’s /nvestigation: (see table p. 119)

1) This mean error has been estimated, and is based on the argument that the
value of the correction, whieh was found by graphic compensation, rests on about
25 occultations, while Auwers has arrived at the result (A.N. Bd. 110, column 336)
that one disappearance at the dark limb yields a longitude, of which the mean

error may be considered to be + Qs,5.

( 119 )

| | Obs. — Comp.
1874 Limb. | Remarks.
ly | Ace
e | |
September 21 I upper | —O 14 —4N7 Very unsteady.
| |
24! Tupper | +0.07 +1.3 | Clouds.
» 26 | If lower | =9198. |) 104
» 27 | I lower | —0.08 +1.9
» 30) IL lower —0 04 —5.1 | Clouds.
October ‘le |p Ut —0.07
» {5 | I +0.35 Very faint, uncertain.
» 20) 1 upper | 0.22 —1.4 | Very unsteady.
» 22 |} I upper 0.13 -+-0.8
» 24 | I upper 0.15 +1.8
» 26 | IL lower —0.08 41.1
» 27 | Il lower —0.12 +1.3
» 28 | IL lower —0.14 42.3 | Clouds.
» 30 | IL upper —0.14 —0.5
Mean value: —0°O4 —0"4

These results have not yet been published, but have been lately
communicated to me by Dr. E. F. van pe Sanpe Bakuuyzen.

It will be desirable also to consider the other determinations of
the longitude of St. Denis de la Réunion. Dr. E. F. van pr Sanne
Bakuuyzen kindly communicated them to me. These determinations,
whose results only we shall mention for brevity, were made by
Lord Linpsay and Dr. Copenand on the one side and by Messrs.
Low and Prcuite on the other side, on their respective observing-
stations Belmont and Solitude, both on the isle of Mauritius, the
differences of longitude of those stations and St. Denis being deter-
mined by transportation of chronometers.

Lord Linpsay and Dr. Copruanp') found for Belmont :

1) Dun-Echt Observations. Vol. HL. p. 171.

by means of 52 chronometers on the home

voyage *) 3'50408.03
from observations of the moon :
from 11 occultations (7 disappearances and

4 reappearances at the dark limb) 3'50™40s.60 + 08.33

from 12 culminations of the moon 42 6
Assigning to these two results weight 2 and 1,

we get as mean result 3450™41s,27
Reduction on St. Denis flag-staff determined

by transportation of chronometers — 8™533.41
Hence longitude of St. Denis flag-staff E. of Gr.

from the chronometers 354146s.62

from observations of the moon 47 .86

The German observers found for the longitude
of Solitude *)

from 6 culminations of the moon 3550™39s.52 + 38.29
from 3 occultations 40-33) 25-94
whence in the mean By OKO) NS} Se tL (55)
Now Solitude is situated 0s,89 west of Belmont,
Belmont 8™53 ,41 east of St. Denis,
hence Solitude SiO2 OD a Poe te
Hence longitude of St. Denis (flag-staff) E. of Gr. 3'41™475.61

Combining all these results, omitting only that from the chrono-
meters (comp. footnote) we have :

by means of the longitude of BuLMonr, Weight

(observations of the moon) 34147586 + 08,90 1,25
by means of the longitude of Souirupr,

(observations of the moon) AT 361 2220077025
determination by GERMAIN (culminations

of the moon) : al 40) == OG lee

determination by Ouprmans and Bak-
HUYZEN, (occultations with corrections
according to AUWERS) : 47 996 = 0 933 0eI6

Adopted longit. of St. Denis flag-staff 3>41™475,69 + 0s,44 4,38

!) Unfortunately the outward voyage has not yielded any result, because the rates of
the chronometers after the landing could not be determined, as it had been neglected
to wind them up. And this accident also takes off much of the value of the home
voyage, because through this the difference between the rate at sea and that on
land could not be eliminated. Auwers has already made this remark in: Die
Venus-Durchgiinge 1874 und 1882. Bericht tiber die Deutschen Beobachtungen,
Vol. VI p. 265, and therefore we shall also leave this result out of account.

2) A. Auwers. Die Venus-Durchgiinge 1874 und 1882. Bericht tiber die Deut-

schen Beobachtungen Vol VI.

The longitude of our station now being determined as well as
possible, I shall proceed to communicate our contact observations of
Venus and the Sun during the Transit on December 9".

Our place of observation was on the battery, in the immediate
neighbourhood of the pavilion of the heliometer; its longitude must
therefore be accepted to be 3'41™47s,81 + 0°,26 = 3541™48s,07.

The ingress took place very early in the morning, the sun being
only five degrees above the horizon. Unfortunately at sunrise the
sky was not quite clear. In the east, a few degrees above the horizon
there was a dark stratus, and it was to be feared that at the instant
of the second contact the sun would just be behind it. So it happened,
and this was the more unfortunate as the station Réunion had
expressly been chosen for the observation of that contact.

At the first contact the sun’s limb was very unsteady. At 5'38™20s
mean time St. Denis, I thought that I saw an impression on the
sun’s limb, which I held to be made by Venus. A passing cloud,
however, prevented me from seeing whether I had been right.
When, after a minute the sun reappeared, I could not distinguish
the impression on the limb any more. At 5'41™20s it could however
be seen plainly. The place where I then saw it was exactly the
same as that where I had thought to see it 3 minutes earlier. However,
as Venus had moved on 6" during those 3 minutes, the observation
of the first contact must be considered as having failed. The mean
between the two instants mentioned is no more than a very rough
approximation.

As said already, the second contact was missed.

But both Mr. Sorrers and I observed the two last contacts.

The formulae, given in the Nautical Almanac of 1874 on p. 484
for the calculation of the contacts, are:

For the first external contact:
t= 1345m58s — [2.5773] 0 sin /— [2,7049] @ cos Lcos (4 + 136°39'.9),
for the first internal contact :
t= 1415™24s — [2,6992] 0 sin /— [2,7462| @ cos / cos (A + 147°55'.7),
for the second internal contact :
¢=17557™26s + | 2,8253] 0 sin 1+ [2.5265] 0 cos 1 cos (4— 55°37'.8),
for the second external contact :
£=1826™54s + 22,7374 o sin] + | 2,5014} 0 cos Lcos(A —_ 37°50'9);

The times are given in Greenwich mean time; and @ stands for

the radius, / for the geocentric latitude and 4 for the longitude east
of Greenwich of the place of observation.

If in these formulae we substitute log 9 sin / = 9.5488 (—),
loy @ cos l= 9.9707* (4); 2 = 55°26'95 and add to the obtained
times the adopted longitude of the battery 3'41™485.1, the error of
which does probably not exceed one second, as appears from the

preceding investigation, we obtain the following results.
St. D a NS ie ea ti.
M. T. St. Denis Ons10! | S, |

Con- |M. T. Greenw. Obs -Comp.

II 147 58 43,5

21 40 31,6 | 21 39 46,2
IV |48 28 93, 4

22 10 11.5

— [m1 4s,0
—0 59,0

tact | comp. comp. 0. | Ss.
I | 13h55md55s.0 | | 7Th37M438,1 | 17h39m50s : | | +1 m1 7s |
IT |44 26 49,3 | 1SRISM eines | missed H |

Neither Mr. Sonters who observed with the telescope of the helio-
meter, nor myself who used the Fraunhofer telescope of Mr. pr
Bravrort’) have seen anything of the so-called black drop. The
former telescope was provided with the strongest eyepiece, magnifying
86 times, the latter with one magnifying 121,5 times.

I shall say only a few words here on the observations with the
heliometer and the photoheliograph.

The heliometer made by Maerz at Munich has caused me through
its numerous imperfections much trouble and numerous investiga-
tions relative to the instrument proved later to be valueless. Only
a few days before the transit took place we detected a defect in
the construction of the instrument, which rendered the adjustment of
the parallactic stand illusory, so that all the measured position angles
were unreliable. Nevertheless I have made complete sets of obser-
vations with the heliometer, viz. distances of the ‘‘Perseus-stars’, for the
determination of the scale value and during the transit two sets of
eight distances between the limbs of Venus and of the sun. This was as
much as the cloudy state of the atmosphere prevailing during the whole
transit would allow to do. The first set was made in the ordinary
manner, the other along the most advantageous chord. (Versl. en
Meded. Kon. Akad. Amsterdam, Nat. Afd. 2° Reeks, Vol. 1X, p.127).

The division errors of the scales must still be determined ; I hope
to do this soon and then to revert to these measurements.

As to the observations with the photoheliograph, unfortunately the
atmosphere, even in the moments that it allowed measurements with
the heliometer, had a very bad effect on the clichés made. The

1) In my previous paper I have erroneously mentioned Mr. Sroop as the owner
of this telescope; he possessed it in 1835, when Kaiser used it for his observations
of the comet Halley.

( 125 )

limbs of Venus are generally so ill defined that there is no question
of making microscopic measurements. Dr. P. J. Kaiser, assisted by
Mr. M. B. Rost van Tonninern, has done everything in his power
to succeed, but specially in making photographic observations we
are powerless against atmospheric conditions.

I cannot finish this paper without expressing our thanks to the
Dutch and Dutch East Indian Government, who have assisted the
expedition to Réunion as much as they could, also to the Teyler
Society, the “Hollandsche Maatschappij der Wetenschappen”, both
at Haarlem, to the ‘“Bataafsch Genootschap” at Rotterdam and to
Mr. pe Beavrort, who contributed efficiently (the TryLur Society and
Mr. pe Bravrort by lending the photo-heliograph and a telescope) to
procure the necessary means to the expedition. We have also cordially
to thank the Governor of Réunion, (Mr. pr Lormrt)'), the Maire
of St. Denis, (Med. Dr. Le Stnkr), the director of the “Banque de
la Reunion’, (Mr. Briper), who often assisted and advised us, and
further several other inhabitants of St. Denis, who opened their
houses to us. Among these I mention Messrs. Brrtuo, Hucor, pn
Touris and Przzant. In Mr. Cuainimy, watchmaker, we fortunately
found a clever instrumentmaker, who several times made the necessary
reparations to our instruments.

I must also mention that Mr. Sorrrrs (engineer of the Geographical
service at Java) during the passage from Batavia via Aden to Réunion
suffered already from the first attacks of a liver-complaint, which
a few years later, April 10 1879 carried him to his grave. At
St. Denis he was sometimes, for several days unable to assist me at
the heliometer, then Mr. Baknuyzen obligingly took his place. On
these occasions he was looked after with the greatest care by the
military surgeon Mr. Muircec,

Lastly I mention the valuable assistance in several respects given
by the “amanuensis” of the expedition Mr. T. F. BLanken.

1) Searcely had we cast anchor in the harbour of St. Denis, when the harbour-
master arrived in a boat to offer us in the name of the Government his assistance
to carry ashore the passengers, the luggage and the instruments,

Proceedings Royal Acad. Amsterdam. Vol. VIII.

i

( 124 )

TABLE Ia.
Results for the correction of Germain’s value for the eastern longi-
tude of St. Denis-Réunion, obtained from occultations, using
disappearances and reappearances indistinctly.

be 3) Fo
2 | Name or apparent |3/# &
No] 1874 | 5 lees AL | ¢. | GAL : Ge
2 | place of the star, ||° 3
co) S12 2,
a
4 |Sept.19} O. | Arg. Z. 223 No. 75 |83| D | +4864 | 0.70 | 44915 +4854 14.43
2) » » | O. |Cordoba XVIII. N°.77/8 | D | —6.85 | 0.60 | —4.11 —3.95 9.36
Syl: Syl) Oh » » NY1589/9 | D | 48.294 | 0.74 3.40 +11.14 91.83
Ay Dal Os » » N°1612'8 | D | +8.56 | 0.60 5.14 +11 .46 78.80
5| » 24] O. | Arg. Z. 311 No. 72 |9 | D | —6.02 | 0.90 | —5.42 —3.12 8.76
6] » » | O. » » » » 75 |83} D | —6.75 | 0.995| —6.72 —3.85 | 14.75
7| » 22) O. | 33 Capricorni 53] D | 43.42 | 0.29 | +0.90 +6 .02 10.54
9| » » | O. | Arg. Z. 255 No.27 |7 | D | —6.75*| 0.50 | —3.38 —3.85 7.4
AO} || sees | RO; » » » » 32/8 | D}| —0.42 | 0.63 | —0.26 +2.48 3.88
(15) den nO: » » » » 34/8} D} —1.37*| 0.89 | —1.22 +1 .53 2.08
ADH yey Os » » » » 3d(7}| D| —4.99 | 0.97 | —4.84 —2.09 4.06
14) » 26) O. | 73 Piscium 63) # | 4+3.13"| 9.91 | 42.85 ).03 33.09
16 |Oct. 2) B. | 53 Geminorum 6 | R | —2.07*| 0.28 | —0.58 0.83 0.49
; aoa
Ai or SIRO 14990 51 387 73| # | 44.70] 4.00} 44.70] 44.60 | 21.46
419} » 45) O. | B. A. ©. 5800 63) D |—13.14 | 0.27 1) —10.24 28.34
20} » 46} B. | Arg. Z. 223 No. 47 |8 | D | —4.99 | 1.00 | —4.99 —2.09 4.37
c 18u 3m 38,135 AS = ¢ a
22} » »|B |} ogooysgn74 (9 | 2| 40.13] 0.40} 40.05) -++3.03 3.67
23] » » |B.O.| Arg. Z. 223 No. 49 |8 | D | —4.09 | 0.95 | —3.89 —1.19 41.35
94) » » |B.0.) » » » » 5219| D| —4.03 | 0.515} —2.08 —1.13 0.66
95) » » |B.O.. » » » '» 51/8] D| —3.59 | 0.49 | —1.76 —0.69 0.24
184 6m 418,75 Pare ms
27 | > >| Bi | (osogisgna) 82] 2.| —b:53 | 0.99 || 5.47! Sakeg ual iteaee
98} » » | B. | Gould 24851 183] D | —1.00 | 0.87 | —0.87 +1.90 3.44
; 19u2m35s.76
29| » 17] 0. t oSuITae) 93 D| $0.02} 0.19} 0.00] 49.92 | 1.62
30| » »|0.| Arg. Z. 24 No. 9 [ga] »| —6.95 | 0.58 | —4.03/ —4.05. | 9.54
Sil Sy ese OF » » Ql » 12 /84| Db | —2.40 | 0.35 | —0.84 -+0.50 0.09
Sai ed? -di)(20: » » » » 44 [83] D| —6.41 | 0.62 | —3.79 | —3.24 6.39
34] » 18) B. » » 239 » 103 8 | D | —6.47 | 0.95 | —6.15 —3.57 42.40
35] » 19) B. » » 247 » 99 9 | YD} —2.79 | 0.98 | —2.73 +0.41 0.01
30] » » | B. | x Capricorni 6 | D | —4.46 | 0.97 | —4.33 —1.56 2.36
Om 94s: 7:
371 > »| B. (apis rane 811 p | —9.09 | 0.94 | —8.54| 6.19 | 36.02
38} » 22) B, | 24 Piscium 63] D | —2.18 | 0.73 | —1.59 +0.72 0.38
21.80 |417.89 4A7.38
—81.14 10) | ———
a m?— 13.91
—63.25 m= $3972
21.80 |_——-|_™__3.72__+ o6g9

—2890 |V21.80 4.67 —

* These occultations have been calculated after Brssex’s method. The modification of this
method, to which I referred in the footnote of p. 605 of my previous paper, may be found, as I
perceived afterwards, in Cuauvenxr, I p. 556, in § 344. It consists in this that the coordinates 2, y, z,
§, 4 and § are not computed from hour to hour but only for the instant of the occultation, the
Greenwich time of which is found by means of an assumed value of the longitude.

The differences between these results and those after my method were:

No. | R | 5 | 7,
9 lw 66 | iam | — 0865
ul 167.7 | 12 5.5 | 40.43
14 16 41.1 | 1316.8 | — 0.46
16 15 36.8 13 56.6 — 0.80
22 15 5.1 | 550.38 | 40.17

It appears that all the differences remain below one second of time.

( 125 )

TABLE Ib.
Results for the longitude of St. Denis de la Réunion, from dis-
appearances and reappearances separately.
The 3 reappearances give >G= 2.19 ©GAL=-+ 3.97 hence A Lre= +13.81

The total sum was 21.80 — 63.25
Hence the disappearances alone 19,51 — 67.22 A Lypis = — 3.43
Mean — 05,81
N°. € | Ge
Disappearances,
4 45807 , 17.99
2. Ao 7.02
3 [414.67 | 100.78
4 1441.99 | 86.26
ay a) 6.04
6 |— 3.32 | 10.96
7 |4-6.55| 12.44
9 |— 3.32 ebd
10 |4+ 3.01] 5.74
44 |42.06] 3.77
12 |— 1.56 2.36
19 |— 9.74 25.46
20 |— 1.56 2.43
22 |+ 3.56 5.07 | 97 m?— 358.07
23 |—0.66| 0.42 Me 49:95
24 | 0.60] 0.19 ee ongd
25 |— 0.46 0.01 (not. used)
27 |— 2.410 4.37
98 142.43} 5.43
99 14 3.45| 2.96 Uh aaa
“ee a ig 29 m?— 363.87
Bina A203) (P< OF37 ce oa
32 |—2.68| 4.45 ee ee
34 | 3.04] 9.19 pepe ae
35 |+ 0.64| 0.40 =.
36 |—1.03| 4.03 a= 0.64 Y=+08.80
37 |— 6.66 | 30.12 m2 i :
38 [1.25 | 1.14 | 3419 — OT VS 20.30
Reappearances, 2m?— 5.80 5 91
14 |+4.32| 1.58 mi— 2.90 read ttn
16 |— 3.88 4,21 m = + 1870 ki see
1418 |— 0.11 0.01 (not used)

(126 )
TABLE I,

tesults for 4 “ after the application of the correction for the
radius, disappearances and reappearances together.

No.| AL | Corr. | 44 G | GAL é G2
corr.
4 | 44°64) 40°85 | 42°49 | 0.70 | 44.74 | 14964 16.07
De | Oe Sau) eters) 7200.60) |S eas aaa 7.64
3 | 48.24] 40.77 | +9.01 | 0.74 | 46.67 |441.16 92.17
4 | +8.56 | +0.81 | +9.37 | 0.60 | +5.62 |441.52 “' 79.63
5 | ~6.02 | +0.74 | —5.28 | 0.90 | —4 75 | —3.43 8.82
6 | —6.75 | +0.80 | —5.95 | 0.995] —5 92 | —3.80 44.37
7 | +3.12 | +0 90 | +4.02 | 0.29 | 44.47 | 16.47 11.04
9 | —6.75 | +3.75 | —3.00 | 0.50 | —1.50 | —0.85 0.36
10 | —0.42 | +0.74 | 40.32 | 0.63 | +-0.20 | 12.47 3.84
44 | —1.87,| -1.42 | —0.95 | 0.89] —0.92 | =11.90 3.91
49 | —4%.99 | $0.78 | —4.21 | 0.97 | —4.08 | —2.06 AI
44R| 43.43 | —1.24 | 44.89 | 0.91 | +1.72 | 44.04 14.85
46 R| —2.07 | —1.60 | —3.67 | 0.98 | —1.03 | —1.52 0.65
18 R| 44.70 | —0.77 | +0.93 | 1.00 | 0.93 | 43.08 9.49
49 |—13.44 | 41.39 |—11.75 | 0.27 | —3.47 | —9.60 24.88
90 | —4.99 | +0.75 | —4.24 | 1.00 | —4.24 | —2.09 4,37
92 | 10.43 | 40.97 | 44.10 | 0.40 | 40.44 | 43.95 4,22
93 | —4.09 | +0.81 | —3.98 | 0.95 | —3.42 | 1.43 4.22
9% | —4.03 | $0.88 | —3.45 | 0.515) —1.62 | —1.00 0.51
95 | —3.59 | +0.90 | —2.69 | 0.49 | —1.32 | —0.54 0.14
97 | —5.53 | 40.74 | —4.79 | 0.99 | —4.74 | —2.64 6.90
9g | —1.00 | +0.84 | —0.46 | 0.87 | —0.14 | -11.99 3.45
99 | +0.02 | 41.06 | 41.08 | 0.19 | +-0.21 | +3.93 1.98
30 | —6.95 | +0.80 | —6.45 | 0.58 | —3.57 | —4.00 9.28
34 | —2.40 | 0.92 | —1.48 | 0:35)| —0:52 |-10.67 0.16
32, | —6.41 | 40.79 | —5.32'] 0.62] —3.30 | —3.17 6.23
34 | —6.47 | 40.92 | —5.55 | 0.95 | —5.27 | —3.40 10.98
35 | —2.79 | 44.06 | —4 73 | 0.98 | —1.69 | 40.49 0.18
36 | —4.46 | +0.99°|. —3.46 |.0.97 | 3.36 | —1.34 eGyP
37 | —9.09 | 44.05 | —8.04 | 0.94 | —7.56 | —5.89 32.61
38 | —2.48 | 0.77 | —1.41 | 0.73) | —4.03'| 0.74 0.40
21.80 /--18.70 30 m2? = 374.43
—65.58 m?— 12.48
46.88 m= 3853
gee Saas la 3= 0.572, V = 40° 76

Utrecht, June 24, 1905. :

( 127)
Chemistry. — “On some derivatives of Phenylearbamic acid.”
By Dr. F. M. Janeur. (Communicated by Prof. P. van Rompurau).

The following contains a crystallographic description of some deriva-
tives, chiefly nztroderivatives of phenylcarbamic acid C, H, . NH . COOH
which have been kindly presented to me by Prof. van RompurGu.
The substances belonging to this series, which have been investigated are:

Phenylearbamic Methyl-ester.
Methylphenylearbamic Methyl-ester.
1-4-Nitromethylphenylearbamice Methyl-ester.
1-2-4-Dinitromethylphenylearbamic Methyl-ester.
1-2-4-6-Trinitromethylphenylearbamic Methyl-ester. (a-Modijication).
1-2-4-6-Trinitromethylphenylearbamic Methyl-ester. (8-Modijication).
1-2-4-Dinitromethylphenylearbamic Aethyl-ester.
1-2-4-6-Trinitromethylphenylearbamic Aethyl-ester.

In addition, a description is given of 1-2-4-6-Methylphenylnitramine
m. p. 127° C., which has been obtained from 1-2-4-Dinitromonomethyl-
aniline m.p. 178°C. by means of fuming nitric acid, which aniline
is the product of decomposition of the two Dinitromethylphenylcarbamic
esters on heating with strong hydrochloric acid *), and which has been
already described by me in the Zeits. f. Kryst. Bd. 40 (1905). p. 119.

1. Phenylcarbamic Methyl-ester.

Cah Nie CO2O(CH,); map. 47 - C:

The compound crystallises best from
alcohol and always in the form of color-
less, elongated, rectangular little plates,
which are very poor in combination
forms.

Rhombic bipyramidal.

Gs 01,5952): 1

The relation }:¢ cannot be determined
on account of the absence of planes
from the zones of (100,001), and of
(001,010).

Forms observed: a = {100}, strongly
predominating often vertically striped ;
p = {110}, very lustrous; m= {120},
narrow or totally wanting and sometimes
as strongly developed as p; 6—= {010},
indicated only a few times; c= {001},
Fig. 1. reflects well.

1) van Rompurcu. On the action of nitric acid on the esters methylphenylamino-
formic acid. Proc. 29 Dec. 1900, Vol. IIL p. 451.

( 128 )

Measured : ‘alculated :
Wp ——1 00) 1G Oy" 57 2551 _
a:m = (100): 20)= 38 31 38°34!
@ 2c — (400) (001) — 904 S080
m:p ==(120):(410) = 19 34 19 24
pap Clo) (410) = 64 10 64 10
p: 5 = 10): (010) == 32/40 ay)

Completely cleavable towards {001} and towards {100}.

Orientated extinction on all planes in the vertical zone. The optical
axial plane is {001} with a as acute diagonal. The axial angle is
small, the dispersion fair with @ >v around the a-axis.

The sp. gr. of .the crystals is 1,251 at. 19°; the equivalent
volume 120,7.

2. Methyl-Phenylcarbamic Methyl-ester.
C,H, . N(CH,). CO. O(CH,) ; m.p.: 44° C.

The compound crystallises from
alcohol in large colourless crystals,
which are frequently in clusters,
often exhibit rather opaque planes
and possess a peculiar camphor-
like odour.

Rhombic-bipyramidal.

a:b:¢= 0,8406 : 1 : 0,3320.

Forms observed :

b = {010}, strongly predominating;
m= {110}, and g= {011}, both
well developed and yielding sharp
* reflexes ; 7 = {201}, fairly lustrous.

Different crystal-individuals ex-
hibit not inconsiderable differen-
ces in the angular values.

1

i}
1
'
'
.
'
‘
1
'
'
‘
'
1
$
h
‘
'
‘
'
1
‘
‘

A

- ‘
.
‘

Measured : Calculated :
6m = (010): 410) = * 492 5774 a
b: is = (010):(011)=—=* 71 38 =
rir = (201):(201)—= 108 25 103272382

Paine == (201) GdlO) = Giles 61 40'/,

( 129 )

Very completely cleavable towards +.

The optical axial plane {001} whilst 6 is the first diagonal. The
axial angle is small, the dispersion strong and perhaps abnormal.
It was not possible to properly characterise it with the means at
my disposal.

The sp. gr. of the crystals is 1,296, at 19°; the equivalent
volume 127,31.

Topical axes: %: yp: @ = 5,1358 : 6,1099 : 4,0569.

On account of the symbol {201} the relation 6:c = 1 : 0,6640 has
been taken.

8. 1-4-Nitro-Methyl-Phenyl-Carbamic Methyl-ester.
C,H, (NO,) . N (CH,) .CO .O(CH,); melting point: 108° C.
(4) (1)

This compound erystallises from alcohol or benzene in the form
of small delicate needles, or large, pale-sherry coloured, somewhat
flat crystals, which, however,
are very poor in planes, and,
therefore, do not allow of a com-
plete parameter-determination.

Monoclino-prismatic.

a:b = 0,6640: 1.
== 10758".

Forms observed :
c={001} generally strongly pre-
dominating ; m = {110} well
developed ; 6 = {010}, narrow,
Often the planes of mand 6 are
curved and the crystals exhibit
Fig. 3. greater anomalies in the angular

values.

The habitus is mostly broadly flattened towards c, sometimes c
and m are equally large and the habitus consequently becomes
rhombohedric.

Very completely cleavable towards {001}.

The optical axial plane is probably {010}; on c one optical axis
is visible on the border of the field of vision.

The sp. gr. of the erystrals is 1,522 at 14°; the equivalent-volume
== 137,98.

{ 130 )

4, 1-2-4-Dinitro-Methyl-Phenylcarbamic Methyl-ester.

Oz

w

Cleavable
On {010},

== 1695320

Topical axes: ¥

C,H, (NO,) . (NO,

Mec
(4) (2)

m-p:; 9" C:

N(CH,) . CO . O\CH,);
(1)

The best crystals are obtained from
xylene. They are of a pale yellow colour
and have the appearance of small, thick
parallelogram-shaped crystals.

Monoclino-prismatic.
@ 101.6 O59 Ae AAO Sitio:
= 88° 437/,’.

Forms observed: 4 = {010}, predomina-
ting and very lustrous; 7 = {101}, broad
and sharply reflecting; w= {111}, also
broad and very lustrous ; 0 = {111}, some-
what smaller than 7 but giving a good
reflection ; g = {011}, small and approxi-
mately measurable. The crystals are broadly
flattened towards 6.

Measured: Calculated :
ry — (111): 101) =* 32°44 =
> — (111): (471) =* 64 223 =
:w — (114): (111) =* 86 373 ==
:@ = (010): (411) = 57 54 57° 49!
0 = (010) 3G) — omen 57 16
:q = (111):(011) = 42 35 (about) 42 54
[0 == (ai). (ait 58 26 58) 12
:r = (411):({01) = 74 12 74 24
towards {141}.

the angle of extinction with regard to the side 6:
is 22°; an axial image could not be observed.
The sp. gr. of the crystals is 1,506, at 14°; the equivalent volume

:~:@ = 4,4794 : 5,8963 : 6.4123.

( 131)
5. 1-2-4-6-Trinitro-Methyl-Phenylcarbamic Methyl-ester

CH,(NO,). (NO,) . (NO) N(CH.) . CO! O(CH,);-m. p. 118°C.
4 (2) (1)

This compound oceurs in two modifications.
a-Modijication.

This a-modification is the one usually deposited from the ordinary
solvents, alcohol, acetone, benzene ete. The crystals described here

have been obtained from acetone. They are colourless or of a pale
sherry colour and very lustrous.

Monoclino-prismatic.

a:b6:¢e= 0,5758 ; 1 0,8382.
B= 75°41".

Forms observed: m — {110}, broad and very lustrous ; ¢ = {001},
ideal reflection; g— {011}, large and very lustrous; w= {121},
generally broader than q, sometimes also narrower or even completely
wanting ; @ = {100}, lustrous but narrow ; 7 = {101} is often wanting
but reflects well; 0 = {121} very narrow and dull.

Measured: Calculated :
:¢ = (100): (001) = * 75°41’ a

: m = (100) : (110)

2g a
= &

=D ONO —
CGE (O01) (O11) = * 39) 5

( 132 )

Measured: Calculated :

ie een — (SCD) (0 (01) a CTE I 77°32’
Mig == AOR (OL) = ole 7 61 39
no —= G10) G2) 2630 26 27
ao == 000) 21)" 47551), 48 0
ao :¢ = 42a) (OOD ==) 5/43 57 42
Ong — 4A2i)\e (Old) = 735, 16 35 12
mg —= (dO): (Ol) =" “81 40 81 59
70 — dO) 1Aiod)— v34: 397). 34 37'/,
Crore) (Old) == 546 1912 46 22
c:r =(001):(101) = 65 387/, 65 36
a:r =(100):(101)= 38 511/, 38 43
m:r = (110):401)= 47 2 47 3
negu== Ol) (Old) == = VOR o2 (is bes P/F

No distinet plane of cleavability was found; perhaps there is one
present parallel to m.

The symmetric extinction on {110} with regard to the side
110): (110), ete. amounts to about 18°; on a and ¢ it is normally
orientated. The average refraction is a trifle greater than that of
«-monobromo-naphtaiene.

The sp. gr. gravity of the crystals is 1,612, at 19° ; the equivalent
volume is 186,10.

Topical axes: 4%: W: w : = 4,2360 : 7,3555 : 6,1655.

5). Trinitro-Methyl-Phenylcarbamic Methylester.

B- Modification. When long kept, the erys-
tals of the a-modification turn a little
darker, somewhat more orange-yellow. The
symmetry and all the angles of the ¢-modi-
fication are, however, preserved.

Sometimes, the alcohol deposits long
needles together with crystals of the a-
modification, These needles have an orange
colour; at about 105° they again turn
yellow and then melt just a little below
118°. Although it is not as yet quite clear
in what relation these needles stand to the
crystals, it is nevertheless certain, that they
represent a second less stable modification
of the compound. The meltingpoint of the
crystals of the «modification obtained from
various solvents, or after heating in diffe-

( 133 )

rent ways, fluctuates between 114° and 118°. A further investigation
will be necessary to see what really takes place here.

Rhombic-bipyramidal.

& G—06596)1.

The relation 6: ¢ cannot be determined, for want of the necessary
terminal planes.

Forms observed : m = {110}, broad and lustrous; ¢ = {001}, very
sharply reflecting; a= {100}, narrow, well reflecting; p= {310},
very narrow and yielding bad reflexes.

Measured: ‘aleulated :
a:m = (100) : (110) = * 33° 244’ aa
m:m = (110):(110) = 143 114 113°41/
Tice — dal) (OO 90s 1 90 O

~
S
~

pu 110)610) =~ 120 17% (about) 21) (0
ple (310): (100) —=* 13) 2; “@bout) 12 24

Perfectly cleavable towards {001}.

The optical axial plane is {001}; the first diagonal is the a-axis,
The apparent axial angle is in ¢-monobromonaphtalene about 86°;
extraordinary strong dispersion with @ >, round the first bissectrix.
Orientated extinction everywhere in the vertical zone.

The sp. gr. of the needles is 1,601, at 19°; the equivalent
volume = 187,32.

6. 1-2-4-Dinitro-Methyl-Phenylcarbamic Ethyl-ester.

C, H, (NO,)4) . (NO,)2) . Nay (CH,).CO.O(C,H,); m. p. 112°C.
a This compound erystallises
ee from a mixture of benzene and
ligroine in the form of large,
corlourless, very lustrous erystals
represented in fig 8.

Monoclino-prismatic.

@: 0b 6 =O1652528 7 O!7035:
B= 6959e

Forms observed: c= {001},
predominating and very lustrous;
6 = {010}, about as broad as ¢
and sharply reflecting ;m = {110},
well reflecting and _ properly
developed, sometimes with deli-
cate striping parallel to m:c;
q = {011}, narrower but readily
measurable ; 7 = {101}, very dis-
tinctly developed and yielding

Fig. 8.

sharp reflexes ;

altogether.

( 134 )

Measured :

710) = *63° 14’

7m = (1:00)

Je = 410) : 001) 73
ee —— Old): (O01) == 33
26. = (011): (010) =. 56
Os == (410) - (010) = 58
Ce "all ) (GED (0) eels
0 = (120))- (O10) ==) 39
‘:¢ = (101): (001) = 58
>: m = (101): (410) = 58
5 = (10) ea O1e) a7
:y = (011): (101)= 63
:b =(001):(010) = 89

12
583 (cirea)

41} (cirea)

(oe)

5

p = $1203, very narrow and duli; often absent

Calculated :

Very perfectly cleavable towards {001}; like “glimmer” the crys-
tals may be reduced to very delicate lamellae.
On {001} orientated extinction; on {110} the inclination
elasticity-axis towards the vertical axis amounts to 19°;

27°

plane is, probably, situated perpendicularly to {010}.
a-monobromonaphthalene etched figures were obtained on mm, ¢ and

D

?

The

Sp.

volume = 184,12.

Topical axes: y%: ap: w = 4,9130 :

7,5296 : 5,2970.

with regard to the side 6:c, in the acute angle £.
by

which are in accordance with the indicated symmetry.
er. of the crystals is 1,461 at 19°C.; the equivalent

of the one
on {O10}:
The axial
means of

7. 1-2-4-6-Trinitromethylphenylcarbamic Ethyl-ester. .

C,H, (NO,) . (NO,) . (NO,). N(CH,) -CO.0(C,H,); m.p. 65° C.
6 (4 D2) Oa

( 135 )

The erystals which have been meas-
ured are derived from a mixture of
benzene .and ligroine.

Delicate, very transparent, flat, pale
sherry-coloured needles which possess
a strong lustre.

Monoclino-prismatic.
GnOi6 = 059759): 1: OL3929
i= OUR

Forms observed: b= {O10}, yielding
ideal reflexes and well developed :
o = {121}, a = {100}, and ¢ = {001},
very narrow and dull; m—= {110},
broad and very lustrous; g = {O11},
well developed and yielding sharp
reflexes; an orthodome {ho k! is indi-
cated but not measurable. The needles
are elongated towards the c-axis and
somewhat flattened towards {010}.

Measured: — Calculated:
fe: mM (110) > AO) 83> 90! =
m:q = (110):(011) = *60 2 at
Og) = (01d): (O11) 39548 —

m:b = (110):(010) = 48 3 48° 43!
bg =-(010): OFL) = 705 *6 70. 6

m:g = (110): @11)=119 58 .119 58
m:o = (110): (121) = 64 46 65 2

Oge— C21); OM nn 32 De OS)
q:m= (011): (110) = 87 42 87° 28
Md = A0)) 300) 4 ar 41 574

A distinet cleavability was not observed.

The angle of extinction on 6 is 9° with regard to the vertical
axis, in the acute angle @:c; on m it is about 65°.

In oil of cloves, solution figures were obtained as represented in
fig. 8; they are in accordance with the found symmetry.

The sp. gr. of the crystals is =1,471 at 14°; the equivalent
-volume = 194,42.
Topical axes: x = yw: w= 7,9976 : 8,1950 : 3,2198.

( 136 )
8. 1-2-4-6-Trinitrophenylmethylnitramine.

C,H . (NO,) . (NO,) . (NO,).N (NO,) (CH); m. p.: 127° C.
(6) (4) @) ()
The compound is obtained from benzene + acetone in the shape
of small, very strongly refracting, pale sherry-coloured needles, which
possess a strong lustre and are, geometrically, very well built.

Fig. 10.

Monoclino prismatic.
a:6:¢c= 2,7823 :1 : 3.5242.
B=Wprolee
Forms observed: c = {001}, most strongly developed of all;
a={100} and r= 101}, both strongly reflecting; ¢= {O11} some-
what more opaque.
Measured : Calculated :
a:c = (100) : 001) = * 75°313 —
a:r = (100): 401) = * 43 363 oe
eg = (001) 2011) 73340 _

c:r = (001): (101) = 60 53 60° 53’
q:q= (011): (011) = 32 32 32 40
a:q==(100):(011)= 85 54 85 58}

A distinct cleavability was not found.

Optical axial plane is {010}; on {001} one axis is placed nearly
perpendicular. The double refraction is moderate and negative ;
extraordinarily great dispersion with @ > v.

The sp. gr. of the little crystals is: 1,570, at 19°; the equivalent
volume = 182,16.

Topical axes: 4: py: @ = 7,4485 : 2,6772 ; 9,4347,

(137)
Chemistry. — “On the presence of lupeol in some kinds of gutta-

percha.” By Prof. P. van Rompureu.

An investigation of the so-called resinous constituents of various
authentic kinds of gutta-percha, made first in conjunction with
Dr. Sack and afterwards with Dr. v. p. Linppn, has shown that some of
them contain various cinnamic esters of alcohols which seem related
to cholesterol. One of these cinnamic esters, which appeared identical
with Tscuircn’s’) crystal-albane, and occurs as a beautifully crystallised
compound m.p. 241° (corr.) I have submitted to a closer investigation,
with Dr. v. p. Linpen. On saponification, an alcohol was obtained
melting at 210°, which on being treated with benzoyl chloride and
pyridine yielded a benzoate melting at 264° (corr.). The melting
points of these two last substances agree exactly with those of lupeol
and its benzoate.

Lupeol has been discovered by E. Scuuizn’*) in the skins of lupines.
At my request Prof. Scuunze was kind enough to present me with
a quantity of lupeol and its benzoate for the purpose of compa-
rison, for which I wish here to express my best thanks. The alcohol
being mixed with the lupeol, the melting point was not lowered;
neither was this the case with the benzoates.

In addition to its occurrence as a cinnamic ester, lupeol also
seems to occur as an acetate in a substance related to gutta-percha,
called “djelutung”, the product of the milky juice from some species
of Dyera, which is known in European commerce under the
name of “bresk” or Pontianak; this has been shown to be probable by
Mr. Conren, who is making a study of this article in my laboratory.
In a consignment of “bresk” for which I have to thank Messrs.
Wuiszk & Co., of Rotterdam, the amount of lupeol appeared to be
rather considerable, thus enabling Mr. Conen to make a study of this
otherwise somewhat inaccessible product. On oxidation with chromic
acid a beautifully crystallised ketone (m. p. 169°) has already been
obtained, which also yields with hydroxylamine a crystalline substance.

Mr. Conzn, who intends to further investigate these substances,
has also found in the “djelutung”’ the substance melting at 235°,
which I had found previously in the gutta-percha from Payena
Leerii, and which has been characterised as the acetic ester of an
alcohol melting at 195°.

1) Arch. d. Pharm. 241, 653.
2) Zeitschr. f. physiol. Chemie 15, 415; 41, 474.

(138 )

Chemistry. — “On the action of ammonia and amines on allyl
Jormate”’. By Prof. P. van Rompurcu.

The great ease with which ally] formate is saponified by alkalis
induced me to try whether ammonia, which, as is well-known, acts
upon esters of fatty acids but slowly at the ordinary temperature,
would not react equally readily on this ester. The result surpassed
my expectation.

If gaseous ammonia is passed into allyl formate still containing
a little allyl alcohol, it is rapidly absorbed whilst the liquid gradually
becomes so hot, that it is necessary to connect the flask with a
reflux apparatus in order to prevent loss of ester. If the contents
are heated to 120°, when the increase in weight slightly exceeds
1 mol. of ammonia to one mol, of ester, the excess of ammonia
passes off with the allyl alcohol and if now the residue is distilled
in vacuo, a fine yield of formamide is obtained (b.p.1o 113°), which,
after a single freezing, melts at 2°.4 *).

If, however, dry ammonia is passed into pure allyl formate,
hardly any action is noticed in the first hour. The ammonia is but
slowly dissolved so that the concentration is only very small, but
eradually as the reaction proceeds the gas is more eagerly absorbed
and the temperature rises increasingly.

If, therefore, we wish to prepare in a short time large quantities
of formamide by means of this method it is advisable to add to
the allyl formate a few percents of allyl alcohol, although this may cause
a slight diminution in the limit value ’*), as happens, generally, in the
formation of amides from esters.

I showed many years ago that allyl formate may be readily
prepared by heating the diformine *) of glycerol, obtained from glycerol
and oxalic acid. As we can now obtain commercial formic acid of
ereat concentration (99—100 °/,) at a very low price, large quan-

1) Francumonr (Rec. XVI, 137) found the melting point at 3°; other observers
state a lower figure.

Pure formamide may be distilled by rapid heating without perceptible decompo-
sition (b.p. 219°); at least with such a preparation | did not succeed in demon-
strating the formation of ammonia and carbon monoxide, which are readily obtained
from an impure product.

2) Bonz, Zeitschr. phys. Chem. II, 865.

5) I think it is not superfluous to point out that the theory recently defended
by Ner (Ann. 335, 230) that the formation of allyl alcohol from glycerol and
oxalic acid must be explained by a dissociation of diformine into formic acid and
propargyl alcohol is based on an error, The main product of diformine on heating
is allyl formate.

( 139 )

tities of allyl formate may be prepared in a still more convenient
manner by heating equal parts by weight of glycerol and formic
acid. The temperature is kept for some time at 125°, during which
dilute formic acid distills over. Gradually it is raised to 240° and,
with a quiet evolution of CO, containing a little CO, a mixture
passes over consisting of allyl formate, allyl alcohol and a very
little formic acid. This is again submitted to distillation, the portion
boiling up to 100° being collected. After treatment with dry potas-
sium carbonate, the liquid is again distilled and the portion boiling
below 85°, which mainly consists of allyl formate, is collected
Separately. ;

This ester may also be procured by distilling allyl aleohol with
twice its weght of formie acid and collecting the portion passing
over below 85°. The product is then treated with dry potassium
carbonate and once more rectified.

After it had been ascertained that ammonia acts so very readily
with allyl formate it was decided to try the action of amines also.

The investigation showed that amines of the fatty series, primary
as well as secondary ones, readily react with the same. Benzylamine,
phenylhydrazine and piperidine also seemed to react but no reaction
could be observed with aniline.

If we mix one of these amines with allyl formate, in the majority of
cases a rise in temperature does not necessarily take place immediately.

This rise rather varies in the different cases; whilst its maximum
is sometimes reached fairly quickly, sometimes only after a lapse of
about 20 minutes.

The reaction appears to be of such a nature that when working
at a constant temperature we can ascertain its progress by means
of a quantitative determination of the absorbed amine.

I intend making a series of experiments with different amines. The
following contains a brief description of some qualitative experiments.

If methyl- or ethylamine is passed into allyl formate these sub-
stances are absorbed with great evolution of heat and the amides
formed are left behind after distilling off the allyl alcohol.

5 grams of propylamine being mixed with 10 grams of allyl for-
mate the temperature rapidly rose from 19° to 60° and propylform-
amide (b.p. 219°—220°) was formed.

5 grams of isopropylamine being mixed with 10 grams of the
ester the temperature slowly rose to 50°, whilst a good yield of
isopropylformamide was obtained (b.p. 203°—204°),

10
Proceedings Royal Acad. Amsterdam. Vol VIII,

( 140 )

5 grams of isobutylamine being mixed with 7 grams of allyl for-
mate the temperature rose to 75°. The — not as yet described —
isobutylformamide boiled at 229°.

5 grams of allylamine being mixed with 10 grams of the ester the
temperature rose to 65° and the allyl alcohol being removed by
distillation, allylformamide was obtained (b.p. 220°.5).

With benzylamine (5 grams) and allyl formate (5 grams) the
temperature rose from 19° to 55°. The alcohol being distilled off
and allowed to cool to the ordinary temperature, there remained
in the flask a solid mass, m. p. 62°. The melting point was not
altered after recrystallisation the compound from petroleum ether,
which is very suitable for this purpose.

Benzylformamide was first obtained by HoLLuman’), who described
it as a substance melting at 49°. This statement is probably due toa
clerical error, at least, a specimen prepared by Prof. Horteman and
kindly presented to me by Dr. Virmevien of the Groningen Labora-
tory did not begin to soften until 59°. Boiling petroleum ether
extracted a substance melting at 62°, which on being mixed with
my own product did not alter its melting point.

Phenylhydrazine gives no rise of temperature with allyl formate,
but on being kept for a day, an abundant quantity of formylphenyl-
hydrazine forms, m. p. 145°.

With secondary aliphatic amines there is less heat evolved in
the action on allyl formate.

Dimethylamine readily forms dimethylformamide. The action of
7 grams diethylamine on 10 grams of allyl formate causes (in about
20 minutes) a slow rise to 33°. Diaethylformamide was readily obtai-
nable in a pure condition. Dipropylamine (5 grams) mixed in a
Weinnoip flask with 5 grams of the ester caused a slow rise to
35°.5. The dipropylformamide obtained boiled at 211° (corr.).

Judging from a preliminary experiment, diisopropylamine seems to
react less readily; 3 grams of both compounds being mixed, only a
slight elevation of temperature was noticed. This reaction deserves
in particular a closer study.

With diisobutylamine the evolution of heat is also trifling; only
3° rise 10 grams of each substance being mixed. All the same, a good
yield of diisobutylformamide was obtained, which boils at 227°—228°
(corr.) and which, to my knowledge, has not yet been described.

Methylbenzylformamide (5 grams) with allyl formate (5 grams)
gives a rise to 55°. The product formed has not yet been solidified,

1) Rec. 13. 415.

( 141 )

Piperidine (10 grams) with allyl formate (14 grams) gives a rise
from 10° to 83°, and a very good yield of the formyl derivative,
b.p. 220°.

The boiling points of the substituted formamides exhibit peculiar
regularities to which I hope to refer later on.

The dialkylformamides and formylpiperidine have acquired some
importance owing to the interesting researches of Bouvnaunr ‘), who
used them as a starting point in the preparation of aldehydes ; the
above described simple methods of preparation may perhaps prove
to be of service.

Chemistry. -— “On the action of hydrocyanic acid on ketones’.
By A. J. Unter. (Communicated by Prof. P. van Rompuren).

Although it is stated in every textbook on organic chemistry that
ketones may combine with hydrocyanic acid, the conditions under
which this addition takes place have up to the present not been
studied, and only those few cyanohydrins which are solid and may
consequently be readily purified by recrystallisation have been isolated
in a pure condition *).

Three methods of formation of these substances are known:

1st. Action of dilute or anhydrous hydrocyanic acid on ketones,
either by heating the mixture for some hours in scaled tubes at 100°
or by simply leaving the two components in contact with each other
at the ordinary temperature for several months.

2d, Action of nascent hydrogen cyanide on ketones, for instance
by very slowly dropping fuming hydrochloric acid on potassium
cyanide covered with acetone.

34, By double decomposition of the so-called bisulphite compounds
of the ketones with a solution of potassium cyanide.

A closer study of the nitriles of the oxy-acids was made in conse-
quence of an observation made by Prof. van RomsBureH *) as to the
action of solid potassium carbonate on a mixture of dry acetone and
hydrocyanic acid; a small quantity of this salt caused the mass to
boil and the temperature to rise to 70°.

The same phenomenon is caused by potassium hydroxide, potassium
cyanide, ammonia, amines, in fact by all substances whose aqueous

1) Bull. Soc. chim. [3] 31, 1322.
2) Acetonecyanohydrin, obtained from Kallbaum, seemed to contain much free
hydr ocyanic acid.
8) Meeting 27 June 1896.
10*

( 142 )

solutions possess hydroxyl ions; the presence of water greatly favours
this catalysis.

If an attempt is made to isolate the evidently formed cyanohydrin
by distillation under reduced pressure it is again resolved for the
ereater part into its components. If, however, the action of the
potassium earbonate is stopped by means of a few drops of sulphuric
acid, the mixture on being fractionated in vacuo first yields a
distillate consisting of hydrocyanic acid and acetone and then the
nitrile; by a second distillation this may be obtained in such astate
of purity that silver nitrate with nitrie acid no longer gives a
precipitate of silver cyanide.

Traces of a base are, however, sufficient to again partially resolve
the pure nitrile into its components, which in this case is, of course,
accompanied by a fall in temperature.

Theory demands that the same equilibrium should be reached
whether we start from one mol. of acetone plus one mol. of hydrogen
cyanide or from pure cyanohydrin. In order to check this it is not
necessary to determine the equilibrium both ways by analysis; the
easiest plan is to measure some physical constant; for this I chose
the refraction.

Found, starting from a mixture of acetone (1 mol.), hydrogen
cyanide (1 mol.) and a trace of potassium hydroxide np‘? = 1,39721.

Found, starting from the pure nitrile and a trace of potassium
hydroxide np** = 1,39818.

It having been thus ascertained that it makes no difference from
what system we start, it became important to express the equilibrium
in figures.

For practical reasons I always started from the nitriles; about
one gram of the compound and 0,2 milligr. of potassium hydroxide
(in a 10°/, solution) were introduced into a tube, which was then
sealed and immersed in a beaker containing a solution of silver
nitrate acidified with nitric acid, and the whole was then suspended
in a thermostat for some hours.

If now the tube is broken the nitric acid at once neutralises the
potassium hydroxide and the free hydrocyanic acid will be precipi-
tated as silver cyanide. The liquid is decanted, the precipitate is
dissolved in potassium cyanide and the silver deposited electrolytically
in the usual manner. In this way it was found that one mol. of
acetone and one mol. of hydrogen cyanide combine at O° to the
extent of 94,15°/,, at 25° to the extent of 88,60°/,.

For ethylmethylketone these values are, respectively 95,57°/, and
90,36°/,; for diethylketone 95,90°/, and 91,29°/,.

( 1438 )

It is my intention to also determine this equilibrium in the case
of other aliphatic and aromatic ketones and also aldehydes.

The investigation of Urrcn’s diacetocyanohydrin and the products
of the action of gaseous hydrochloric acid on oxynitriles quoted by
Pinner *) but as far as IT know not further studied, has already been
taken in hand.

In the light of the above results I have examined the different
modes of preparation of the oxynitriles more closely.

Method 1. Dry hydroeyaniec acid mixed with dry acetone and
kept for six months in a well-closed steamed flask is still completely
unchanged. On mixing, a slight rise in temperature took place.
That, however, no trace of the addition product has been formed
may be proved by first determining the total percentage of hydrocyanic
acid by means of the well-known titration with silver nitrate and
then by ascertaining the amount of free hydrocyanic acid in the
same way as was done in the determination of the equilibria. We
will then find the same figures. After six months the mixture still
showed the same refraction, which also proves that no change had
yet taken place.

The reason why previous investigators obtained eyanohydrin all
the same may be safeiy attributed to the fact that there were still
present traces of moisture and that minute traces of alkalis from the
glass vessel considerably accelerate the reaction.

It is now also obvious why the methods 2 and 3 should lead to
a good result as the alkaline potassium cyanide is always present in
excess. It need hardly be said that the formation of nascent hydrogen
eyanide previously looked upon as the most important factor in
method 2 has nothing to do with the real reaction.

Although former investigators *) have not succeeded in preparing pure
cyanohydrin by the second method, nothing is easier than the isolation
of the pure nitrile by distillation under reduced pressure, if only care
be taken to have a slight excess of hydrochloric acid present after
the reaction has taken place.

The following are the chief properties of the nitriles, as yet
investigated.

Dimethylketonecyanohydrin is a perfectly colourless liquid practically
odourless. Sp. gr. at 18° 0,9342. Decomposes on distillation at the

1) B. B. 17, 2009.

2) Urecu, Ann. 164, 255.
Tiemann u. FRiep.anper, B. B, 14, 1970.

( 144 )
ordinary pressure, b.p. at 23 mm. 82°, m.p. — 19,5°, n§> = 1,40526.

Lthylmethylketonecyanohydrin, colourless liquid with a faint ketone-
like odour. Sp. gr. at 18,5° 0,9324. Boiling point at 20,5 mm. 91°.
Does not solidify in a paste of solid carbon dioxide and acetone.
mes = Aion

Diethylketonecyanohydrin, colourless, a somewhat stronger odour than
the former nitrile. Sp. gr. at 18,5° 0,9300. Boiling point at 18.5 m.m.
97.5°, does not solidify in a paste of carbon dioxide and acetone.

ni, = 1,42585.

University Org. Chem. Laboratory, Utrecht.

Chemistry. — “The molecular rise of the lower critical temperature
of a binary nuixture of normal components.” By J.J. van Laar.
(Communicated by Prof. H. A. Lormn'z).

1. In the “Chemisch Weekblad” of April 8 1905 (II, N°. 14)
I derived an expression for the so-called molecular rise of the lower
critical temperature, viz:
f. (=) = 2V6p — (1 + w),
1 OA)
in which 6 represents the ratio of the two critical temperatures

gal

2 : b,
and w the ratio —.
1 b,

In this I started from the approximate assumption, that the critical
temperature of a binary mixture may be represented by the simple

expression

yy

Fl aed
be,
The formula found is at any rate more accurate than that of
van ’T Horr, according to which the molecular rise would be constant
(Chem. Weekbl. of Nov. 21st 1903 (I, N°. 8)), and I adduced a few
examples to show that the expression found by me represents the
experimental results of CENTNERSZWER') very accurately — provided
the molecular weight of the solvent SO, is doubled.
Bicuyer in his thesis for the doctorate *) came to pretty much the
same result with regard to CO, as solvent. He, too, had to double
the molecular weight of CO, in order to get sufficient concordance

1 Z. f. Ph. Ch. 46, 427—501 (1908).
*) June 1905, p. 125—130.

( 145 )

with my formula for the substances examined by him (except for
naphtaline and chloro-nitrobenzol).

Now Bicuner thinks the assumption of a (CO,), bimolecular at
the critical temperature very doubtful, and Kupnen too recently called
my attention to the fact that according to /zs measurements ') of the
vapour pressures of liquid CO, at different temperatures, the vapour
pressure factor / presents a perfectly normal course, in opposition
to what the measurements of Ruenautt at O° and 10° C., and those
of CaILLETET at —50° to —80° give for it’).

Nor has the assumption of a bimolecular (SO,), really any foundation.

Now, just recently *) I have examined the accurate course of the
plaitpoint curves for binary mixtures of normal substances, so that
it is now possible to derive a more accurate expression than the
above, in which the critical (plaitpoint) temperatures of the mixture
were identified by approximation with the temperatures of the coin-
cidence of the inflection points of the successive w-curves. That this
was, of course, not true, was sufficiently known, and that the difference
ean be considerable has been more than once emphatically stated by
vaAN DER Waats. One look at the plate adjoined to my paper mentioned
above shows at once how perfectly different the course of the plait-
point line — also at the beginning, at 7’, — can be.

It will appear from the following derivation that the values found
from the above approximated formula should be more than doubled
in many Cases.

Kerrsom has already derived *) a general expression for the molecular

ee lan ali
Kise
Ne dee

and as in bis final expression, viz.

T, \ dz

i but as he used the law of the corresponding states,
0 °

there oceur all kinds of quantities, which have either to be determined
experimentally, or have to be calculated from the equation of state,
I preferred to derive the required expression directly from the relation
found by me for the course of the plaitpomt line for mixtures o
normal substances.

1) Phil. Mag. 61, Vol. 2.

2) See my paper in the Arch. Teyler (2) 9, 3¢ Partie, p. 54.
3) These Proc. of June 1905, p. 33 et. seq.

4) These Proc.; Comm. Leiden N° 75, p. 6.

( 146 )
2. This relation was the following: ')

(12) | (1 2eyr— Bed —a2 | Ve —"| Bal 9 OAV +

a(r—t) (31) |= 0. > Pra bord orspon re, We)

In this 0=av—8Va=(b,Va,—b,Va,) + av—b); a=Va,_V a,
and B= b, — 0,.
In the derivation it was only assumed that. a,, =Va,a, might be

2
put, so that the quantity @ may be represented by baa) Va,+2Va,

This is the on/y simplifying assumption.

We now proceed to make the above given expression homogeneous
in the way of p. 35 et. seq. of my last paper. (These Proc. June 1905).
By considering only the case 6,=6, more closely (which was
sufficient for our purpose), we simplified this expression considerably
in the paper mentioned, but now we shall put the quantity @ not
= 0, so that a new variable quantity must be introduced.

Let us put as before:

Va, b,
YI ® ; al

But now also:

then we get:

b
Veg tae 3 2s 0 tne).

a

Hence after re by # see. a’v' (1) passes suceessively into:

(-N-2
(Na tee “\+" 2 Q le

a a v a v(1 — 2)
and

(: — now (p + ) t — 2) — 3a (1 — 2) no | +

+(p+2) (1 —w(1 + n)) E (: —no(g+ ») (2 — 2nw o+9)+
(g + «)? (: —o(1+ n)) (: — 3w (1 + n))
— : | =0%

v(1— a2)
1, c. p, 33, formula (2), Cf, for the derivation; These Proc, of April 1905,

( 147 »)
For small values of x this becomes:

y*(1 -0\( 1-80 +n) )
==)

“2

(l—nwg)*+¢(1 «| 1—nwg)(1—2nwg) +

As viz. @ approaches then to ‘/,, 1—w(1-++nz) is replaced by
1— ao, but 1 —3w (1+ nz) has been retained. Further introduction
of w —'/, yields:

oh v(t = 3a(1 +n)
1 /.ngp)+), 9 |aa— ng) —'/, ny) ‘|=

a“

from which follows:

"hg" (1 — 80 (1+ m0) 1 —?/. ng)
iain ( /s"@) — 3(1—'/, ng) (1 — ?/, ng),

wv “hs YZ
or, after division by — ?/, g°:
30 — 1 (l—*/.ng)’ , 3
— + 3en = — aE ng) (lL — 7), np)i-
wv wae yg IB gp ‘ ;

If we now put w—'/,(1-+ 4d), we get:

beat 3 9
cf ee + 551m) (1 —"hng)—n, « (la)
as we may put 3wn—n. Thus we have separated in the first
member the only term in which numerator and denominator approach
to 0, whereas in the second member all infinitely small terms have
been neglected by the side of those of finite value.
Formula (1a) indicates, in what way the volume v varies in the
neighbourhood of «=O with «2, when we viz. vary the temperature
in such a way that we remain in a plaitpoint.

3. Let us now introduce the temperature.
For this the relation holds: *)

9
nt =<|'« (1 — «) 6? + a(v — Dat Spe ee)
wv
Here 6 is again = av — BYa. Reduction gives successively :
D) 2 2 b 2
apa") @( —a(a -="*) + Ae -*) |
v v a a v is

and

RT= ol #0 —9(1—nw(y + ) ++ ¥( —w(1 +n)) |

1

1) 1c. p. 33.

( 148 )

wo Paes Va :
as = while — and — are replaced by their values (see § 2).
v ) v a 3
1
Now
8a 8 ay’
RE
27b, 27 »b,
hence:

=", T, aI ah (1 — no (yp + »)+ (p +2)? (3 —ao(l +n) |

If we now put 7=— 7,(1 +7), o='/,(1 + 9), this becomes for
small values of wz:

Oa ed ee (1425 )a—oy (1-2) |
ia y

1-w

in the second member of which only terms of finite value and those
of the order « remain. We draw attention to the fact that according
to (1a) d is of the order «2. Further substitution of wo—'/, 4+)
yields:

1+d[ zk
Lltr=*/,——| «(1 —1/, ng)? + 4/,92(142 All — 6) (1 — na) |,
oC
as 1—o=*/,—"/,d=*/,1—/,8), 20 eee
The last expression becomes now :

14 = (1 +o See +( +22 —¢—ne)|,

or if we neglect terms of ae order than the first:

Ce = — d=) +6.
9"

And now it proves, that the terms with d vanish, so that we

Jd
do not want the value of — from (dq) for the calculation of the limiting
wv

: T .
value of the relation —'). For the sake of completeness we have,
v
however, calculated this value, as it may be of importance for some
problems to know in what way v varies with x in the neighbourhood
of the lower critical temperature (remaining on the plaitpoint curve).

i) This is, of course, in connection with the fact that at the critical temperature

of the first component the spinodal line touches the line =O, and — as the
spinodal curve is vertical at that place (i.e. // to the y-axis) for very small values
_of 2 — a change of v will therefore only bring about a change of temperature

(and so also of the plaitpoint temperature) infinitely smaller than the change of
temperature, brought about by a change of x.

( 149 )
So we find finally:
e __ @—*/,ng)’ 2

= SS aPpS—=H5-9 ol 6 oo & vo (Pe)
# “1 69"

which is the required expression, by means of which the limiting

T
value of — at «=O may be calculated for every given value of

wv

g and n.

4. Now it remains only to express the relations found in the
ordinary variables.
These are viz. (see § 1):
ts (7) 2 Ps =i
1s b,
Now the quantity ¢ introduced by us in § 2 and 3 is represented
by :
Var Ven VAR IES oa 1
ween Ge -17 0, Vb Op
while m is given by:
ame p : B —_ b,—6,;

n= :o = = = tie
x Vv b, b, ~

Formula (2a) passes therefore into:

1 BoLY
t ( i! Te

ae (VY Op—1)—

g —

+ AVY Gyp—1) —(y—1),

or
="), | orn -- Yr | + 2VOp — (1 +9),
aay
or finally, as t=

1

) =2V 60-94"). }vor—1)— "9-1 ~ (8)

fh Mele
DG ae He Nudes

The original expression, derived on the assumption that /R7,
may be approximately represented by = must therefore (see § 1) be
completed by a term: :

‘1 \e~—1) — fs].

This is the correction which must be applied, and it is easy to
see, that it can considerably modity the original approximated

expression.

( 150 )

Let us now introduce the ratio of the critical pressures of the
two components, viz.

Ps
— = 2%.
Pr
: : Orne: : 2
Evidently the relation iss — exists, which changes (3) into:

B= i(s 30 Gee mc |

= sie ye We = ie

an 6? Le al Ne

ra

le oo) are ic
=—6y—-+ co v=)

ry ¢ 7

NE ahle Lane TO 1 ( 1/3 1 1?
Tg = 42= VV 5 vz)- 1, i (8a)

1

or

being the final expression for the molecular rise of the critical tem-
perature on the side of the lower critical temperature.

Now a case of frequent occurrence is, that the critical pressures
of the two components differ (ttle. If these pressures are the same,
a —=1, and (8a) becomes:

ASO Oa), A) 2 oe) ee)

whereas the former, approximated expression (see § 1) for this case
1 ell 1

fhe
» for the case x=1 the former expression must be multiplied

la al

would yield: (p is thn =@) A=@—1=

by @= ae in order to yield the correct expression.
1

A few instances will prove that it is no longer necessary now to
double the molecular formula of the solvent.

As a is near 1 in most eases, and the formula (34) varies very
little with changes in the value of a, we shall use the formula
A =6(6—1) for convenience, the sooner as the values of 7, (the
critical temperature of the dissolved substance) are all unknown, and
can be given only by approximation.

Let us first take the four substances which CENTNERSZWER’s expe-
riments induced me to calculate in the “Chemisch Weekbl.” (Le. p.
227—228). We shall now calculate the values of 7’, from the values

( 451 )

of 4 found experimentally, and see if the values found in this way
are about the double of those of the (absolute) melting temperatures *).

K 4 T Melting

found | calculated) caleulated| — point POE

Anthraquinone 3,58 2,46 1060° 560° 1,9
Resorcine 2,36 212 910° 480° 1,9
Campher 1,53 1,83 790° 450° 4,8
Naphtaline 1,45 1,80 770° 30° 2,2

1,95 average

The values of 7’, are calculated from 7’, = 6 7, where 7, = 480°,
being the critical temperature of the solvent SO,.

So we find really a value in the neighbourhood of 2 for the ratio
between critical temperature and melting temperature. We call attention
to the fact that 2,0 is found as mean value for this ratio for bi-
and tri-atomic substances; for miulti-atomic-substances this mean
value rises to 2,3. There are, however, substances, where the ratio
mentioned falls to 1,4 or rises to 3,5. The values caleulated by
means of the formula 4 = 6 (6 —1) are therefore in any case not
in contradiction with what experience teaches us.

In the second place we shall consider in the same way five
substances, which have been examined by Bicuner only recently.
(See Thesis for the doctorate, p. 128—129). The solvent was CO,,
of which 7’, = 304°.

‘ : 7, | Melting hae
found | calculated) calculated) — point znotent

Naphtaline Jao AHS 650° 300° 1,9 (in the preceding

: table 2,2)
CHCl, | 265 | 9290 670° 335° | O41
C,H,Br, 2,87 2,27 690° 360° 1,9
CHBr, | 2,32 2,10 640° 280° 23
o-C,H,CINO, | 3,87 253 770° 305° | 25

2,14 average

1) See my paper in the Bonrzmany-Festschrift (1904), p. 322—324,

(152 )

Here too, we find therefore values for the ratio in question, which
are not in contradiction with its empirical value.

Doubling the molecular formula of the solvent is therefore no
longer necessary, and we may, therefore, say that the formula found
by us (8a) or approximated (3) represents the molecular rise of the
lower critical temperature very satisfactorily.

Finally I may point out, that the experiments — of CENTNERSZWER
as well as those of Bicunrr — are not so accurate that the difference
between 1,9 and 2,2 for naphtaline is of much importance.

The reason of this is easy to see; it is exceedingly difficult
to observe the critical plaitpoint temperature accurately. For it
is required for this purpose, that the corresponding volume be
accurately known beforehand, and that the volume of the tubes used
be chosen accordingly. Else, of course, not the plaitpoint temperature
sought, is found, but another temperature, situated more or less in
its neighbourhood. And this too can be a source of inaccuracies ').

From all that precedes it sufficiently appears that van *r Horr’s
assertion that the value of A is constant, and equal to about 38, is
altogether incorrect. For the value of 4 is quite determined by the
ratio @ of the critical temperatures.

If 6 should happen to be in the neighbourhood of 2,8, then
A =6(8@—1) will lie in the neighbourhood of 2,3 1,3 =3.
And now it has been very misleading, that really for the examined
substances the values of @ lie nearly all near 2,3. (For the five
substances mentioned examined by bicuner the mean value of @ is
2,25, for the substances investigated by CENTNERSZWER this is also
the case). If 6 = 3, we should find about 6 for A, so this is twice
as much! Hence there is no question of constancy.

1) Also CentNeRszwer calls attention to this in his paper (Z. f. Ph. Ch. 46,
p. 427—501 (1903). See specially p. 446, 459, 464—466, 469—470, 489—492 and
497—499. It appears from these passages, how much trouble he has taken to
determine the exact ‘Fiillungsgrad”, and in this way to get as near as possible to
the critical plaitpoint temperature. As the determination of the rise of the critical
temperature was only of minor importance to Bicuner, the values given by him,
cannot — as he himself states — lay claim to the accuracy reached by GentNerszwer.

( 153 )

Chemistry. — “On the six isomeric tribromoxylenes.” By Dr. F. M.
Jager and J. J, Buanksma. (Communicated by Prof. A.
HOLLEMAN).

The six isomeric tribromotoluenes were prepared in 1880 by
Nevite and Wintner ') and again in 1903 in a different manner by
JaknGER*) with the object of studying the connection between mole-
cular and crystallographic symmetry with isomeric benzene derivatives.
In order to be able to extend this study to another series of com-
pounds with an analogous chemical character we have now prepared
the isomeric tribromoxylenes and give a short review of the mode
of formation of these substances; we intend publishing a more extended
report later on in the “Recueil”

Tribromo-o-xylenes.

These substances are prepared by starting from the orthoxylidines
1-2-3 and 1-2-4 according to the subjoined scheme :

CH, CH CH;
ANC: Br \CHs Bu CHs
Al | —> = (|56° => = {see |
NH, NH; Br
Ve
CH, CHa CH,
VANG6 aoNGH: 7 NGH:
2. Pe ta FN68euf Ct ee [41059
SUA BNA Br VAs
NH, NH, ‘Br

The orthoxylidines were treated in glacial acetic acid with the
calculated amount of bromine and in the dibromoxylidines thus
obtained the NH,-group was replaced by Br according to SANDMEYER’S
method. The tribromoxylenes thus obtained were purified by distilla-
tion in steam.

Tribromo-m-xylenes.

3. 2-4-6-tribromo-m-xylene was prepared in different ways.

a. Starting from symmetrical xylidine,

CH, CHs CH,

AN Br/ Br BZ Br

195° >

NH, a NH, a die: \ Aol
Br

The sym. xylidine was converted into tribromo-sym.-xylidine and

1) Ber. 13. 974.
2) Dissertation, Leiden, 1903.

( 154 )

the NH,-group was then eliminated by means of amylnitrite with
addition of finely divided copper.

b. Starting from 4-6-dibromo-2-amido-m-xylene prepared according
to ‘Auwers'), we also obtain 2-4-6-tribromo-m-xylene m.p. 85° by
replacing the NH, by Br according to SANDMEYER.

c. Acetoxylidide 1-3-4 yields on treatment with bromine and water
dibromoacetoxylidide *). If this is boiled with hydrochloric acid, so
that the acetyl group is eliminated and if in the dibromoxylidine so
obtained the NH,-group is replaced by bromine 2-4-6-tribromo-m-
xylene is also obtained.

CH, CH;
Br” \ Br Br Br
| | —> | 65° =
Noes NE:
Nit Nit,
CO
CH,

4. «a. In order to arrive at 4-5-6-tribromo-m-xylene 4-6-dibromo-
2-amido-m-xylene was converted by means of bromine into 4-5-6-
tribromo-m-xylidine and from this substance the NH,-group was
eliminated by diazotation and boiling with alcohol.

CH, CH; CHs
Bry7 \NHy. --BrY NINES Br
120°] —> — [190° => ~~ | 105°
\ fous Br\_ /CHs Br\ /CHs
Br Br Br

4. Starting from 6-bromo-4-amido-m-xylene m.p. 96° we obtain
by bromination 5-6-dibromo-4-amido-m-xylene m.p. 35°, which is
converted by means of the SANDMEYER reaction into 4-5-6-tribromo-
m-xylene.

CH; CHy CH;
Br Br \ Br/
| 96° =>) 35° e ese 405°
CH; Br\ /CHs Br CH;
Me Cs SS

5. After many failures to prepare it differently, 2-4-5-tribromo-m-
xylene was finally made in the following manner.

CH; CH; CH;
/ \NE, 7 \NNE; 7 \Br
gine] > = | 500 | = >| 87° |
\ fos BEN NAD

Br Br Br

1) Ber. 32. 3313.
2) Genz. Ber. 3 220.

( 155 )

Starting from 4-bromo-2-amido-m-xylene prepared according’ to
N6LTING’), we obtained 4-5-dibromo-2-amido-m-xylene by bromination
and from this 2-4-5-tribromo-m-xylene was prepared according to
SANDMEYER.

6. Finally, tribromo-p-xylene was prepared according to the sub-
joined scheme.

CHs CH; CHs
7 NNE; A \NMp JNBr
ies sc] => |o°| —- | soe
My we co f Ba Va Br
CH; CH; CHs
Consequently all six tribromoxylenes had been obtained.
CIs CHs CH; CHs CHs CH,
Br/ \CHs Z\Nch BrA\Br BrA\ Yo NB a 4 eer
86° | | 1059 | 85° | 105° | 87° 89°
ee BN 7B \ 7 CHa BA Cis BeX CHa Br\ 78
br Br Br br Br CH;

We wish here to express our thanks to Prof. FRaNcnimonr, who
kindly presented us with the specimens for this research.

Zaandam Rye
Zaandam, ywyi 1905,
Amsterdam,

Meteorology. — “Oscillations of the solar activity and the climate”.
(Second communication *). By Dr. C. Easton. (Communicated
by Prof. C. H. Win.)

(Communicated in the meeting of May 27, 1905).

At the end of the first communication on this subject the suppo-
sition was started, that the 11-year oscillation of temperature with
regard to the eleven-year cycle of the sun’s activity generally was
accelerated in the cold, retarded in the warm part of the larger
oscillation. In order to investigate this matter more thoroughly, I
proceeded as follows:

1) Ber. 34, 2261.
) See for the First Communication: Proceedings of Noy. 26, 1904, p. 368.
In that paper p. 372 read 89-years instead of 178-years. Furthermore strike out
_ what has been said on p. 369 about the experiment of Savéur.
1G
Proceedings Royal Acad. Amsterdam. Vol. VIII

(156 )

The observed maxima of the eleven-year cycle were placed one
beneath the other, beginning with the greatest positive deviation of
the observed Maximum and ending with the greatest negative one.

Afterwards the cold-factors found for these periods were put in
their places on both sides of that line of the Maxima, the dates
being recorded accurate within a quarter of a year. Of the series
thus obtained those before 1750 were provisionally left out of
consideration: they are much less reliable, according to Wor and
Newcoms. The rest was divided into three groups (taking as limits
the deviations + 0.4 and — O4 years): group A showing as a
rule a strongly positive deviation from the maximum, group B
containing small deviations in both directions, in group C the
deviation being on the whole strongly negative. The mean deviation in
group A (38 eleven-year cycles) is -+- 1.5 years, that in group B
(3 cycles) 0, that in group C (6 cycles) — 1.7 years. For each
eroup the cold-periods which fell in the same vertical column,
were combined and the three rows of numbers thus obtained were
smoothed.

The following rows (Table I) show the result; J/ means the place
of the observed maximum, m and m, the calculated normal minima
on both sides.

TABLE I.

Acceleration or retardation of the cold wave with regard to the sun wave.
A 8 100 4 0) 0 3) ie 5) 2) Gr Sie 5 en ee ec nD eT
B 8- 316 99 7. 6) 320 ysl 2er0s 10> sb 241055 aon OMS
C 4 7 (9 AO 21 (26° 45) 16S 5) AS 7 ee SiO ei

m M my,

No conclusions can be derived with any certainty from this table.
Here again we find some indication of a distribution of the winter-
cold, within the 11-year cycle, different for the warmer and the colder
periods. Still however the curves b and C rather suggest a correlation
of the minimum of the sun wave with the Maximum of the cold-
wave. An other investigation seems to point in the same direction.
It was made by the method explained on p. 372 of our First Com-
munication); it merely differs in so far that only the cold-factors
since 1615 were included. We thus obtain the following table as
a counterpart of Table IL of the First Communication:

1) Table IL of the First Communication is based on a 356-year period; of such
periods, 3 are available. Each value given in that table represents therefore a
frequency for 3 years. The values that follow hereafter represent a yearly frequency.

( 157 )
TABLE IL.

Wintercold and phases of the 11-year suncycle since 1615.

(Groups from cold to warm),

A B C D
m 4.33 0.414 0 24 —
ap 0.77 0.25 0.44 0.26
M 0.42 0.36 0.24 =
dpy 0.27 0.46 0.21 0.20
dpy 0.32 0,53 0.24 0.13

There is, it is true, a strong elevation about the time of the
minimum in the coldest group and a small elevation at ap in the
warm one, but nothing is left of a curve corresponding with the
di-year one. This table does not confirm Table IL of the First
Communication. For the rest its value is only small, beeause the
material, though purer, is so scanty: in 26 eleven-year cycles only
51 observed cases of severe or very severe winters are available.

The telluric perturbations thus seem to be so strong that, though
they cannot cause the disappearance of the greater fluctuation, they
practically abolish here the 11-year cycle, such at least is the case for
the data at present under discussion. With regard to the supposition
at the end of the First Communication, our conclusion therefore
must be that the result of a more thorough investigation of all the
available data is negative.

We have already remarked that there appears only a very inecom-
plete correlation between the frequency of the sun spots and the
oscillation of the temperature. The correspondence became only some-
what more apparent if, for the cold winters, we combined four 89-y.
periods in one of 356 years. However the indication of parallelism
obtained between the periodicity of the severe winters and that of
the quantity 1/—m of the sun’s oscillation, leads us to consider more
closely the other elements of the sun’s oscillation. For a long time
it has been well known that the maximum of an eleven-year cycle
as a rule succeeds the minimum the faster, the higher the wave.

As the amplitude of the sun’s period must be restrieted within
certain limits, this might be supposed to mean that the steepness of
the ascending phase represents the only variable element in the sun’s
oscillation and that the minimum thus would remain constant, at

dL

( 158 )

least would not be displaced systematically. In that case indeed a very
high wave of the sun spot curve would have to coincide with an
abbreviation of the normal ascending phase. When we examine
the oscillation more closely, however, it soon appears that the place
of the minimum varies systematically. We are thus led to inquire,
whether for a comparison with tellurie oscillations just this accelera-
tion and retardation do not present real advantages over the other
elements of the sun’sactivity. As by no means the observed deviations
of the maxima and minima always agree in amount and direction,
I investigated separately :

1. the deviations of the maximum,

2. those of the minimum,

3. those cases in which the deviations of Jf and m have the same
sign. Where such is not the case, we should not attribute it to errors of
observation without additional proof. However, case 3 will show the most
pronounced deviations of the oscillations. The investigation was made
first using the whole of the available materials ; second excluding
all periods before 1750 (that is of the least reliable observations).
As the result did not deviate very strongly, only those of the last
mentioned investigation are here communicated.

Meanwhile it seemed desirable to use not only Nrwcomp’s list,
but also the data as given by Prof. A. Wo rer '), because relatively
small deviations in the observed values may already have an appreci-
able influence. In this case also the difference of the results was
fairly small.

For the following tables I used Wotrrr’s data as a basis; only
in the 1V" it was deemed necessary to communicate also (in paren-
theses) the result obtained from Newcome’s data.

In the tables II] and IV, column 1 shows the groups of the periods,
arranged according to the amount of the deviations beginning with
those that are largest positive; column 2 contains the numbers of 11-year
cycles; 3 the mean amount of the deviations; 4 the quantity M/—m;
5 the length of the period; 6 the mean of the Relativ-Zahlen (accord-
ing to the smoothed table of Woxrrr), in parentheses I placed the
means of the highest elevations of the curve; 7 the cold-factors
(yearly frequency). Table V column 3 shows the mean deviation as
computed from the two phases.

1) A. Wotrer, Astron. Mitteilungen, XCIII, 1902.

( 159 )
TABLE III.

11-year periods, arranged according to the deviat. of the max.

1 9 3 4 5 6 7
M M—m L R.-Z. Cf.
A | 4 + 1.4 | 5.9 4453 32.8 (64) 0.33
B 4 — 0.4 4.4 41.2 47.7 (103) 0.31
4 — 3.2 3.5 Adie 60.3 (128) 0.59
| | |
TABLE IV.

ll-year periods, arranged according to the deviat. of the min.

3 4 5 6 a
1 | 2 m M—mn L R.-Z. CK
a hee 0.8 Pe) 10.5 46.5 (91) 0.27
| | fal 0.7) | (4.5) (10 9) | [43.2 (90)] (0.29)
|
B 4 — O04 4.7 Ae 2 44.8 (90) 0.38
| (0) (4.3) (40.7) | [54.4 (111)] (0.40)
Cc 4 — 2.2 4.4 Als Ut) 52.5 (114) 0.57
(— 2.0) | (4.9) | (1.8) [43.6 (94)] (0.51)
TABLE V.
11-year periods, arranged according to mean deviat. of both min. and max.
rl 9 3 4 3) 6 7
m, and M, M—m. L R.-Z, ch
A 2 + 1.6 5.8 10.6 35.4 (66) 0.35 |
B 2 = e0 3.9 40.7 54.2 (144) 0.50
c 2 = 40 3.4 1.4 66.3 (142) 0.70 |
!

These tables show that the variation is least apparent in the length
of the period, furthermore that the deviations in the position of the
maximum of the period (as was to be expected) agree well with
those of the Relativ-Zahlen and of the quantity 1/—m ; finally that
this correspondence is less satisfactory for the deviations in the position
of the minimum, which however agree well with the modification
in the cold factors. In Table III, it is true, the largest cold factors
coincide with the largest deviations of the other elements of the

( 160 )

sun’s activity, but the course is irregular (the list which also contains
the periods before 1750 shows the same peculiarity).

It appears from Table IV that the correlation of the course of
the coldfactors with the deviations of the minima is evident according
to Nuwcoms’s data as well as to those of Wo.rerr.

When we consider only those periods however, where the devia-
tions of m and J/ are of the same sign, we find correlation between
all the elements of the sun’s activity and the cold factors. (That the
extension is so small, proves that the acceleration of J/ and m is rather
to be explained as an acceleration of the whole period). Unfor-
tunately, if we leave out of consideration, as in Table V, the data
which must be considered insufficiently reliable, the materials becomes
so limited that the result can prove but little, and can only be con-
sidered as a strong indication.

In the preceding investigation the periods have not been chrono-
logically arranged ; it was not possible therefore to find any evidence
of a periodic modification in the deviations of the sun’s oscillation.
According to our former results, however, we may expect that these
deviations will generally correspond with the great periodic wave.
To test this point the 11-year cycles have been arranged according
to the adopted 89-year period ; the deviations have been compared
with the cold factors found for each period. An arrangement corre-
sponding with the computed maxima seemed preferable to an arran-
gement corresponding with the minima, because of the previously
indicated acceleration of the strong cold waves, beyond the observed
solar minimum.

In Table VIA the vertical columns represent the eight 11-year
cycles contained in the 89-year period; the first begins in 1648 ;
in the 238° and 24" square however I placed the periods 1626—1637
and 1637—1648, the periods since 18594 being of course not yet
available. In each square the uppermost number represents the
deviation of the maximum, the second the deviation of the following
minimum according to Nrwcoms. Where the sign + or — follows
ihe number, the deviation amounts to at least half a year, either in
the positive or the negative direction. A O indicates a smaller devia-
tion. The lower number gives the total of the cold factors between
{wo consecutive maxima. Those phases to which Nrwcoms assigned
a weight smaller than 8, have been placed in parentheses.

Table VI contains the same data according to WoLreEr; however,
in the last square but one I have here written down the observed
phases 1894 and 1900,

( 161 )
TABLE VI.
Deviations of M and m and cold factors in the 11-year cycles, arranged according
to the 89-year period.

A. Newcomb.
I Il Ill IV V VI Vil VIII
| ee Rees
(40.2) 0} (0.4) 0 \(-+-4.0)+\(+-2.9) +) (—0.3) 0 1.4 + 2.7 +}-+0.8
5i3}0 ant (+1.9) +/(+0.8) +) —1.8 — Ty .0 4- Ty + |-+0.8 ai
4 40 2 3 | 4 3} 4 4
0.9 4.4 | 1.4 See —4.4 —|}—0.6 —|41.4 +
1 6 rE Wp 0 alin, iW —1.9 —|—4.14 —|—1.5 —|—0.6 —|+0.9 +
4 | 3 7 33 9 5 6 Al
14.9+4+/—0.8—|/—05—, 0 0O alts 0 |+0.2 0 |—0.5 a +1.9 ria
0.4 0 }—0.5 —|-+0.6 +]-+0.4 0 0.4 0}+0.2 0 1.0 + | -+0.8
4 5 | 4 3 6 3 4 1
B. Wolfer.
\(-—0 6) —(—O 6) —\(-++8.3) +)(4-2.2) +)(—0.9) a 0.5 +1424 a +0.3 0
\(—0.41) 0}(—0.2) 0)(4-2.4) +-\(-+-1 .0) (ey Ty 2 \(4+1.6) ++\(-++-0.9) +
ee He J s
0.4)0 0.9 + 1.0 —1.9 —|—4.2 —|}—5.6 —}/+0.40 0.5 +
f 08) 4 20 0) ay, a —2.1 —|—4.1 —| —1.6 —}— 0.5 — 1.4 +
29) —0.9 —|—1.14 —/|—0.2 0|—0.8 —|41.4 0.5 + |(4+41.0)+
0.6 —1.0 —|+40.4 0 |+40.4 0 1.0 +)-+0.5 1.3 + )-+41.0 +

In order to bring into greater evidence those cases, in which the
deviations are most decidedly indicated, I give in table VIIa, for
the whole of the three last 89-year periods, the numbers of + or —
(following the amounts) which remain, when I take together the
signs of the same vertical column.

Furthermore I computed the sum of the amounts in each ver
tical column, both exeluding and including the values in parentheses,
taking then the mean amount for each phase (VII, b* and bb).

Table Vile gives the elevations of the Relativ-Zahlen-curve
computed from Worrkrr’s table in the following way: the mean
was taken of the three highest yearly means on both sides of the
Maximum ; the series of numbers thus obtained (nearly two complete
series since 1750) were placed one beneath the other; then the
means Of these values were again computed.

The curve c* was then obtained by placing against the rotation

( 162 )

number of the period the means of the last mentioned numbers
taken in pairs (viz. the means of the maximum at the beginning
and at the end). For the curve cb the preceding maximum value
was written down in the same place.

In table VIId have been inserted cold factors as combined in each
column for the three last 89-year periods.

Table Vile? shows the sum of the cold factors between the com-
puted maxima of the 11-year cycles, obtained by addition of all the
89-year periods elapsed since 848. They are nothing else than the
totals of Table I in my first communication about this subject (in
e> I gave a greater weight to the most recent data, see p. 163).

Figure I is a graphical representation of the curves of Table VII.

Fig. I. 89-year period.
a—c: Sun’s activity.
d—e : Climate.

( 163 )
TABLE VII.

Deviation in the elements of the sun’s activity, and 89-year oscillation of the climate.

I Il It IV V VI VIL VIII
a
Direction of deviation. BSP fp he | Sie 0 |3— 0 Omer
b? +0.9] 0 |} 0.3|— 0.7|\— 1.6|— 0.6/+ 0.6/4 4.0
Dyk 0.6 |— 0.4/4 1.2/4 0.4/— 1.4/— 0.6/4 0.6/4 1.0
Amount of deviation. +
ce 79 90| 89] 105] 406] 77] 50] 52
Rel, Zahlen 65 94) 87] 92| 118] 9%] 61] 40
d 19 18 | 43 9} 419/ a] 1 3?
Culd factors
3 Per. of 89 years.
a 31 M5} 38| 96| 39) 97] 49 9?
aa eee o7 30| 98| 20} 31] 92] 4] @
0 < s .
All the 89-year Per. | | |

All the curves show a corresponding course; the correspondence
of the curve of the cold factors with that of the deviations im time of
the sun waves is certainly as well indicated as that with the deviations
in height. As was to be expected the correspondence does not extend
to details, but the strong depression in an interval of 8 eleven-year
sun cycles is apparent in all the curves.

In conclusion a few words on the apparent general increase of
the coldfactors during the last centuries, which might be inferred
from table I in my First Communication. Does it justify us in
assuming a secular refrigeration ?

In my opinion there is no reason for such a conclusion; it seems
more probable that it must be explained provisionally at least by
the incompleteness of the dafa for the earlier centuries. The numbers
of winters which have a weight 5 according to KOpprENn are distri-
buted as follows over the different centuries (the material for the
XIX‘ century is not homogeneous with that for the preceding ones),

Before the year 800. . . . . 4 winters.

In the IX century. . 7 rr
ee OD x » c 9 . . 5 ”
9099 XI ” . 7 ’ C 7 ”
Op XI 2 C d s . rd ”
oa ROIs Sareea UO) %
$9099) XIV ” s 5 o * 6 ”
Sh on XV 2S a Noowetae ee oS: *
EES XaV/ lame, OE eel, a
ee!) NOV SIT aes io A eg Shea
ae XV eo Be, AS) _

gah (CLO) eee es Satay (3) Weegee

( 164 )

There is no evidence of a smooth course, but rather a sudden
increase about the XIV" or XV" century. If there was any question
of slowly increasing refrigeration, this would appear most clearly
between the XV and XIX" century, the material being more
complete during this time.

For the three 356-year periods treated in Table I, First Commu-
nication, the numbers are: 25, 37 and 66. When treating this table
with the weights 4, 6, 10 for the 1st, 2°¢ and 34 “ereat period” we
obtain for the totals (see Table VII and fig. 1, f 6):

I (848—1203) 16 48 20 24 28 20 0 08
If (4204—1560) 18 69 108 36 54 36 48 18
HT (1561—1916) 24.0 18.0 15.0 14.0 23.0 16.0 11.0 4.0

Total vs ta 2 80" 285-20) a eee

In the recent material the depression about the middle of the
89-year period is pretty evident; we must remark however that it is
not to be found in the even 89-year periods of Table I; the totals
for the even periods separately become:

1a A411, AD AO eas.

This may be attributed of course to a 356-year period, in which,

as appears from the table, the third 89-year subperiod is strongly

anomalous.

I will now try to summarise the main results of the investigation
contained in these two communications.

We have seen — particularly when investigating the 11-year
cycles, leaving out of account their connection with any longer period —
that the purer our data the more evident the correlation between
modifications of the sun’s activity and the deviations of the climate. In the
same degree however these data become more scanty and the acci-
dental deviations might perhaps have a predominant influence for
this reason.

In these circumstances the first of our final conclusions must be this :
Our data are insufficient for any rigid proms
In regard to each several part of the investigation, we can at most
qualify them as strong indications.

For some points, however, these indications taken together are so
strong that they may be considered convincing. Such is to my opinion
first, the existence of a fluctuation, both in the activity of the sun
and in the climate, larger than the well known 11-year cycle of
the sun spots.

This conclusion, though very probable, would not seem to be demon-

{ t65))

strated, if it rested exelusively on the data about cold winters.
(Table | First Communication).

A purely accidental coincidence becomes inadmissible however, now
that we have found a similar fluctuation in different elements of the
sun’s activity, the more so, where the possibility — nay probability —
of a causal connection between the two phenomena is obvious. This
conclusion is strengthened by the correlation between the sun’s
activity and the temperature in tropical regions, found by KéppENn and
NorDMANN.

On the other hand the exact nature of these fluctuations cannot
yet be established.

The parallelism of the frequency of cold winters and the Relativ-
Zahlen is most strongly marked, if we take as a basis the period of
356 years=32 eleven-year cycles (see fig. I of our First Communication).

Moreover we find both in the sun’s activity and in the climate
indications of a shorter period. Meanwhile it is only the 89-year
periodicity which appears clearly in our data.

The matter may perhaps be cleared up by hypotheses about the
physical cause of the oscillations (hypotheses into which I have not
entered here). Therefore, what may be considered sufficiently demon-
strated about the nature and the length of the periodicity, comes
to ths: Retardation and weakening ofthe 11-year
suns oscillation together with adiminution of
tne number of cold winters every 89 years.

Moreover it seems sufficiently certain that strong deviations from the
“normal” 89-year oscillation occur at the same time in the sun’s activity
and in the climate. They are perhaps caused by the existence of a
still longer period (see for instance the considerable acceleration and
increase of the sun’s oscillation in the latter half of the 11 89-year
period — second half of the 18'® century — and the exceptionally
high cold factors of that time).

Finally we may conclude from the whole of our investigation, and
this is perhaps the most important conclusion: that in connection with
meteorological phenomena, not only the frequency of the
Sun spots; but also other elements. of the sun’s
aetivity curve deserve our attention.

Of course the possibility of a prediction of certain characteristics
of the weather, long in advance, with a considerable degree of
probability, is contained in our result.

Also its importance for explaining geographical and geological
phenomena is obvious. I do not now wish to enter into details about
these matters.

( 166 )

Chemistry. — “On colorimetry and a colorimetric method for
determining the dissociation constant of acids.’ By Mr. F. H.
KispMan Jr. (Communicated by Prof. 5. Hoogewerrrr).

On colorimetry.

During the last few years I have been obliged to undertake a
large number of colorimetric determinations, which had to be made
as accurately as possible.

The impossibility of making really accurate colorimetric deter-
minations without taking a number of precautions, made KNEcHT ')
utterly reject this method of working. As KNecut’s method (titration
of the colouring matters by means of titanous chloride) is not appli-
cable in all cases, it was thought that an effort to improve the colori-
metric method, would not be undesirable.

PRINCIPLE OF THE COLORIMETRIC METHOD.

Starting from the supposition that on diluting a solution of a
colouring matter, neither the amount, nor the nature of the colouring
matter present, undergoes a change, the principle of the colorimetric
method is as a rule indicated as follows:

If we evamine in transmitted light two solutions, containing the
same colouring matter, the concentrations will be iwversely proportional
to the heights of the layers of the same colour.

This formulation will be found in OstwaLp, Handbuch fiir Physiko-
Chemische Messungen*) and in Herrmann, Coloristische und Textil-
chemische Untersuchungen *),

The first supposition cannot at all be accepted as being generally
correct; in fact, in the practice of colorimetry the circumstances, in
which it is correct, occur but rarely.

In future those solutions of colouring matters, where these suppositions
are permissible and which may, therefore, be determined colimetrically
without precautionary measures, will be styled directly measurable.

If the colouring matters under examination are not electrolytes,
their nature and amount will suffer no change by dilution. Such
colouring matters are, therefore, directly measurable.

But with acid, or basie colours, or their salts the case is different,
as these can but rarely be determined directly. The cause may be
found sometimes in the electrolytic dissociation, in other cases in a

1) Journal of the Society of Dyers & Colorists 1904, p. 242.
*) Ibid. p. 179.
8) Ibid. p. 63,

( 167 )

hydrolic phenomenon, which plays its part. If we have a solution
containing acid colours, whose anions possess a different colour from the
undissociated acid, the solution will exhibit a mixed colour composed
of the colour of the anions and that of the undivided acid. This
phenomenon may be readily demonstrated by means of the acids of
the following colouring matters: Methylorange, metanilyellow and ben-
zopurpurin 4 B.

That the change in colour, which these acids undergo when their
solution is diluted, must really be explained in this manner, is proved
in the second part of this paper, where an application is made of
the fact that such dilute solutions may be restored to their original
colour by addition of dilute acids ’).

This explanation disposes of the theory of Kiistmr?) and of that
of Guaser*) as to the indicator methylorange and it appears indeed
that the methylorange-acid is, for an indicator, a fairly strong acid.

This phenomenon also occurs with salts of acid colours, therefore
when testing the so-called acid and directly-dyeing technical colours.

Such a case has been mentioned by C. H. Sturrer *), who noticed
it when testing solutions of tsonitrosoacetophenonsodium. He found
that these solutions assumed an increasing yellow colour on increased
dilution and he rightly attributes this to the more powerful electro-
lytic dissociation caused by the dilution. In this case the ionisation
in /10 solutions had proceeded so far that a further dilution caused
no further visible change in colour.

If however we want to measure solutions of benzo-pure-blue, benzo-
azurin or allied colouring matters, it will be noticed that in solutions
containing from 0.1—0.05 gram in a Liter (approximately W/8000—
N/6000)the phenomenon is still of such an interfering nature, owing
to the great difference in shade of colour, that a direct measurement
is impossible. Siurrer’s dissertation only reached me when my researches
had already been brought to a close.

In theory, analogous phenomena are possible with basis colours
and their salts, but I have not as yet met with any such instances
and in fact, have not searched for them.

When salts of very weak acid colours are tested, the hydrolysis proves
very troublesome, if the colour of the anions and that of the acids

1) Compare A. A. Noyes and A. A. Biancuarp Journ. Americ. Chem. Soc. 22
p. 726 and Central Blatt 1901. I. p. 11 n® 15.

2) Kiisrer, Zeitschrift fiir Anorganische Chemie 8 p. 127,

3) Guaser, die Indicatoren.

4) CG. H. Sturrer. Het mechanisme van eenige organische reacties. Academisch
Proefschrift, Scheltema en Holkema. 1905,

( 168 )

should be ‘different. As the solutions get more diluted, they exhibit
colours approaching the shade of the colouwr-acid.

A very striking example, which lends itself well for practical
demonstration, is furnished by sodium carminate. Sodium alizarate
may be also used, only the solutions, on being diluted, soon become
turbid, owing to the slight solubility of alizarine.

From these examples it follows that the fundamental principle
of colorimetry ought to be expressed as follows:

Solutions of the same colouring matter, when tested colorimetrically,
evhibit in layers of the same thickness the same intensity of colour if
they possess the same concentration.

THE COLORIMETER.

This apparatus must be so constructed that the liquid under exa-
mination may be brought to practically the same concentration as
the standard liquid.

The apparatus best suited for this purpose is that of SaLLERON ?),
modified by Koppescuaar’). In this colorimeter the most concen-
trated of the two solutions is diluted with water until it has the
same colour as the weaker solution. From the amount of water added,
the desired concentration is calculated. It is a matter of indifference
whether the most concentrated or the most diluted solution is used
as the standard liquid.

As it is not possible, when using the colorimeter of SALLERON-
KorprscHaar, to make rapidly successive readings of a quantity of
solution to be measured, the apparatus, which I am now using and
which is represented in the annexed drawing, is perhaps to be pre-
ferred. It is constructed from a colorimeter of C. H. Wotrr*). The
tube containing the standard liquid, the standard tube S is connected
by means of a small horizontal tube / with the glass cylinder A
in which a plunger C' is suspended. This plunger can be moved up
and down by means of a cog-wheel device along the standard B. Jn
this way the level of the standard solution may be raised or lowered :
by providing 6 with a scale, the position of the liquid may be read
off on the same.

The actual colorimeter stands in the dark chamber D. It consists
of the standard tube S and the tube containing the liquid to be
measured, the measuring tube M.

1) Zeitschrift fiir Anal. Chemie 11 p. 302.
2) Zeitschrift fiir Anal. Chemie 38 p. 8.
3) Dingl. 236. 71.

(170 )

The illumination of such a colorimeter is generally effected by
means of a mirror placed below the tubes, which reflects the light
from the sky. Owing to the clouds, this illumination may be very
irregular, therefore artificial light is preferable. Incandescent light
is very satisfactory. With artificial light, however, a mirror cannot
be used, as small displacements of the lamp greatly affect the illu-
mination of the two colorimeter tubes; instead of a mirror, a piece
of ground milky-glass is then employed.

Above the tubes is placed the optical arrangement, which serves
to create a field of vision, which is divided into two parts, one of
which is illuminated by the rays, which have traversed the standard
tube and the other by those, which have traversed the measuring
tube.

In principle, it is preferable to make both halves of the field of
vision exactly the same shape, as they are then observed under
exactly the same conditions. These conditions are not satisfied by
the prism-system of Lummer and Bropuun, which has been applied
by H. Kriss to the Woxrr-colorimeter *), The field of vision is here
a circle surrounded by a ring. This may, perhaps, partly explain
the less favorable report of the Pbhotometer Committee of the Nether-
land Society of Gasmanufacturers *).

The prism-system of Fresnen, generally met with in colorimeters,
suffers from the drawback that it is liable to give way, when being
cleaned, and cannot then be again properly joined together. This
creates in the field a heavy black line of junction, which greatly
impedes an accurate observation. A prism made from milky-glass *)
is not advisable on account of the transparency which causes the
two halves to illuminate each other in the neighbourhood of the
line of junction. An equality of colour is then noticed before it is
really a fact.

I use a prism of polished telescope-metal with angles of 45°
illuminated by two little mirrors also at angles of 45° placed above
the tubes. The line of junction is then hardly visible and the prism
is proof against the influences of a laboratory atmosphere.

The apparatus is now used as follows: The standard tube and
the vessel A are provided with standard liquid, and fixed in such
a manner that the height indicated on the rod 6 really corresponds
with the position of the liquid in S. The standard liquid would
have to be more diluted than the solution to be measured. A known

1) Zeitschrift fiir Instrumentenkunde 14. 102.

2) Report of the said committee 1893.

8) Ostwald I.c. p. 180.

quantity of the latter is then introduced into the measuring tube J/
and when the colours are equal, a reading is taken. In many cases
it is not possible to take a reading, owing to the difference in shade
of colour of the two liquids, but still we are able to see at which
heights of the standard liquid this is decidedly darker or lighter
than the measuring liquid.

The measuring tube is then filled with water up to the average
of those heights and definitive determinations are now made. The
difference in concentration between the two liquids is now in most
eases so slight, that a difference in shade is no longer perceptible.

In any case it is desirable to dilute the contents of the measuring
tube up to the height found and to take a fresh reading, even when
determinations may be readily made without dilution.

In the case of acid colours this mode of working can sometimes not
be applied. The measuring liquid should then be gradually diluted
until, the colours being equal, the height of the measuring liquid is
about the same as that of the standard liquid.

In the case of such small differences in the concentration it may
be safely assumed that the concentrations are inversely proportional
to the height of equally-coloured layers.

The great advantage of this method of working is this, thatat the
final determination a series of readings can be taken, also that the
standard liquid can be alternately changed from darker to equality
of colour and from lighter to equality, as is done in polarisation.
This renders each determination very certain.

The readings may be rendered much more delicate by placing a
coloured piece of giass on the ocular. It is necessary to choose such
a colour that the rays of light, transmitted through the measuring
liquids are also transmitted through the coloured glass. Atrial with
a pocket spectroscope or a consultation of ForMANEK’s work ‘Der
spectralanalytische Nachweis kiinstlicher organischer Farbstoje”, renders
the choice easy.

These glasses are readily made by dyeing old photographic plates
with basic colours, which is easily done in the cold.

A colorimetric method for determining the dissociation constant

of acids.

Acid colours whose anions possess a colour different from that of
the acid itself, and which we will call ¢ndicator-acids, may be used to
determine the dissociation constant of the indicator-acids themselves
in the first place, and also of all other colourless acids, if we have

12

Proceedings Royal Acad. Amsterdam. Vol. VIII.

at our disposal a colourless acid the dissociation constant of which
is known with certainty.

PRINCIPLE OF THE METHOD.

If the aqueous solution of an indicator-acid is diluted with water
the colour will change in the direction of the colour of the anions.
If for the dilution of an indicator-acid solution an isohydrie solution
of a colourless acid is used, the degree of dissociation will not alter
and the colour of the solution will remain the same.

If the solution of the indicator-acid is diluted with water, we may
titrate back with an acid of which the concentration of the H-ions
is larger than that of the solution of the indicator-acid, until the
original-colour is restored. We have then prepared from the water
and the acid solution a mixture, which is isohydrie with the solution
of the indicator acid.

Starting from an acid with a known dissociation constant — stan-
dard acid

and an arbitrary solution of an indicator-acid we may
in the same manner determine the dissociation constant of a second
colourless acid, by preparing as directed, from the standard acid as
well as from the unknown acid solutions, which are isohydrie with
the same solution of the indicator-acid. The acid solutions are then
mutually isohydrie and the caleulation of the dissociation constant
is readily made from the above data.

THE OPERATION.

The above described colorimeter is best suited for this method.
The solution of the indicator-acid is introduced both in the standard
tube and the measuring tube. The amount of indicator-acid does not
matter, provided the quantity of it, in both tubes, is exactly the
same. After most carefully adjusting the colours, the contents of the
measuring tube are diluted with an accurately known volume of
water, say, acc. If now of a standard acid the dissociation constant
is Ay and if from this is prepared a solution of a dilution v4, we
then titrate with this solution the contents of the measuring tube until
the colours are again the same. If this should require 4 ce the dilution
at which the solution of the standard acid is isohydriec with the
given solution of the indicator-acid is :

atb

IEE SB Seiee
i A e

If now the dilution of the indicator-acid is known, or if we have
found in the same way the dilution at which an unknown dissocia-
tion constant yields an isohydric solution, then, calling both dilutions

(173)

Vy, the unknown dissociation constant Ay is found by the following
caleulation :
If in the solution of the standard acid we call the dissociated
part « then
a :
ra = Cn,
the concentration of the //-ions, therefore,

Ka 4
Y —_— _ —_— 1 SS
CH 2 ] a0 Vie VA 7 |

For the acid with an unknown dissociation constant A’, we may
calculate the same from

F: Kp oa A 4 oy Tie
, == — a ——— .
gremee Kp Vp z

as Cy and Vx are known. It is, however, simpler to make the
calculation as follows :
From

2 KV ee @
are and — = ©
= tp V
we find:
: KE 7
(CP SS SOK
V

As both acids are isohydric in dilutions of, respectively, V4 and
Vp, Cy will be the same in both, therefore
KA Spree Kp Lae
a (HO a kG
Val a

2

from which

Ka= Kz . — . —

Test EXPERIMENTS.

In order so show the accuracy of the process, I have made three
determinations. In the first one I have determined the dissociation
constant of benzoic acid, taking the constant of salicylic acid as
known. In the second experiment I have determined the dissociation
constant of anthranilic acid, using the figure obtained for benzoic acid.
In a third experiment, the dissociation constant of propionic acid has
been also determined with the aid of benzoic acid.

Determination of the dissociation constant of benzoic acid.

Given :

KE = 0.00102 U; == 150
v1, = 100

(oi74, )

Indicator: metanilyellow-acid. The change in colour of this indicator
on dilution was not strong enough. Therefore, the solution was mixed
before use, with a few drops of hydrochloric acid, which caused the
difference in colour to be more decided and more readily observable.
This addition may be made without fear, for the colour is used as
the indicator for the concentration of the hydrogen-ions and an equal
concentration of the hydrogen-ions always gives the same colour. Only
we must take care to use the same indicator solution for the whole
series of determinations.

Found:
15 ce of indicator + 50 ce of water. Colour again restored with

Nec salicylic acid solution, therefore :

50
i +O 5< 150 = 988.5 = V,

15 ce of indicator + 10 ce of water. Colour again restored with average
22.5 ce of benzoic acid solution, therefore
10 + 22.5
22.5

0 0,

from which :

0.00102 =| — 14% Ee | 0.0006294
Cpe == (I), Z
2 (5 00102 983.5 s '

= 0.00109 , Ltt He mnnnened 083.5 eee
plement eT No atl E= (NNIe :

By the electrolytic process the value of A, = 0.00006. ')

Determination of the dissociation constant of anthranile qaerd.
Given: Ky, = 0.00006 v7, = 200
Vq = 100

Indicator : methylorange-acid.

Found:

15 ce of the indicator in both cases diluted with 50 ce were titrated
back to the original colour.

For this was required

Benzoie acid : 0.87 ce

Anthranilic acid: 1.56. ,,

1) Nernst. Theoretische Chemie, p. 404.

(eon)
From which :
Vee o00'— 11690
er OIsTiae tt
50 + 1.56
i < 100 = 3306
: por ;
0.00006 gre
CH= =| = jl a LA 4 ae SI | = 0.00004749
2 0.00006 < 11690 z

and

3306 1 — 11690 s< 0.00004749
11690 © 1— 3306 & 0.00004749
The electrolytic process gave 0.0000096. ')

Kg = 9.00006 .

= 0.0000089.

Determination of the dissociation constant of propionic acid.
Given: K,=0.00006 v,= 442.5
1020

Up

Indicator: methylorange acid.

Found :

15 ce of indicator were diluted with 25 ce of water; the colour
was restored on adding 1.5 ce solution of benzoic acid.

Vj hd 7816

15 ce of indicator were diluted with 10ce of water; the colour
was restored on adding 6 ce solution of propionic acid.

10 6 :
V,= ae >< 1020 = 2720
6
; 0.00006( ba Nene ce fit BroOGRCA eT
ee art es BIE, —— (1) 626
Ht 2 0.00006 s< 7816

and

2720 1 — 7816 X 000006261

7816 1 — 2720 X 0 00006261
Found by the electrolytic process: 0.0000134.
This method may, perhaps, prove useful in cases where the

electrolytic method meets with difficulties, for instance in the deter-

K,, = 0.00006 ><

— 0.0000128 ’).

mination of very small concentrations of hydrogen-ions or in the
determination of the concentration of hydrogen-ions in presence of
other cathions. I intend making further experiments in that direction.

Laboratory, Netherland Technical School.
Enschede, 15 May 1905.

1) Osrwatp. Zeitschr. fiir Physik. Chemie 1889, p. 261.
2) Osrwatp. Zeitschr. fiir Physik, Chemie 1889,

4

( 176 )

Mathematics. — “On the number of common tangents of a curve
and a surface.’ By Dr. W. A. Vursivys. (Communicated by
Prof. D. J. Korrrwse).

§ 1. Let €, be a plane algebraic curve of class 7, and )S, an
algebraic surface of class m,. Every tangent of C, touching S, is a
tangent of the section s of the surface S, with the plane V” of C,.
Conversely, each common tangent of C, and s is a common tangent
of C, and S,. The curves C, and s being of the class 7, and m,
respectively, they have 7,m, common tangents. Hence, S, and C,
have 7, 7m, common tangents too.

Let the plane V" of C, oceupy the particular position of touching
S, in d points of ordinary contact and in x points of stationary con-
tact, the class of the section s is now

m, — 2d — 3y").

Hence, the curves s and C, have now
, (m, — 2d — 34)
common tangents. Every tangent of C, passing through one of the
points of contact d and y is a common tangent of C, and S,, without
being a common tangent of C, and s. In §8 will be proved, that,
if D, be the developable formed by the tangents of C,, each generating
line of YD, touching S, in a point ¢ counts for two common tangents

,

of C, and S, and each generator of ), touching S, in a point x
counts for three common tangents of C, and S,. If C, be a plane eurve,
the developable D, is the plane V counted 7, times. Every ordinary
contact d gives thus 27, common tangents of C, and S,, and every
stationary contact y% gives 37, common tangents of C, and S,. Thus,
the total number of common tangents is

rr, (m, —2d—3y) + 2d7,+ 347, =7, m,.

If the plane |’ of C, and the surface S, touch along a line, then
every tangent of C, touches S, and the number of common tangents
becomes infinite. This case presents itself if S, be a developable and
V one of its tangent planes. Every tangent of C, touches S, twice,
if S, be a torus and J” be one of the planes touching S, along a cirele.

If C, be a curve in space the number of common tangents of C,

and JS, is still 7,m,, where 7, is the rank of C,. This will be proved
first for some special curves and surfaces and afterwards for the

general case.

§ 2. Let S, be a cone with vertex 7’; the projection of a common

1) Verstuys, These Proceedings, May 27, 1905.

Civ?)

tangent of C, and S, on an arbitrary plane V7, not passing through
the centre of projection 7’, is a common tangent of the projection p,
of C, and of the section s of S, with J). The converse is equally
true. The class of p, and s being 7, and m, respectively, the num-
ber of common tangents of p, and s and thus of C, and S, is 7, my.

If S, be a developable D,, a tangent ¢ of C, touches D, if a
tangent plane of YD, pass through ¢, and conversely. Let D', and
C’, be the polar reciprocals of C, and D,. To a plane of D,
passing through a tangent ¢ of C, corresponds a point of C", on a
generator ¢ of D’,, and conversely. The number of these intersections
of C’, and D', is 7, m,, for the curve C’, is of order m, and the
developable D', of order 7,. Then, since there are 7, m, planes of
PD, passing through tangents of C,, these are 7, 7m, common tangents
of C, and §,.

We can show in a very simple way, that if the curve C, be an
arbitrary algebraic curve of rank 7,, and the surface JS, be an
arbitrary algebraic surface of class m
tangents is still 7

the number of common
,m,. For the tangents to S, form a complex of
order m,, and the tangents of C, form a ruled surface of order 7,.
Now according to a theorem due to HatpHen ') the number of their

2?

common rays is 7, m,, Which proves the proposition.

§ 3. Some theorems concerning the contact of a developable with
an arbitrary surface will be deduced from the theorem proved above.

Let (, be a twisted cubie C* and D* the developable formed by
its tangents. Let S, be an arbitrary surface of order n,, having a
cuspidal and a nodal curve respectively of order r, and §, and let
D* and S, have an ordinary contact in d and a stationary contact in
y% points, whilst none of the tangents of C* is an inflexional tangent
of S, and C* does not touch jS,. The number of common tangents
of €* and S, is now also for this particular position 7, m, or 4 m,.
These common tangents of C* and S, are: 1st the tangents of C*
touching the curve of intersection s of D* and S, and 2°¢ the
tangents of C* touching JS, in the points dandy where the surfaces
D* and S, touch. Let every common tangent of C* and S, passing
through an ordinary point of contact d count for «common tangents
and let every common tangent through a stationary point of contact
x count y times.

The number of common tangents of. C* and s will be

4m, —wd— yy.

1) R. Sturm, Linien Geometrie, | p. 44,

(4"8 )

Let A be the developable formed by the tangents of s. Let / be
a common tangent of C® and S,, touching C* in F and s in P.
One of the tangents of (C™* consecutive to / meets S, in two real
points of s, consecutive to P, as / is supposed to be no principal
tangent of S,. The osculating plane J” of C* in F& contains therefore
four consecutive points of s, so it is a stationary plane a@ of s in P.
Consequently the plane |’ is also a stationary tangent plane of
along the generating line 7. So C* has in f three consecutive
points in common with A’ and no more.

The 8, points where C* meets S, are cusps @ of s'), so they are
triple points on the developable J.

Each of these 3n, points @ counts at least for three points of
intersection of C* with A. Each of these points 8 counts for not
more than three points of intersection, as we have assumed that C®
does not touch S,, and the tangent in 8 to C* does not lie in the
triple tangent plane of A’ in 8, which triple tangent plane coincides
with the osculating plane of s in Bg, i.e. with the tangent plane of
S, in p.

The curve C® meets AK only in the 4m,—«d—ya points R
and in the 3n, points B, as every tangent to s lies in an osculating
plane of C*, and through a point of C” no plane can pass osculating
C still elsewhere. The order of A’ or the rank of s is

y= 4m, + 38n, — 2d — 3x”).

So the number of points of intersection of C* and XA is

3(4m, + 38n, — 2d — dy).

As the only points of intersection of C* and A are the points

R and @ counted three times, we find the relation
3(4m, + 3n, — 2d — 3x4) = 3 X 3n, + 3(4m, — rd — yy)
from which ensues
Pac — os

or in words:

If the developable D* and an arbitrary surface S, have an ordinary
contact, two consecutive generating lines of D* touch S,.

If the developable D* and an ordinary surface S, have a stationary
contact, three consecutive generating lines of D* touch S,.

These theorems hold good too in the case that the developable is
a cone °*).

1) Verstuys, Mém. de Liege. 3me série, T. VI, 1905. Sur les nombres Pliické-
riens etc.
2) Verstuys, These Proceedings May 27, 1905.

5) Verstuys, These Proceedings May 27, 1905.

(179 )

The two theorems mentioned above and their reciprocals and some
special cases will now be treated algebraically.

§ 4. Let C, be a rational twisted curve of rank 7, and Sa surface
of order n, possessing no multiple curves. Let
au +- by + cz +-d=0
represent the osculating plane of C,, @,6,¢,d being integer rational
algebraic functions of ¢. Differentiating we find for an arbitrary tangent
of C, equations of the form
a,e+b,y+e,24+d,=0,
ae+tbyteae+d,=0.
Solving y and z in function of v and ¢ we find:
Ax + B De+H

— =

G Ce

Rae pede ee aA

in which A, B,C, D and £ are functions in ¢ of order 7,. If we
substitute the values (A) in the equation of the surface /S, we arrive
at an equation (4), which is in « of order m and in ¢ of order
nv,. Wor every value of ¢ this equation (4) furnishes the 7 values
of « belonging to the points of intersection of a tangent / to C..
If two of these values become equal, the tangent / will meet the
surface Sin two consecutive points and as S is supposed to have
no multiple curves the tangent / will also be a tangent ofS. Those
tangents of C, are excluded which are at right angles with the Y-
axis, all points of intersection with .S possessing the same «; so all
roots v coincide, without the points of intersection coinciding. Every
line being at right angles with the \-axis meets the line at infinity in
the plane «= 0. So the number of these particular tangents of (©, is 7,.

The equation (4) has two equal roots in wv for a certain value of
t, when this value of ¢ causes the discriminant of (4) to vanish.
The discriminant is in the coefficients of (2) of order 2 (7—-1) and as
the coefficients of (4) ave of order 7,7” in 7, the discriminant is of
order 27, 7% (n—1) in ¢.

By a parallel displacement of the axes the plane =O can be
made to pass through one of the tangents of CL which is at right
angles with the Y-axis.

Writing ¢+ q for ¢, we can take g in such a way that this
tangent of C, lying in «=O corresponds to the value f= 0. The
equation (B) has then passed into an equation (47, where for /= 0
all roots w vanish.

The first equation (A)

( 180 )

Awz+t B Cy—B
= — or -?#& = ——_—_

CG A

must now pass into «=O for t=O, so that C and #2 must contain,
after the change of variables, ¢ as a factor, A not being divisible
by ¢# As the projection on the plane «=O of the tangent lying in
this plane can be any arbitrary line and as C’ vanishes for ¢= 0,
PD and / must also vanish for ¢= 0. In the equation (5) the coef-
ficient of a will be divisible by ¢ and the coefficient of a divisible
by tn,

According to SaLmon') the discriminant of equation (57) will
be divisible by ¢@—D. For each one of the 7, particular tangents
of C, which are at right angles with the Y-axis n(m—1) roots of
the discriminant of equation (2) become equal. That discriminant
possessing 27°, 2 (n—1) roots, there are left 7, (7—1) roots, to each of
which corresponds an equation (4), possessing two equal roots. So
there are 7,n(n—1) tangents of C,
possesses no multiple curves the class m is 2 (n—1). The number of

which also touch S. As S

common tangents of C, and WS is thus as before mentioned

rm.

§ 5. So far we have supposed that C, occupies no particular
position with respect to JS. For particular positions of C, two or
more of the common tangents of C, and S can become consecutive
tangents of (,. Let ¢ be a tangent of C, touching S in P, and leta
tangent of (, consecutive to ¢ be also a tangent of S. The developable
PD, formed by the tangents of (, and the surface S will touch
in P. We shall now investigate when the contact is ordinary and
when stationary.

For simplification I assume for C, the twisted cubie C*

=o 64 6 ate)
The equation of the developable D, or D* is now:
27 —10(@ a P) ys + il y* ok 4 (a “he py 2—3(a+ py — 0,
or
OSS = Sy eeten > ee eel)
4 p

If we choose for point P where the surface S touches D* the

origin of the coordinates the equation of S is
OS 2 aa? 2 hay + by? + ete 3 = (CB)

The surfaces D* and S have stationary contact in the origin when

1) Modern Higher Algebra, § 111, note.

(181 )

a( i —— fj = QRS) te oeeecrsee eee (C)

P.
The equation of an osculating plane of C* is

hd

d8(¢ +p)? 4+ 38yt—72-=—0.
The equations of a tangent to C* are
&—Aew+tpytt+y—90, (w+p)l—2yt+-=0,
or
y=2(@4+ pt—f, z=3(e@+ p)?—2?.
Substitution these values of y and ¢ in the equation of .S, we find
an equation of order 2 in wv
O=a,+a,e¢+ a, 2° + a, x + ete.
where
a, = 3 pl’ + 4 bp? ? — 2 — 4 bpt* + ete.,
a, =4hpt4+ 30 — 2ht? + 8 dp’ — 4b 4 ete, . (D)
a,=a+4ht+ 4b? + ete.

The discriminant of this equation is of the form

a, p +a, yw’, ”).

As a, and a, contain respectively @ and ¢ as a factor, whilst »
and y are in general not divisible by 7, the discriminant is divisible
by # or the discriminant has two roots ¢=0. As to every root
of the discriminant (except the particular 1 (”—1)-fold ones) corre-
sponds a common tangent of C* and |S, the X-axis counts here
for two common tangents of C* and 5S, or the two consecutive
tangents of (* lying in the common tangential plane of D and 3S
both touch also JS.

The discriminant is a determinant, which gives when developed
according to the elements of the first two columns

{2 na, a, — (n—l1)a, tg, +n? a,’ 9, + na,4,¢, +4,°9, - (£)

§ 6. If the N-axis does not coincide with one of the inflexional (or
principal) tangents of S in the origin P, then the )’-axis can be taken
so that 4 =O; to this end we have but to take for )-axis the diameter
of the indieatrix conjugate to the \-axis. The expressions for the
coordinates of a point on C® will not change if we now also
take for plane «=O the plane determined by the new Y-axis, and
one of the two tangents of C" meeting the Y-axis outside P, and for
plane at intinity the osculating plane of C* in the point where C*
touches the new plane «= 0, whilst for plane y= 0 is taken the

1) Satmon, Three Dim. § 204.
2) Satmon, Modern Higher Algebra, § 111.

(ean)

plane determined by the X-axis and the point where C* touches 7 = 0.
When =O the terms of the lowest order in ¢ in the coefficients
a, a, and a, are respectively of order 2, 2 and 0.

The terms of the lowest order in ¢ of the discriminant appear in
the first term of the equation (#7) at the end of the preceding §,
namely in the term 2na,a,g,. So the terms of the lowest order
in ¢ are

Ca (8p + 4bp*) ?
where C represents a constant. The discriminant possesses three roots
¢=O or the X-axis counts for three common tangents of C* and JS, if
a (3p + 4bp*) = 0
or if
== 0% 3 4bpi— 0) p= 0:

If 9 +46) —0, the surfaces D* and S have according to (C) a
stationary contact, as 4 is also equal to nought. The origin Pis now
an ordinary point (not a parabolic or double point) on the surface
S and the common tangent (the X-axis) does not coincide with one
of the inflexional tangents of S in P.

This furnishes the theorem :

If an arbitrary surface S and a developable D* have a stationary
contact in an ordinary point P of both surfaces and the. generating
line | of D* through P is neither of the two inflecional tangents of
S in P, then l counts for three common tangents of the cuspidal
curve C® and of S.

If @=O0 the surfaces D*‘ and S have according to (C) still a
stationary contact, as still 4 = 0. The origin P is now a parabolic point
of SS whilst the Y-axis is the only inflexional tangent. The coefficients
a,,d, and a, all contain the factor @. So the discriminant possesses
the factor @4, so that now the discriminant has four roots ¢—= 0. So

0?

the X-axis now counts for four common tangents of C* and S.

If p=0, then C* touches S in the origin ?, whilst the oseulating
plane of C* in P coincides with the tangent plane of S in P.
The terms of the lowest order in ¢ in the coefficients a, a, and a,
are now respectively of order 3, 2 and 0. So the discriminant (/)
is divisible by @, so that C* and S now have in the origin P three
common tangents. Writing in the equation (4) of the surface S for
the coordinates of a point on C® the expressions 7=7,y=?,2= #,
we obtain an equation in ¢, containing @ as a factor. The curve C*
has thus in the origin only two points, but three tangents in common
with S.

If h=a=p=O, then C* touches the surface S in a parabolic

( 183 )

point P, the tangent in 2? to C* coincides with the principal tangent
in P of S, whilst the osculating plane of C* in P coincides with the
tangent plane of S in P. From the expressions (PD) for a,,a, and
a, follows that the discriminant (/’) is divisible by #, so that C* and
S have now four common tangents in common in the point P.

If fi —— p30) then €* touches S still in a parabolic point;
the only difference to the preceding case is that C* no longer
touches the principal tangent. From the equations (/)) and (/’) ensues
that C* and S possess only three common tangents.

If kh =b=Oand p Z O, then P is a parabolic point for which the
principal tangent does not coincide with the tangent to C°.
(D) and (/7) ensues now readily that the Y-axis counts but for two

From
common tangents of C* and 5.

§ 7. When the X-axis coincides with one of the principal tangents
of S in P then the axes cannot be taken in such a way that 4 = 0;
but we have now a@—0O. The terms of the lowest order in ¢ in the
coefficients a,, a, a, (D) are now respectively of degree 2, 1. 1.
So the discriminant (/’) is only divisible by #, so that now the X-
axis counts for two common tangents of C* and S. The X-axis itself
has now with S in P three consecutive pots in common, so it
counts already for two common tangents. A tangent of C® following
the X-axis does not touch S any more.

The term of the second degree in ¢ of the discriminant (/’) has
now for coefficient 16 Ch? p*, where C' is a constant. So the diserimi-
mant has three roots ¢==0, when h=O or p=O. The case h=0
is just the one treated in § 6.

If p=a=O then C* touches in P one of the principal tangents
of Sin P, whilst the osculating plane of C* in P still coincides
with the tangent plane of S in P?. Out of the expressions (7) for
a,,a, and a, it is evident that these coefficients are respectively
divisible by #, @ and ¢. So the discriminant (/) is divisible by ¢ or
it has four roots ¢=0. The X-axis counts thus for four common
tangents of C* and S. By substitution of e=?t, y=?, z= in
the equation (4) of the surface S we find that C*® and S now have
in the origin three consecutive points in common.

§ 8. Let G,
developable formed by its tangents and let DY, touch the arbitrary
surface S in P. Let / be the generating line of D, touching JS in

P and let Ff be the point, in which it touches C,. Let V be the

now be an arbitrary twisted curve and PD, the

( 184 )

osculating plane of C, in A. Through R and five points of C, con-
secutive to & a twisted cubie C* can be brought, on the condition
that R, / and V are an ordinary point, an ordinary tangent and
an ordinary osculating plane of C,. The developable D* formed by
the tangents to C* and the developable D, bave in common the
line / and four consecutive generating lines.

If 7 must count for 2, 3 or 4 common tangents of C* and_S,
this is also the case for C, and S. The theorems proved in § 6
and 7 for C* hold good for any twisted curve. This gives rise to
the following theorems :

If the developable D, corresponding to curve C, touches any
surface S in point P whilst the generating line 1 of D, through P
is no injlexional tangent of S, the line 1 counts for two or for three
common tangents to C, and S according to the surfaces having in P
an ordinary or a stationary contact.

If the point of contact P of D, and S bea parabolic point on S,
then | counts for four or for two common tangents of C, and S
according as the inflewional tangent of S in P coinciding with lor not.

If the point of contact P of D, and S be a hyperbolic point on
Sand if the tangent | of C, coincides with an injlewional tangent
in the point P of S, then | counts for four or for two common
tangents of C, and O according to R coinciding with P or not.

Tf C, touches S in P, whilst the osculating plane of C, im P
coincides with the tangent plane of S in P, then the tangent lin P
to C, counts for four or for three common tangents of C, and O,
according to l being an injlevional tangent of O in P or not.

The theorems proved here for curves in space hold with a slight
modification (see § 1) still for plane curves. They can be easily
proved by taking for C, first a parabola p* after which they can
be directly extended to an arbitrary conic section and after this to
an arbitrary plane curve.

Delft, June 1905.

Physics. — The shape of the sections of the surface of saturation
normal to the a-axis, in case of a three phase pressure between
two temperatures.” By Prof. J. D. VAN DER WAALs.

In these Proceedings of March 1905 I have (fig. 4, 5 and 6)
represented in a diagram some sections of the (p, 7’, )-surface normal
to the Z-axis for three temperatures, at which three phases can
exist simultaneously. The three temperatures chosen were: 1s" the

( 185 )

temperature which we might call the transformation temperature and
which I shall indicate by 7%, (fig. 5), 2"¢ a temperature a little
below the transformation temperature (fig. 4) and 3" one a litile
above 7%,.

In the case that these sections are known for all possible tem-
peratures, the saturation surface is of course quite determined and
known, and so all other sections e.g. those normal to the .-axis,
are also determined. But it appears from the given figures, that
though the realizable part of the saturation surface has a compara-
tively simple shape, the non-realizable part has a fairly intricate
course — and that it is necessary to know also that intricate portion
if we wish to get an insight into the course of the part that is to
be realized.

To the intricacy of the hidden part it is due that though all the
sections normal to the v-axis are given by those normal to the 7*
axis, the shape of the (p, 7’),-sections will not always be easy to
derive. Now that I for myself have obtained an insight into the
course of these sections I have thought it not devoid of interest to
try and make clear the properties of this curve by means of a
series of successive ligures.

If we wish to represent these (jp, 7’), figures in a diagram, all

the surface must of course be known — in other words according
to the course of our derivation from the (p,)7 sections — a// the

(p, 2)r sections must be known.

Between two temperatures which are known by experiment, see
fig. 4, 5 and 6 Le., such a (p,.x)r section has two tops, viz. P and
Q. If 7 is raised, the part that has P as top, is narrowed, and
the part that has @ as top widens, and the reverse. This property
is perhaps not quite fulfilled in the schematical figures of the paper
mentioned, but it follows immediately from the fact that with con-
tinued rise of temperature the top P vanishes, whereas with sufficient
lowering of 7’ the top Q vanishes. Let us call the temperature at
which P vanishes 7’, and that at which @Q disappears 77. I choose
these symbols 7, and 7%, because I think of the mixture of ethane and
alcohol as an example for the shape of the (p, 7,.x)-surface discussed
here. Of these mixture the plaitpoint circumstances have been deter-
mined by Kugnen and Rosson. At 7’, the whole top the plaitpoint of
which is P, will have contracted, and the onty trace left on the
outline of the (p, .x)-figure of the complication found at lower values
of 7, is a point, at which the tangent is horizontal, while at that
place there must be an inflection point in the (p,.)-curve, which
has for the rest a continuous course. For 7’ equal to 7), this is the

( 186 )

ease for the point Q vanishing on the outline. Just as experiment
yields the values of 7, and 7), it also gives us the values of 2,
and vq at which the tops ? and Q will disappear. For temperatures
higher than 7, and lower than 7), the (p, “)r-curves have lost
the complications which they had for values of 7’ between 7, and
7. Only at temperatures which lie little above 7’, or little below
7,, there is still a deviation to be found from the well-known
looplike shape of these figures, as there are inflection points to be
found. So at 7, and 7, the complications whieh I shall eall exter-
nally visible complications, have disappeared. But before we can
say we know all the particularities of the whole (p, 7, v)-surface,
among which I also reckon the /idden complications, the question
is to be settled whether the disappearance of the external complica-
tions involves the disappearance of the hidden complications, whether
perhaps the hidden complications may continue to exist long after
the external complications have disappeared. Figures (1) and (2)
make clear between which two alternatives a choice must be made.
According to tig. (1) the disappearance of the external complications
would involve the disappearance of the hidden ones. According to
fig. (2) the hidden ones continue to exist when the external ones
have disappeared. And even when 7’ rises above 7, they are still
there. At higher values of 7’ the hidden complication gets detached
from the outline. The spinodal curve — —W— retains its maximum
and minimum, and there are still two plaitpoints, viz. at this maxi-
mum and minimum. And only at a certain value of 7’ lying above
7, that maximum and minimum have coincided to a double point
and the hidden complication is about to disappear.

For the point Qa similar question occurs. Have all the complications
disappeared at 7%, or is it required that 7’ descends below 7), before
the hidden complications have also disappeared on this side?

[ must own that I have long been in doubt on this point, as will
appear when we compare the answer I shall now give to this
question with remarks I made previously on the experiments of
Kunnen and Rosson.

According to Kortrwre’s result a double plaitpoint will always
originate on the spinodal curve. but in itself this does not seem
decisive. For according to both figures, to fig. 1 as well as to fig. 2,
a double plaitpoint disappears or appears on an existing spinodal
curve. But in fig. 1 this takes also place on an existing binodal
curve. And now it is Korrewsne’s opinion, that such an appearance
of a double point, viz. on an existing binodal curve, would be such
a special case that we must not conclude to it but in the utmost

¢ £87 )

necessity. This is in fact an argument that speaks for fig. 2, but
which did not seem to me perfectly conclusive. For who warrants us, that
these very special circumstances do not occur here? It is chiefly to
decide this point, that I have also examined the course of the (p, 7’),-
lines. And this examination has taught me, that the particularities
which occur in these lines, do not clash with the assumption which
leads to fig. 2 — whereas we should be confronted with difficulties,
when we concluded to fig. 1.

Then fig. 3 is drawn up on the supposition that there are still
hidden complications beyond the values of 7 and 7',. In this figure
is drawn in the first place the projection on the (7) x)-plane of the
phases coexisting at the three phase pressure, viz. the continuous
eurve DEAC. So this line represents the locus for the points A'A A" of
the figs. 4, 5, 6 of the paper of March 1905. The value of 7’ for
the point / is therefore 7, and for the point A, 7’ has the value
of 7. That this broken line consists of three almost straight pieces
is not essential, but it Aas been assumed that it does not change its
direction continuously at the points / and A.

In the second place the projection of the plaitpoint line has been

given by: __.—_. It consists of a piece which may be considered as
the projection of the points P of the figures of March 1905, i.e. the
left part up to the point /. The part lying on the right from the
point A represents then the projection of the points @Q of the figures
le. Every part of this line lying between /# and A is projection
of the hidden plaitpoints.
_ As we make one double plaitpoint disappear at 7’ > 7’, and
the other at 7’ < Y,, this middle part starts on the left still
running to higher values of 7, (the piece HZ) and on the right
there is a piece mA, that also runs to higher values of 7. The remaining
part of this plaitpoimt projection curve, viz. the piece J/m descends
therefore with increasing value of a. That this plaitpoint curve
possesses a maximum and a minimum value will be shown presently.
This middle piece is the locus of the plaitpoints # of the figs. 4, 5,
6l.c. The part between # and J/, and also the part between A
and m is the projection of the higher plaitpoimt of the hidden
complication in the cases that this complication still exists either
above TJ, or below Ti.

In the third place the three phase pressure is traced. In the points
of the line D/ thinner lines have been drawn parallel to the p-axis,
increasing in length as we reach the point /. The three phase

pressure itself is denoted by ——-— -W—. We must, of course, take
care that points of the branch of the three phase pressure lying
13

Proceedings Royal Acad. Amsterdam. Vol, VIII.

( 188 )

above EA, and also of the branch lying above AC must fulfil the
condition that for the same value of 7 the pressure must have the
same value for the three branches.

In the fourth place for some values of 7’ sections parallel to
the (p,v)-plane are given and those parts of these sections are
drawn which correspond to the pieces A’PA and AQA" of the figs.
4,5,6 le. We must then, of course, take care that the maxima of
the curves fall above the projection of the plaitpoint curve. It is
hardly necessary to remark that at any rate as long as 7’ lies between
7, and 7, the plaitpoint pressure for the left-hand branch, and
also for the right-hand branch is greater than the three phase pressure.
But if we want to compare the value of the plaitpoint pressure and
that of the three phase pressure a¢ the same value of 2, we have
to carry out another construction. Let G be a point of the projection
of the three phase pressure. Let us draw the line GH parallel to
the Z-axis, then #7 (a point of the projection of the plaitpoint curve)
has the same value of xv, and so above H/ a point must be sought
of the plaitpoint curve itself. How high this point lies depends on the
value which the plaitpoint pressure has for this value of a. In the
point Hf a somewhat thicker line has been drawn parallel to the
p-axis, whose length would have to denote the value of this plait-
point pressure. This length is left undetermined in the figure —
but is clear that it will be smaller than the amount of the three
phase pressure for the same value of «. For at the value of
T, as it is for the point G, the pressure above G in the section
for the chosen value of w is equal to the three phase pressure.
The value of 7 for the point // is smaller than that for G. Between
these two values of 7’ the (p,7’),-section of the (p,7,7)-surface has
a continuous course, and in such a (p,7’),-curve the pressure rises
with the temperature. Only in the case that a maximum in the (p,«) 7-
curve occurred, the pressure above /7, so the plaitpoint pressure could be
smaller than that above G. But in our diagrams we shall assume the
more general case. Themodifications which would ensue from the
assumption that in the region discussed here a maximum pressure occurs,
would render numerous new figures necessary, and it will not be
difficult to give them when the more common case has been understood.

According to fig. 3 there is in our case a maximum and a

Aus : z ., aT pi
minimum for 7',;, so that there are values of # for which v= 0.
av

2

For a plaitpoint (: =) is equal to 0, because it is a point of the
pr

ae
! : aS =
spinodal curve, and at the same time 7s is equal to 0.
& if
Pp

( 189 )
The differential equation of the spinodal curve is

So dv dy
= la dp — =I, 6 6 - (ll
le ee Ge ie = le eat oe

The differential equation of the plaitpoint curve is

13C Pv ks
=) de + (<a) p— (3) GE 0s. =e oe Ee)
da‘ Jy PT dx*® Jy T du* ]yT

From (4) follows:

d*y
2 Ades ]oT dx, pr
Sone av
dx?

in (2), we find:

Ip
If we substitute this value of - PP

d?v hat
du* },7 \ dx ]yT

= ei
da J yt dy dy av d'y
Gay e 0 oe - Gd r oe
and
ay as
a an Ey ill, (4)

de) d®v dy ze) ay oe
du pl dx? pl da* pr da* pv
: ’ 1 dT
From this equation (3) follows that ap) can become O when
dx},

dx* },1

found for substances, for which no three phase pressure occurs when
there exists a minimum critical temperature. It is wellknown that
in this case the binodal curve splits up, and that there is a point of
inflection for the isopiest in this point. There is a double plaitpoint
also then, which originates or disappears at a certain temperature ;
but though we can speak of a double plaitpoint, the value of

d'
( =), is not =O then.
da* pL

ats :
In the case under consideration the value of i) is equal to
av

T

dv 4 dp 5 b6 =
(=) = 0. In this case (2) is not equal to O. A similar case is
’ ade Jy]
y p

0 in the point at which a double plaitpoint appears or disap-

AS ee

(190 )

pears, as may be derived from the figs. 1, 2, 3, l.c. Between

certain values of p and at suitable values of 7’ there are isopiests,
ae ‘ é 3

on which —— is four times equal to 0. On such isopiests— is
de") 7 da*)7

4P

€
three times and ~-— — twice equal to 0. We now can choose
de") T
the value of p and 7’ such, that these two points in’ which
doo. mee : pee caS
; is 0, coincide. As then two values of # in which a I)
da pr da pik
also coincide, such a point is a plaitpoint. For such points

aS PS ds : :
- and {| — and | — is equal to 0. These three equations
dx? }yT da? Jy T dx* } 7

determine then the value of 2, p and 7, at which such a double
plaitpoint appears or disappears.

14
If in (3) and (4) we put the quantity (=) = 0, then both
pr

Ak

dT dp 4 s ‘ ;
G ) and (@) will also be equal to O, from which follows that
av J pl av J pl

I /
not only the plaitpoint temperature, but also the plaitpoint pressure
will present a maximum and a minimum. As we only assume the

dp

Pp. oF : :
case that ip 8 Positive, there will be found at the same time a
€

’

maximum value or a minimum value for the two curves. In the
points “ and A there is therefore no maximum or minimum for
the plaitpoimt curves, and this is also to be expected for the curve
of the three phase temperature, though this perhaps might call for
further examination. For the properties which are to be derived by
us this is, however, not of great importance.

Let us now proceed to describe the properties of the sections of
the (p, 7, x)-surface normal to the w-axis or in other words the
course of the (p, 7’),-curves.

We remark then in the first place that for values of « below
wp and above ac the (p, 7’),-curves will present their usual shape
without any complication. For values of « between wp and xg and
also for values of 2 between «4 and 2c there is a complication in
these (p, 7’),-lines. For values of « between xp and «xz the three
phase temperature lies higher than the plaitpomt temperature; the
reverse is the case for 2 between v4 and vc. On such (p, T),-
curves the usual plaitpoint occurs, but at a plaitpoint such curves,
considered in themselves, do not present any particularity. But a
point also occurs on them at which the three phase pressure is reached,

(491 )

and at such a point the curve suffers an abrupt change of direction.
As for every value of # the line DHAC is met only once, this
sudden change of direction occurs only once in a (p, 7’),-curve.
- This determines the external course of such a section sufficiently.
Beyond the point of change of direction the points for which IV,,
and V,, are equal to O will give rise to a maximum value and to
a critical point of contact. But we contine ourselves here to the
modifications which are the consequence of the three phase equilibria.

In the points, at which such an abrupt change of direction
occurs, a part of the internal or hidden course of such a (p, T),-
curve begins and the series of figures (a, 6, c, d ete.) indicates
this hidden course for the values of v, for which the three phase
curve is met. Seen on the (p, 7’),-curve such a point presents itself
as a node. The part of the curve coming from below continues
through the node, also the part coming from above, while there
is a third part which joins the points, where this onward course
stops. The temperature of the node is, therefore, quite determined
by the point at which DAC is cut by a line parallel to the 7-
axis with the given value of # as abscis. But the size of the hidden
part is very different. As it has quite disappeared beyond ap and
ve, it is but small for values of # only little greater than ap or
only little smaller than ae. But chiefly the different hidden parts
are distinguished by the occurrence or non-occurrence of a plait-
point and when it occurs by the place where it oceurs.

In what precedes it has already been remarked that the plaitpoint
does not lie hidden for values of # beyond wz and v4. But for all
values of « between xz and w4 it lies on the hidden part, so on that
which might be called the loop when the (p, 7’),-curve is drawn.
This appears at once when the (p, «)7-figures are consulted l.ec. But
depending upon the value of « the plaitpoint can have three different
places. It may either lie on that part of the loop which may be
considered as the continuation of the lower part of the (p, 7’),-curve
— oor it may lie on the branch of the loop joining the points at
which the onward course from below and above stops — or it may
lie on the part which may be considered as the continuation of
the part coming from above.

The first case occurs for « between vy and awy,, the second when
a lies between x and «7, and the third ease when «x lies between
am and x4. So if we have drawn a (p, 7’),-curve, e.g. one of the
figures of the series (a,6,c,d ete.), and when we proceed in the
same direction in such a part, also following the loop, we follow
the motion which the plaitpoint has when a changes continuously.

(192 )

A plaitpoint always being a point where the stable and unstable
region meet, it would be incorrect to speal of stable, metastable and
unstable plaitpoints. But when we pay attention to the coexisting
phases in the neighbourhood of the plaitpoint, the preceding names are
appropriate for such phases according to the described situation of
the plaitpoints. As long as the plaitpoint lies on the external part
of the (p, 7, 2)-surface, the coexisting phases in its neighbourhood
are stable; as long as it lies on those parts of the loop which
may be considered as a continuation of the external branches, the
coexisting phases in its neighbourhood are metastable, and when
the plaitpoint lies on the remaining part of the loop, the coexisting
phases in its neighbourhood are unstable.

In the series of the figures (a,b, c,d ete.) is, besides the loop
of the (p, 7’),-curve and the place of the plaitpoint, also the shape
of the spinodal curve indicated. This spinodal curve is the section of
the spinodal surface with the plane which has the chosen value of z.
All the points of the loop which lie below the spinodal curve represent
unstable phases and those which lie above it, metastable or stable ones.
Thus e.g. in fig. 4, in which the plaitpomt lies on the retrograde
branch of the loop, the spmodal curve is a curve which cuts the
loop in two more points. In concordance with the figures 4, 5, 61. e.
are the points of intersection indicated by the letters D and C. By
raising the temperature in these figures, the point C is moved to the
left, and when the temperature is lowered, D moves to the right,
which makes it possible for them to come into the chosen «-plane.

If from a (p, 7’),-curve for a chosen value of x the curve is derived

. : . (ap
which belongs to a value of «+ dz, the value of (72) must
be known for every value of 7.

dx
must have the same value for p. If we draw both the (p,7),-curve

and the curve Cola aa. as has been done in the figures 4, 5 and 6,
there will be intersection of these two (p,7’)-curves in all the points in

lp
If (2) is=0, the (p,7),-curve for the values # and «w+ dz,

. dp 5 z < 5
which ( P) = 0. In the figures mentioned the curve for 2 + dz is
Av 4h
represented by . —., and now the two (p,7’)-curves will cut every-
where where the spinodal curve cuts the first (p, 7’)-curve, according to
She dp as
the property that for coexisting phases {—~-} =Owhen{-—~}] =0.
ss av) T de” ]yT
Also in the point where the spinodal curve touches the curve (p, 7’),, 80
in the plaitpoint, such an intersection of the two following (p, 7’),.-curves

( 193 )

takes place. This may be assumed as already known from the properties
of a (p,7'),-curve, when there are no complications by hidden
equilibria. It might possibly be expected that in a plaitpoint, where

besides ie ") , also (3) is equal to 0, double intersection and so
dx?) 7 dx
P

ee
contact would take place. If, however, we develop the equation
; _ (dp :
which teaches us the value of AE VIZ.
ax T

dp as
v peli (O55) |
Et dx, /T ei dx?) p1

for the case of a plaitpoint in the form :

dv,\ («,—2,)’ dp (7,—2a, as
= («,—w,)
dz,7 pT 2 dx, dat,! pr

as
dp duke
Al 3 = (a,—w 1 See
dz.) T d*v,
= 2

. : F j 4 dp
it appears that in the case of a plaitpoint, the quantity { —
av

or

r
is only once equal to 0 on account of the factor #, — 2.

It may be remarked here for the better understanding of the series
of figures (a,b,c ete.) that the first set of four viz. a to d holds
for values of v lying between a point halfway vg and wy, and the
point / itself, « moving to continually smaller values. Fig. d
holds for zy. The second set of four values holds for # between
2 and xp, and Fig. g is the representation for 7’= 7;,.

The remaining figures (4',c’ ete.) hold for values of 2 lying on
the right side. Fig. g' is the representation for 7’= 7;, on the right
side and fig. d' holds for «= «4.

Physics. — “Vhe (T,x)-equilibria of solid and fluid phases for variable
values of the pressure’, by Prof. J. D. van DER WaAAts.

In two communications (October and November 1903) I discussed
and represented in diagrams for the ease of equilibrium between a
solid and a fluid phase 1st the (p, z)-figures for constant value of
T and 274 the (p, 7')-figures for constant value of x. So only the

(194 )

treatment and discussion of the (7°, 2)-figures for constant value of
p was left. I have not given the third communieation, in which these
last figures were to be discussed, first because they could be derived
directly from the other communications on the discussed equilibrium
and secondly because I would not make it appear as if I attached
too much importance to the hidden equilibria, showing the continuity
between the equilibria which can be observed and which seem
discontinuous without the hidden ones. However, some noteworthy
particularities would have presented themselves, and so, induced by
questions of Dr. Smits on subjects in which suchlike particularities
occur and at his request, I will briefly discuss at least the principal eases.

The differential equation (see the preceding communication) has
the following form for constant value of p:

Let us think the (7x),-curve of the fluid equilibria inter se construed,
If the second component is more volatile than the first, the two bran-
ches of this curve descend. In fig. 1 this (7,v)-curve is closed on the
side of the second component, and it is therefore assumed that the
pressure chosen lies above the eritical pressure of this component.
Further in fig. 1 the curve of the fluid phases has been drawn, which
coexist with the solid body. For so far as these phases lie within
the curve of the fluid equilibria, they are not to be realised or with
difficulty. In fig. 1 it has been assumed that the circumstances are
chosen in such a way that this curve passes the region of fluid equi-
librium twice, as is the rule for lower pressures and so also for
lower temperatures.

If the value of the pressure increases, and so also the value of 7’,
the curve of the fluid equilibria inter se ascends, while its form is
modified at the same time. The curve of the equilibria with the solid
phase ascends also with p, but in a smaller degree, at least on the
side of the liquid equilibria.. Now in fig. 1 we have ascribed such
a value to the pressure, that there are still two different three phase
equibria, while in fig. 2 a value is ascribed to p, at which the solid
body coexists with a plaitpoint phase of the fluid equilibria — so
that above that pressure the curve of the equilibria with the solid
phase passes only once through the region of the equilibria inter se
of the fluid phases. In fig. 3 p has ascended so far that there is
again equilibrium between the solid body and a plaitpoint phase.

For still higher value of p there are no longer three phase equi-
libria and the curve for the equilibria of the solid body with a fluid

(195 )

phase has got quite detached from the curve for the equilibria inter
se of the fluid phases. £
In the two previous communications on this subject |. ec. I discussed

2

the values of = and ws, and I refer to them for more particulars
die") T
on the course of the (7'),-curves.

I will only discuss here one more particular point occurring in
these curves for the case that the two surfaces of equilibrium, viz.
that for a solid phase with a fluid phase and that for the fluid
equilibria inter se, get detached at a value of 7 and p below that
of the plaitpoints.

The figures 4, 5, 6 and 7 refer to this. In this case the two intersections
of the curve of equilibrium between the solid and the fluid phase with
the region of the fluid phases will meet at a certain pressure in a point
of the spinodal curve on the liquid side, and then concur to one
single curve. In this case the meeting point becomes a double point,
and at still higher values of p a part is detached, as it is drawn in
fig. 6, and at still higher value of p it has contracted to a single
point lying on the spinodal curve on the vapour side.

It has further been assumed in these diagrams that the two
branches of the spinodal curve, also if we have to do with a (p,z, 7’)-
surface, lie inside the surface. At high temperatures and in the neigh-
bourhood of a plaitpoint this is, of course, the case. At lower tem-
peratures, however, the branch of the spinodal curve, which lies on
the liquid side in a (v,7,7’)-surface, moves to the vapour side in a
(pz, 7')-surface, and gets even far outside the surface. In the same
way the branch of the spinodal curve, which lies on the vapour side
in a (v,v,7')-figure, moves to the liquid side in a (p,,7’)-surface,
and at low temperatures it has even got outside.

This is the consequence of the fact, that the value of « and 7
determines a phase indubitably only when moreover the value of v

7

is given. If the value of p is given, then three different phases may
be indicated by this value. I shall, however, not enter more closely
into the treatment of the complications which are the consequence
of this, here. I shall only just mention that the point where the hidden
equilibria disappear in fig. 7, can lie in quite another place than is
the case in fig. 7.

(196 )

Physics. — “On the hidden equilibria in the p-x-diagram of a
binary system in consequence of the appearance of solid
substances.” By Dr. A. Smits. (Communicated by Prof. J. D.
VAN DER WAALS).

1. Some time ago Prof. van pER Waats') showed that in a
binary mixture (A, 4) one of the two p-7-lines for the three phase
pressure, viz. that which runs from the eutectic point to the higher-
melting substance (2), can present some particularity.

It proved, namely, that at the triplepoint this p-tline must have
the direction of the melting line (of B). -

As for most substances 7; > vs, or in other words the substance
expands when melting, increase of pressure causes as a rule a rise
of the melting point.

If in the p-tprojection, (Fig. 1), the triplepoint of the substance
B is denoted by d, then the melting line dg will in most cases
run from the triplepoint to the right, and as the three phase line
cd must have the direction of dg at d, this three phase pressure line
will have to present the particularity, that it does not only possess
& maximum of pressure, but also a maximum of temperature, as is
represented in Fig. 1 in an exaggerated manner.

2. As has been shown before?) and will be further discussed in
the second paper, in a dissociating composition the same course may
be found on a larger seale, and it is this part of the three phase
line, that is the most important in compositions. Therefore it seemed
desirable to me to examine the p-v-diagrams from the triplepoint to
higher temperatures for the simple case discussed just now, and to
treat the compositions afterwards.

3. The p-r-diagram at the triplepoint, and at a temperature
slightly higher are presented in Fig, 2.

As appears from Fig. 1, we find a double section of the three
phase line for the first time at the triplepoimt, when coming
from lower temperatures, and a consequence of this is, that
besides at the triplepoint pressure, three phases may also occur
at a much higher pressure. This case is more closely defined in
Fig. 2 by the p-v-diagram corresponding to the temperature ¢,.
The lowest three phase pressure or the triplepoint pressure is repre-

1) These Proc. Vol. VI, p. 280,
2) Baxnuis Roozesoon, Zeitschr. f. phys. Chem. 4. 31.
STORTENBEKER, A > . P Saye

(197)
sented by the point g, where the liquid curve acg, the vapow
curve weg and the solubility isotherm /ceg meet.

The second three-phase pressure is represented by the curve ec s
and lies considerably higher. The coexisting phases have here different
concentration, and are denoted* by e,¢ and s. Here cis the saturated
liquid phase and e the vapour phase which coexists with the solid
phase s.

If we now start from these three phases, and lower the pressure
at constant temperature, we reach the region for solid 4 -+ vapour,
as Fig. 1 represents. So below the three phase pressure ecs and
above the triplepoint pressure of 6, only vapour ean occur by the
side of solid 4 in stable condition, and the curve representing the
vapours which can coexist with solid 6 is the lowest branch of the
continuous solubility isotherm, viz. eg.

The line ac denotes the unsaturated liquids coexisting with the
vapours lying on the line we. The lines eg and eg represent meta-
stable conditions, viz. supersaturated solutions with their coexisting
vapours. The curve c /, the upper part of the solubility isotherm
represents the liquids coexisting with solid DB.

The second p-«-diagram, represented in Fig. 2, corresponds with
a somewhat higher temperature ¢,. We are now above the triplepoint
temperature and the lowest three phase pressure is not a triplepoint
pressure now, but perfectly comparable with the highest three phase
pressure. A result of this is, that we get below the lowest three
phase pressure a reflection of what takes place above the highest
three phase pressure. The solubility isotherm /, ¢, e, e', cf,’ cuts, as
it were, a portion out of the region for liquid ++ vapour, on account
of which between the two occurring three phase pressures indicated
by the curves e,c,s, and ec, s',, only solid B can coexist with
vapour in stable condition. The line c', /,', like ¢, /, represents now
the liquids coexisting with solid , and the portion e, ¢', the vapours
coexisting with solid 2.

As to the whole course of the solubility isotherm it may be observed
that this line has now two maxima, two minima and four vertical
tangents. For the portion /, ¢, e, the points of contact of the vertical
tangents lie in the metastable or the stable region, whereas those
for the second piece ¢', c', /', are situated in the unstable region.

At higher temperatures the three phase pressures draw nearer and
nearer to each other and coincide finally, as may be seen from fig. 1.

When we examine this change in the p-v-diagram it appears that
the points e, and c, move downward and at the same time to the
right, whereas the points e’, and c’, move upward and to the left,

( 198 )

As to the points @, and g,, they move upward with increase of
temperature.

The result of these shiftings must of course be, as has been just
said, that at the maximum temperature of the three phase line (see
Fig, 1) the two three phase pressures become equal, and the points
c, and c,, like e, and ¢’,, coincide; then the solubility isotherm does
not cut the liquid line any longer, but just touches it in the point
where c, and c’, have coincided, after which no three phase pressure
is possible and the solubility isotherm has got detached from the
liquid line.

Though all this seems very simple, the representation of the inter-
mediate stages presented some difficulties, which Prof. v. p. Waas
was kind enough to remove by allowing me to examine some 7*2-
diagrams corresponding with pressures respectively smaller, equal
and larger than the three phase pressure, in which exactly the same
succession of states occurred *).

What has been drawn in accordance with this in the figs. 3, 4
and 5, may be brought into words in the following way :

When the two three phase pressures have drawn so near, that
the two branches of the solubility isotherm would touch, intersection
takes place, and we get a curve as is indicated by /f, ¢, ee, ¢,/",
in Fig. 3. Immediately afterwards, i.e. at somewhat higher temperature
a portion gets detached, as represented in Fig. 4 and we get two
solubility isotherms; one is fee's’ and the other forms a closed
curve ec'd, a part of whieh (ec’) runs through the stable region.

At the maximum temperature of the three phase line the two
three phase pressures ¢cs and e'c's' have coincided, as is represented
in the diagram corresponding with the temperature ¢,, in Fig. 4,
and the solubility isotherm /,¢,7/;' no longer cuts the liquid curve,
but only ¢owches it in the point ¢,.

With the exception of this one point it runs therefore wholly
through the stable region. The other closed branch no longer cuts
the vapour curve, but only touches it in e,; further this branch
as a whole has contracted through the shifting of d to d,. This
second branch of the solubility isotherm lies therefore at the maximum
three phase pressure partly in the unstable region, partly in the
metastable region.

At a temperature slightly above the maximum temperature of the
three phase curve ¢,, the solubility isotherm ///' is quite detached
from the liquid curve, as is represented in Fig. 5; in the same

1) See the foregoing paper by vAN DER WaAats.

( 199 )

way the closed branch has got detached from the vapour branch,
and has further contracted. At the temperature /, the distance between
the solubility isotherm /, 7", and the liquid curve has increased, and
the closed branch has vanished, after having contracted to a point.

4. In the figs. 6, 7 and 8 I have drawn the v-v-sections of the
y-a-t-space diagram for the case discussed here ; they follow immediately
from the v-x-sections, given by me last year ').

The section indicated by continuous lines in Fig. 6 holds for the
triplepoint temperature ¢,. There aech is the region for liquid + va-
pour, ehe the three phase triangle, hed the region for solid
B-+ vapour, be fm the liquid region and chn/ the region for solid
B + liquid.

The curve for solid-fluid or the solubility isotherm dec / meets
the metastable vapour branch ed exactly on the line for the substance
B, because at the triplepoint temperature of B the vapour which is
in equilibrium with solid 4, is perfectly the same as that which is
in equilibrium with liquid A.

At a somewhat higher temperature the curve for solid-fluid cuts
the vapour and liquid curve twice each, just as was the case in
the p-x-diagram. The curve for solid-fluid has then a shape as is
indicated by the dotted line /, c, ¢, e, ¢c,/", in fig. 67). Just as in
the p-x-diagram, this line cuts a piece out of the liquid-vapour region ;
in consequence we get two separated regions for liquid and vapour
viz. a@,e,¢,6, and e',d,h,c,. From this particular situation ensues
further the existence of two three phase triangles, viz. c, e, g, and
c,e,g,, between which is situated the region for solid 6 -+- gas
9g, ¢,¢;, two liquid regions b,c, /,m and h, cf; and two regions
for solid 6 -+- liquid viz. g,¢,f, and g, ¢, fy’.

With increase of temperature the line solid + fluid assumes the
shape of a loop, as is indicated by the line fee'ec'/’ in fig. 7; 0n
this follows immediately detaching ofa part, splitting up into two bran-
ches, viz. into the line 7", c,¢,/, and the closed line e, ¢', 0. At the
maximum three phase temperature (fig. 8) the line /¢/" touches the
liquid line and the closed line eoe¢ touches the vapour line. Above
this temperature the two lines get detached from the liquid, respee-
tively the vapour line and the closed line e, 0, e, disappears as a point
in the metastable region.

1) These Proc. Vol. VI, p. 484.
*) At a temperature, only very little higher than the triplepoint temperature,
a part of the branch e'; c, f'; will fall outside the line for B.

( 200 )

5. Fig. 9 represents the most interesting part of the projection of
the p-t-a-space diagram on the p-7-plane for the case that the plait-
point curve meets the solubility curve, as with ether and anthra-
quinone. In this fig. the possibility has moreover been assumed,
that the second plaitpoint temperature 7, of a saturated solution lies
above the triplepoint temperature ¢,. Fig. 10 represents for this case
the p-v-sections corresponding with the temperatures ¢, and ¢, (see
fig. 9).

The section for ¢, differs from the second section in Fig. 2 only
in this, that the liquid branch passes continuously into the vapour
branch with the point A’ as plaitpoint. If we now pass on to lower
temperatures, the downmost three phase pressure becomes smaller
and the upmost greater, while the plaitpoit pressure diminishes. In
consequence of these last two changes the points e, ¢ and K get
nearer and nearer to each other, and when we have descended to
the temperature ¢,, the points e, c and A’ have coincided or in other
words the upmost three phase pressure has become a plaitpoint
pressure; this circumstance is accounted for in the p-a-section, corre-
sponding with the temperature ¢, (Fig. 10).

At the temperature ¢,, the triplepoint temperature, for which no
p-w-section is drawn here, because it immediately follows from that
for ¢,, the remaining downmost three phase pressure has become
triplepoint pressure, and the points é,, c, and g, have coincided.

Below this temperature the p-v-sections over a certain temperature-
range consist only of a solubility isotherm of the shape of e, gf;
in Fig. 10, as has been discussed before.

That the case assumed in Fig. 9 is not often to be realized, is
obvious, but that it is a possible case, is, in my opinion, not doubtful

Amsterdam, June 1905. Chemical Laboratory of the University.

Physics. — “Contribution to the knowledge of the pa- and the pT-
lines for the case that two substances enter into a combination
which is dissociated in the liquid and the gasphase.” By Dr.
A. Smits. (Communicated by Prof. J. D. van DER Waats).

The purpose of the following paper is to give a connected repre-
sentation which is in logical connection with the p,v,-diagram which
has been recently drawn up by Bakxuis Roozrsoom and in which it is
assumed that only the components can oceur as solid phases, for the
most important particularities of the equilibria between a vapour,

( 201 )

liquid and solid phase for the case that the last is a dissociable
combination.

Some of the points mentioned in what follows had already been
given by him’), but it was not till now that they could be
combined into a connected whole through the knowledge obtained
during the last time (see inter alia the preceding paper).

1. For the case that two substances A and B enter into a com-
bination, I shall distinguish three cases.

dst that the vapour tension of the combination lies between that
of the components. (fig. 1).

2°¢ that the vapour tension of the combination is smaller than
that of the components. (fig. 2).

3d that the vapour tension of the combination is greater than
that of the components. (fig. 3).

2. If we bear in mind that for the case that the combination
does not dissociate, a p-x-section for the system A-+- AB+ B is to
be considered as a junction of the two systems A-+- AB and AB-+- B,
it is not difficult to combine the p-x-seetions for the system 4-+- AB-+ B
for different temperatures into one diagram.

If we first examine case 1, where the vapour tension of AB Jies
between that of the components, it may be observed, that in the
system A-+ AB, A is the substance with the higher and AB that
with the lower vapour tension, whereas in the system AB-+-B, AB
has the higher and 6 the lower vapour tension. Bearing this in
mind, we get, led by the diagrams given by me before *), to fig. 1,
in which the hatched regions am LL, 1, HE, ee,e,dc,c,c,dc E'L’, é
and HH, TU U,1,a'm',m',m',m’ indicate the vapours and liquids which
coexist with solid phases (A, Ab, AB and JS) at different tempera-
tures. We shall call these regions henceforth the three phase regions ;
they have as base the line which joins the points # with #, res-
pectively £' with £#’, and are bounded on one side by a vapour
line and on the other side by a liquid line. The latter has already
been called solubility curve before. At and below the eutectic
temperature the p-«-section is simplest and it consists for each of the
two systems of two lines representing the vapours which can coexist
with solid A or solid AB respectively with solid AB or solid B.
Such a p-z-section is found in the lines 40 y, LH’, k', where it must
be expressly stated that og. and gl’, do not form a continuous
curve, but are two separate branches, which cut at q.

1) Rec. Tray. chim. 5, 335 (1886).

2) These Proc. Vol. VI, p. 484.

( 202 )

In the point 0 coexists a vapour with the two solid phases A and
AB and the point L’, denotes the composition and the pressure of
the vapour phase which can coexist at the eutectic temperature
with a liquid 4’ and two solid phases AB and B.

In the second p-a-section the curves /, /, y, e, g, e and ’, / represent
vapours coexisting with solid phases; the lines me and m'c' denote
the liquids coexisting with the vapours e/ and e'/', while the lines
ms, cf, ¢,f' and m's' represent liquids coexisting with solid phases.

What change this section is subjected to with rise of temperature
is so easy to follow from the diagram, that it does not call for a
discussion.

It may only be observed here that for the case that the combi-
nation reaches its critical point sooner than the components, we get
somewhat above this temperature a p-x-section which consists of two
loops 6, and 0',y, in which the vapour branches have continuously
passed into the liquid branches, and an opening has been formed
between the two loops. This, however, does not complete the p-a-
section, for, solid phases may still oceur by the side of liquids at
higher pressures, though above the critical temperature of A B and
the melting point of A and 5, when viz. A, 6 and AB melt with
increase of volume, as generally happens. Thus the lines £,s, 9; f,,
g;f', and #,s', represent the liquids which can coexist with a
solid phase at the same temperature.

After this discussion of fig. 1, it is not necessary to give a fur-
ther explanation of figs. 2 and 8, which represent the second and
third case, as these figures do not present any essential differences
with fig. 1.

3. It is more interesting to see what happens, when the combi-
nation AB dissociates somewhat. In this case the total p-2-section
is no longer to be considered as two separate p-v-figures joined, but
as one whole and we arrive therefore at the conclusion, which sounds
rather paradoxical, that the characteristic feature of a combination
becomes apparent only when the combination is somewhat decomposed
into its components. All the curves which meet at an angle in the
figs. 1, 2 and 3 at the place of the line for the combination, now
pass continuously into each other. This applies, therefore, not only
to the gaslines, but also to the lines which bound the three phase
regions.

With regard to the gradual transition of these three phase regions,
it may be observed, that it does not take place at the point where
the three phase region lines cut the line for A 5, but always left

( 203 )

or right of this line, dependent on the mutual influence of the
vapour tension of the components.

In order to elucidate this important point I have indicated in the
figs. 4, 5 and 6 what shapes the three phase regions hatched in the
figs, 1, 2 and 3 can assume in the neighbourhood of the line for
the combination, in the case that a slight dissociation takes place in
the liquid and gas phase. It is not improbable that there will also
be dissociation in the solid phase in this case, but this is not taken
into account here, in the first place because it is most likely exceed-
ingly slight, and in the second place, because the diagram becomes
much more intricate, when this dissociation is taken into account.

Fig. 4 corresponds to fig. 1, fig. 5 to fig. 2 and fig. 6 to fig. 3.
For fig. 4 it may be remarked, that for the case that the vapour
tension of the combination lies between the vapour tensions of the
components, the liquid and the vapour line draw very near to each
other somewhat past the line of the combination, on the side of the
component with the smaller vapour tension, but that they do not
reach each other, so that there remains a gap between them, from which
follows, that in the series of liquids and vapours which can coexist
with solid A, not a single point can be pointed out where vapour,
liquid, and solid phase have the same concentration, nor is there a
point where a vapour and a liquid phase have the same concentration.

This latter is only the case when the vapour tension of the combination
lies between those of the components, for when the vapour tension of
the combination is smaller or greater than those of the components,
we get according to a rule of Gises a three phase region with a
minimum, fig. 5, or with a maximum, fig. 6, and at the place of
this minimum or maximum the concentration of the vapour and
the liquid phase must be identical.

If we now discuss figs. 4, 5 and 6 at the same time, we may
remark, that there for a special temperature the situation is indicated
of the two three phase pressures ees and c,é,s, and the continuous
line for solid-fluid or solubility-isotherm. This line must have an
horizontal tangent at the point where it cuts the line for the com-
bination. For this case is viz. wy «xs, or the concentration of the
fluid phase is the same as the concentration of the solid phase and
then it appears from the equation drawn up by van per WaAats for
the equilibrium solid fluid:

dp _w&s—x¢ 07g
dar Vs cally

dp
a5

that
0.

14
Proceedings Royal Acad. Amsterdam. Vol. VIII.

( 204 )

If we now examine the changes with rise of temperature, it is
noteworthy, as will presently become clear from the p-/-lines, that we
have here to deal with changes quite analogous to those discussed in
the preceding paper. The three phase pressure lines ec s and ¢, @, 8,
approach each other and coincide at CS. This coincidence takes
place at the maximum temperature of the p-é-line for the three phase
pressure. One of the two three phase pressure lines sec and s, e, ¢,
must therefore pass through the positions indicated by ec, and e'c,
before it coincides with the other. Now it follows from the three phase
pressure line e',c', that the gas phase e’, has the same concentration
as the solid combination, while the liquid phase c', has another
concentration. The three phase pressure line e'c’ on the other hand
indicates, that the liquid phase c' and the solid phase have the same
concentration, while the gas phase e’ has another concentration.

We see therefore, that the three phases solid-liquid-gas have not
the same concentration at any pressure, but that they become two
and two equal. We may express this also in this way: When a
combination dissociates in the liquid and the gas state, then this
combination has no longer a triplepoint, for this point has split up
into two other points where vapour and solid respectively liquid
and solid get the same concentration.

After I had written this paper, I found to my surprise, that
already in 1897 van per Waats had arrived at the same result by
a way which I intended to discuss later on and for which the diagrams
had been already drawn ’).

4. In order to make the difference between a non-dissociating
and a dissociating combination more apparent, we consider the case,
that we bring the dissociating combination AB in solid condition
into a vacuum, and make the temperature rise continually.

The solid substance AB will, when the volume is not too large,
coexist with its vapour, till the pressure has become equal to that
indicated by the point e,'; at this moment a third phase appears,
viz. liquid. As the total concentration must always be that of AB,
it is necessary that the concentration of the vapour phase lies on
one side, and that of the liquid phase on the other side of the line
for AB; as follows from the figures 4, 5 and 6 this is really the
case. With rise of temperature three phases continue to exist for
some time side by side and the vapour and liquid phases coexisting
with solid AB lie on the same horizontal line. The liquid phase
lies on the liquid branch c'c,' and the vapour phase on the vapour

4) Verslag Koninkl. Akad. 21 April 1897, 482.

( 205 )

branch ¢'e,', while the solid phase lies naturally on the line GG",

With a suitable volume the three phases may now continue to

exist over the whole range of pressure and temperature indicated

by e,'c, but then the vapour phase must have disappeared when

we have reached the point c', for in the point ce’ the liquid has the
same concentration as the solid combination.

Beyond the maximum, respectively minimum in figs. 5 and 6 the
concentration of liquid and vapour is, of course, always different,
just as in fig. 4, and as the line for the combination never coincides
with this maximum, respectively minimum, when there is a maximum
or minimum, in the above mentioned case the two phases will
necessarily only get the same concentration, when two phases
coexist.

At c', we leave the three phase region, which we had entered
at e,' and coexistence of solid A 6 -+ liquid at higher temperatures
can now only take place under higher pressure.

Van per Waats has called the temperature of the point e,' the
macimum sublimation temperature and that of the point c', the
minimum melting point temperature or the melting point proper of
the combination.

Further it is worthy of note that as appears from the figs. 4, 5 and 6,
the highest three phase temperature cannot be reached when we
start from pure AZ, as this temperature corresponds with a total
concentration which contains more A than the combination.

It is obvious from the foregoing, that the distinguishing feature
between a non-dissociating and a dissociating substance is this, that
whereas for a non-dissociating substance the three phases can only
exist at one temperature, they can coexist for a dissociating substance
over a certain temperature range. We may express this also in this
Way: a non-dissociating substance has only a three phase point,
but a dissociating substance a three phase ine.

Before leaving figs. 4. 5 and 6 I will point out, that the solubility-
isotherm at the maximum three phase temperature has a shape
which is indicated by the line f'’C/f',. This line touches the liquid
branch at C and has an horizontal tangent at the point where it cuts
the line for the combination. At higher temperature the solubility
isotherm gets detached from the liquid line and moves as a whole upward,

5. It is obvious that the two three phase pressure lines ecs and
é,¢, 8, must coincide at the maximum three phase temperature, but
where the place of coincidence in the three phase regions must be
drawn is a point which calls for further elucidation from the sub-
joined p-t-projections.

( 206 )

In the figs. 8, 9 and 10, IT have drawn projections for the cases
1, 2 and 3. Fig. 8 corresponds to case 1, fig. 9 to case 2 and fig.
10 to case 38. The meaning of the different lines is indicated by
letters; thus A denotes the solid substance A, B the solid substance
ZB and AP the solid substance AB, LZ denotes liquid and G gas.
Vand LZ’ are the two eutectic points, a and a’ the triplepoints of
pure A and B.

The p-t-projections show further that we have here a three phase
line for AB+ L + G@ consisting of two branches passing continuously
into each other, and that we must find here the phenomena discussed
in the preceding paper, on a larger scale. As already appeared from
the p-a-sections, the triplepoint has split up into two points / and
f". In the point # there is contact with the line for AB-+-L and
in F" with the line for AB-+ G.

Though for the systems A B+ LF and A B+ G we have to deal
with systems of two components in two phases, they behave as
monovariant systems, because the concentration of the two phases is
identical. This is however not the case for the system liquid A 4+
vapour, and this is the reason why the line for Z + G, which begins
in F’, cannot be represented ; at constant temperature, the pressure
is here still depending on the volume.

The point & does not correspond to the maximum three phase
temperature, and that is the cause of the analogy with the case
discussed in the preceding paper.

For the cases 1 and 2 the three phase line must have the direction
of the melting line at the melting point #, as van per Waats has
proved. As in most cases v; > v,, the melting line runs from the
melting point to the right. This involves the necessity that the point
I’, (see fig. 8a) lies below R, i.e. at a pressure smaller than that
corresponding to the maximum three phase temperature. If we there-
fore proceed to higher pressure, the succession is: /’, F, R. It is
also obvious from the relative situation of these three points, that
in figs. 4 and 5 the pressure corresponding to the maximum three
phase temperature 4 CS, must lie somewhat above the melting point
pressure e'c'.

Let us now consider the rare case that vj < vs, so that the melting
line runs to the left, as is represented in fig. 86. Then, the point £
lies above the point R, and the succession towards higher pressure
is F', Rk, F. A consequence of this situation is, that in this case the
two three phase pressure lines ecs and c,é,s, (figs. 4 and 5) must
coincide between the pressures corresponding tot #’ and F. This,
however, not being the only modification which oceurs in figs. 4 and

( 207 y

5, I have represented in fig. 7 the figure into. which fig. 4 is
changed when wy < vs.

We see then, that something peculiar appears i.e. a continuous
closed solubility isotherm Geefe,e,, which contracts more and more
with rise of temperature, and disappears from the stable region at
the maximum three phase pressure.

When the lower three phase pressure line c,¢,s, has ascended to
é,'¢,', or in other words, when we consider the temperature of the
point £', the solubility isotherm has assumed the shape indicated by
the line e,e,'c,'f,¢,. At this temperature the minimum of the solubility
isotherm, which was still in the stable region before, gets into the
metastable region.. If we raise the temperature up to the melting
point of AL or to the temperature of the point /, the former phe-
nomenon is repeated with respect to the maximum, which just below
this temperature was still to be found in the stable region. Accordingly
at still higher temperature no maximum or minimum occurs in
the portions of the solubility-isotherm passing through the stable
region, and we only retain the lines ¢,c', and e,e',. These lines
become smaller and smaller with rise of temperature, and the two
three phase pressure lines e,¢, and e',c’, approach each other more
and more, till they have coincided at the maximum three phase
temperature in “SC. The branches of the solubility isotherm touch
at this temperature exactly in the points 4 and C. With further rise
of temperature they retreat altogether to the metastable and unstable
region, after which they disappear’).

In the case that v7 >v, solid AB ean still coexist with liquid
above the maximum three phase temperature, viz. under higher
pressure, but this is not the case when w<v,, which also follows
uready from the figs. 8a and 8d.

We see further that in fig. 8 two three phase lines occur with a
maximum temperature, in fig. 9 only one, and in fig. 10 three.

Fig. 9 shows, that the case 2, where the vapour tension of the
combination is smaller than that of the components, is certainly the
most interesting case, as it can yield a three phase line with a
minimum and two maxima, which had been unknown up to now.

As to the situation of the two branches of this peculiar three
phase line we may still remark that only one of the possible cases
is drawn here. These branches may pass through each other or
partially coincide, but this does not make any essential difference.

Remarkable complications present themselves, when it is assumed

') Later on I hope to discuss the course of the solubility isotherm in the meta-
stable and unstable regions between the temperatures of F’' and F more in details.

( 208 )

ihat meetings take place between solubility lines and plaitpoint lines,
but to this I hope to come back afterwards.

In the lower parts of figs. 8, 9 and 10 the projections of the
solubility lines on the 7-2-plane or the melting curves under the varying
three phase pressure are represented. From this we see that these
lines, Commencing at the components or the combination, must not
always be drawn descending from the beginning, but that they will
often first ascend, and descend after having reached a maximum.

These particularities have disappeared when we trace the melting
curves at constant pressure or a t-v-section at constant p.

Amsterdam, June 1905. © Chemical Laboratory of the University.

(August 17, 1905).

KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM.

PROCEEDINGS OF THE MEETING
of Saturday September 30, 1905.

DOCH

(Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige
Afdeeling van Zaterdag 30 September 1905, Dl. XIV).

Gio Wee ian arsS:

W. Einrnoven: “Analysis of the curves obtained with the string galvanometer. Mass and
tension of the quartz wire and resistance to the motion of the string”, p. 210.

Sypney J. Hickson: “On a new species of Corallium from Timor’’, (Communicated by Prof.
Max WEBER), p. 268.

J. D. vAN per Waats: “Properties of the critical line (plaitpoint line) on the side of the
eomponents", p. 271,

J. D. vAN per Waats: “The properties of the sections of the surface of saturation of a binary
mixture on the side of the components”, p. 280.

J. D. van per Waats: “The exact numerical values for the properties of the plaitpoint line
on the side of the components”, p. 289.

D. J. Hursnore Por: “Bork’s centra in the cerebellum of the mammalia”. (Communicated
by Prof. C. WinkLer), p. 298. (With one plate).

G. van Risnserk: “The designs on the skin of the vertebrates, considered in their connection
with the theory of segmentation”. (Communicated by Prof. C. Winker), p. 307.

J. P. van per Srox: “On frequency curves of meteorological elements”, p. 314.

N. L. Séuxcey: “Methan as carbon-food and source of energy for bacteria”. (Communicated
by Prof. M. W. Betserincx), p. 327. (With one plate).

C. Scnovte: “Determination of the Tromson-effect in mereury”. (Communicated by Prof,
H. Hae), p. 331.

D. Mot: “Ester anhydrides of dibasie acids”. (Communicated by Prof. A. P. N, Francui-
MONT), p. 336.

L. van Irarue: “Thalictrum aquilegifolium, a hydrogen cyanide-yielding plant”. (Communi-
cated by Prof. P. van RompurcH), p. 337.

P. van Rompvrcu: “On the action of ammonia and amines on formic esters of glycols and
glycerol”, (IL), p. 339.

J. C. Kiuyver: “A local probability problem”, p. 341.

W. Karreyn: “A definite integral of Kummer”, p. 350.

Z. P. Bouman: “An article on the knowledge of the tetrahedral complex”. (Communicated by
Prof. JAN DE Vries), p. 358.

C. A. Pexernarine: “On the excretion of creatinin in man. Report of a research made by
C. J. C. van Hoocenuuyze and H. VrenrrLorcu”, p. 363.

Kk. Sissincu: “On the theory of reflection of light by imperfectly transparent bodies”. (Com-
municated by Prof. H. A. Lorentz), p. 377.

W. A. van Dorp and G. C. A. van Dorr: “On the chlorides of maleic acid and of fumaric
acid and on some of their derivatives”, p. 387.

The following papers were read :

15

Proceedings Royal Acad. Amsterdam. Vol, VIII.

( 210 )

Physiology. — “Analysis of the curves obtained with the string
galvanometer. Mass and tension of the quartz wire and resist-
ance to the motion of the string.” By Prof. W. Etnrnoven.
(Sequel to former communications on the string galvanometer).

(Communicated in the meeiing of April 22, 1905).

CO NTE Nas.

1. Introduction.

2. The principles of the method.

3. The mass of the string.

4. The resistance to the motion of the string.

5. The acceleration.

6. Analysis of some curves.

7. Absolute measurements of the mass of the string and of the

resistance to its motion.
8. The tension of the quartz wire.
9. The practicability of the galvanometer for special purposes.

1. Introduction.

When recording the movements of the quartz wire in the string
galvanometer with the object of becoming acquainted with various
irregnlar oscillations of electric tension or current, we may in
many cases see in the directly recorded curve an accurate or
nearly accurate image of the oscillations sought. These cases are
met with if the deviations of the quartz wire are aperiodic and
quicker than the oscillations to be recorded. As an instance we
mention here the galvanometric tracing of the human electrocardio-
eram, the various tops of which are reproduced in the exact pro-
portions by the recorded curve, as was shown on a former occasion’).

When very rapid oscillations are to be recorded, it is necessary
to make the deflections of the galvanometer also of very short
duration, which can easily be done by giving a great tension to the
string. In doing this, the movements of the string must in some
way be damped, however, in order to avoid oscillations. The condenser
method can render good service here *), especially if it is applied in
such a way that the motion of the string is nearly brought to the
limit of aperiodicity.

1) Vide: these “Proceedings” 6, p. 107, 1903.
2) Vide; these “Proceedings” 7, p. 315, 1904,

Gaia)

We must remark, however, that the galvanometer loses in sensi-
tiveness when its deflections become quicker. If the time of a
deflection which is always kept at the limit of aperiodicity is
reduced a times, the sensitiveness diminishes qa’ times.

Now it sometimes happens that very rapid alternations of current
must be measured, which are so small, that they require a sensitive
position of the galvanometer in order to be perceptible and measur-
able and in these cases the condenser method can no longer be
applied.

If under these circumstances the deflections of the galvanometer
are less rapid than the oscillations of the current which must be
studied, the curve, recorded by the quartz wire, will no longer
render these oscillations accurately. We must then, as has also been
done with many capillary-electrometric curves, calculate from the
properties of the instrument used and the data of the directly
recorded curve, the value of the real oscillations. This calculation
may lead to the construction of a new curve, which now in all
particulars is the correct expression of the oscillations.

The manner in which capillary-electrometriec curves may be cal-
culated, is known '). Do the curves, recorded by the string galvano-
meter, fulfil the same conditions and is it possible to apply the same
calculation to them?

The answer to this question must in general be negative. We
shall try in this paper to analyse the curves of the string galvano-
meter and to show that they admit of equally accurate constructions
as the capillary-electrometric curves, but that the manner in which
they are calculated is different.

In this connection some properties of the galvanometer, more
especially the tension and mass of the string and the resistance to
the motion of the string must be discussed. We shall try to express
the value of these three quantities in absolute measure, by which a
clearer light will be shed on their significance for the practicability of
the instrument generally and for the modifications which might
advantageously be made in order to render the galvanometer service-
able for special purposes.

2. The principles of the method.

In the analysis of the curves, recorded by the string galvanometer
we start from the assumption that the resistance offered by the air

') For literature on this subject vide: Pruiiger’s Archiv f. d. ges. Physiol.
Bd, 99, p. 472, 1903,

15*

to the motion of the quartz wire, increases proportionally to the
velocity of motion of the wire.

This assumption, which is justified on theoretical grounds, will later
appear to be also practically proved by the results of analysis.

If we assume it to be true, we must also expect the motion of
the string to be completely determined by the laws that govern the
motion of an oscillating body with electromagnetic damping. For
electromagnetic damping, as in our case the air-damping, develops
a resistance, increasing proportionally to the velocity of the moving
body. In the string galvanometer both electromagnetic and air-damping
occur. They can be expressed in the same units and their combined
effect is always equal to the effect of either of them, if only in the
latter case the amount is made equal to the sum of the two, taken
separately. In other words: if 7a represents the resistance through
air-damping *) and 7, the resistance through electromagnetic damping,
we may reckon for the total damping resistance :

Pao Ri oo Sd eo o to “oo (ll!

The laws of motion of an oscillating body with electromagnetic
damping are e.g. extensively dealt with in Konnrauscn’ textbook *).
They agree with the laws determining the charge or discharge of a
condenser through a conductor having resistance and self-induction,
which are found in various textbooks on electricity *). The formulae
given in the above-mentioned works form the foundation of our
further calculations.

If we denote by:

m the virtual mass of the image of the string,

r the virtual resistance, damping the motion of the string,

c the sensitiveness of the galvanometer for constant currents,
two conditions may present themselves. In the first the motion is
oscillatory and

Am
r< sone asl gs gon gel 55D

c
in the second condition the motion is aperiodic and
4m

r> eee gh 2 ee

c

1) Besides the air-damping and the ordinary electromagnetic damping still other,
very feeble, damping influences are active, which may however be neglected here.
They will be dealt with in chapter 7.

2) Lehrbuch der praktischen Physik, S. 448. 1901.

5) Vide e.g. J. A. Femina, The alternate current transformer. London. I, p. 370, 1890.

(213)

The units in which m, 7 and c are expressed and the meaning
which must be attributed to the words “virtual mass of the image
of the string’ and “virtual resistance” are dealt with in the following
considerations on

3. The mass of the string.

In practice it is impossible or scarcely possible to find the mass
of the string by direct weighing. For this the weight is too small.
Also a calculation of the mass from the dimensions and the specific
eravity of quartz and silver does not lead to the desired result, since
the measurement of the diameter of the fine thread and of the thick-
ness of the still much thinner layer of silver cannot be carried out
with the necessary aceuracy, the more so because the layer of silver
often shows inequalities and its thickness is irregular.

So we must proceed in another way and we reach our purpose
in the simplest way if we start from the “virtual mass of the image
of the string’. The image of the middle of the quartz thread, which
is projected with great magnification, moves along a straight line
over a millimetre scale each time the string deflects. Instead of this
image we’ imagine a material point, acted on by a force which is
equal to the force that moves the whole. string. The acceleration
experienced by the fictitious material point from this force, determines
what we call the “virtual mass of the image of the string’.

The unit in which this virtual mass must be expressed depends
of course on the units chosen for the force and the acceleration. As
the cause of the motion of the string is the electric current passed
through the galvanometer, the moving force can be expressed in
units of intensity of current. We choose for this purpose the micrampere.

The unit of acceleration is determined by the units chosen for the
distance and the time. In accordance with the millimetre scale along
which the image moves, the distance is expressed in millimetres,
while it is found practical to take the millimetre also as the unit
of time. For the deflection of the quartz thread is recorded on a
plane which is moved along with uniform velocity perpendicularly
to the direction of the motion of the image, so that in the recorded
curves, from which we must always derive the data for further cal-
culation, time is represented by a length.

The above mentioned units of force, distance and time form a
system which may be called the millimetre-micrampere or [m-—uA]
system. Expressed in this system the unit of virtual mass m is the
mass which experiences from a force of 1 micrampere an acceleration
of 1 mm, distance per mm. of time.

( 214 )

The unit of virtual resistance (or better of the virtual resistance
coefficient) ') 7 is the force of 1 micrampere resisting the motion of
the string when the image of the string moves with a velocity of
1 mm. distance per 1 mm. of time. The unit of sensitiveness ¢ is
the sensitiveness with which a force of 1 micrampere causes a
deflection of the image of 1 mm.

These units are constant as long as the field intensity H7 between
the poles of the electromagnet, the length / of the quartz thread,
the magnification 6 of the image of the string and the speed V of
the recording plane remain unchanged.

If the string is strongly stretched and is suddenly passed by a
constant current, it will perform damped oscillations. We have then
the condition which we will in the first place consider more closely
and which is expressed by formula (2):

Od ee

To this case also the following formulae (4) and (5) apply:

ey, a Ae Cer es [2

2 Ae
tat| 148, ere es (5)

Here means the period which we would obtain without damping,
whereas 7’ represents the real period with damping.
Vi = eens Oe ace Ont koko. (CO)
a,

5 : 4 a 3 :
k beine the damping ratio, £ = ——=— = ete. in which a,, 4,, 4
=) cao) a a 1 2 3
3

2
are the deflections with damping.
From the formulae (4), (5) and (6) follows:
Re
2 SopNaaT ya Dy amiien ycx:-.((C)
As the period 7’ is expressed in millimetres, the value of m changes
if the velocity V of the recording plane is changed. In our measure-
ments J)’ was nearly always 500 mm. per second. Only with a few
photograms the sliding frame had a different velocity, so that with
these a reduction will have to be applied in order to render the
values of m comparable.

1) The word ‘resistance coefficient’? expresses the meaning of 7 better than the
word resistance. But the latter is simpler and is also used as a coefficient in the
theory of electricity, where it denotes ohmic resistance. In this paper we shall
repeatedly use “resistance” instead of “resistance coefficient”,

( 215 )

In order to express im, in the millimetre-micrampere units for a
photogram of which the speed of the sliding frame is V, mm. per
second, mz being the known value for a speed Vz, we use the

Veena
Me = mg X Va seis dM nSt ts nase te 5

relation:

From what precedes we see that the measurements required for
a ealeulation of the value of m when the value of |’ is known,
are limited to:
the sensitiveness ¢,
the deflections «,, @,... ete. and
the period 7”.

The measurement of the sensitiveness ¢ presents no difficulties.

The intensity of the current is known from the electromotive force
of the source and the resistances used. The source consisted of
storage cells, the electromotive force of which remained very constant
and might be put at 2 Volts, while the resistances were taken either
from manganin resistance coils or from a_ graphite resistance of
Siemens and Hatske which had previously been checked.

The deflection of the quartz thread is measured on the photogram
on which the network of square millimetres is found. With the
magnifying glass it is easy to estimate 0.1 mm., so that a deflection
of 30 mm., which is often used, is known with an accuracy of 0,3°/,.

If the permanent deflection is «mm., the intensity of the current
=7 micrampere, then the sensitiveness is e=— millimetres per
micrampere.

The values of the oscillations a,,a@,, etc. are, like the permanent
deflection, read off directly from the network of square millimetres.
But as these values are smaller than w and the absolute error in

each measurement remains unchanged = 0.1 mm., the accuracy of
a a :

the value found for : =——=—... ete. is not great. Moreover a
a a
2 8

Pubs , . a a
distinct difference is often found between — and —*, so that we are
a, a,

F A a
obliged to calculate a tiean value, e.g. by putting = =F

a,
Fortunately the value of h, as it occurs in our measurements, has
only a small influence on the final result, relatively large variations

in & searcely causing any difference in the calculated value of m.

Caley

Of the greatest importance is the measurement of the period 7
which in formula (7) occurs squared. The total amount of 7’ is
always very small, in some cases even less than 1 mm. so that it
is desirable to carry out this measurement with an accuracy of a
very small fraction of a millimeter.

For this purpose I first used an excellent astronomical instrument,
destined for the investigation of celestial photograms, and which
was put at my disposal by the kindness of the director of the Leyden
Observatory *).

Later I used exclusively an ordinary microscope stand, pattern
I¢ of the firm Cari Zeiss. The stand is provided with the large
movable cross table of the same firm. On this table a small wooden
board is fastened, which follows the cross movements, being always
supported by a smooth plane on which it can slide easily. On this
board, which has in the middle a spacious opening, the photographic
plate to be measured is placed.

As object-glass a, is used with a focal distance of 40 mm., while
for the eye-piece that of Ramsprn is used, also made by Zuiss, with
micrometer screw and drum reading. The eye-piece is so adjusted
that the direction of motion of the measuring wire coincides with
the direction of the abscissae of the photogram. After the microscope
has been sharply focussed on the photogram and at the same time
care has been taken that the image to be measured and the image
of the crosswire lie in the same optical plane, the microscope tubes
are firmly screwed on to the stand by means of a clamp, which is
expressly made for this purpose. The tubes must be quite immovably
connected with the stand. Also the eye-piece is screwed on to the
microscope tube so that it can be touched with the hand without
observing the slightest displacement of the crosswire with respect to
the image to be measured. For measurements requiring some accuracy
these arrangements are indispensable.

With the microscope tubes pushed in, 1 scale division of the drum
corresponds to a little less than 5 on the photogram, and since 0.1
of a scale division can easily be estimated, the error in the reading
is less than 0.5 uw. It would be a mistake, however, to suppose that
also the result of the whole measurement can now be known with
the same accuracy. For there remain errors of a different origin.

So e.g. we start in our measurements from the shadow lines
which in the photogram form the net of square millimeters, a base

1) I wish to express here my best thanks to Prof. vAN pE SanpE BakHuyzeNn
for his kind assistance.

( 247 )

which in itself is not free from errors. The ordinates of the net
are recorded by a quickly revolving disk with spokes, casting 500
shadows per second on the photographic plate. The variations in
the rotational speed of the disk cause only small errors. Reserving
a more extensive description of this time-marker for a later occasion,
we may here mention that the rotational speed is very constant and,
in all probability, always agrees to 0.1 °/, with the pre-determined
value. But we have no guarantee that the mutual distances of the
spokes, although carefully marked off on the lathe, come up to the
same standard.

The greatest errors are probably made by the setting of the cross-
wire. To be sure, surprisingly sharp settings can be made on the
fine, sharp top, representing the turning point of a movement of the
string, so that a greater error than lu need not be made, but often
it is necessary to mark the exact middle of a broader top; in which
case the error becomes larger, of course. With many of our measure-
ments of the period 7’ the error must be estimated at 1 to 2°/,.

We shall now give the results of some measurements.

With a speed of the sensitive plate of V—=100 mm. per second,
some photograms were taken of string n°’. 10.

The sensitiveness was c= 10.92 mim. per micrampere,

the period T= 043 mm.
the damping ratio [p= say}

With these data the value of m is calculated from formula (7) at
3.76 XK 10—-+ [mm — pA}.

For a comparison with following results this value is reduced by
formula (8) for a speed of the sensitive plate of 500 mm. per second.
We find then m= 9.4 & 10-3 [mm — uA].

Another series of measurements with the same string gave, at any
rate as far as the second figure, exactly the same result. The speed
of the sensitive plate was now 500 mm. per second.

Besides ¢ was = 3.5 mm. per micrampere,
Te = 1.16 mm. and
k == Oo,

from which we calculate again m = 9.4 10-?[mm — uA).

The agreement in the two final results is the more remarkable
because of the great difference in the sensitiveness c and the period 7’.

String 13 is thinner than 10, having besides a smaller conductive
resistance. The measurements with sudden passing of a constant
current, show that with V = 500 and c= 5.69, the value of 7’ is
1.32 and that of & 3.1. From this we calculate from formula (7)
that the mass m is 6.9 10-3 [mm — uA]

( 218 )

Finally we mention still two series of measurements with a still
thinner string, n° 14,

In the first photogram we tind with V = 500, and e=5,75 that 7’is 1,
and # = 3,8 from which is caleulated m= 3,7 K 10-3 [mm — uA].

In the second photogram V is unchanged but ¢ = 3,15, 7’= 0,705
andy 386:

From this we calculate m= 3,5 « 10—* [mm — WA]. So we may
take for the mass of string n° 14 the mean value which is
m = 3,6 < 10-3 [mm — WA).

4. The resistance to the motion of the string.

The resistance to the motion of the string can be determined in
various ways. We shall begin with a description of the method which
may be considered to give the most accurate results.

Let the quartz thread be only moderately stretched, and let it
suddenly be passed by a current of constant intensity. Let the
deflection of the thread be recorded on a rapidly moved sensitive
plate, so that a curve is obtained, which is schematically represented
in the following figure 1.

‘ig. 1.

From A to B the quartz thread is in the O-position. At B the
constant current is made, by which the image of the string is moved
upward until if reaches its second position of equilibrium about D.

(219 )
The fundamental formula, found in the above-mentioned text-books *),
which represents this curve, runs:

Elie Gp dq 1

ane —— 1) etunsh tok Pew oher ua: aco ee (IG
dt? * m dt me q @)

Here m, 7 and ¢ have the same meaning as before, namely m the
virtual mass of the image of the string, 7 the virtual resistance,
damping the motion of the string, and c¢ the sensitiveness of the
galvanometer for constant currents. ¢ means the time and q the
distance of the image of the string from its second position of equi-
librium or in other words: the distance of any point p of the curve
from the line CD.

All units are expressed in the millimetre-micrampere or {mm—wA |
system.

Calling 0 the radius of curvature in any point of the curve, we
have:

: : BNI ae <c:
and further putting the tangent of the angle of inclination =
¢

Wwe may write:

3
1 + v2
g=cerv + em ( <. 5 or 16 ad oy fg (GUID)

Here @ is positive when v increases, negative when v decreases
with increase of ¢.
For the case that we may put @ = formula (11) simplifies into:
TSO 6 oo hoe so oo (le)
This case must present itself somewhere in a point s of the curve.
From £ to s the curve is concave upward, from s to D concave
downward, s itself being the point of inflection. For the point s
@=o, so that for this point formula (12) applies. We write it in
the form

(PSR 5 a ANG ote Soto ACLS)

In order to determine the value of the resistance * by means of
this formula, the sensitiveness c of the galvanometer must be known,

1) Vide Kontrauscu |.c. p. 450 and Fiemine |. c. p. 368.

( 220 )

while two quantities must be measured with respect to the points s
of the curve, viz. its distance g from the second position of equili-
brium and the tangent v of the angle of inclination.

These measurements are made with the same microscope stand I+,
already mentioned in the preceding chapter, on the table of which
the photographic plate can be moved by screw motions in two
mutually perpendicular directions.

For object-glass a, is again used or another more strongly refract-
ive lens, while for an eye-piece an arrangement is used which must
be explained now.

At the outside of a Huyarns eye-piece n°. 2 a pointer is fastened.
When the eye-piece is turned round the optical axis, this pointer
moves over a circular dial, screwed on to the microscope tube and
divided into degrees. In this way we can read off how much the
eye-piece is turned in the microscope tube. Moreover in the focal
plane of the ocular lens of the eye-piece a fine crosswire is found,
which is so set that the crossing point lies at the centre of the field
of view. Hence, when the eye-piece is turned in the microscope
tube, while the crosswire turns, the crossing point will remain
immovable. The whole arrangement, of which the principal parts
have been taken from the analyser of a Zuiss polarisation microscope,
works very accurately and enables us to make a number of measure-
ments in a short time. This is done in the following way :

A photographic negative or better still a diapositive is placed
under the microscope, so that one of the directions of motion of the
cross table coincides with the direction of the abscissae of the photo-
gram. Next the whole microscope tube is again immovably screwed
fast at such a height that the real image of the figure to be measured
lies in the same plane with the wires of the eye-piece. In this way
one can by means of the screws of the cross table easily and quickly
cause the image of any point of the photogram to coincide with the
crossing point of the eye-piece.

One begins with placing an absciss in the crossing point, and then
turns the eye-piece so that one of the two crossed wires, e.g. wire
A coincides with this absciss. Then an arbitrary point P of the
curve is placed at the crossing point. The eye-piece is now turned
again in such a way that the wire A forms a tangent line to the
curve at P. The angle through which the eye-piece must be turned
in order to get from the first into the second position, is the angle
of inclination of the curve at P. It is read on the graduated circle
in whole degrees, tenths being estimated. So a tangent can be drawn
at any point of the curve after it has been brought to the crossing

point of the ocular wires, and the angle of inclination of the curve
at this point can be measured. The method which also deserves
recommendation for the measurement of other, especially capillary-
electrometric curves, leaves little to be desired as to ease and
quickness. Its accuracy can be judged from the following results.

In the reticular scales, which together with the movements of the
quartz thread are photographed on the plates, the ordinates are not
perfectly perpendicular to the abscissae, which must be ascribed to
the circumstance that the image of the slit, formed by the cylindrical
lens on the photographic plate, is not perfectly perpendicular to the
direction of motion of the sliding frame. I did not carry out my
original intention of correcting this error by a calculation, since
the deviation only amounts to 0°.3.

Besides we can, when measuring angles of inclination less than
45°, take an absciss as base, when above 45° an ordinate, by which
proceeding the deviation becomes of still less importance.

But from measurements of this deviation by means of the turning
eye-piece, if appears that the measurements of angles can be made
with errors not exceeding 0°.1. This will be seen from the following
table.

TAB aki I
Number | Direction | Direction | Difference Deviation
in direction from the
of the of the of the between average
H abscissae value of
plate. | abscissae. | ordinates. | ordinates. CRIIR7G
|
A 22, 110°.8 21°.05 89245 + 0°.05
A 37 200°.6 |/411°.0 89°.6 — 0°.1
A 123 123°.0 33°.3 Sora 0
A 124 122°.8 33°.0 89°.8 + 0°.4
A 132 422° .9 33° .2 Soeei 0

No nearer explanation is wanted. In the last column but one we
find the results of the measurement of the angles under which in
various photographic plates the abscissae are cut by the ordinates.
Taking for the average value an angle of 89°,7, we see that the
greatest deviation from this value is only 0,°1.

It must be remarked, however, that the direction of the tangent
in a point of the curve cannot be determined so accurately as the

(222, )

direction of an absciss or an ordinate. The stronger the curvature of
a curve and the more rapid the changes of curvature take place,
the more uncertain the determination of the direction of the tangent

at any point becomes.

The point s at which the curve under examination has no curva-
ture, is easily found under the microscope.

A number of points in which the curve is cut by the abscissae
or ordinates, are successively brought to the crossing point of the
wires in the eye-piece and in all these points the angles of inclination
are measured in the above described manner. It is noticed that with
increase of the abscissae the angles of inclination first increase and
then decrease. At the point of transition must be situated the required
point at which the curvature of the curve is 0 or @ =o.

In this way the angle of inclination in the point sought, is directly
known, while the distance of this point from the second position of
equilibrium of the quartz thread can at once be read off on the
reticular scale. Millimetres are directly indicated, tenths must be
estimated.

A curve written by the quartz thread has in many eases a fairly
considerable breadth. We can then consider it as a double curve, and
carry out the measurement of the angles of inclination as well at
the upper as at the lower side of the image of the quartz thread,
in this way applying a control by which the accuracy of the final
result is enhanced.

Before mentioning the amounts found by direct measurement, we
must dwell a moment on the velocity of motion of the sliding frame.
Most photograms were taken with a speed V of 500 mm. per second.
In order to render values of v that are expressed in the millimetre-
micrampere system, but were measured with different values of V,
mutually comparable, they are all calculated for V = 500. For this
purpose we use the formula:

Va
"A faa ts Jol sa? ow pit asia foe ca ee mmm Re
in which 7, means the virtual resistance of the quartz thread with
a speed of the frame of V., mm. per second, rz being the corre-
sponding value with a speed V3.

We must further take into account the circumstance that the reticle
en the photogram does not always consist of accurate squares. As
was already mentioned in the preceding chapter, the rotational velocity

( 228 )

of the spoked disk, by which the ordinates are recorded, is very
constant. The error in the absolute amount may be estimated at no
more than 0,1 °/,. But the velocity of motion of the frame cannot
be so accurately regulated. The speed of a frame, once moving, may
be considered perfectly constant, but between photograms, taken at
various times, the speed may differ fairly much. Hence the distances
between the ordinates in one plate are somewhat greater than in
another.

The distance between the abscissae is absolutely the same in all
plates. We could not expect otherwise, since the abscissae are recorded
by the shadow of a glass scale; mounted in a fixed position at a
few millimetres, distance from the movable frame. The glass scale,
containing very fine division lines, was expressly made for our pur-
pose by the firm Zutss.

The mutal distances of the abscissae and also of the ordinates are
measured under the microscope by means of the millimetre divisions
with nonius, which are found on the above described cross table.
For 30 scale divisions along an ordinate we invariably find 30,1 mm.
so that 1 scale division along an ordinate may always be put
= 1,0033 mm.

The scale divisions along an absciss in the middle half or middle
third part of the same plate are also equal, as may be seen e. ¢.
from the following measurements of photogram A 34, see table II.

It must be remembered that the nonius did not enable us to read
with a greater accuracy than O,1 or 0,05 mm. so that for these
comparatively rough measurements the mutual distances of the

TASB He aie
Number Reading Length of 10
are the on the scale divisions
millimetre | along an absciss
ordinate. scale. in millimetres.
ee
0 | 61.2 —
10 Bail 6) 9.7
20 41.9 9.6
30 32.2 9.7
4O 92.5 9.7
50 12.8 9.7
60 jadita) 9.65

( 224 )

ordinates must be considered as perfectly equal. More accurate
measurements have already been mentioned in the preceding chapter
and need not be discussed again.

From the table follows that in the examined photogram 1 scale
division along an absciss = 0.9675 mm. Since the length of a scale
division along an ordinate has already been given in millimetres we
ean directly express a scale division along an absciss in scale divisions
along an ordinate. Calling this value d, we find in photogram A 34
d= 0,964. In other photograms we found values for d, varying
between 1,033 and 0,901.

If d deviates somewhat considerably from unity, the value of the
tangent of the angle of inclination will have to be corrected. If the
tangent of the measured angle of inclination is v, we shall have to
take the product vd instead of v in the computation of 7.

Formula (138) is then replaced by

Ppesn na

cvd

. (45)

We now give some results of measurements, for convenience’ sake
assembled in the following table III.

TAC Ee Un
Number | Number | ¢ V Ta
of the | of the |. 2%... | in millime- | | in milli- | in [wm—#A]
fin milli-) v d

quartz | photo- | atres | tres per | metres calculated

thread. gram. “S- | micrampere. | per sec. | for Y= 500.
10 50 | 93.5 | 1093 | 6.02 |1.032| 400 0.0193
413 A 37 21.8 798 1.637 | 0.975 500 0.0174
14 4123 | 23.6 | 1133 4.322 | 0.984 500 0.0160
14 A 124 21.6 563 3.006 | 0.901 500 0.0155

During the recording of the curves to which the numbers of this

table refer, always a great resistance — more than 1 Megohm —
was inserted in the galvanometer circuit, so that the electromagnetic
damping might be neglected and the resistance rq to the motion of
the string was only caused by the air. We see that the value of
ra, Which for all the has been calculated for V = 500,
decreases in the order of the numbers 10, 18 and 14 of the quartz
threads. In the same order also the thickness of the silvered strings

curves

diminishes. The diameter is:

for string 10 without silver 2,4 4, silvered 3,0 u
a Sag io a is 1.6 nu, Pe 2.5 u
6 » 14 p 3 ist (B. 169%

We leave unsettled what function the air-damping is of the diameter
of the string, but wish to point out here particularly the influence
which the damping may experience from the layer of silver on the
quartz threads being more or less smooth. Especially string 14 has
an almost perfectly smooth coat of silver, while the skin of string
10 shows distinctly visible inequalities.

We will now investigate whether the amount of 7, for a definite
string may be regarded as constant, however much the tension of
the quartz thread and thereby the velocity of its deflections may
be moditied.

For this it is necessary to measure 7, once more with a strongly
stretched string and oscillating deflections, but the methods which
are at our disposal for this purpose do not nearly furnish such
accurate results as could be given above. We mention here two
methods of measuring.

In the first place the same method as above may be used, but
this time it is applied under unfavourable conditions. As the string
makes rapid movements, the tangent of the angle of inclination
becomes large, so that its amount cannot be determined with the
desired accuracy. Besides the uncertainty in the value of q also
increases very much,

Of the curves, by means of which in the preceding chapter the
mass of the string was calculated, plate A 61 of string 13 shows
the greatest value of 7’, viz. 771,32 mm. I therefore have chosen
this curve for an attempt to determine the value of» by formula (15)
oe

cod

A resistance of more than 1 Megohm was inserted in the galvano-
meter circuit so that the electromagnetic damping might be neglected
and consequently r might be put = 7g. The following values
were found:

is) mM.

¢ = 5,69 mM. per micrampere.

vy = 88,1

d= 0.979.
As the quartz thread with rapid motion writes a fine line on the

sensitive plate, the angles of inclination can be measured with great
16

Proceedings Royal Acad. Amsterdam. Vol. VIII,

( 226 )

accuracy. An error of 0,1° is possible, however, in the measured
value of the angle. And if the real angle is 0,1° greater than the
value found, the tangent must be taken 104,2 instead of 88,1. So
an error of O,1° in the measurement would cause here an error of
over 18°/, in the final result. Putting the uncertainty in the value
of gq at 1,5 mm., also the probable error, caused by this alone, is
found to be 17 °/,.

The measuring arrangement which we used, does not enable us
to make the result more correct.

In the second method of determining the value of the resistance
to the motion of the string with a strongly stretched, oscillating
string, we make use of the following formula: *)

ap rT 5
NE = 7 te As one eee ce
in which the letters have the same meaning as before, viz.

k; = the damping ratio,

ry =the resistance sought,

7’ = the period of the oscillatory deflections,

m =the mass of the string.

We write formula (16) in the form:

4m lank
eS He Net sea peer i -((1E7/))
and substitute for # the value which we found, according to formula
(7) of the preceding chapter, to be
[2
¢ {39.544 (lyn hb)

ea)

a

We then find:
AT lgn k
—— eee eee 8 6S)
6 $39,544 (gn kh)?
In order to obtain comparable results, we always calculate 7 for
a speed of the sliding frame = 500 mm. per second, using again

=

formula (14) 73 =7, ae
V,

From the two last mentioned formulae the value of + is now
caleulated for four out of the five curves which in the preceding
chapter served for the determination of m. During the recording of
these curves a great resistance of more than 1 Megohm was each
time inserted in the galvanometer circuit, so that the resistance to

1) Vide e.g. Konurauscu |. c. p. 448.

( 227 )
the motion of the string was again caused by friction of the air

only. Henee we may identify 7+ with 7g. The results of the calculation
are found assembled in the following table LV.

MAB) SEV.

Number | Number i! c v ra
of the of the lie: +); |in millime-jin millime-| in [wu—v4],
quartz photo- TENT oe Lg tres per tres per calculated
thread, gram. metres. micramp, sec. for V = 500.
| | u
10 46 | (4.49) 0.43 10.92 100 (0.0207)
“5 418 | (0.604) 1.16 3.0 500 (0.0196)
13 A 51 (1.13) 4.32 5.69 500 (0.02355)
14 | 4 429) || (4.335) 1.— 5.75 500 (0.0199)

Also these results lack the accuracy
the motion of the string

©

which can be obtained when
is slow. The difficulty lies in the value of

k/ which, as was remarked before, presents a great uncertainty.
In the following table V the average values 7 are given as they

TABLE V,
Number Re rsabie Ta Ratio of 7a with
of the oo nan | with strong strong tension
: tension of the : t

quartz eaves tension of the) to 7a
thread. feeaa |quartz thread.| with feeble tension.

10 0.0193 (0.0201) 1.04

13 0.0174 (0.0210) 1.24

14 0.0157 (0.0199) 4.27

were found by calculation according to the three methods mentioned.

We see that the value of ra is a little greater with a greater
of the thread. we must infer that the
resistance, caused by the friction of the air, does not increase quite
proportionally to the velocity of motion of the string, but that with
very rapid motions of the string the increase of the resistance of the

tension quartz From this

ar is a little greater.

Practically, however, this result has no influence on our further
calculations. The increase in the value of 7, may be ealled very
The in of 1:1,27, while

16*

small. ereatest increase was the ratio

( 228 )

tensions of the quartz thread were used, varying from 1 to 324.
That with smaller tensions of the string than were used above,
the proportionality between the velocity of the string and the
frictional resistance of the air, is maintained, has been sufficiently
proved on a former occasion’). A standardising curve with a sensi-
tiveness of 1 mm. detlection for 10~—!° Ampere, was recorded on
a photographic plate, which moved with a velocity of 10 mm. per
second. After the sensitiveness had been reduced ten times and the
speed of the plate had been ten times increased another standardising
curve was written. This latter shows the same shape as the former,
both curves can be so superposed that they coincide over nearly their
whole length.

So we may say that the tensions may vary from 1 to nearly 3000,
the value of 7q@ being only slightly altered.

Let us now consider the amount of the electromagnetic damping,
expressed by 7. The value of 7, is nothing else than the pondero-
motive force which the string experiences by its motion in the mag-
netic field when the velocity of the image of the string is unity.

If this ponderomotive force is rendered in the | mm—wA)| system,
it is expressed by a certain number of micramperes with the given
leneth of the thread and the given intensity of the field.

The caleulation can be made in the following way.

The electromotive force /, generated in a wire, which is perpen-
dicular to the lines of force of a homogeneous magnetic field, and
is displaced with a velocity of v7, centimetres per second in a direction
which at the same time is perpendicular to its length and to the
lines of force, is

H=v, lH xX 10-8 Volt,
in which / means the length of the wire in centimetres and H the
intensity of the field in C. G. 5. units.

When the middle part of the string moves with a velocity v,,
the average velocity of all the parts of the string may be put

9
—v, = 0,687 v,, so that for the electromotive force e, generated by
a4
a deflection of the quartz thread, we may write
e\= 05631 vl SG NO SS alt ee reer ote (LCS)
Moreover we have

Coole
v, =v x 100’ (20)

1) See these “Proceedings” 6, p. 107, 1903.

( 229 )

in whieh, as above, v, means the velocity of the middle of the
string expressed in centimetres per second, 7 the velocity of the
image of the string, expressed in millimetres distance per millimetre
of time, V" the velocity of the frame in millimetres per second and
6 the magnification used.

Substituting in formula (19) the value vy as given in formula (20),
we have

OL(sait/ h Wal <=)

e= — Volts.

b

Calling the resistance in the galvanometer circuit 7 Ohms, the
intensity of the current /, caused in the cireuit by the motion of
the quartz thread and expressed in micramperes, is

Fen’ SOS ORGS ae Se S38

w wh

micramperes. . . (21)

The ponderomotive force, experienced by the string, is expressed
by Z{mm—pA}.
Hence if we put in formula (21) > =—1, / becomes =7, so that
we have
0}63'7 Vil << 10553

"= ——Imm—wA] ... . (22)
wh ;

From the above formula (22) it appears that, in order to calculate
r,, we must know V,/,2,4 and H. Of these quantities the four
first mentioned can be measured with sufficient accuracy, while on
the other hand the value H/ of the intensity of the field presents
some difficulty. Opposite the middle of the string there is an opening
in the pole shoes of the electromagnet, by which in that place the
intensity of the field, although not zero, yet becomes much smaller
than in any other place of the slit-shaped space, in which the quartz
thread moves.

Measurements of the intensity of the field in different parts of
this slit-shaped space showed that in those places where the intensity
is greatest if may be put at about 22500 (C. G. 5.), the current
of the field magnet being regulated at 2.7 Amp. With this current
of the field magnet nearly all our observations were made. The
average intensity of the field, caleulated over the whole length of
the slit, appeared to be about 10°/, less, i.e. about 20250 (C.G.38.).
3ut this value for /7 is inadmissible since the places of least intensity
are found at the middle of the slit where the lines of force exert
the greatest influence on the motion of the string. For H we shall
have to reckon considerably less than the average value of 20250

( 230 )

(C. G. S.), and since sufficiently extensive measurements of the
strength of the field in all parts of the slit are wanting, /7 would
have to be estimated approximately. Rather than doing this we shall
follow an entirely different way.

It is possible, namely, to get to know 7 in a different manner.
For this the resistance to the motion of the string must be measured
twice, first when a very great conductive resistance is inserted in
the galvanometer circuit and then after the conductive resistance in
this circuit has been made as small as possible. In the first case the
resistance to the motion of the string consists of air-damping only, in
the second case of air-damping and electromagnetic damping combined,
The difference between the two values gives us the value of the
electromagnetic damping. Remembering formula (1) we write:

17 i Phe eC ole eae ac (22)

When the value of 7 has become known by means of 7 and 7a
H can be calculated. For this purpose we write formula (22) in
the form:

I rybu X 1000

0.637 Vi

(24)

We. have = 660, V=500 and /=12.7, so that we may
write for //:

(ih —— Nae eed. GO 6 6 a. 6 (25)

HT was computed for string 138. Plate A384 shows a eurve,
recorded, when suddenly a constant potential difference was established
between the ends of the quartz-thread, the galvanometer circuit being
closed by a small external resistance, the amount of which may be
neglected. Analysis of the curve shows that g¢ = 24.8, c= 89.4,
vy = 0.98 and d= 0.964, from which we derive by means of the
formula =) that 7 = 0.0294.

cod :

Formerly we found 7, to be 0.0174, from whieh follows that
” = 0.0120.

The conductive resistance of the quartz-thread is 29000 Ohms,
and from this we compute by means of formula (25) that 7—= 17600
KOR Cerca

Two tables follow now. In table VI are found for three quartz-
threads the data, enabling us to calculate the value of 7 by formula
(15). The speed of the sliding frame is always 500 mM. per second,
the galvanometer circuit being closed by a small external resistance,
the amount of which may be neglected.

( 231 )

AV AIBA IDS Wile

Conductive

Number | resistance | Number | q cml. || | | 7,
of the w of the |., mille} 22 milli- boy in
quartz- of the photo metres per
thread. quartz- gram. | metres. | micramp. | | | [mm—p 4].
thread, | | |
10 LC OD 0Nn ODE em men so) aed Oc A OOS ln OSOS12
| | | | | |
| | :
43 | 9000 | 434 | 94.8 89% | 0.98 | 0.964 | 0.029%
| | | | é
14 | 17800 | A 132 | 26.4 582 | 2.33 | 0.927 | 0.0210
| | |

In table VIT we find the values of 7q from table Ill together with
those of » from table VI. The difference between the two values is
indicated in the last column but one by 7; the last column gives
the value of 7, as ealeulated by formula (22), for H the value
17600 [C.G.5.] having been taken.

The accordance between the values of these two columns is very
satisfactory and may be considered as a proof of the accuracy with
which in general the resistance to the motion of the string may be

ies
ron (15).
The ratio of the directly measured values of 7@ and 7 is
for string 10 as 1 : 1,615
: Ble ee SSS)
” 0 ae erase

determined by means of the formula 7 =

TVA BLES Vale

Number | w = . | rb ete ne,

3 | ; } Ta ? lealeulatedateons calculated from the
of the | in | ah : aq length of the string
; er -asured, | measured. | Me aoa ol! Cand=the: field

string. | Ohms. measured.) values of r and 7a aa :
Boe) : intensity.
i i
= oo
410 | 40000 | 0.0193 | 0.0312 | 0.0119 0.0108
143. | 9000 | 0.0174 | 0.0294 0.0120 0.0120
44 | 47800 | 0.0157 0.0210 | 0.0053 0.0064

To conclude this chapter we give some remarks concerning the
condition for which the motion of the quartz-thread just reaches the
limit of aperiodicity. In this condition we have the relation :

4m

Saeed ints ek go (OG)

c
,

According to this formula one would be induced to determine the
value of ¢ from m and 7, but it will appear in chapter 6 that for
small tensions of the quartz-thread the virtual mass of the image
of the string is not a constant value.

Hence it is not or hardly possible to derive from measurements
that were performed with other tensions of the string, the value of
m for the ease that the limit of aperiodicity has been reached. And
if m is unknown ¢ cannot be caleulated, of course.

So if one wants to know the sensitiveness. for which the limit
of aperiodicity is reached, one is obliged to determine this directly
by experiment. The results of a number of such determinations
which, as will be understood, were only made in a rough way, are
found united in the following table VIII.

From the data of the preceding table and from the values of 7
it would be possible to caleulate the values of m.

re

Similarly those of the time-constant') T by the formula oe

These caleulations, however, must be omitted, since c has a far
too small degree of accuracy here, to attach any importance to the
results.

TAS Bash. VI

Sensitiveness ¢ for the limiting

Number value of aperiodicity.
of the with air-damp- with air-damping
and electro-
string. ing only. magnetic damp-
; ing combined.
10 (120) (50)
13 | (130) (45)
14 (115) (55)

5. The acceleration.

When analysing a curve, recorded by the capillary electrometer,
if we wish to know the potential difference which at a certain
moment exists between the mercury and the sulphuric acid, we have,
besides the properties of the instrument and the speed of the recording

1) See Fremine, |.c. pp. 377 ff,

( 283 )

plane only to take into account the velocity of the motion of the
meniscus. When analysing a curve, however, recorded by the string
ealvanometer, the analogous data are not sufficient in general. It
will often be necessary to take into account not only the velocity
but also the acceleration presented by the image of the string.

This must be aseribed to the fact that in the capillary electrometer
the resistance to the motion of the meniscus is very great ') compared
with the mass of the mercury thread, so that this mass may be
neglected, when it is desired to calculate the existing potential differ-
ence from the velocity of motion, whereas with the string gaivano-
meter the resistance to the motion of the quartz-thread is very small,
and hence the mass of the thread in many eases has a distinct
influence on the velocity of its deflections.

These considerations may be succinctly rendered by formula (11),
already developed in the preceding chapter:

ro|

1 32)
g = erv + em ( int Z oe igs be oo cela)
9

If 7 is very great compared with m the second term behind the
= sign may be dropped and the formula becomes

=O 0 5 ene eo oo ee ()

This formula (12) can be applied as well for the analysis of
capillary electrometric curves as for curves of the string galvanometer
for which v is small and o large.

On the other hand, for moderate values of » and » the mass m
may no longer be neglected, so that then analysis of the curve will
only be possible if besides the velocity also the acceleration can be
measured. This acceleration, expressed as virtual acceleration of the
image of the string in millimetres distance per millimetre of time, 1s

1+7°)°
nothing else but esas
Q

Assuming as known the general conditions under which a curve
is written by the string galvanometer, and also the distance from any
point of the curve to the zero line, one will have to measure the
tangent 7 of the angle of inclination and also 9 the radius of curvature,
in order to calculate the potential difference which existed between
the ends of the qnartz-thread at the moment, that the arbitrary point
mentioned was recorded. Under unchanged general conditions each

1) On the influence of frictional resistance on the movement of the meniscus in
Lippmany’s capillary electrometer, see these “Proceedings” II, p. 108, 1899.

( 234 )

point may be said to be fully characterised by its distance from the
zero Jine and the values of » and 9.

The distance from the zero line may very easily be determined
by the presence of the square millimetre net, while in a preceding
chapter it was pointed out how v is measured. So we have only
to deseribe the best way of ascertaining the value of the radius of
curvature @.

Three different methods were tried for measuring 9 of which one
only proved practicable. The other two will only be briefly mentioned.
First a reduced diapositive was made photographically of a large
drawing on which a number of circles with different, accurately
known radii were represented. On the diapositive the radii vary
systematically from 0,5 mm, to a. It must be so laid on the curve
io be measured that one of the circles coimeides with the curve in
any point of this latter. By direct comparison the value of @ in that
point will then be known.

In the second method three points of the curve are measured,
situated at small but mutually equal distances. Calling / the distance
of the two extreme points and p the distance of the middle point
from the straight line that joins the two extreme points, the radius
of curvature at the spot where the measurement is made, is

k? +4?

— we

N 8 p

Here / represents the chord and p the height of the cireular are

under measurement.

( 235 )

The third method, the only one that could be applied with good
results, as was remarked above, consists in measuring the angles of
inclination in two points of the curve, situated near each other.

Let p, and p, be two near points of a curve, the radius of curv-
ature of which keeps the same value g in all points between p, and
Ps the angle of inclination at p, being represented by @ and that
in p, by B.

MX is an absciss in the coordinate system which was recorded
as a net of square millimetres together with the curve, but has been
omitted in the figure, while M7), p,q, and p,q, are ordinates.

It is seen from the figure that

Me Ma,
gin ¢ = —" and sin i ds ‘
Q
Putting 177, — Mq, = 9 we have
od
o=- et ore ee OCD

sin B — sina

The value of ® can be read off in a simple manner on the net
of square millimetres, while the angles « and 8 must be measured
by means of the eye-piece with cross-wires. This arrangement and
the accuracy that can be obtained by it, have already been dealt
with in the preceding chapter; we now put the question in what
eases the determination of @ may or may not be practically useful.

Let us once more consider formula (11)

3
1407)?
= cae on ae be Oe Od ag oe gl((I10)

this time as the expression of a curve, representing the damped
oscillations of a strongly stretched quartz-thread. For each reversing
point the value of 7 must be put = 0. Hence for a reversing point
the formula becomes

Ue ON here PP eney cine a) rs 1 aif aise (28)

in which the sensitiveness ¢ is an accurately known quantity. So it
would only be necessary to determine g and @ in order to be able
to calculate at once a value for m from every reversing point.

But here the practical difficulty lies in the quick variations which
@ shows already for moderate values of g. The time 9 has now to
be taken so small that it can no longer be measured with sufficient

( 936 )

accuracy, at any rate with our measuring arrangement, when the
microscope has been mounted with the cross-wire eye-piece. And
hence o itself becomes inaccurately known.

So we conclude that measuring @ has no practical value when
we want to know the value of m for a strongly stretched oscillating
string. Moreover this value has already been determined for this
case in a satisfactory manner by the method described in chapter 3.

3ut the measurement of @ does obtain practical value when we
want to know the virtual mass of the image of a string, which has
written a curve with a feeble or moderate tension of the quartz-
thread '). When analysing different curves it is not sufficient to use
the calculated real mass of the quartz-thread, since, as has already
been mentioned and will still more clearly appear in the following
chapter, the virtual mass of the image of the string is very con-
siderably modified by changes in the tension of the quartz-thread.

We finally remark that when the velocity v is great, also the
angles @ and 8 become large, by which the difference of the sines
diminishes for the same difference of the angles. This causes a
diminution of the accuracy with which @ can be known.

Also when 0 becomes very great the determination loses in accuracy,
since then for an equal value of v the difference between sim a@ and
sin 8 greatly diminishes. But this drawback is of no practical conse-
quence, as in the analysis of a curve the value of @, as soon as it
gets beyond a certain limit, may be put 2 without a large error.

6. Analysis of some curves.

We give in this chapter the results of the analysis of some curves,
written, when a known, constant potential difference was suddenly
established between the ends of the quartz-thread.

The first of the curves to be dealt with was recorded with a
rather feeble tension of the quartz-thread, i.e. with a rather sensitive
position of the galvanometer. 1 mm. ordinate = 1,87 >< 10~% Amp.
or the sensitiveness ¢ = 535. The speed of the sensitive plate is
17 =500 mm. per see., so the value of 1 mm. abseiss = 2 0,

We call 0 the moment when the electric current is started.
Now the angles of inclination of the curve are measured at t= 146,
26, 36, etc. In the following table IX, in the first column the
values of ¢ are expressed in thousandths of a second and in the

1) How a separate calculation of p can be avoided here, will appear in the
following chapter.

( 387 )

TABLE IX. (String 10, Plate A 22).

Al 2 3 y | 5 | 6
t : | : | Dilference
1 Y 'between the
in thou- “ Par in’ milli- in milli- measured
C

sandths of
a second.

0
0
1

1
it

0.
. 0682

0

. 762
.863
O00

O83
238

236)

vd}
1389
AOS
O28
945

926
2856
.798
.770
. 705

D.O8O
O15

Raye
D4
OL

ABA
51
423
.398

368
364

23850)
311

.303

0.279

ASI
135
O945

Seeger

re OL OLolc Oa~1~10
CS STRO EL, DOeweie

ero
mawe

Naar

oo

~1— to

capt | and caleu-
| ‘lated value of
Calculated. | q
in mm.

metres,

Measured.

94 6 | —
94.9 | 478 — 6.4
93.8 ey | — Or
DEY ac} 93.4 0.1
Dd2)a7/ O)3} ge) il a
1.4 20.7 = On7
(20.7) he (E0)57)) (O)
OA | 90.7 0.6
18.9 18.4 — 05
Wis 17.9 0.2
16.6 16.4 = (>
15.5 £505 0
14.5 15.0 O25
i) 13.6 0)
1 (ea 9 13.0 0.3
4.4 = 1.0
seal — 0
10.4 — 1.0
9.7 = 0.6
94 -—— O.5
8.5 | — 0.6
8.0 | — 0.4
Fiat || = 0.6
eA — 0.5
6.5 = 0.6
6.2 — 0.5
5.8 — 0.4
5.4 | —- 0.6
5.4 0.8
4.7 | — O.5
Lil | = 0.7
4.4 _ | 0.6
2.8 = LE eS
2.0 - — 0.3
ale — Ou
10 — O41

( 238 )

second column the values of the tangent v of the angles of incli-
nation existing at these moments.

In the third column the values of the product rcv are given,
calculated in the following manner.

If of the first part of the curve the concave side is directed
upward, the concave part of the second half is directed downward.
At the point of inflection @ = @.

Here by formula (13)

i emer ace Sl)

or

so that from the values here given for g and »v, the value of re
can be calculated. For any other point of the curve the constant
value vc is then multiplied by the value of v for that point.

In the fourth column the values of q, i.e. the distances of the
image of the string from the second position of equilibrium, are
given as the results of direct measurements.

In the fifth column the values of g are given as calculated from
the formula

ty/? — tga
o

gq = cerv + em Mee er nop ore (ei)
while in the sixth column the differences between the measured and
calculated values of q are given.

The above formula (28a) requires some explanation.

3
eh (ae
It is obtained by replacing in formula (11) the value ——
tyB — tga
by - ;
a
3
1 + v2)2 dq.
As was remarked above or a is nothing else but the
oO at

expression for the acceleration. Since we used as the only method
for measuring @ the measurement of two angles @ and 3, see fig. 2,
we can also, these angles bemg known, find an approximate expres-
sion for the acceleration by means of their tangents.

The velocity at the point p,, fig. 2, is given by tya, at the point
p, by tg. The difference in velocity is tgg—tga. Assuming the
acceleration to be constant during the time %, it is expressed by

( 239 )

tg — tga
o

By using formula (28) instead of (11) we considerably simplify
the calculations. In the first place it becomes unnecessary to look up
the sines of @ and 8, while the tangents of these angles are already

known, since they were wanted for the determination of crv. And
3

then it is not necessary to calculate the values of (1 -+ v*)2.
The data, serving for the determination of the acceleration, have
been collected in table. X. One also finds in the sixth column the

TABLE X (string 10, Plate A 22).
- —~-— ——— == -— —
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8
| wise
Se eon |
t Sn} Difference | Algebraic
| between iga—tgx sum of the
: in 5 mee) and) tell (/aueerc neem values
in thou tyz gp YP—tyx measured of the two
sandths iyi : s s values of i ceding:
of a. | milli- values of ¢ in preceding
ean Pain eeciimetres columns in
ae ~ metres. millimetres, | ” millimetres.
Kee
1 1 |(0.696)1)) 0.836 | OMGT | =H il — || aA
| |
eee ele 10762 | 1.000 | 0.238 | — 9.3 U2 3 <5 Sl
3 1 | 0.863 | 4.083 0.220: | — 6.6 6.7 0.1
4 DEOL SES!) AC 2380) uO. Sie eee! Dall 14
| |
64 SS a a (O}eL ee =eOse (0) | = OF
| |
8 a se} Se (0) 0.6 (0) | 0.6
| | | |
10 2, A SORE 1.403 |— 0.066 | 0.2 = (0-4 — 5
} | |
12 | 2 | 4.139 1.028 |— 0.055 0.8 = 06 0.2
14 | 2 | 4.403 | 0.945 |— 0.079 0.6 =T028 052
16 9 1.028 0.926 |\— 0.051 0.5 05 0
| |
| | |
18 2 | 0.94 | 0.856 |-- 0.045 1.0 = ORS 0.5
i}
20 2 | 0.996 0.798 |— 0.064 ONG) 1086 0
22 2 0.856 0.770 |— 0.043 (Oe i ae 0.3
|
| |

1) z has been calculated here by the formula z= 24 — 8, in which y repre-
sents the angle of inclination of the curve at the time ¢=1.

( 240 )

difference between rcv and the measured value of g, in the seventh

‘ ty — tya é P
column the value of ca +5 -— and in the last column the algebraic
sum of the values of the sixth and seventh columns.

If ow measurements had an absolute precision the values of the
last column would all be = 0.

In the calculation of the tables IX and X, the correction has been
neglected which must be applied if a scale division along an absciss
is not equal to a scale division along an ordinate. Thus we assumed
that the net of square millimetres consists of real squares or in other
words that d= 1. This has no influence on the calculated value of
rev, since the correction of 7 compensates that of v; but it has an

. ty — tga ; .
influence on the values of 5 . But the differences are not of

such an order of magnitude that the correction would be necessary ;
the general form of the curve remains unaltered.

The results of the measurements and calenlations which are given
in figures in the preceding table IX, can be illustrated by means of
a diagram. In fig. 8 which corresponds to table IX, the net of square
millimetres is represented at about twice its natural size. A scale
division along an absciss = 26; a scale division along an ordinate

= 1,87 & 10-9 Ampére.

So5Bo
a Beas
REFEREES
N Saeenre
ROCCO
Coon

SEP TY

aries

7

ae
TEE

Vig. 3

String N°. 10, Photogram A 22, Tables IX and X.

Absciss 1 scale division = 27; ordinate 1 scale division = 1,87 >< 10~" Amp.

( 244-)

The regularly bent line of medium thickness represents the recorded
curve. At the moment ¢—O the constant current is made. In the
case of an ideal, absolutely accurate analysis we should find two
straight lines, one of which would rise vertically from A to B, while
the other from £6 to C would be horizontal. The results of the actual
analysis according to column 5 of table IX are represented by the
thick line, while the thin line represents the values of rev according
to column 3 of this table.

For the virtual mass m two different values have been used ; for
the first 4 thousandths of a second m has been put 0.0567 which
is 6 times the amount found in chapter 3. At f=66 and t= 8a,
m has no influence, since at these times @ may be put = o. Begin-
ning with f=—106, m has again been reckoned but this time with
a value 0,0187 which is twice the value of chapter 3.

If a single value is given to m the results become much less
satisfactory and the question arises whether the whole analysis must
not be considered worthless now that it appears to be impossible to
assume a constant value for m.

In reply to this we remark in the first place that, as will pre-
sently appear, the variation in the value of m is only of account
with a great sensitiveness of the galvanometer, i.e. with a feeble
tension of the quartz thread. Besides, even with the most sensitive
position of the galvanometer still an important part of the analysis
can be usefully applied. For practically a curve, such as is obtained
e.g. in many electrophysiological investigations, will consist of parts
of various curvatures and will always show a number of points for
which @ may be put =o or the acceleration may be put = 0.
In all these points m need not be taken into account. Since r ean
be measured with great accuracy the analysis is in all respects
satisfactory here.

Moreover the analysis can be applied wherever the curvature
and at the same time the angle of inclination are not too great —
in our case already in all points that are recorded later than 0.004
second after the starting of the current

as will appear from table
IX and figure 38. For each delinite tension of the quartz thread a
definite value for m be taken.

The reason why in general m is represented by another value
for different tensions of the quartz thread will be discussed in chapter
8. But here we must ask why m can also vary when the sensitiveness
of the galvanometer and together with it the tension of the quartz
thread remain unchanged. For an explanation of this unforeseen
and somewhat disappointing phenomenon we have in the first

ls

Proceedings Royal Acad, Amsterdam. Vol, VIII.

( 242 )

place sought for errors in the measurements, which might be occa-
sioned by the not sharply defined edge of the quartz thread being

TABLE XI (string 14, Plate A 132).

— —- — —-
1 2 3 4 5 6
Difference
J q q between the
in thou- 5 pene in milli- in milli- measured
sandths of metres. metres. and calcu-
lated values
a second. Measured. | Calculated of q
in millim.
0 — — 32.7 — =
1 1.66% 18.8 31.9 27.0 — 49
9) 1.872 ive? 31.0 29.9 —1.4
3 2.069 93.4 30.0 30.8 0.8
4 9 2415 25.0 99.2 29/37 0.5
5 2,.290 25.9 28.0 28 . 4 0.4
6 BBY) 26.4 26.9 26.4 — 0.5
— (2 332) (26.4) (26.4) (26.4) (0)
7 2.332 26 4. 95.9 26.4 0.5
8 2.204 25.0 25.0 94.5 — (0.5
10 2.087 23.6 23.0 23.0 0
42 1.881 PAGS OAR 20.6 — 0.5
14 AIDS 19.8 49°5 49.2 = (0.8)
16 4613 18.3 17.9 17.9 0
18 4.511 Pea 16.5 16.7 0.3
20 1.418 16.0 15.4 d5e5 0.4
22, 1.280 14.5 13.9 414.0 0.4
24 4.179 13.3 42.8 13.0 O22
26 ahaakilah 12.6 Atal 67) 1283 0.6
28 1.032 (Bee 1027 11.4 0.7
30 0.942 10.7 9.9 — 0.8
32 0.848 9.6 9.0 —_ 0.6
34 0.821 9.3 8.2 — ala
36 0.751 8.5 Te) _ 1.0
38 0.676 bad 6.9 —_ 0.8
40 0.635 7.2 6.3 _ 0.9
49, 0.563 6.4 5.8 — 0.6
44 0.504 yer 529) _— 0.5
46 0.456 5g 4.8 — 0.4
48 0.423 4.8 4.4 _— 0.4
50 0.382 4.3 4.0 — 0.3
52 0.350 4.0 Bad) — 0.3
56 0.304 3.4 Bio — 0.3
60 0.254 929 2.6 = 03
70 0.4151 ih AVevi} — 0
| |

(e213))

TABLE XII (string 14, plate A 132.)

if | 2 | 3 | 4 | 5 | 6 7 8
t | 3 | ee tga—tya | Algebraic
: rev osm) | sumlofsthe
inthou-| ™ an | Panto tge@—tgz | and the values of the
sandths || .). 4g ; Ss measured | in two preceding
jg, | ee | values of ¢ | columns in
second. metres. | | in maillime: | millimetres, | Millimetres.
4 4 |(1.488)')| 4.872 0.384 — 13.1 8.2 — 4.9
2 1 1.664 2.069 0.405 — 9.8 8.7 —14
3 1 | 1.872 2.215 0.343 — 6.6 7.4 0.8
4 1 2.069 2.290 0.221 — 4.2 4.7 0.5
3) 4 2.215 2.332 0.417 — 2.1 2.5 0.4
a ees = = (0) == 05 (0) == 085
7 = = = (0) 0.5 (0) 0.5
Samii 2 2.332 2.087 |— 0.122 0 — 0.5 — 0.5
10 | 2 | 2,204 1.881 |— 0.161 0.6 — 0.6 0
12 | 2 | 2.087 | 4.753 |— 0.167 0.2 — 0.7 — 0.5
die 2s N45 884 1.613 |— 0.134 0.3 — 0.6 — 0.3
16 2 | 1.753 1.511 |— 0.121 0.4 — 0.4 0
18 2 | 1.613 1.448 |\— 0.097 0.6 — 0.4 0.2
20 YN ats lal 1.280 |\— 0.115 0.9 — 0.5 0.4
22 2 | 1.448 1.179 |— 0.119 06 — 0.5 O04
24 2 | 4.280 1.411 |\— 0.084 0.5 — 0.3 0.2
26 2 |4.179 | 4.032 |— 0.073 0.9 033 0.6
28 PA la leaulsl 0.942 _— 0.084 1.0 — 0.3 0.7

photographically distorted where the curve bends. But the errors so
caused are far too small to explain the matter; moreover they are

>

1) z has again been calculated here by the formula 2 =2y — B, see the note
at the foot of table X,

aE

( 244)

to a great extent eliminated if the measurements at the lower side
of the string are controlled by measurements at the upper side.

The most likely explanation has to be sought, in my opinion, in
the lack of homogeneity of the magnetic field. The middle part of
the quartz thread is placed between the objectives of the microscopes,
where the magnetic field is only very feeble compared with the
field in which the other parts of the thread are placed. The pondero-
motive force which causes the quartz thread to deflect, when it is
passed by a current, is consequently smaller in the middle of the
string than at the two ends. These latter, so to say, draw aside the
middle part and so it can be understood that with feeble tension of
the quartz thread the displacement of the middle part lags.

The stronger the tension of the quartz thread the more regularly
it will move over its whole length. So we may expect that for a
less sensitive position of the galvanometer the values which we must
take for m will be more equal among each other.

In tables XI and XII and figure 4 belonging to them, we will
first give the analysis of a curve written by string n’. 14. The tables

a er
TT TT

7

+t
El bahe}
Bis

aye aif fiale

SA

A

eoee

scaeiiaeit
al A

FEC

|

[ein] ae

cH
GS
Load

String N°. 14, photogram A 132, Tables XI and XII.

Ahbsciss 1 scale division =2z7, ordinate 1 scale division = 1,72 ><10-9 Amp,

( 245 )

br diagram require no nearer explanation since they are in every

0

respect analogous to those of string n°. 10 that were discussed above.

We have here again J7 = 500 hence 1 mm. absciss = 20. Further
¢= 582, so | mm. ordinate =1.72 X 10° Amp. The value of m

90

has been put during the first 5 thousandths of a second at 0.037
which is rather more than 10 times the value found in chapter 3.
At t= 66 and t=T7., m has no influence. beginning with t= 85,
m has been taken into aecount again, this time with a value of
0.00688 or rather more than 1.9 times the value of chapter 3.

Some other curves, also recorded with a sensitive position of the
galvanometer show after analysis diagrams that agree completely
with the two above deseribed diagrams 3 and 4. So they require
no nearer elucidation here.

We will not omit, however, to give the results of the analysis of
a curve recorded with a less sensitive position of the galvanometer.
The numbers are collected in tables XIII and XIV, which like the
corresponding diagram 5 have been arranged in the same manner
as the preceding tables and diagram. They represent a curve written
by string 14, with 1 megohm in the galvanometer circuit. Here we
have c= 115,2, hence ordinate 1 mm. = 8.67 KX 10—* Amp., while
absciss 1 mm. is again = 2 0.

The value of im can be kept constant here at a value which is
1.45 times greater than the value which would hold for strong
tension of the string.

We see that most of the examined points are calculated with an
error, smaller than 1 mm. and that the correction is already pretty
accurate after lo. After 16 the error amounts to 1.4 on a total
deflection of 30.6 mm., i.e. 4.6°/,. This proves that by means of
analysis of the curve with a sensitiveness of the galvanometer
e= 1152, for which 1 mm. deflection corresponds to a current of
8.67 > 10~° Amp., the real intensities of the current can be known
beginning at Jo after starting the current, and then progressing
0.5.6 each time.

In all probability these times can still be materially shortened
when the speed of the photographic plate is increased. With the
curves mentioned in this paper a speed of the sliding frame of
500 mm. per second has chiefly been used, but it is evident that
with improved mechanical appliances it will be possible to attain
greater speeds. We have lately succeeded in obtaining very regular
speeds of 1 M. per second,

At the end of this chapter we remind the reader that an analysis
of the curves is only necessary when it is desired to measure very

t
in thou-
sandths of a|

second,

isu)
ot

o ow @ oo

10
41

( 246 )

TABLE XIII (string 14 plate A 125).

to

AS

in milli-
metres.
Measured.

30.

28.

6

4

6

0

SS)

5 6
Difference
qd between the
in milli- |measuredand
pss calculated
metres. ralnecton
Calculated.

: ite
in millim.

—
ot
bo

=
=
~1 ow

o wo oo
Sy tls), Sy ee)

0.8
0.6

feeble currents in very short times. As soon as the galvanometer
may be less sensitive, by applying a method of damping which was
formerly described, curves may be obtained, directly recording the
accurate intensity of the current in less than 1o,

( 247 )

TABLE XIV (string 14, plate A 125).

|
1 2 3 re floes 6 7 8
= 7 — Su. =
s | | Difference
t | between gp—tgz | Algebraic
; in rev and the) “’~ 5 ___| sum of the
in thou- | | tg@—tgx | measured two preced-
tg% ty 3 Ca 5
sandths | ini I? 5 values of in ing columns
Sigae | || | q in
second. | atres | in Se millimetres. | millimetres.
res.
= |
0.5 0.5 | (9.36)!)) 12.74 6.70 —— 110); 4.0 — 62
1 0.5 10.78 13.45 4.74 — 42 48 — 1.4
25 | — = = (0) (0) (0) (0)
2 0.5 | 413.45 | 40.89 | — 4.52 2.4 — 2.7 — 0.6
2.5 /0.5 | 412.71 | 10.02 | — 5.38 1.4 — 3.2 — 1.8
3 0.5 10.89 8.71 | — 4.36 Py | — 2.6 (054
Seon |,0.5 10.02 7.50 | — 5.04 2.7 — 3.0 — 0.3
4 1 | 10.02 5.700 | — 4.382 2.5 — 2.6 — 014
5 it 7.50 4.504 | — 3.00 2.3 — 1.8 0.5
6 4 5.700 | 3.078 | — 2.622 2.7 je vdeo Ave
i 1 4.504 | 2.951 | — 2.253 Ns 0.6
8 1 3.078 1.688 | — 1.390 | Ay — 0.8 0.9
9 1 | 2.251 4.163 | — 1.088 1.6 — 0.7 0.9
10 1 4.688 | 0.740 | — 0.948 1.4 — 0.6 0.8
44 1 1.163 |(0.439)?)| — 0.724 1.0 — 04 0.6

1) Calculated, see the note at the foot of table X.
*) B has here been calculated in the same way as the first « of the table,

+4 ae
al
q t+ =
nl +
— |
+ - H
le Ste } 4
all 3 {it 4
| oul!
leh >
Sse ee
imircl Toma il eal ill eile let
Isletas ot ese iE
a Le Sats tM aad
+ t++-+4++ = $—+—+ ie) | + Hie
jemlieuilaal sp . +$—— t a t—+ +—|
EASE BS HE Et Sa ts ate
cleat NESS Lets el eS a - J
es Safa 2 clase sas
JB ic ea en
Al _— = E zi ] fz
EERE AEE eee
(cafeterias
J aaa || am aa tt zl

Fig. 5.

String N°. 14, photogram A 125, Tables XI[L and XIV.
Absciss 1 scale division = 2c, ordinate 1 scale division =8,67 10-9 Amp.

7. Absolute measures of the mass of the string and the
\

resistance to the motion of the string.

As soon as m, the virtual mass of the image of the string is
known in millimetre-micrampere units, it is not difficult to calculate
the real mass of the string in grammes. In order to do this we
must first in formula (7) express the values of 7 and c in the
ordinary units of the (C.G.S.)-system and then the circumstance has
to be taken into account that although the middle of the string and
consequently the image of the string performs rectilinear oscillations,

( 249 )

yet the motion of the quartz thread as a whole has a more com-
plicated character.

Calling m, the real mass of the string in grammes, 7’, ihe period
in seconds and c, the sensitiveness in centimetres deflection of the
middle of the string per dyne, we have

IE c a?
De TO >< |= SK 7 >< Raber ct af (29)
1

2
Benger es ; ;
The factor eo introduced by the circumstance we mentioned

already above. The method by which the amount of this factor is
calculated will be given later, after the tension of the string will
have been dealt with. We now proceed to a nearer discussion of

7

anid
Gf C, :

T represents the time in millimetres while the velocity of motion
of the sliding frame is /’ mm per second. The time in seconds is
consequently

the values of

eee
Sn 0A
or
ES Se Gy
fn V

In order to determine the value of —, we must take into account

>

a3!
the magnification used, 6, the strength of the magnetic field H and
the length of the string /.

_Hf is expressed in (C.G.S.) units and / in centimetres.

c, as was mentioned before, is the sensitiveness, expressed in
millimetres deflection of the image of the string per micrampere,
c, being the sensitiveness, expressed in centimetres deflection of the
middle of the string itself per dyne.

The force deflecting the quartz thread when a current of 1 micram-

|
pere is passed, is —— dynes. Hence

107
fl
c OX For ~ 10h
or
ec Hib er
Rica ee aD

By means of the formulae (29), (80) and (31) we can now express
m, in m and find ;

( 250 )

nm =— m xX —

j 2 mT tty oaanor (24)
The accuracy with which m, can be calculated in grammes depends
of course in the first place upon the accuracy with which m is known
in| [mm—uA) units and further upon the accuracy of the values of
H,/,b and V. The latter quantity occurs in formula (82) squared
and so would have a preponderant importance. But the time-record-
ing arrangement which we used, works, as we saw formerly, with
so great an accuracy that we may neglect errors in the value of V.
Also / and 6 can be measured with sufficient accuracy, while for
HT the value has been taken which we found in chapter IV, namely
17600 [C. G.S.].
The error in the absolute value of m, I estimate ata few percent.
In formula (32) we have.

H = 17600,
PTO
b = 660,
Vi==(500;
from which follows thatm, = 7,28 X10-4m. . .. . . . (83)

In chapter II] we found that:

for string N°. 10 m= 9,4 10-3 [mm — vA]
” »” ? 13 i 6,9 Dx 10-3
” ” ” Ne — 3,6 aX 10-3

By formula (83) we calculate from this the mass of the strings in
absolute measure :
for string N°. 10 m, = 6,85 x 10-® gram.
»» Parnas oer) to Oe UO os
y a ey SS FOS < Me

We passingly remark that for recording sounds we use a very
light short string: a 2,5eM. long, 1 thick quartz thread, of which
the weight may be estimated at about 1,5><10—* grammes.

From the length /, the diameter d of the naked quartz thread and
the specific gravity of quartz s, the weight of the quartz may be
calculated as

? 2?

39 2?

3)

d?
j= ls,
4
This weight is only inaccurately known on account of the uncer
tainty in d. But combined with the value of m, it may serve us
to obtain a rough idea of the relative weights of quartz and silver
in the string. Calculated in this manner we find this ratio

( 251 )

string 10: 1 quartz to 3.5 silver

> 13 » iL o> 9 6.4 2?
ee eee ee es

We now proceed to express the resistance to the motion of the
string in absolute measure. According to the definition formerly given
ry is the virtual resistance to the motion of the string in micram-
peres, when the image of the string moves with a velocity of 1 mm
distance per 1 mm time.

We call 7’ the resistance to the motion of the string in dynes when
the middle of the string moves with a velocity of 1 eM. per second.

The above mentioned unit 7 refers to a strength of field H, a
length of the quartz thread /, a magnification > and a speed of the
writing plane J’.

Hl ;
Since the force of 1A is equal to 107 dynes, we may write:
Hl 106
2o 120 = SS
107 V
or
Hib -
th ==) >< 106V ° e . . . . . ‘ . ( )
Substituting in this the above values for H, /, 6 and V, we get
O20 5 ay ie eae heme (OD)

It is unnecessary to give here the absolute measures of the elec-
tromagnetic damping. These were discussed in chapter IV where
they served us for accurately determining the value of H.

On the other hand the absolute measures of the air-damping 7'¢
may find a place here.

In chapter IV the air-damping was found

for string N°. 10 rg = 0,0193 [mm — uA]

a ye tor iO OMAL ae ,

a Lr OO Sse te ar
By formula (35) we calculate from this

for string N°. 10 7g = 0,00569 dynes.

% eo ona OO0SIS, 5;

” ale yO OLDE ois ae

It would be desirable to compare these values with those that
might be calculated by means of the kinetic theory of gases. But
then we ought to bear in mind that we have combined in 7,’ other
causes Of damping besides the air-damping.

These causes are threefold :

( 252 )

1 If the magnetie field is non-homogeneous, there may arise
vortex currents in the layer of silver during a deflection of the
quartz thread.

2. If the string is para- or diamagnetic, it may by its motion
induce currents in the iron of the pole shoes.

3. Also a non-magnetic string will induce a movement of electricity
in the pole shoes when it is passed by a current and moves.

But all three causes are so small that they may probably be
neglected compared with the air-damping.

8. The tension of the quartz thread.
S.

In order to calculate the tension of the quartz thread under various
circumstances, we begin with assuming a special case, namely that
ihe thread is strongly stretched and is placed over its entire length
in a homogeneous magnetic field. A constant current passed through
the galvanometer causes a permanent deflection of the thread which
assumes the shape of a catenary.

Calling w, the deflection of the middle of the thread and 7, the
ponderomotive force experienced by the thread, the tension is:

i,l 2
rr eco (Gh)

1
Here S and 7, are expressed in dynes, while the deflection w, and
the length 7 are given in centimetres.

Now
SoS Oy eee a ee tate Ste

in which ¢,, denotes the sensitiveness of the galvanometer, as was
already mentioned with formula (29), expressed in centimetres
deflection of the middle of the thread per dyne.

From formulae (386) and (387) follows that

l
Se | olen) ve oe
And from (81) and (88) we derive that the tension is
FIl*b
= —_—. io 1 RE eee)
8<10%

Substituting again for H, 7 and 6 their values, viz. H = 17600,
: 1

/=12.7 and = 660, we find S= 234 x — dynes, which result,
=

‘alculated in grammes, gives for the value of the tension

1
Si= 02389 56—crammes! 8 (40)

C

We see from these formulae that the tension is inversely proport-
onal to the sensitiveness.

For a sensitiveness c= 1 the tension would be 259 milligrammes.
Assuming with Turrirati’) that a thin quartz thread has a tensile
strength of 100 kilogrammes per mM? section, when using a string
of 2.39 uw? section or 1.75 w diameter, the sensitiveness of the galvano-
meter may be diminished to c=1, i.e. to 1 mm. deflection for
1 micrampere without the thread breaking. The strongest tension we
applied with string n°. 14 corresponds to a sensitiveness of 1 mm.
deflection for 3 <x 10-7 Amp., hence ¢ = 3.3, while the diameter of
the string amounts to 1.7 u.

From these data it appears that the maximum tension used by us
is still 3 times smaller than the tensile strength of the string. We
remark here that this limit has only been calculated for the quartz,
while the silver which might also contribute something to the strength
has been left out of account.

String n°’. 10, when uncoated, has a diameter of 2.4. From this
we calculate that it will bear such a tension that the sensitiveness
of the galvanometer is reduced to a minimum of min = 0.529. The
maximum of the practically applicable sensitiveness is Cmax = 10°.

. &nax
The ratio

= 1.89 & 10° indicates the possible variation in sensi-
Cmin

tiveness, which may undoubtedly be called enormous.

The value found for ¢max gives rise to some remarks about the
corresponding tension Spine According to formula (40) we should
find for c= 10° a tension of 2.89 & 10~6 grammes, an absurd value,
since the weight of string n°. 10, here used as an example, amounts
to 6.85 & 10-6 grammes, i.e. nearly three times more, and since of
course the tension in a_ vertically stretched string cannot possibly
be less than its weight. But this absurd result is easily explained by
remembering that formula (40) only holds if the quartz thread is
strongly stretched and so behaves as a string, which was premised
in the calculation of the tension.

From the results obtained we must conclude that with feeble
tension the quartz thread no longer moves like a string. There are
sufficient data to prove that the motion of the quartz thread does
not even completely agree with the vibration of a string when the

1) Philosoph. Magaz. Vol, 30 (5), p. 99. 1890.

( 254 )

sensitiveness has been diminished to ¢ = 100 and hence the tension
is over 300 times the proper weight of the thread.

We must not lose sight of the fact that besides the stretching
arrangement there are various forces which act on the quartz thread.
If the thread is so thin that its elasticity may be neglected while
only gravity has to be reckoned with, it will assume, when entirely
relaxed, the shape of a catenary. If it is paramagnetic it will be
bent towards one of the poles in the strong magnetic field which is
never perfectly homogeneous. And if its elasticity may not be
neglected, it will assume shapes, determined by the position and the
direction of the extremities at the places of attachment, while also
a slight torsion about the longitudinal axis may make itself felt.

When the tension of the thread is gradually increased by screwing
the upper end of the thread upwards, it is easy to observe with the
microscope the moment at which the thread is pulled straight.

Before the thread is straight its middle will be displaced nearly
in a horizontal plane when the stretching arrangement is screwed
up. After the thread has been pulled straight the middle is displaced
upwards on account of the elongation of the thread, this. displacement
being half that of the extremity of the thread. The thread once having
been pulled straight, already a small increase in tension will force
it to move like a stretched string.

The results of the calculation about the tension of the quartz thread
are in agreement with the relation existing between this latter and
the distance of the two extremities of the thread. Of most of the
quartz threads it might be assumed that they were just on the verge
of being stretched when their sensitiveness corresponded to about
1 mm. deflection for 10~§ Amp., or ¢ = 100. In this position very
small changes in the mutual distance of the extremities cause already
ereat differences in the sensitiveness. But when the mutual distance
of the extremities has once been so far shortened or lengthened that
the sensitiveness has thereby become either considerably increased,
e.g. to 1 mm. deflection for 10—!° Amp, or considerably reduced,
e.g. to 1 mm. deflection for 10~7 Amp., conditions are altered. Then
great displacements can be given to the upper end of the quartz
thread by means of the stretching arrangement, causing only relatively
small changes in sensitiveness.

Further the sensitiveness of the thread, once stretched, shows an
increase which is inversely proportional to the increase in length.
So e.g. with string n°. 13 the sensitiveness will be diminished for
an extension of 100 from c=100 to ¢=10, and for another
100 uw from ¢ = 10 toc = 5. The sensitiveness is inversely proportional

( 255 )

to the tension so that we may say that the increments of the tension are
proportional to the increments of the length, which must be expected fora
stretched elastic thread. For increments in length of ratios 0: 1: 2 the

1 1 1 1
incre 3 Sl are as | — ]: —:— or as {| —]:1: 2.
increments in tension are as Ga 10'S s (=)
2

‘ Fu 4
We now proceed to derive the factor 3 in formula (29) and sup-

pose again that the string is strongly stretched and is placed over
its whole length in a homogeneous magnetic field.

According to the laws, obeyed by the vibrations of a string, we
have :

Alm,
>

Tv,

in which +r, denotes the period in seconds if no damping were
present, while, as was mentioned before, S represents the tension
in dynes, / the length in centimetres and m, the real mass of the
string in grammes.

l
Now by formula (88) we also have S= ae that we may write
Cy
Alm, l
Ane arti
or
ad 41
Un Soe 0 . . . . . . . .
: 32¢, SS
— 2
From formula (4) we know that t= 2a Vime or m= Tein from
mC

which follows, having regard to formula (41), that
m, NG ae

= aes
m T Cy 8

. T .
and since — = ai we may also write
T

INES yO IRIs
eX T OSES
1

This formula is identical with formula (29) which proves that the

na?

factor sought by us is indeed

We make a short digression here about the calculation of the
sought factor for the case that the motion of the quartz thread
deviates from the vibration of a string. We shall still assume, howe-
ver, that the thread is over its whole length in a homogeneous
magnetic field.

( 256 )

In the first place it is easy to tell when the sought factor must
be equal to unity. The stretched thread ought then always to move
perpendicularly to its length, and ought to execute in its entirety
exactly the same movements which in reality are only executed by
the middle of the string.

In the second place we shall make the calculation for the case
that the two halves of the thread after deflection, form the two sides
of an isosceles triangle, while we assume that the movements of
the middle of the thread are the same as those of the middle of a
real string. The sought factor then gets the value */, and is found
in the following manner.

The kinetic energy of the thread is caleulated while it is in the
phase of its quickest motion. Let the velocity of the middle of the
thread then be v, and let the mass wm, be distributed evenly over
the whole length of the thread. Under these circumstances and with
the assumption that the two halves of the thread always remain
straight lines, the kinetic energy is

zmv,?

: (42)

Let the first mentioned imaginary thread for which we found the
factor 1 have a mass a, and let it execute the same movements as
the middle of the last mentioned thread. Then its energy in the

same phase of motion will be

E,= en Se, Ses

Call the permanent deflection w, and the total ponderomotive
force f, then the work done by the ponderomotive foree when a
deflection has been made, is

in the first case tara
in the second case Eien
so that [OR ORO Sir ce Se
From formulae (42), (43) and (44) now follows that
amv," mv,"
Berean

3

and consequently that # =>.

9. The practicabihty of the string galvanometer for special purposes.
/ YO. A es f /

For some purposes it may be desirable in order to judge of the

( 257 )

practicability of a galvanometer to know its normal sensitiveness.
This is calculated by the formula *)
A a
Deen ee eee CS)
10¢ low
where #, denotes the normal sensitiveness,

1 the deflection in millimetres,

t the period of a whole oscillation (~) in seconds, cal-
culated for undamped vibrations, *)

T the current in amperes,

v the microscopic magnification, and

w the internal conductive resistance in ohms.

On account of an interesting paper by Warrer P. Waite") we
remark that the quantity normal sensitiveness is sharply defined in
a formula and hence need not give rise to misunderstanding. The
quantity mentioned may be very useful for forming an idea of
changes which may eventually be made in an existing galvanometer
or may advantageously be applied in the construction of a new
instrument. In the normal sensitiveness one has a valuable, important
datum about a galvanometer, but it is obvious that the practicability
of the instrument is still far from being determined by it.

For when judging of the practicability a number of other properties
play an important part, as e.g. the internal resistance that can be
reached in practice, the amount of the damping, the constancy of
the zero point, the proportionality of the deflections to the currents, ete.

The normal sensitiveness is

formuoneaded OP Se oe 8 es ee LOS
5 Wepee operate bs ts ys, Se ne ee ee ecahOe
- ee ule row. ae oneal OSs

a , 20 (passingly mentioned in chapter 7) 2,1 « 10°.

If we could succeed in making an aluminium wire of 1 u diameter
and 12.7 em. length, i.e. of the same length as each of the three
first quartz threads, we should obtain a galvanometer of which the

1) See formula (5) in Ann. d. Physik. 12. p. 1063. 1903.

?) In Ann. d. Phys. I.c. we denoted by ¢ the duration of a complete oscillation.
This qualification implies that our formulae only hold for periodic motion. We
think it desirable to mention expressly here, that the period ¢ must be calculated
for undamped oscillations.

*) Water P. Wurre. Sensitive moving coil galvanometers. The Physical Review
vol. 19, n°. 5, p. 305. 1904.

18

Proceedings Royal Acad. Amsterdam, Vol. VILL,

( 258 )

internal resistance would be 5180 ohms and the normal sensitiveness
35 « 105.

We will now discuss some other conditions by which the practi-
cability of the galvanometer for various purposes is determined.
Galvanometric methods may be divided into:

I. Those in which an oscillating deflection is required.
Il. Those in which the deflection must preferably be aperiodic.

The first methods are subdivided into I A, those with a slow
period as in an ordinary ballistic galvanometer, and I 5, those with
a quick period, as in the optical telephone of Max Wien and the
vibration galvanometer of Rupmns.

The methods mentioned under | A are applied for the measure-
ment of capacities and of small times by PovitLer’s method, in general
always when small quantities of electricity have to be measured.

Now the properties of the string galvanometer enable us to measure
these small quantities of electricity also with an aperiodic deflection.
If the electric current is only of short enough duration the deflection
of the string is im fact proportional to the quantity of electricity
passed.

For the smallest quantity of electricity which can still be demon-
strated we found on a former occasion’) as the result of a rough
‘calculation 5 Xx 1O~! ampere-seconds, corresponding to the charge
of a sphere of 4.5 em. radius at a potential of 1 volt. This calculation
was for a deflection of O.1 mm. of string n°. 10. For string n°. 18
the actual measurement was made. The sensitiveness appeared to
be a little greater still: 1 mim. deflection for 4>< 10—-" coulombs,
so that with this thread a quantity of 4 10—!? coulombs can be
demonstrated.

But the sensitiveness for small quantities of electricity would still
be considerably increased if the damping of the motion of the string
could) be removed or diminished, e.g. by enclosing the string in a
vacuum. We should then obtain a slowly oscillating quartz thread
which would be thousands of times more sensitive than the most
sensitive ballistic galvanometers now existing.

I 5. The string galvanometer can very well serve as an optical
telephone or as a vibration galvanometer and so advantageously
replace the telephone as well for measurements of self-induction as
of electrolytic resistances.

1) See these “Proceedings” 6, p. 707, 1904.

( 259 )

For the first purpose I used it with good results *) when instead
of the silvered quartz thread a thin metal wire was stretched between
the poles of the electromagnet. It appeared to be very easy to make
the period of the vibrations of the string agree with that of the alter-
nating currents used. In a few seconds one has increased or dimin-
ished the tension of the string accurately to the desired amount and
for my purpose neither the sensitiveness nor the certainty of the
reading left anything to be desired.

If it should be necessary to increase the sensitiveness, a vacuum
could be applied, by which one would be enabled to obtain less
damped vibrations even of the lightest quartz thread. It must be
remarked that a vacuum is not always necessary for obtaining little
damped vibrations, especially when alternating currents of very short
period, e. g. of O.00L second and less are used. For the greater the

(2)
tension of the quartz thread, the smaller the damping ratio becomes.

II. The methods in which the deflection of the galvanometer must
preferably be aperiodic are distinguished as II A, those with slow,
and Il 5, those with quick deflection.

Il. A. Of those with slow deflection we choose two examples : the
measurement of currents with great external resistance, such as is
applied for examining insulation resistances and the measurement of
currents with small external resistance such as thermo-currents.

In both these measurements deflections of long duration, e. g. of
10 to 20 seconds can be used with good result. Here the normal
sensitiveness of the quartz thread in the galvanometer, as it is now
mounted with strong air-damping in the Leyden laboratory, no long-
er plays a part. Under these circumstances the mass has only a
small influence on the movement of the thread and the velocity of
the deflection is chiefly determined by the amount of the damping.
This latter only depends on the friction of the air when insulation
resistances are measured.

If by applying a vacuum the movement of the quartz thread could
be brought near the limit of aperiodicity and at the same time the
deflection could be made slow by sufficiently relaxing the tension of
the thread, an instrument would be obtained by which insulation
resistances could be measured, many thousands of times greater than
is now possible with the most sensitive galvanometers.

In the measurement of thermo-currents some of the good points of

) Ueber Nerverreizung durch frequente Wechselstréme. Ppuiiger’s Archiv f. d.
ges. Physiol. 82,$. 101, 1900. See also “Onderzoekingen” Physiol. Laborat. Leyden.
2nd series IV and Y,

18%

( 260 )

the string galvanometer come out least. Besides the difficulty of the
air-damping one meets here that of the electromagnetic damping,
which soon becomes very considerable.

With unchanged strength of the field the electromagnetic damping
is inversely proportional to the ohmic resistance of the circuit. With
thread n° 10 the air-damping and electromagnetic damping are as
1: 0.6, the ohmic resistance in the closed circuit being 10.000 ohms.
When measuring an insulation resistance the electromagnetic damping
vanishes and the thread will want about 15 seconds for a deflection,
when the sensitiveness is regulated at 1 mm for 10—' amp. When
measuring a thermo-current, for which the external resistance in the
circuit may be neglected and only the galvanometric resistance of
10.000 ohms has to be reckoned, for the same sensitiveness the
duration of a deflection will be 1.6 times greater i.e. 24 seconds.
Putting the condition that the duration of a deflection shall not
exceed 15 seconds, one has to be contented with a 1.6 times less
sensitiveness and obtains 1 mm deflection for 1.6 x 10—-' amp. or for
1,6 X 10-7 volt.

Since a deflection of O.1 mm can still be observed in practice, the
now existing galvanometer will be able to show a P. D. of 1.6 & 10-8
volt when thermo-currents are measured.

The application of a vacuum would only little increase the sen-
sitiveness for small potential differences, and would not reduce
the minimum to more than 0.6 x 10-8 volt. Also using a quartz
thread with smaller resistance will only cause little change in
this sensitiveness. If the ohmic resistance becomes 7 times less the

: : 1
smallest observable difference of potential will become (= Se 0.6)
nu

<< 10-8 volt.

But there are two means for increasing the sensitiveness for a
potential difference, which must be mentioned. They consist in making
the strength of the field smaller and in shortening the quartz thread.

We suppose the string to be placed in a vacuum so that the
damping of its motion is only caused by electromagnetic influences.
It is further assumed that the deflections are aperiodic and so slow
that the influence of the mass of the string on its velocity of motion
may be neglected. If under these conditions the strength of the field
is reduced w times, and at the same time the tension of the string
a? times, for an equal duration of a deflection the sensitiveness will
be a times increased.

But it is easy to show that a useful diminution of the intensity
of the field cannot be driven very far. For it must be remembered

( 261 )

that the influence of the mass of the string on its velocity of motion
can only then be neglected, if a strong damping is present.

If by diminishing the intensity of the field one goes on reducing
the damping, and yet wishes to retain the aperiodicity of the deflect-
ion, one will be at last obliged to make the quartz thread, the
ohmic resistance remaining the same, still lighter than it is already.

We can- generalise these considerations, and at the same time
calculate the obtainable maximum of sensitiveness for thermo-currents
in absolute measure, if we proceed as follows.

We put the condition that the deflection of the thread shall be
aperiodic and that the duration of a deflection shall not exceed a
pre-determined amount, e.g. 10 seconds. The most favourable conditions
are then obtained if the movement of the thread is just brought at
the limit of aperiodicity.

We further assume that of damping influences only the electro-
magnetic damping has to be reckoned, either because the thread is
in a vacuum, or because the electromagnetic damping has so increased
that relatively to it the air-damping may be neglected.

At the limit of aperiodicity the formula, mentioned at the close
of chapter IV, holds :

4m an
Gtr Mei aee et aeheg. TeeMay - Fehces (26)
7
and besides
2m
Uf) as
a”

in which T represents the time constant *).
Both formulae refer to the {mm—wA] system. Expressing 7, in
dynes, mm, in grammes and T, in seconds, we get

(PIE oe
ie Gul OR Edynes > see ta aie (46)
w x
mlH 32 : ae .
c=—— X — X 10-6 b millimetres per micrampere. . (47)
" d
and
m 16
T =— — seconds be <p Oo! ere (Hs)
in elas
From formulae (46), (47) and (48) we derive that
xp
C= — X 1l0'x <b Ca re hate Ys 9
c Hl * Tv oN T: (4 )

1) See Fiemine, |. c, p. p. 377 seq.

Calling c, the sensitiveness for a potential difference, expressed in
millimetres deflection per microvolt, we have
C= Cs Ww

from which follows, together with formula (49) that
1 ; x
to = X WO ees eee

We further derive from formulae (46) and (48) that
mw 851102
aE eras

These last two formulae (50) and (51) furnish us with all the
data for easily examining the influence of various changes in the
cealvanometer on ifs sensitiveness for thermo-currents.

In the first place we point out that making a thread thinner or
thicker has no influence on the sensitiveness cs, if only the product
m,w in formula (51) remains unaltered.

Using a metal wire, the value of mm, reinains naturally the same
however the thickness of the wire may vary, if always wires of the
same metal and of the same length are used. It may be advantageous
to use a heavy, thick wire, since then the air-damping may be
neglected, without the wire requiring to be placed in a vacuum.
Also the practical difficulties of applying a very feeble tension may
perhaps in this case be more easily solved by means of an elastic
stretching arrangement than when a thin wire is used.

In the second place we point out that by formula (50) the sensi-
tiveness c, is inversely proportional to the intensity of the field and
to the length of the wire.

We first give our attention to the intensity of the field and
imagine a thread of constant length 7= 12.7 em. The question how
far the intensity of the field may under these conditions be dimin-
ished, can be answered by means of formula (51).

In order to raise the sensitiveness to a maximum, the field strength
must be reduced to a minimum. If T, and / are constant, then
according to (51) myo must be made a minimum. Using a thread
of homogeneous material, m,i is only determined by the nature of
the material, so that the question about the minimum of /7 is reduced
to the question for what material m,z is a minimum. As far as I
can judge this is the case for aluminium, which has for /—= 12.7 em.
a value of m,204;— 1.394 « 10-*.

Assuming for T, the value 2.5 seconds, the deflection has been
nearly completed after 10 seconds. A distance of 1.85 °/, of the total
deflection remains to be travelled through then. After 12.5 sec. this

0)
a0

distance has been reduced to 0.68 °/,, after 15 seconds to 0.25
of the total deflection.

Now putting in formula (51) T, = 2.5,/= 12.7 and mw as mini-
mum = 1.394 >< 10-3, the minimum of // works out at 940 (C.G.S.)
By formula (50) we calculate from this the maximum of sensitiveness
¢; = 484 mm per microvolt.

Let us now consider the shortening of /. If a limit was soon found
where diminution of H/ ceased to be useful, this is not the case
with the shortening of 7, which may be pushed as far as we like
as long as no practical difficulties are met with. By making / shorter
e.g. a times, as well the mass as the ohmic resistance are each
reduced a times. The value of mv thus becomes a’ times less, so
that T, remains unaltered (formula 51) and the sensitiveness c, (for-
mula 50) becomes @ times greater.

A last remark may follow about the two formulae (50) and (51).
We first assume that they are both valid, and that the values of
mw, / and H have been so chosen that T, = 2.5. We next assume
that the mass mm, is changed, while all the rest of the instrument,
including 7, remains constant, and ask how the movement of the
wire is altered by this. When m, is increased, the motion of the
wire becomes oscillatory. When m, is diminished the motion remains
aperiodic but transgresses the limit of aperiodicity. The duration of
the deflection is lengthened while the sensitiveness remains the same.

This latter case agrees with the actual conditions in the string
galvanometer used by myself. The mass of the quartz thread is in
reality very small. If it were = O the duration of the deflection
would be exactly twice as great as when mm, possessed the desired
value. Hence there is under these circumstances an advantage in
increasing the mass of the wire to a certain value. ‘)

String n°. 18 has a mass and an air-damping which were not
accurately measured, but which will not differ much from the cor-
responding values of string n°’. 10. Its ohmie resistance is about
2 times smaller, however, and amounts to 5100 ohms. With a time
of deflection of about ‘'/, minute the sensitiveness is ¢, = 20 mm.
per microvolt. If I could increase the mass of this string in a prac-
ticable manner, I should) with unaltered sensitiveness bring the

1) The time constant is doubled when m= 0. See Firemine |. ec.

It may be superfluous to remark that for the measurement of insulating resis-
tances increase of 1, will offer the same advantages as were mentioned above
for the measurement of thermo-currents,

(264°)

motion at the limit of aperiodicity, and obtain a time of deflection
of about 15 seconds.

We remark here that thread n°. 18 may easily be so feebly
stretched that its time of deflection becomes about one minute, by
which the sensitiveness is increased to c,—=40mm. per nicrovolt.
Since, as is proved by the photograms, 0.1 mm. can still be read
off, with thread 18 a P.D. of 2.5 % 10-®volt can actually be
demonstrated. Also with this feeble tension of the thread the zero
point remains constant, while the image of the quartz thread remains
sharp over a pretty long part of the scale. It may be considered
remarkable that one should be able to displace so slowly with the
greatest regularity a suspended little thread of only a few thousanths
of a milligramme weight.

Il B. We now come to the methods in which the deflection or
the galvanometer must be aperiodic and at the same time quick.

These methods in the first place find an application in electro-
technics, e.g, for investigating the shape of the oscillations of potential
and current obtained by means of dynamos, interrupters, induction
apparatus, ete. For these purposes the oscillograph is already used
with good results, which instrument possesses a considerably smaller
sensitiveness than the string galvanometer, but yet can be of excellent
service in the measurement of stronger currents.

In the second and for our purpose most important place the methods
mentioned under II 6 find their application in electrophysiology.
Here in many cases the string galvanometer cannot be replaced by
any other instrument.

A number of electrophysiological investigations of the most various
kind can be made with the same string. So in the laboratory
the same string n°. 18 is now used for investigating the electro-
cardiogram, cardiac sounds and sounds generally, retinal currents
and nerve currents. Yet we will briefly discuss here the conditions
which must be fulfilled by a string, chosen from a number of available
strings, in order to yield the best results in a certain electrophysio-
logical investigation.

Let us begin with the tracing of the human electrocardiogram,
The current may here be derived from both hands. The hands and
lower arms are immersed in large porous pots, filled with a solution
of NaCl, placed in glass vessels, containing a solution of Zn SO,.
In the zine sulphate solution are amalgamated zine cylinders, connected
by connecting wires with the galvanometer. Under these circumstances
the ohmic resistance of the human body varies with different persons
from 1000 to 2000 ohms, an amount considerably smaller than the

(265 )

resistance of a thin, silvered quartz thread. Of the formerly mentioned
quartz threads 10, 13 and 14, thread 13 will give the best results
in tracing the electrocardiogram, since of this thread the ohmic
resistance is smallest. ‘To be sure, the normal sensitiveness of thread
14 is about 14 times greater, but the currents, received by this
thread from the pulsating heart, will be about twice weaker on
account of the greater resistance.

Besides having a smaller ohmic resistance thread 13 has over
thread 10 the additional advantage of possessing a smaller air-resistance
to its motion. This latter property here plays an important part.
For in order to obtain deflections of practicable magnitude, e.g. of
10 to 15 mm., the sensitiveness of the galvanometer must be so
adjusted that a potential difference of 10-4 volt in the circuit corre-
sponds to 1 mm. ordinate. For obtaining this the quartz thread must
be rather feebly stretched, so that the deflections are aperiodic and
under these circumstances a diminution of the resistance to the motion
of the string will cause a quicker deflection.

Sticking to the condition that a potential difference of 10-4 volt
shall correspond to 1 mm. ordinate, we trace with string 13 a human
electrocardiogram which is almost absolutely accurate.

With string 10 and especially with string 14, however, curves
are then recorded which require corrections. Although the amounts
ot these corrections are small, and do not go beyond a whole milli-
metre, so that in many cases they may be neglected, it is not
superfluous briefly to remember here the cause of these deviations.
It is found in the relation between the velocity of the deflection of
the galvanometer and the velocity of the oscillations in potential
caused by the action of the heart.

The quicker the galvanometer deflects, the more accurate the
photogram of the oscillation of potential will be.

The sensitiveness of string 14 must be so adjusted for tracing the
human electrocardiogram that about 1 mm. ordinate corresponds to
0.5 x 10-8 amp. Now with this quartz thread the limit of aperiodicity
is in a circuit with small external resistance only reached with an
about four times greater tension of the string. If by applying a
vacuum the resistance to the motion of the string could be diminished
so that the limit of aperiodicity were already reached at the first
mentioned sensitiveness, when tracing the electrocardiogram the
velocity of motion of the string would be considerably increased,
so that then also string 14 might reproduce the oscillations of poten-
tial with almost absolute accuracy.

We now pass to the discussion of a second example from electro-

( 266 )

physiology, the investigation of the action-currents of a nerve. Here
the galvanometer has to fulfill conditions which in many respects
differ from those described above. Choosing as our object the nerve
of a frog, from which the current must be led to the galvanometer,
we shall have to count with a great external resistance, e.g. ot
10° ohms.

Compared with this the resistance of the galvanometer, even that
of thread n°. 14 may be ealled small. The potential difference caused
by the action of the nerve, and available for the current to be
measured, is considerably greater than that which is met with in
the investigation of the human electrocardiogram, but the duration
of a nerve action current is shorter and is measured by only a few
thousandths of a second.

These data show us the way in choosing a quartz thread.

In the first place we easily perceive that the differences in the
ohmic resistance of the quartz threads can only have an insignificant
influence on the intensity of the action current, since the resistance
of the nerve itself in the cireuit is preponderant. Further, the deflec-
tion of the quartz thread must be very quick, hence the tension
ereat ; and since an oscillating deflection must be avoided, it will be
desirable to adjust the tension so that the motion of the string is
brought to the limit of aperiodicity. But even under these cireum-
stances the deflection is not quick enough for accurately reproducing
the action current of the nerve. We must therefore apply means
that enable us to increase the velocity of deflection without the
motion becoming oscillatory. We shall have to try to increase the
damping, and can for this purpose apply with good result the
“condenser method” formerly described by us. *)

So we come to requirements here which are opposed to those
which we had repeatedly to put in the above described methods.
Whereas applying a vacuum had then to be considered an important
advantage, now increasing the damping becomes an urgent necessity.

Under these conditions the conception of a normal sensitiveness
comes out to its full advantage and it may be briefly stated that of
a number of threads of equal ohmic resistance that with the greatest
normal sensitiveness is to be preferred. If the external resistance in
the circuit is great compared with the resistance of the galvanometer,
then of a number of threads with equal normal sensitiveness that
with the greatest ohmic resistance will have to be preferred.

For the investigation of the action current of the nerve of a frog,

1) See these “Proceedings” 7, p. 315. 1904.

( 267)

0

among the three threads mentioned, n'. 14 will have to be preferred,
since the amount of the normal sensitiveness as well as the resistance
of this thread exceed those of the two other threads.

Finally we make some remarks as to the manner in which the
velocity of deflection may be raised to a maximum. A great velocity
is in general obtained at the expense of the sensitiveness. But there
are a number of investigations, notably the recording of sounds, *)
in which the sensitiveness of the string galvanometer may be very
considerably diminished. Even when the string, at the risk of break-
ing, is stretched to its maximum and hence its sensitiveness reduced
to a minimum, relatively feeble sounds can still drive the image of
the string out of the field of vision.

By strongly stretching string 14 we could impart to it an oscil-
latory motion of which the period was 7’ = 1,41 6. If the oscillations
were damped by means of the condenser method, a deflection could
be obtained, requiring a time of 0,86 and proportional to the
current to be measured with an error of 3°/, 7). If an accuracy of
O37,
tion of 2,26. The sensitiveness was here 1 mm deflection for
3 < 10-7 amp.

From the data of the preceding chapter follows that under these
conditions the tension of string 14 can be still 3 times inereased
before its breaking point is reached. Hence if the string is so strongly

was desired, one had to be contented with a time of deflee-

stretched that it is at the point of breaking, its deflections will become
V3 times quicker, so that its oscillations will show a period 7’—=
0,815 o. In practice we have not raised the tension of string 14 so
high, however.

The question how to obtain quicker oscillations was simply solved
by using a shorter wire. String 20, which was already discussed
above, has a diameter of 1 and is 25 mm long. With a practi-
cable tension that could be applied without risk of breaking, it
performed with a sensitiveness of 1 mm deflection for 10—° amp.
oscillations of a period of 0,31 0.

This period corresponds to a tone of 3280 vibrations per second,
about g* sharp or almost the highest tone of an ordinary piano.
We remark that the string can still be shortened and be more
strongly stretched, so that a much higher number of vibrations can
easily be reached, while it must also be borne in mind that a string
with slow deflection can yet very accurately record sound vibrations

1) On the method of recording sounds see these “Proceedings” 6 p. 707, 1904,

2) See these” “Proceedings” 7, p. 315, 1904,

( 268 )

of high frequency. So strings 10, 18 and 14 reproduced with feeble
tension and slow deflection the sound waves of a tuning fork of
2380 whole vibrations per second. The recorded period was about
24 times shorter than the proper period of the quartz thread. If the
same ratio of periods holds for string 20, this latter must be able
to reproduce with ease tones of 77000 whole vibrations per second.
On a following occasion I hope to return to the recording of sounds.
Also a discussion of the practical execution of some of the experi-
ments deseribed above and a description of different designs of the
string galvanometer will have to be postponed to a following paper.

Zoology. — “On a new species of Corallium from Timor.” By
Sypvry J. Hickson, Professor of Zoology in the Victoria
University of Manchester. (Communicated by Prof. Max Weer).

The species of corals included in the family Coralliidae have been
arranged by systematists in the four genera, Corallium, Pleurocoral-
lium, Hemicorallium and Pleurocoralloides.

The genus Hemicorallium of Gray was merged with Plewrocorallium
by Ridley in 1882, and quite recently Kisninouye has called attention
to the difficulty there is in maintaining the distinction between Pleu-
rocorallium and Corallium.

One of the principal characters of Pleurocorallium is the presence
in the coenenchym of peculiar twinned spicules which Ridley calls
“opera-glass” shaped spicules. These ‘“opera-glass’ shaped spicules
are not supposed to oceur in the genus Corallium. Whether future
investigations will support the view of KisHinouyr or not is a
question which need not be considered here, but the absence of
“opera-glass”” shaped spicules in the specimen about to be deseribed
justifies its position in the genus Corallium, that is, to the genus that
includes Corallium nobile the precious coral of the Mediterranean
sea and the seas of the Cape Verde islands and Corallium japonicum
one of the precious corals of the Japanese seas.

3efore proceeding to a description of the new species a few words
may be written concerning the geographical distribution of the
family. Corallium nobile occurs in the Mediterranean sea and off the
Cape Verde islands. Some species attributed to the genus Pleuroco-
rallium occur off the island Madeira, and quite recently a specimen
of Pseudocorallium johnsoni has been dredged off the coast of Ireland.

( 269 )

Off the coast of Japan occurs Corallium japonicum and several
species which would be included on the old system in the genus
Pleurocorallium but ave referred to the genus Coralliium by Kisnrnovye.

Isolated specimens of Coralliidae were also obtained off Banda in
200 fathoms, the Ki islands 140 fathoms and Prince Edward Island
310 fathoms by the Challenger and there is a doubtful record of
a specimen of Plewrocorallium secundum trom the Sandwich islands.
Fisheries of more or less importance have been carried on in the
Mediterranean Sea, off the Cape Verde Islands and off the coast of
Japan, but there is not, I believe, any historical record of a syste-
matic fishery for precious coral in any other part of the world.

In 1901 the value of the coral obtained off the coast of Japan
was over £ 50.000 and it is a fact of considerable interest that
a large part of this was exported by the Japanese to Italy.

The coral Fishery of Japan is of very recent growth for in the
time of the Daimyos the collection and sale of coral was prohibited,
and it was not until the time of the Meji reform 1868 that it
began to assume important dimensions.

That the Japanese of old times valued the precious coral is shown
in the numerous ‘“Netsukes” and other ornaments which are decorated
with it; but the origin of this coral is not definitely known.

On many of the Netsukes the coral is represented in the hands of
darkskinned fishermen, “Kurombo”’; never in the hands or nets of
the Japanese.

Now the art of Japan is quite sufficiently accurate to prove that
the Kurombo were not Ainos nor Japanese, nor Malays vor Euro-
peans; but the curly-hair, the broad noses and other features that
are consistently shown render it almost certain that the Kurombo
were Melanesians or Papuans.

The only regions where such folk live that have hitherto yielded
specimens of precious coral are the Banda seas. As already mentioned
the Challenger discovered precious coral in deep water off the Banda
and Ki islands, but the specimens were “dead” and it was consequently
impossible to determine definitely to what species they belong, but
they were referred by Ridley to the species Plewrocorallium secundum.

In the material that was kindly sent to me by Prof. Max Wrper
from the rich collections of H. M. Stpoca there were a few small
pieces of a beautiful coral which I recognised at once to be a Coral-
liid. There can be no doubt that it was alive when captured by the
dredge and it reached me, not fully expanded, but in a.good state
of preservation.

The locality of this find was station 280 i. e. at a depth of 1224

(270. )

metres in the middle of the strait that separates the E. end of the
island of Timor from the small island Lette or in other words on
the Southern boundary of the Banda Sea.

The axis of this coral is covered with very little or hardly any
crust, is apparently as hard as the best Italian coral and is of a good
colour, although a litthe darker than that which is regarded by the
jewellers as the best quality.

The discovery of this specimen suggests that the dark skinned
“Kurombo” fisherman that supplied the ancient Japanese jewellers
with thei precious coral, lived some where within the region of
Timor. It is of course improbable that they were able to fish in
such a great depth as 1224 metres, but as the species of Corallium
range in depth from 10 fathoms to several hundred fathoms, it is
quite possible that they had knowledge of shallow waters off their
coast where the coral grew abundantly.

It is not for me to suggest that there is a prospect of a valuable
coral fishery in the Banda seas; but now that it is known that
living precious coral does occur in deep water in this region of
the world it would not be a matter of surprise to scientific. men if
it were subsequently found at depths sufficiently shallow to be
obtained by ordinary fishing boats.

The specimen obtained by the Siboga does not agree exactly with
any known Coralliidae in those characters which are used by sys-
tematists for the separation of species and it is necessary to find
a new name for it, and I should like with Her royal permission to
name it Coradlium reginae in honour of Her Majesty the Queen of
Holland whose interest in Zoological Science in general and in the
researches of H. M. Siboga in particular has been manifested on
more than one occasion.

The specimen agrees with other species of the genus Coralliaum
in the absence of the curious “opera glass” shaped spicules and the
presence of spicules of the octoradiate type only in the general
coenenchym.

It differs from Corallium and agrees with many species referred
to the genus Pleurocorallium in having the branches arranged
principally in one plane and the zooids scattered irregularly on
one face or surface of this plane.

The autozooids are indicated by well-defined verrucae projecting
about 1—5 m.m. from the general surface of the coenenchym. ‘bese
verrucae ave large as compared with other species being about 1—4
mm. in diameter. The coenenchym is thin, and the axis hard and

(eid)

either not marked or very faintly marked in some places by longi-
tudinal striations.

The base of the main stem of the specimen is 6 m.m. in diameter
and the primary branches are 4—5 m.m, in diameter.

Some further particulars concerning the anatomy of the species
will be described with illustrations in a future publication. For the
present the diagnosis of the species given above is sufficient. Before
concluding this preliminary note I have, with very great regret, to
record that on Sept. 22°¢ a fire broke out in my laboratory and
some portions of the specimen were seriously burned and _ scor-
ched. Fortunately there is still a considerable fragment that appears
to be uninjured.

Physics. — “Properties of the critical line (plaitpoint line) on the side
of the components.” By Prof. vax pur WaAats.

By Crntnerszwer and Smits’ observations, by a remark of VAN
‘tr Horr and by van Laar’s calculations *) a discussion has been car-
ried on on the rise of the critical temperature of a substance in
consequence of an admixture. In this it has been perfectly over-
looked that already more than ten years ago the principal properties
of the critical line, and also the properties at the beginning and at
the end of this line were discussed and determined by me *).

For normal substances, I found by a thermodynamic method, which
is a perfectly sure way, for the quantity mentioned the formula (9)
(ie p...69)

Oe

0x Ov? = MRT \ da dv
I shall explain further on why I make some reservation for abnormal
substances.

r dx oC Ov?
Ca ab ae a (ee i

And with the aid of the equation of state I derived from (9) for-
mula (11)

“ip ©

a
Lilog— hh
d log i ks nee b i +] ( ° H,)

du, di, 16 dx,

1) These Proc. p. 144.
*) Verslag Kon. Akad. vy. Wet. 25 Mei 1895, p. 20 and 29 Juni 1895, p. 82.

( 272 )

As in this derivation of (11) from (9) the quantity 6 of the equa-
tion of state was supposed constant, (11) must only be considered as
an approximation.

If for the present we keep to this form, (11) may also be written :

Gkit! Celie 9 hile We Giey NE
Tdx, oie du, a all> dx, a 3b a) sth “apa
Tr

And taking into consideration that 6 = ———*__—_ we_ find finally :
8 X 273 p,

dT ai 4 Que Ik Meher, = 9

Tae, Dds \rde 2 pas)

The quantity 7 occurring in this equation, represents the critical
temperature of the unsplit mixture. For this quantity I have already
demonstrated in my Théorie Moléculaire that it may get a minimum
value for some sorts of mixtures — and the observations of KuENEN,
Quit and others have furnished instances of the existence of such
a minimum value. If the admixture should be of such a nature that
such a minimum value existed, it would, of course, be perfectly

yy

5 yy lq . ¢ 7 . .

absurd to substitute 7”, — 2 for an But the existence of such a
= Av

minimum critical temperature is only to be expected, at any rate

only observed, when 7), and 7, differ little. When they differ much,

dT c j ; . é
“ean be represented by 7, — 7), at least with approximation. As

(é vo

6 depends on w linearly at least with some approximation, we may write

Let us with these approximate values compare equation (1) with
Knrsom’s observations on the mixtures of carbonic acid and oxygen ‘).
The critical temperatures of these substances differ sufficiently
to enable us to use the approximate values. 7), (for oxygen)

is namely about half of 7), (that of carbonic acid) — and
so we put for eos the value pr aewcueee 0,493, and for
~ “Rides 304,02 a
154.2 304,02
eee the value zu —-- vee or 0,271. With these values
b da, 304,02
72,98

1) These Proc. VI, p. 616.

we find:
dT 9

=— 0,493 + — (—0,493— 0,0903)? = — 0,493 40,1914 —— 0,302.
Tdx, 16
The value found by Kexrsom for « = 0,1047 is AT = — 8,99.
Supposing this value of 2 small enough to be substituted for dz,, we
dT
find ——— — — 0,284.
Tdx,
For «=0,1994 this value of AZ’ found by Krrsom is equal
to — 18,47; with these data we should tind ——- = — 0,304, so
av

0
perfectly equal to the value calculated by means of (1). We have here
not a molecular increase of the critical temperature, but a decrease,
as indeed, was to be expected, because we had to do with the
addition of a more volatile component.

Though I derived formula (9), on which formula (11) of 1895
and formula (1) of this communication are founded, in more than
one way in my two communications of 1895, I will derive them
once more here in order to have an opportunity to discuss somewhat
more fully some questions which present themselves in the derivation.

For the plaitpoint line the simple relation :

077
dp ts Ow? pr

dT (0*v ,
027? ) oT

074
holds, which, ead not being directly known, may be brought
J
Ov? . do dv o

Pp

under the following form:

028 dv\? & 0" dv | 0’
pe. 7 (%) AG oT da Dee “\ dade T ( pt Oe? oT

ar aR

dun. dv 0°¢ ; =e
The factors of { — and ( — and also { — being finite
da pT da pr da? ) yt

dv
quantities, and on the other hand (Z) being infinitely great, when
a pl

the plaitpoint lies at c—0, we may write for this case:

dv\?
dp Op 078 dur pT
Ah we — SS A Oey Ue aye
dT @ *Ge)a(ae (3)
pr

da?
19

Proceedings Royal Acad. Amsterdam. Vol. VIL.

(-274.)

If we put

07x) 07 yp Oe ;
— —- if — O ra 6 i

Foe i) J, then , because the plaitpoint
is a point of the spinodal line. In the same way:

of dv of

dv cara

v\daJy7r da
because it concerns a ae

If we multiply the numerator and the denominator of the fraction
0? 2
occurring in (38) by e 5 we get:
v

O?w

*(3)+ avr
r= dv) er sD) (3
(= dx? )yT
The value of é =) G
dx?

vy we derive from:
07
(2 = alae Ov

0°
Ov?
and find then:
0*w
zt ai (=) Op) OO oP (= y
Ov? dx? Are Ov? 0v70v Ox Ov 0x Ov? ~—- 0? p (Ow Ov
Ov?

07y 0°
As for the critical point of a component both aaa = ; and — as equal

to 0, the last equation becomes:
Oy
07y\? a) = 07) ae oe os Ow dy
du) \da? dvdv) Op Oxdv Oa Ov?

Ov?

Ow)

The limiting value OST can be found from the equation which

dv?
expresses that the critical point of the component is a plaitpoint, viz. :
Of op Of O*p %).
dv dxdv Ow Ov?

1) In a derivation of the discussed formula in my communications of 1895 I put
=—1())5

of
dv

It would have been more accurate, if I had put this quantity infinitely

of
small compared to .
Ow

( 275 )

Now
OF LOE Ope oer Obs Ow dp
Ov 0a? 0v Ov? 0a? Ov Es Ox Ov Oxdv?

and
Of O*rpd%p d*p d*p > OW dp
Ox dw? Ov? Ow? Ovdv? - Owdv Ou2dv

ety ' . 074p 0? yp 0?y.\? Oty
or taking into consideration = and =)

Ow? Ov? Oxdv Ov?
Oy
of dw? dv® > Ow Op
= lim
dv = (555) 5 0? SES Oadv?
dv?
and
Ow
op af =(2 Vin On? +(e i Op
Ov? Ox du dv 7p? Oxdv ) Oa dv?
(a)
* fo OED OF OD Core:
y equating a ands and ae gat find :
ow d*w
ae lim. ua — Se = x) lim. Fae -f- oan
dxdv 07 dxdv? — \ Ov Ow? — Oxdv?”
Ou? (5)
Ow
For normal substances the limiting value of op? is known.
fe
From p= MRT ((1—a) 1 (1—a) + ale} — il pdv follows:
ow ee Op
te) = MRTI ees ee dv
Oy ao Pl 0?p
= — MRI eae | eee
ep\ MRIT1—22) op |
| aie wai(l=2)te ) J, On? 3
o*w
0u* 1

for «=O we get ——-— =

aes ~ MRT
Ox?

19*

For abnormal substances this quantity would probably be found
of the same value, but this would require a closer investigation,
into which I shall not enter here. For this reason I have made a
reservation for abnormal substances above.

dp
For the value of ($5) of the plaitpoint curve we get now the
us

equation :

& )
, (4) aa (58) 07 Oxdv

aT) ,, OT), | Ov? (Op 1 O°
dxdv) MRT Oadv?

The critical point of the component is an homogeneous phase, in
the same way the plaitpoint is a new homogeneous phase. But the
quantities 7’, v, and «=O increase by dT;,, dv, and dz,. Hence:

pe ae I (st) d
p= (52) +(¥ By de, + ar bs lv,

and & 2) being O, also

. ae Ears Op ic Op Eat.
OM Oey een Ov) 7 dT

comparing with (4) we find see ee sought of

u
dT ia aD ae tL \Oa Jor

(4)

is -
According to the equation of state, supposing 4 constant, we get:
a (08 a 07s 2a . (Op ;
ed () = and Wee oe The value of ()., is equal
ig eae = = za and of (5) equal to
GE bide, ida? Owdv

MRT db anal)
NG TEE. en ato’ 9 o so
ai i(v—b)? da daw\?

da 1 MRT =| 1 \ 1 MRT db

dT "| dwv® (v—b)'da\ MRT (dev? (v—b)* de
Tdz, 2a

Th

dT S MRT v» Z| a lee MRT ~v* “db

ade a (v—b) de QMRT lade a (v—b)*da

If we put, as is found with constant value of 6, v= 3b and

3 8a , :

MRI =o7,° We find the value given above :
alo

a a

_ dlog— d log —

d log I b 9 b?/,

ane daz 16 dx, :
In what precedes the relation between the variation of 7’, and that
of « has been discussed for the beginning of the plaitpoint line. Let
us now proceed to the discussion of the relation between the variation

of p and that of 7.

2

. d : :
From the equation of a given above, we derive:

a
Tdp\. Top 1 fop\ Tde,
p dT aap oT bas Pp Ow peal
: one : : Op : dp
Now in the critical point of a component a7) 38 equal to aT
v a
: : T dp
for the saturated vapour tension. And for numerous substances — —
P a

r

for the saturated vapour tension is about 7 in the critical point.

Op
oe

Tde 1
0 . . *
now being known, we want still the knowledge of —

dT P
: ; T' dp SPF a
for the calculation of a7 for the plaitpoint line.
p aT’),
1 (0p
We can calculate — ae by means of the equation of state.
Pp C)/oT

If we put again 4 constant, we find the value indicated above :
Op Ne eee
P 0x Tue P
a
1 (=) sae
p Ow oT Pp

1
With »= 3b and p= ai we should find for carbonic acid and

MRT db dal
(v—b)? dx Wids yy"
or

Sled: a2 1 da
27 b dxe(v—by? Pertaran Vs

oxygen
1 =) = 8 >< (0,493 + 0,0908) )
p \Oe)or

or

1) See page 273.

(278 )

1 a) as
-- =: = 1/2
P Ow cr

: / ; Tdp ,

According to Kursom’s observations the value of (=) for the
beginning of the plaitpoint line is equal to — 6,3 for «= 0,1047,
and equal to -— 6,08 for «= 0,1995. From this we calculate, with
T ( 07 f , ; 5 els Op
—{— | =6,7 (the value found for carbonic acid) —| — ] = 3,921
P le I P @
and 3,824 — so more than double the value which follows from
the equation of state, when we put there 4 independent of the volume.

The values given by Krxsom for pressure and temperature of the
critical tangent point, and of the critical point of the unsplit mixture,
furnish a means to test the reliability of the value of ~(2) , as

Pp 0x) 7

it has been calculated from his observations.

For the mixture 20,1047, Ap amounted to 9,9 for the critical
tangent pressure and 47’ to — 7,69 for the critical tangent temperature.

If we again write for this homogeneous phase:

ce Vere ees
i — (se), dT + (3),

or
1dp _ T Op ote Op
p da iO day ct Last asap dx oT
we find:
9,9 oa —— 69 1 (Op
72,93 X 0,1047 ° 304,02 x 0,1047 Pp Ow oT
or

1/0
1,297 + 1,62 = ale) = 2.917,
Pp \Ox) or

And from the observations for #« = 0,1994

16,72 = eT 1 (dp
72,93 X 0,1944 ' 304,02 x 0,1994 | p (Z),,
or
; 1 (dp :
1,15 + 1,635 = a(x) = 2,180.
iv oT
For the homogeneous phase of the critical circumstances of the
unsplit mixture which cannot be realized, Kersom found Ap =—5,23
and A7’=— 18,34 by the application of the law of the corresponding
states. From these data we find:

OE — 18,34 1 (dp
72,93 X 0,1047 —’- 304,02 x 0,1047 | p \de)o7r
or

. re 5/707
— 0,685 + 3,86 = — Ge eos
oT

The fact that Ax, AT and Ap cannot be considered as differentials
will undoubtedly contribute to the circumstance that this quantity
shows such different values if calculated from Kursom’s observations.

1 (Op
But though the calculated values for — a) are not the same,
p \Oe).r
it appears sufficiently that the value of this quantity lies in the
neighbourhood of 3, and probably above it. That the equation of
state gives a so much lower value if we put 4 constant, must be
attributed to the fact that the influence of this erroneously introduced
simplification is great here, whereas this simplification caused hardly

gal

c : elndare ‘ 1 /0p
any error in the calculation of ———. The value of —{—] we
dl p \de) 7

found equal to:

a 1 da 82 We? 1 db
vp, a dx 27 (v—b)*? b dz :

With v,=306 we find the value 3 for = , while the second

U' Px
a
al
b 1 dlb ae 2 ee
factor becomes equal to ag -|- aaree But it is sufficiently known
av av

that the critical volume is much smaller than 34, and that the
variability of 6 accounts for it. The same cause to which it is due

TE ODN aay :
that at the critical volume Ca) is found equal to 1-6, instead
Pp

equal to 6 instead of to 3. Let us

of 1+, causes us to find

3
2

vp,

briefly prove this.

ie ah) me ve

In the critical circumstances the value of the first member ig

1 so
about 7 or ait —6. If we use this value, we find for — ap
Dx Un? p \0e Jor

1 /0
double the previous value, i.e. 3,5, The second factor of ale!
p \Oe Jor

( 280 )

will now have to suffer some modification too, and as I shall show in

a

dl —
da by Ela) b Lf ab
ae

1
another communication, be equal to — - : —
7 | a da 6 b dx dx ~ 6 b dx

but the difference is slight — and this factor and others of a similar form

lanl yy

. . . id
oceurring in the value of Fae the value of ie caleulated on the
av ae

supposition of & variable, may be considered as sufficiently accurate.
We must therefore not expect to find a perfectly complete dis-
cussion of the problem in what precedes. If this was wanted, a closer

Op
investigation would be required for the determination of (2 and
z av

Ohi g
——, if we take 4 dependent, not only on «, but also on v—, and

Ow Ov’
b, ON
Ob ==i0u pp a(= )+e(= Yt i,
v

henee put :

while 6, = (6,), (1 — x) + (4,), # is put. But in the following com-
munication I shall show that in this particular case, the component
being in critical circumstances, we can determine the value of these
quantities without entering into a closer investigation.

Physics. “The properties of the sections of the surface of saturation
of a binary mexture on the side of the components.” By
Prof. vAN DER WaAAIS.

I have brought the differential equation of the p,v,7-surface of a
binary mixture into the following form :
07s w
21
0, dp = (¢,—2,) & dz, + —
ai : On? pT Al

& y

0? 0? ; dz.0v

In this equation (5-5) is equal to ( =) 7,00)
pr vT

da, 0a.,7 Ow
“Out
en 07s 0? yp MRT
For, 2, infinitely small v,,=v,—2,, Gals (= ==
and. for w,, we may substitute the molecular heat of evaporation
of the component, which we shall denote by Mr. The above equation
is then simplified to

(as)

MRI Mr —_
(v,—v,) dp= (e,—wx,)da, + — dT.
Eis 1

The properties of the initial direction of the sections normal to the
T-axis, normal to the p-axis and normal to the w-axis are given by this
—f£, ,

— is known.

vy

If x, and v, represent the value of w and of the molecular volume
of the liquid phase, and in the same way 2, and v, these quantities
for the vapour phase, then the equation :

é a
equation and they are known when the value of —

——
ay dx, +

& 2

(v,—v,) dp = MRT a

holds for the vapour phase.

As the difference of the specific volumes of liquid and vapour of
a component is generally represented by wu, v,—v, = Mu, and this
equation might also be written
4 7.

da, + T Qe.

cal

For the section normal to the z-axis, so for the component itself,
we find the well-known equation of CLApEyRon:
. dp

jas

u dT

@,—a
ALCP =— Ll 2

; : ae : @ 4
For this section it is not required to know —, but for the other
| me
z,

sections it is indispensable.
This relation is found by means of the property which says, that

Ow Fee eae
ae: must have the same value for liquid and vapour phase.
x

From :
w — MRT \(1—«)1(1—2) 4 ale} — [ow + F(T)
we find: ;
ow Ki “Op
= MRT 1 — — | — dv,
a T ed 1—wz a

and equating this value for the two phases we get:

@ oe dy = MRT 1+ — sf @ dv
l—z, Ow) yT
nou (®) dv
vT

or

uRT

( 282 )

and so for v, and a, infinitely small :

ap
MRT 1. af

Op
If we represent the mean value of ) between the values
CRY o

Op
v, and v, by (= , we may also write :
vT

Ow
: d
MRT 1 =(v,—»,) ( 2)
fie On) 7

This mean value can also be represented under another form
by the following consideration. According to Maxwell’s rule

Vg
Pe (vs — 2) = { rae,
VY
when p- denotes the tension of the saturated vapour of the component,
from which follows:

Ope d (v, —v ) y Op d (v,—2,)
An (v, —})) =F Pe ‘les =((2 eee + Pe Po ae

or
Ope (””) Op
— (v,—v,) = — dy = (v,—v.) | — ¢
Ow a) =) a a Cae oe

Ope
The quantity — represents the molecular increase of the tension
Oke

of the saturated vapour for the unsplit mixture, assuming for p, the
approximate value
T,—T
Ine = lp, =f

Op
za is found from:
Ow

pe Ow mae a T Ox
For the present let us continue to write:

a One
MRT 1 =v, =v.) =
ee Ou
or
Ope
(vg—v1) =
3, MRT

( 288 )

Let us now first consider the initial direction of the sections
normal to the Z-axis. It can now be found from:

v,) dp 9: One an
log }1 + at | = — + for the liquid branch and
v,—?, dp v,—v 1 Oe
l — for the vapour branch.
of) — aERT da. = — URT ag fr the vapour branch
For very low temperatures we may put PO 1) about equal to
. MRT
unity, and so:
tae =o 8 Olp, FOL,
log = a -
2 Ow T 0x
and
1 dlp __ Ope Olp, Hf OIE
O( —_ — = = — — —
a dx, Ow Ow T Ox
Ope dp dp. :
For the case that ae ae and aa is also equal to O, and it
av av }

1 Qu,
may therefore happen, that the two branches of the p,zr-line touch
in the beginning, and that both have an horizontal tangent.
As condition for this circumstance we have :
Uf Giles dlp,
T de da
which may also be written:

alice /alonda 1 db = 1 da 2 db
J TVs. b drjo> \ a de b dx

or
etal = alt; Alb
hea de —s dw ae dx
or
Re a dlb
fS— | |S
peels dx Tai
Hes tor 7 =} 7,

ay, 1 dlb

da 1 dx =
Pls aie is smaller than unity and for the
MRT G

critical temperature of the component this quantity is even equal to

For higher temperatures

a= i" i v,—?, 2 a ie v,—v, dp
; this case w ay write a oe
n this case we may write 775 ie MRT de,

and we find
a __ Pe
~ On’

and in the same way

The first conelusion we draw from this is, that at the critical
temperature the liquid branch and the vapour branch have always
the same tangent, and therefore touch. The initial direction is given

0; yp Op :
by the quantity = or by (2) . But as at the critical temperature
2 Le) yt

; Op : ; Op
Vv, —=1,, the mean value of | —] is equal to the value which { —
Ow) op Ox )yT

has at that volume equal for vapour and liquid. We have therefore
at the critical temperature :

dp dp aes Op
iz Toe dex, rt \0dw)or

or

1 (dp 1 (dp 1 (Op Olpe : ala. dlb
a ee (
p \d«,)T . p \de)/7 |p \0a st: Ou dx

1

The second conclusion we draw is that at the critical temperature

i ef) ie 6 jal, 1 dlb
p \0u, Spibee | da a 6 ial

which has been put in the preceding communication, but has not
been proved there.

ot . dp dp ; Op
That at the critical point |— ] and {| —)]} is equal to | —
de, ) p dx, ) 7p Ov) 7

we might have immediately concluded, without following the elaborate
way by which we have now arrived at this conclusion. In the same

Ihe : dp se
way that at the critical point aT

Let us first consider a simple suibstanee: - we a pnes from one homo-

geneous phase to another, at which v is increased by dv, and 7’
by dT, then

d Op ' :
and so every —_ ee , also with such a change at which the
fal One;

( 285 )
volume changes, as is the case with saturated vapour. From this
ee lad
follows the well known ae that at the eritical point -($)
P y
Op
or),
If with a binary mixture we oe on one homogeneous phase
to another, at which v is increased by dv, 7’ by dT and x by dz,

then :
dp = (f)., du} @ ipa (5) ma

f)
If (3) = 0, as is the case at the critical point of the compo-

dv aM
0 0
dp = si) dT + (32) oe

nent, then :
also for such variations in which the volume changes.

for saturated vapour is equal to =

The differential equation of the surface of saturation :

w,, aT 0
Uy Spt

veda

holds for the transition of an homogeneous liquid phase to a subse-
quent one and in the same way :

mal:
+ (a, (3 *) ae,
pr

for the transition of an ie [ vapour phase to a subsequent one.
If the first liquid phase and the first vapour phase is the critical
phase of the component, the three last equations must be identical,

w w a5 ( GP Ue == IP Op
and so ~~ = |) or — = :
fe Te, Ol js, Tu a SRN

a—a, (0S v,—a, (0S Op
In the same way ar \aae je aria agen 2 has
Voy Roe © Dae. Be) pT Oe), 7

been proved above as holding for the critical point of the component.
From the general equation :

a —

follows, when v,—v, is infinitely small,

C,— 2, ss Ve— Op
a, MRT \0«)or’

and

( 286 )

w,—v, MRT) _ (dp ‘
vy V,— 7 x a Ox oT

If at lower temperatures the initial direction of the p,z-line is
traced for the liquid phase, and also that for the vapour phase, then
these directions are usually different. Between these two directions
lies the direction for the line which denotes the course of the quan-
tity pe. If this last line is an ascending one, this is also the case for
the two others, and reversely. If the admixture is called more volatile

1) Though it falls outside the scope of our subject, which only treats of
properties on the side of the surface of saturation, I will make a single remark
on the mixtures for which liquid and vapour have the same concentration,
because these mixtures have many propertics which the components also

2

4,
: é ze Op
possess. Also for these mixtures the equation: MRTI— = —— ] dv, or
fp Ox) oT
ty

Ge Op : oe Op
MRT — }) holds. For these cases —=1 and so { — = 0. So for
Be Ox )yT Be Ow) yp

a mixture for which this equation would hold at the critical circumstances,

0 07 0? dp\?
(st) itself would be equal to 0. As also acca 2)

—— =(— ]
Oa Jo Ox? Ov? a5 east be equal to 0,
Op :
also | = | = O, and from:
Ov J oT

follows :

Already in 1895 I made the remark, which follows from this, viz. that for the
point of the plaitpoint curve, at which the line which is sometimes called the line
of Konowatow, meets the plaitpoint line, contact must take place and that just

7

lied,
as for a simple substance — P is about 7.
pal

0
Now I will add that from fe) = 0 follows in the same way, as has been
OY,

derived above, that:
dlog T, 1 dlog 6

- == (1) igh
dx 6) dz

and not

dlog T, PALO
dx 3 Sidinin pel.

as would follow when } is put constant. Already Quriyr pointed out that the last

equation was not satisfied in his observations. According to an oral communication
the equation given here would be in much better harmony with his observations,

( 287 )

than the component, when it causes the quantity p, to decrease, then
wy and un
1 av,
admixture is more volatile than the pure substance and reversely.
In general these three directions approach each other at higher
temperature and at the critical temperature they coincide. An exception
to the rule that with rising temperature the lines approach, must be
allowed for the case that for certain value of 7’ the quantity

0 ,
= = 0. In this case the three directions mentioned coincide at that

the general rule holds, that both

are positive when the

value of 7; as they must again coincide at 7’= T,, and as the
Ope

quantity 5a varies with 7’, they will first diverge up to a certain
av

maximum amount, and finally approach each other again.
The rule about the approaching of the lines might also be
represented in the following manner, which would render my meaning
Op
more precisely. If we write od under the following form, which
vy,
follows directly from the above:
dp Pcte—X%) adlpe
or “MRT a.
da ae veil

dp¢ om pv,—v,) dpe y
da MRT da

lust ees k and ia
MRT dz dpe

or putting

from which follows :

Ca ary dT k

ek—]

due Nak ek—]
ek —

The factor G ) is always positive for & positive, and always

negative for & negative, and is only equal to 0 for £=0; and &

Op
equal to 0 occurs only at the critical temperature and when =~ = 0.
a

fie.
dr

equal to 0. When this quantity is equal to 0, there is a maximum

dU
For all other values of / can —

IT

only be equal to 0, when

dp

or a minimum value for ; * and the variation of this quantity ean
ape
da

‘ : : oe dk
reverse its sign with rising temperature. Reversely when aT cannot

be equal to 0, reversal of sign cannot take place in the variation
of this quantity.

PA%,—2r,) dpe | = dk
MET de follows as condition of wT 0.

From 4 =

fer

Alpe MRT pdvs—v,) dlp,

ae dT MRT dTdx

T,—T dlp, oe dp, fal, dlp. ap GE:
T° de ds Pde “deal — emee

dlp, f aT,

Now [pe = Ip, —f

If we put in which 7, can have all possible values

da T- dx’
‘ dk :
from —o to +o, then a 0 may also be written :
a
Td Pe (v, ai 1
MRT Se ee
a) ae oe
MRI Tae ee

The first member of this equation is always positive, at lower
temperatures nearly equal to 0, and at the critical temperature in-
finitely great. So the second member must also be positive. Or, if
this equation is to be satistied 7, < 7, but positive. In all the cases,

; ; ‘ : dk
therefore, in which 7’ is negative, WT cannot become equal to 0,
and no reversal of sign takes therefore place in the course of
dp
Ghee se
— with the temperature.
dpe

dz
So the reversal of sign only occurs, when in the equation

dlp, Va diky

da es da

T, lies between 0 and 7). The two extreme values give for 7, = 0

(( 289 )

(

the value of =" =0, and for T, = 7; the value of -— 5 ia
ne value o dg 0» anc or £, = 4; the value o Tide = 9 6 de

dp
so the well-known limits for mixtures for which ee can be equal to 0.
Vol
For the initial direction of the section normal to the p-axis, the
following equation holds :

dpe
AGE _ Ar vi—a eee pore 1
TENG , coat =
and
u dpe
RT dx

Lake RE ae
EN aa, eae ‘

Both yield at the critical ne eae of the components :

1 dpe
GN he IIR dpe _ u dpe eo dx 1 1 (dpe
eT ae Tad fal Reals aE r da fap a7 Me Cada) x
P dT

According to results obtained before, we may also write:

Ib VEGHE Ge ai: elerd |
PT; dx , ae Fins T,dx , a5 6 b dx, Vi

Physics. — “The exact numerical values for the properties of the
plaitpoint line on the side of the components.” By Prof. van
DER WAALS.

In my two previous communications, inserted in the proceedings
of this meeting, viz. I on the properties of the plaitpoint line on
the side of the components and II on the properties of the sections
of the surface of saturation on the side of the components, it has
again appeared, that the thermodynamic treatment of such problems
enables us to find a complete general solution — but also that if
we want to compute numerical values in special cases, the know-
ledge of the equation of state is indispensable. In some cases it will be
sufficient, if we make use of an approximate equation of state; but
as soon as the density of the substance is comparable to that in the
critical state, the numerical values calculated by means of the
approximate equation of state can deviate strongly from reality.
This is specially the case with quantities which either refer to the
volume, or are in close connection with it. Thus it is known,
that already the critical volume of a simple substance is not

20

Proceedings Royal Acad, Amsterdam. Vol VII.

( 290 )

equal to 936, the value furnished by the equation of state, in
which 4 is put constant, but that this equation is found rather
nearer to 26. This may be accounted for by taking into account
that & is variable and decreases with the volume. In a mixture 6

= :; eon:
also depends on the composition. Accordingly the quantity qq 8 an
a

intricate expression for mixtures, and must in general be distin-
db
ouished from (=) . If the way in which 6 depends on volume and
v
composition, was accurately known, then there would not be left
any difficulties but those of toilsome and intricate calculations. But
it is sufficiently known, that the way in which 4, even for a simple
substance, depends on v, has not yet been fixed with perfect cer-
tainty, and that in any case the knowledge of the numerical values,
which occur’ in given forms of 4, is wanting. These considerations
led me to believe that this would be an objection to deriving theo-
retically the properties of the beginning of the plaitpoint line with perfect
certainty — and also to determining the numerical values exactly.
It has however, appeared to me that the knowledge of how 6
depends on w and v is not required for this exact determination;
but that for this purpose it suffices to know two quantities which
have been experimentally determined for the critical state of a
simple substance.
3 , T (0p T dp . hed
Let us call f the value which — (eles has in the eriti-
: pP On) s Pp di
cal conditions of the component, and x, the critical coefficient, so that
MRT, = x (pv), .

MRT Ca MRT : a :
— -, follows = and ae

From p=

v—b v plv — b) pe
The equality of MRT =x pv =f (v—) p, gives the value
v= Pale b
ip—te

for the critical volume, in which we have to keep in view, that
now that 6 is put variable with the volume, 4 represents the value
which this quantity has in the critical state. With f=7 and

15 v 28

8

- C . >

%—= — we find — 3° whereas with f=4 and x= 5 we should
4 t :

Pee

v : : r .
find the value as 3. For carbonic acid Krrsom has found f= 6,7

v 6,7
and * = 3,56, from whieh would follow = 314 = 2,134,

(291 )

If in MRT =x pv we put the value of v, we find:

xf
MRT = pb ——.
j-—*

8
With f—4 and «= a the factor of pb = 8, and with f= 7 and
Wey 2 : aes ,
x —=— this factor is found to be only slightly different viz. 8 ra
For the calculation of the value of 4 in the critical condition we
get therefore:
MRT f—x ae ee
= Yr

jo. Coif ; px 273 xf

a
If we put the value of v in the equation —-=/—1, we find:
pr

: : : 8
The factor of 6’, which with f= 4andx = = has the well known

. : ‘ 15
value of 27, is found shghtly above 27,8 with f= 7 and =
If in MRT =x pv we substitute the values found for p and v,
we find:
Mee Se
b f(f—})

>

: Pa hei eee
If we again put f=4 and Bea AE find MRT = ane with

15 a 1
"= 7 and « = — the factor — is equal to ———-;also this va if-
F=T anc 4 th 5 18 equal to sT76° also this value dif

geet
27 meno Ibe
For the calculation of a with the critical values of 7’ and p, the

fers but little from

formula:
__ (MRT) f-1
Ce P xe
holds.
yaa 27 ‘Lae ‘
The factor ra 8 equal to 64 237 with f= 4andx—-.. With
Pg. 2 for ae ’ : 96 1
s(n die —— if is again only slightly different, viz. — = ——.
: : 225 2,34

( 292 )

Op
For the critical condition @ must be 0. From this follows:
We y

MRT 1 0b 9 a
Cape ethyl 1a
and after substitution of the values found for MRT and v
a xf)
Ov ip
5 F 8. 0b
With f=4 and x= git follows naturally that = 0, whereas
Vv

i)
with f=7 and x= 3 it follows that:

0p
In the same way ie 2) must be O in the critical state. From
vw) >

this follows:
Oia * (f—1) (f—x) ( i) 4)
b
Ov? ite

8
With f=4. and x= 3 this value is of course equal to 0. With

: 15 :
j= Cand on we find:

po = 0.1887 3
— b=, = 0,182 ).

aA)

€
Let us now proceed to calculate the value of = at the begin-

Tdx
ning of the plaitpoint line. We have the formula:

02p 1 Op 2
IRE (ae ),4 MRT con

Ldx, — 078 y
Ov?

} Op 0?p <
and have therefore to determine Gar and ( ry for the critical
B) oT v

condition, but on the supposition that 6 varies with the volume and that

070 : peise
1) This high value of — b ee supports the hypothesis that 6, in its dependence

Ov?

on the voluine, has a more intricate form than is represented by a series of ascending

ah
powers of ( eb

( 293 )

)
qa has different values depending on the variations of the volume.
av

Now

WURT g
(32) <- i: @ dao M1) a
erie

Ow (v—b)? dav ov

1 da MRT (0b v3
ade a 0a), (v—b;?

db
If we call 7 the value denoting the change of 4 with change of
av %

av, when we make also the volume vary in such a way that the
mixture is again in the state which may be called the critical state
of the unsplit mixture, ak

as ‘Ob Ob (dv
dx AG 4 a6 i

: if dv db i dA
and v, being = b6—— ,|—}) =— ——., when / and ~ are con-
f-*’ dz }, dx j—2# s

stant, which is the case when the law of corresponding states is
fulfilled. We find then:

0b ie OO) ie \ elo
een ie

We have to know:

0b
(Ge aS
(= v' MRT 2! J x) 7 1d

a dv). (v—b)y? as (v—)? | ee b du
.
; 0b
When we substitute the values found above for JR 7" v and 1 —
f-2 1 db
we find * ate 5 ie and so:

Les __a({lda f—21 2

p \0r) 7 tla dx f—1b dex
or
u
1 (Op } ab 1 db
-(~) =-(-pni— +5
p \da)or du f—1 da

This value is in a high degree dependent on /f.
Op any
With f= 4 we find Lae : l 1 =

> \0a T da 3 bda ;
1 ie dT. 1 db
With f= 7 on the other bands —{— =— 6 = =
J ae : sae Ow ‘ T da 6 bd

(292 )
In the preceding communication | have concluded to the same

Op dpe 4
value from the equality ot (5) and Pras the critical circumstances,
. Ones; da

can Pe ee ea :

and by means of the empirical formula — Eee =f Bass For from this
formula follows

dpe dp, Roe

peda = pda Tda
or

1 dpe dT, 1 db elie

pack) 3 ODE; (dle. Ides
or

1 Op 5 add bie Ie eho
| eae = — (=) ine :
0 \da Jor T,dxe — f—1 bdx

But we could arrive at this equation in a much simpler way still.

From:
asia Op a Op i Op a
sa Ow vr a OT S Over

follows, when d7’ is put equal to UT. (taking for d7, the variation
of the critical temperature of the unsplit mixture)

1 dp 1 Op idp \ di,
de = \ Gs) pan uae) eee

1 Op Push 1 dp, : dT,
p\Or)or — p, da / T da

And this equation is not only preferable because it is shorter, but
also because it is independent of the circumstance whether the law
of corresponding states is applicable or not. The value of / in this
derivation is that of the component.

: Op : 0*p
Besides {| —'} we have to determine the value of [ - _
Oa )y? 0200 ) 7

and so

For this quantity we find:

07d da
0% MRI Ob\ (db dwdv dat
=| = — 1 — — MRT —— —
Gee Tea Slee ey te
or
Op \ 2a(1da MRT 2° 1 0b \ (0b “i ERE a®. hash
dwdv)p vila dx a (v—b)? Ov a 2a (v—b)* dvdv ;
076
In this expression only the quantity =, is unknown. We deter-
Lov

mine it from:

( 295 )

0b ob Ff | db
ie = iL ee oe ete \
G v Ov WiC? \ da

From this follows:

0° oe 1 db
Ovdv Ov? f- 0b dx
. : db
If we substitute the values given above for MRT, (2 -5)
(2)
075 . ; q 0*p : ‘ f
and — 6 — in the expression for | ——] , we find for the value of
Ov? Oxdv T
1 db (f—2) : f ;
the second term sare a2 ae and for the value of the third
) av
t i 1 db f-—4
erm + ——*-—_.,
b dx f
0p
The value of { ——) is then found equal to:
0x00) 7
0?p __ 2a | 1 da 1 db
dvdv) 7 v8 {a dx 6 dx
or
0?p 2a dl,
Ov0v pr Ti dx d
0p
dade dT,
and for 7? we find the simple value , ; SO exactly the same
O76 T dx :
Ov?

value as follows from the equation of state, in which @ is put
constant. This gives rise to the conjecture that this relation might
be found merely from thermo-dynamic relations independent of the
knowledge of the equation of state, and this is indeed the case,
. Op mi eee ian
Let us consider the quantity Ale It is equal to O in the eritical
State of the component. Let us pass from this homogeneous
critical phase to another in which the volume has changed with
dv, the compositien with dr, and the temperature with d7’.
Let us put d7’ again equal to d7., so let us assume that the
mixture with dv molecules of the second kind is again in an homo-

Op\ =
geneous critical phase, then ie) is again equal to 0.
OWGh é

Op 0*p 0*p 0?p E
—— —- == dv —=— la: == VL
e ian ae gar Gal a lank: ;

From:

( 296 )
*) 0p
follows, because d a and . sr are equal to 0:
( FT
0? Pp dT,
dvd T ee
de
Ov
mecne
wi ~~ Oe?”

And from this we find again now only from thermodynamic

and from the relation:

follows:

relations what we have derived already above.
As we found, also by means of mere thermodynamics :

Op 5 (ail i Gha:
= —— ? ———————- — —
Oe Jy TP JI Gh ip Goysalee

we may put without making use of the equation of state :
re abe fp? | dP. \ dp; }

Tde, Tyda unpot (Pete f pdx
Ov?
2h} 2
The factor ——*—~— may be reduced to a simple form, but for

MRT 7

v

the determination of the value of this factor it is required to know

: ees MRT O76 ce ay
the equation of state. If we write /p = , and — = — 2—,this
v—b Ov? v
f ae MRT 2a MRT Ob 2a
acto yecomes equa to C= ees and as (cere —— 7G Ee oF
Op
follows from ae = 0, we get:
(D)
ftp’ 1 ie
7 O76 an 0b 2x (f—1)
MRT — 1—=—
Ov? Ov
15

so with f=7 andx = the value of this factor becomes equal

t es H ]
0 7,: Hence we have

dT __ afi, 49 ( dT, 1 ap, :

Tde, = GPs 45 T, dx 7 & pide

(* Ld s Y . Ip >
If we introduce the quantity 4 instead of — we find:
piu

( 297 )

dT aus f-l1 aus 1 db le

Tde, = lida Pa 2x Toda ft jal Bde \

O

‘ 8 ]
With 7 =4 and x = — we find acain — = —
J 3 = 2% 1

, but with f= 7
1B

A? Ox
values for carbonic acid) the value is not appreciably different from

dT 1 db
0,8. If we calculate with a, = — 0,493 and —-— => — 0,271,
4av

and «= rises to 0,8. With 7 = 6,7 and x = 3,56 (Keesom’s

b da

— 6,7 and x= 3,56 the value of a we find for this value
— 0,259. Though 0,259 is smaller than the values calculated from
Keesom’s observations, 0,284 for. 7 = 0,1047 and 0,304 for 2 = 0,1994,
we must not forget that the ecalewlated value would hold for the
limiting case, viz «=O; and the fact that for Ae =O a smaller
value than 0,284 would have to be expected is at least in harmony
with the cireumstance that the amount is found higher for a higher
value of «.

It is evident from all this that though we cannot do quite without

the equation of state for the caleulati for the plaitpoint
line, yet it is not necessary to know the form of the quantity 0.
T dp ae ty
For the calculation of the quantity — = for the beginning of the
P (

plaitpoint line we have oe the ae

the relation :

T dp 1 & Al! Op
p dT pl 7 OL);

] Op Tdx,
»\ da) Tr dT

or
1-1 a
T dp ; ie ie dat (F—1) 6 de de
€ wale eka a ies id =n esterday)
Tide t Tde ' (f—1)b de

or in numerical value for the mixture of oxygen and carbonic acid:
(T d, = 5,7 | 10,408 0,047)
pape) Sree Gey 1 67 se ee
pd), =——/0,259
With the mixture «= 0,1047 Kunsom has found — 6,3 and with
v= 0,1995 the amount found was — 6,08.

( 298 )

5 LGD dT , ‘
If we take the product of —~, and ; we find the value of
p di Ta
1 dp pouee : ee ee T dp dT
—— for the beginning of the plaitpoint line. As both ——~— and ~——
P du P dl Tdx

are negative for the mixture of carbonic acid and oxygen, the value of

I GON. ae
= is positive.
P da pl

Anatomy. — “Bork’s centra in the cerebellum of the mammala”.
By D. J. Hursnorr Por (from the laboratory for Psychiatry
and Neurology at Amsterdam). (Communicated by Prof.

WINKLER).

In his well-known researches about the cerebella of mammalia *),
Bonk concludes: that “the Lobus anterior cerebelli does contain the
centre of coordination for the muscle-groups of the head, the Lobus
simplex the centrum of coordination for those of the neck; the non-
symmetrical centre of coordination for both left and right extremity
is situated in the Lobus medianus posterior, whilst each of the
Lobuli ansiformes is the seat of one of the symmetrical centra,
respectively for both right, and for both left extremities.” *) .

Within the same line of research, Van Rignperk*) at Luctanrs
laboratory in Roma, experimenting on two dogs, extirpated a portion
of the cerebellum, with the aim of taking away the right part of the
Lobus simplex. —

The secondary symptoms, which were observed during the first
days after the operation, having passed away, the animal experi-
mented upon continued shaking its head, as if it meant to say “no”

This symptom resembled very much a trouble in the coordination,
and such being indeed the case, it would have confirmed the hypo-
thesis of Botx. Therefore it was important to determine with as
much exactness as possible, which portion of the cerebellum had
been removed.

To this purpose the preparation, fixed in formol, was offered

1) Prof. f. Dr. L. Boxx. Das Cerebellum der Siiugetiere. Perrus Camper, Vol III,
part. I, Amsterdam.

2) Prof, Dr. L. Bork. Over de physiologische beteekenis van het cerebellum. De
Erven Boun, Haarlem. 1903.

3) G. A. vAN Ruwperk. Tentative di localisazione funzionali nel cerveletto. Archivio
di fisiologia. Vol. I. Fase. V

( 299 )

for examination to Professor Wuxkrirr, who had the kindness to
leave its further elaboration to me.

Since the sections, that should be made subsequently, were to be
stained after the Wricrrr—Par method, the cerebellum was immedia-
tely after its arrival in Amsterdam refixed in Muninr’s liquid. It
was only when this had been performed that the photographs were
taken (fig. I and II).

The white spots that are seen on the figures, were caused by
celloidine, by means of which the pieces were pasted together. It
was necessary to do this, because the cerebellum, was received here
being cut into three pieces.

In the middle of the surface of the cerebellum we observe a cavity.
If this cavity is divided into four parts along the longitudinal axis of
the cerebellum, nearly one quarter is lying to the left of the median
line, two other quarters are lying in the right median part, and another
quarter (probably the smallest one) is lying in the right lateral part.

The form of this cavity on the surface of the cerebellum, as far
as it is lying in the median portion, is nearly that of a truncated
isosceles triangle having for its basis the paramedian line.

The greater part of this triangle (nearly three quarters) is lying
in the right half, and only one fourth in the left half of the cerebellum.

What imports most now is to find out to which subdivision of
the cerebellum belong the convolutions from which van Riunperk
has extirpated this small piece.

In fig. I and II, next to the defect (fig. Il sub 1), our attention
is drawn immediately by a deep furrow (fig. Il sub 2a), that has
become to all probability more clearly visible by the process of
fixation, than may have been the case during life.

The sulcus primarius is the furrow penetrating deepest to the
medullary nucleus and continuing forward till near the sinus Rhom-
boidalis, causing in this way the lobus anterior and the lobus
posterior to be connected, for by far the greater part, only by a
ridge of medullated nerve-substance. We may therefore safely assume
that, the cerebellum having been eventually shrivelled, this suleus,
lying between two portions so deeply divided, will become more
distinctly visible.

At first view therefore we might hold the furrow indicated sub 2a
fig. Il, to be probably the suleus primarius. Such being the case,
all that is lying before this furrow would be lobus anterior, all
that is lying behind it lobus posterior.

On examining the anterior portion, this is found to consist of two

( 300 )

paris, that may be discriminated with sufficient distinetness (fig. TI,
3 and 4). What we find indicated sub 38 is a coniform swelling,
consisting of a succession of folia, separated by sulci running in the
direction of the margo mesencephalicus.

Accordingly it does not offer any difficulty to recognise in this
portion the lobus anterior.

The case is different however for the folium behind this part
(fig. IL sub 4) lying before the suleus mentioned sub 2a.

it would belong to the anterior lobe if this furrow were indeed
the suleus primarius; but as it is lying behind the suleus sub 26,
the direction of which is totally different from that of the other
sulei of the lobus anterior, the question arises whether this convolu-
tion sub 4 does indeed belong to the anterior lobe.

The direction of this gyrus is totally different from that of all
the other convolutions lying before it, because it does encompass
the basis of the coniform swelling. Whilst the convolutions in the
anterior part are ranged regularly behind one another, the convolu-
tion sub 4 does diverge from that arrangement, because the former
convolutions are implanted in this latter. Relying only on the diffe-
rence in direction between these convolutions, one would be inelined
to consider as the sulcus primarius rather the suicus sub 26 than
that sub 2a.

We see however, that the convolution sub 4, like that of the
coniform swelling, is running from the right to the left. It is unin-
terrupted and the initial direction of this curved convolution is
likewise towards the margo mesencephalicus. This — in addition
to the fact, that Bonk in his description of the cerebella of different
mamimalia, likewise reckons the lower and more deviating con-
volutions to the lobus anterior — supports the opinion that the sulcus
sub 2 is not the sulcus primarius, as we might suppose, if relying
only on the difference in direction between the convolutions sub 3
and 4.

The macroscopical description will therefore have to leave unde-
cided the question, whether the convolution sub 4 must be reckoned
to the lobus anterior or to the lobus posterior.

Nevertheless it is of the greatest importance to delimitate exactly
to which portion of the cerebellum this convolution belongs,
because it has become evident from the figures I and IJ, that on
the surface it is precisely in this convolution that the greater part
of the defect is situated. The examination of sagittal sections of the
cerebellum will have to decide this question.

All that is lying behind the anterior lobe belongs to the posterior

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lobe. This posterior part, with the exception of its first convolutions,
is divided into one median and two lateral portions by the sulci
paramediani, which run parallel to the median line.

Consequently all that is lying between the two paramedian sulei
forms the median part of the lobus posterior, all that is lying
to the right and to the left of them forms the lateral part of
this lobe.

In figure If sub 11 we find the suleus intereruralis. This furrow
is lying in the middle of the lobus ansiformis, (fig. I). The conyo-
lutions that start originally from the median line as crus primum
(fig. IT sub 9) and bend gradually, when arrived in the ultimate
lateral part, to return thence as crus secundum (fig. II sub 10) to
the median part, take before reaching it another bent, this time
straight backward; to continue further as lobus paramedianus (fig.
II sub 12) parallel to the suleus paramedianus.

In deseribing further the lobus posterior we will confine ourselves,
for the sake of convenience, to the left half.

Of course in fig. I and II all that is lying to the left of the
suleus paramedianus belongs to the lateral part, and consequently
may not be reckoned to the lobus simplex, the convolutions of this
latter, according to Bo.k, continuing without any interruption from
the right to the left.

Applying this test to fig. I, we find that the extreme end of the
left suleus paramedianus is stopped by a convolution indicated sub
5 fig. II.

Thence it might be concluded, that the convolutions indicated
sub 5 and 6 in fig. I, accordingly lying above the sulci paramediani
and below the lobus anterior, form the lobus simplex.

On a closer examination of the lowest of these two gyri, i. e.
the gyrus sub 5, we find however, that to the left of the white
spot (the end of the dotted line), in the lateral part of the gyrus
therefore, a narrow furrow may still be observed, that does not
continue to the median line. According to Bonk therefore, this con-
-volution does not belong to the lobus simplex, as the incomplete
furrows in this lobe, like those in the lobus anterior, ought to start
from the median line.

The cause, why the interruption of this gyrus sub 5 is not, as
usually, visible on the surface, must be sought in the fact that the
suleus paramedianus disappears in the depth, and does not therefore
penetrate into this convolution on the surface.

The last convolution, sub 6, fulfills in every respect the conditions
claimed by Bork for the convolutions of the lobus simplex ; as it

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lies directly behind the lobus anterior, and continues without
interruption from the right to the left, whilst incomplete furrows,
not starting from the median line, do not occur in it. Moreover
the convolutions forming the crus primum, le adjacent to it and
originate in it.

This convolution sub 6 thus forming the lobus simplex on the
surface, does not continue very far on the more lateral part, as
may be seen in the figure. It is therefore little developed.

Now if we follow to the right the course of the convolution
sub 4+ fig. II, about which it is not yet decided whether it
belongs to the anterior or to the posterior lobe, we find that this
convolution loses itself in the cavity sub 1. It may be said _there-
fore that the operation has extirpated at least on the surface, a
part of the left median portion and the whole of the right median
portion of this gyrus.

I have stated already that the cavity broadened to the right, and
attained its largest breadth in the prolongation of the sulcus para-
medianus. By the photograph sub I it becomes evident, that in this
place the lesion extends over the convolutions lying before and behind.

A convolution of the lobus anterior lying before the convolution
sub 4, and a convolution of the lobus simplex behind it, we may
therefore assume, in as much as it is allowed to draw conclusions
from the macroscopical aspect, that in the right median part at
least one convolution of the lobus anterior and one of the lobus
simplex have been injured. It must remain undecided whether the
principal defect, situated in the convolution sub 4 fig. H, ought to
be reckoned to the lobus anterior or to the lobus simplex.

It was on purpose I did not hitherto say anything about the
macroscopical deviations in the right lateral portion, because, as was
stated before, the whole of it was divided from the left portion of
the cerebellum and having been thrivelled in the course of the
elaboration, it no longer fitted exactly unto the median part, as
may distinctly be seen in fig. I and II.

In order therefore to avoid eventual errors I neglect the macros-
copical description of the lateral portion of the posterior lobe.
This omission does not involve unsurmountable difficulties, because
the deseription of the sagittal sections remains still to be given, and
by means of these latter we shall have to find out which portions
have been destroyed and which have been left intact.

The cerebellum having been fixed for some time at the laboratory in
Muuier’s liquid it was inclosed in celloidine and cut in serial sections,

( 303 )

Photograph III has been taken of a section on the left border of
the defect, i.e. on the spot where the lesion begins on the left.

Photograph IV has been taken of a section from the right median
part, directly adjacent to the median line.

Photograph V represents a section from the middle of the median
portion.

Photograph VI represents a section very close to the prolongation
of the suleus paramedianus dexter, but still within the median portion.

Photograph VIII represents a section from the left lateral, being
the un-injured portion. It corresponds with the place in the right
lateral part, represented by photograph VII.

If, by the aid of photograph III, we try to delimitate the exact
situation, especially of the different convolutions around the sulcus
primarius, it does not present any great difficulty to know which
is the lobus anterior and which the lobus posterior.

The furrow, lying opposite the sinus Rhomboidales (R.), is the
suleus primarius (s. p.). All that is lying before this sulcus, to the
left of it in fig. IL, belongs to the lobus anterior, all that is
lying behind it, to the right im the figure, belongs to the lobus
posterior,

The strongly developed anterior lobe is divided into four lower
lobules, which I have indicated sub 41, 2, 3 and 4, conform to
Bouk’s description.

Accordingly these numbers correspond with the lobes, designated
in the human cerebellum as Lingula, Lobus centralis and Culmen.

For the posterior lobe I likewise followed Bork’s division, and
accordingly designated the folia by a, 6, ¢ and d; a corresponding
with Nodulus, 6 with Uvula, ¢ with Pyramis and d with Tuber
vermis, Folium cacuminis and Declive. This latter would be the
Lobus simplex.

The rationality of Boxk’s division is demonstrated clearly by this
preparation, as the medullary rays of the folia are all of them
separately implanted in the medullary nucleus.

The sinus Rhomboidales, the roof of the fourth ventricle, is desig-
nated sub R. Opposite to if, accordingly in the figure straight above
it, and separated from it only by the medullary nucleus, we find
the sulcus primarius (s. p.).

As we stated before, it could not be decided with any certainty
from the macroscopial description whether the sulcus primarius was to
be sought for sub 2a or sub 26 (fig. Il), and consequently the
situation of the defect could not be precisely defined ; it is therefore
necessary to determine with the utmost exactness in their mutual

( 304 )

relation the respective situations of the sulcus primarius, the adjacent
gyri and the lesion.

To this purpose I have designated in fig. III sub @ and sub 8 the
two convolutions lying next to the suleus primarius on the surface.

Looking ad @, which represents the first convolution of the lobus
posterior on the surface, we find that it consists of a secondary
radius medullaris ending in a bifurcation on the surface. Such not
being the case with the adjacent secondary medullary rays of ¢,,
this convolution may be likewise easily recognized in the next figures.

The same thing may be said for 8, which represents the most
posterior convolution of the lobus anterior on the surface. For we
observe that the medullary ray of the lobule N°. 4 divides itself
into two portions: the posterior one 8 being the prolongation of the
thick primary radius medullaris and therefore easily recognised.

In photograph III, representing, as may be remembered, a section
taken from the place where to the left the lesion begins, we see
clearly that not the entire convolution 8 has been destroyed, but
mainly that portion of it that is lying next to the suleus primarius.
The convolution of the lobus posterior, lying behind it, has not
been injured at all, its surface, on the spot where this convolution
bends inward towards the suleus primarius being distinctly visible.

This spot (@) being of importance in order to determine whether
the lobus posterior, in case the lobus simplex, has been injured,
I have designated it likewise on the other photographs (1V, V and V1).
Anticipating for a moment on the subsequent description of these
photographs, I may state that they show clearly that this spot on
the surface, where the convolution bends inward, presents nowhere
any trace of lesion.

The direct conclusion to be derived from this fact is that the lobus
simplex has not been injured av ITs SURFACE.

As to the convolution 8 however matters stand differently.

In fig. IL already we may see that from 4, representing the pos-
terior folium of the lobus anterior, the posterior secondary convo-
lution 8 in the upper part has been almost entirely destroyed. Only
a small piece of its most anterior portion remains.

In the direction of the medullary nucleus the lesion extends only
over the upper third part of the sulcus primarius.

The secondary radius medullaris is still distinetly visible at the
spot where it is united to the anterior convolution.

Surveying the successive aspects of the lesion in the figures IV,
V and VI, we find that in IV a very small remnant of the convo-
lution sub p still subsists, whilst the secondary radius medul-

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laris has been cut off almost up to the place of bifureation of the
primary radius.

The lesion itself penetrates inward, a little into the medullary
nucleus, moreover the secondary and tertiary lobules, lying adjacent
to the sulcus primarius, are for the greater part, if not wholly,
destroyed.

This becomes still more evident from the fig. V and VI, where
aul that belongs to the convolution ?, has been destroyed. The
lesion itself penetrates still deeper, in fig. V it has nearly cleft the
radius medullaris, in fig. VI it has done so entirely.

We may therefore conclude: that in the median portion, more
especially in its right half, the posterior lobule of the lobus anterior
has been seriously injured, that even nearly the whole of it has been
removed.

I stated already, that from the anterior portion of the lobus
posterior, i. e. the lobus simplex, nothing has been destroyed on the
surface, as the place where it bends inward sub @, remains visible
on all sections in fig. III, IV, V and VI. Deeper however, the case
becomes different.

In tig. IV we observe that all secondary lobules, lying adjacent
to the suleus primarius in the depth, have been completely destroyed.
In fig. V there has been removed still more, nearly the whole of
the secondary radius medullaris having been extirpated. In fig. VI
it is entirely destroyed, whilst moreover the primary radius medul-
laris of the small lobe c, has been completely divided from the
medullary nucleus.

We may thence conclude that, though the lobus simplex in its
median portion is not injured at its surface, on the contrary in the
depth, in the portion adjacent to the sulcus primarius, it has been
entirely destroyed, even those convolutions that remained intact on
the surface in the paramedian area (figure IIL sub @) having been
divided from the primary radius medullaris.

Considering next the lateral portion, fig. VII enables us to survey
the situation and the division of the folia under ordinary cireum-
stances. The suleus primarius (s. p.) still subsists, as the convolution
sub 4 fig. II, the last convolution of the lobus anterior, is removed
considerably sideways. All that lies before this suleus, accor-
dingly to the left in fig VIII, belongs to the lobus anterior. Con-
sequently the small lobe sub 1 is the last folium of this lobe. All
that lies behind the sulcus primarius, thus belongs again to the
lobus simplex sub 2.

Considering next fig. VII, it is shown thereby that on both sides

21

Proceedings Royal Acad, Amsterdam. Vol. VIIL.

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of the sulcus primarius the secondary lobules have been all destroyed,
and moreover from the lobus anterior even nearly the whole of the
radius medullaris.

Accordingly we find for the /ateral portion the same result as
for the median right half, i. e. that besides the greater part of the
lobus simplex, also a part of the lobus anterior has been destroyed.

Originally we intended not only to find out by means of the
sections, which portion of the cerebellum had been taken away by
Van Risnperk, but likewise to demonstrate the microscopical changes
subsequent to the lesion. To this purpose we used the Wuicrrt-PaL
method of staining.

Unfortunately however, on microscopical examination, it was shown
that nearly the whole mass of medulla had taken a granular aspect.
It was therefore impossible to study the nerve-fibres, and any secon-
dary degeneration they might have suffered with the method of
Waricert-Pan, and for Marcni-preparation the cerebellum proved unfit.

Nevertheless one fact remains worthy of attention: in the part of
tig. VI, designated sub a, accordingly in that portion, separated by the
operation from the central medullaris originating in it, the radius
medullaris not only is stained black as distinctly as the other secondary
medullary rays, but moreover the PurkmJe corpuscles and their
ramifications in this portion (stained by means of osmium acid),
do not present any changes worth mentioning, if compared to those
of the other, un-injured lobules.

SUMMARY.

A. According to Botk’s theory, the Lobus simplex is the seat of
an unsymmetrical centrum of codrdination for the muscle-groups of
the neck.

B. Operating on a dog, van RinseKk extirpated a part of the
cerebellum, about the Lobus simplex. In consequence of this opera-
tion, its secondary symptoms having passed away, the animal retained
a continual movement of the head as if it meant to say “no”.

C. Investigations at the laboratory in Amsterdam taught us that
the operation had destroyed :

a. in the left median part, next to the median line, a small
superficial portion of the last gyrus of the Lobus anterior ;
b. between the median line and the paramedian line to the
right 3
1st. nearly the whole of the last gyrus of the Lobus anterior;
21. nothing from the Lobus simplex at its surface ;

D. J. HULSHOFF POL. BOLK’s centra
I

back

Proceedings Royal Acad. Amsterdam. Vol.

front

in the cerebellum of the mammalia.”
ll

2a 2b

back front

back front

back

Vill.

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3", nearly the whole of the Lobus simplex in the depth,
whilst, towards the paramedian line, likewise those por-
tions of gyri, which remained intact on the surface, have
been divided from the primary radius medullaris.

c. In the part, situated to the right of the paramedian line,

as far as the lesion extends ;

1s*. nearly the whole of the posterior folium of the Lobus
anterior ;
2-(, the greater part of the Lobus simplex.

Physiology. — ‘The designs on the skin of the vertebrates, considered
in their connection with the theory of segmentation.” By
Dr. G. vAN Runperk. (Communicated by Prof. C. Winker).

That there exists some connection between the distribution of
pigments in the skin and its segmental innervation will be evident
to any one who has made some investigations into the questions
concerning the theory of segmentation. Different authors have made
numerous unconnected researches about this subject. SHERRINGTON *)
has pointed out that the stripes of the zebra are ranged in segments
on neck and trunk; whilst he identifies the cross-stripe over the
shoulders of the ass with its dorsal axis-line for the anterior extremity.
WINKLER *) has drawn attention to the fact that deep-coloured rabbits
often show white spots, presenting a marked conformity in distri-
bution and extension with the analgetic areas that are produced
when one or two of the posterior roots of the spinal nerves have
been cut through. It may therefore readily be assumed that these
white spots find their origin in the fact that either one or several
segments lack the faculty of producing pigment. ALLEN *) has
demonstrated that certain series of spots on the skin of the squirrel
correspond with the points of entrance into the hypodermis of series
of skinbranches of the intercostal and other homologous nerves. Two

1) CG. S. Suerrineron, Experiments in examination of the peripheral distribution
of the fibres of the posterior roots of some spinal nerves. Philosoph. Transactions
of the Royal Society. London, vol. 184 B. p. 757.

*) C. Winxter, Ueber die Rumpfdermatome. Monatschrift fiir Psychiatrie und
Neurologie. Bd. XIII, 1903, h. 3, S. 173.

) H. Auten, The distribution of the colour-marks of the mammalia. Proceedings
of the Academy of Nat. Sciences of Philadelphia, 1888, p. 84 et seq. — See also,
Science. 1887.

Dies

( 308 )

years ago I myself) collected some data, by means of which
[ endeavoured to justify the hypothesis, that in two species of sharks
(Scyllium Catulus and Se. Canicula) the deep-coloured transversal
stripes correspond alternately with groups of five and of three
segments, producing more pigment than the other segments.

My present endeavour will be to demonstrate some systematical
views concerning the colourmarks on the skin of the vertebrates
(birds excepted), on the basis of the rather extensive and detailed
knowledge we possess about the segmental innervation of the skin.

To that purpose the first thing to be done is to define as clearly
as possible the object of my investigations. In every animal the
so-called “design” in its widest acceptation, originates in the con-
trasting effect of at least two colours or tints. Generally the next
proceeding is to select one of these colours, that is then regarded
as the “design” in a narrower sense, whilst the other colour is
called the prime-colour. This choice is principally determined by
esthetic motives: its criterion being either the difference in extension, —
the less extensive colour being then taken for the design —, or
else the difference in tint, the lightest tint being then regarded as
the prime-colour. The irrationality of this method is evident, as has
already been pointed out partially by J. Zmnnuck *) a disciple of
Ermer. For when comparing together a few cartoons of equal size,
on which are designed respectively: a small black figure with a
large white margin, a small white figure with a large black margin,
a large black figure with a narrow white margin and a large white
figure with a narrow black margin, — nobody will think in
either of these cases of taking the margin for the design: the figure
remains the figure, whether it be large or small, black or white.
Consequently it is neither by its tint, nor by its extension that the
“design” ought to be disiinguished, but only and exclusively by its
significance. Applying this test to the distribution of colourmarks
on the skin of animals (the design in its widest acceptation) we will
accordingly have to determine at the outset in each case, which
biological, morphological and physiological significance must be
ascribed to the different portions of the design.

Their biological significance may be neglected here; necessarily the

1) G. van Ruperkx, Beobachtungen iiber die Pigmentation der Haut von Scyllium
Catulus und Canicula und ihre Zuordnung zu der segmentalen Hautinnervation
dieser Thiere. — Perrus Camper, Vol. III, 1904, part. 1, p. 187—173.

2) J. Zennecx, Die Zeichnung der Boiden. Zeitschrift f. Wissenschaftliche Zoologie,
Bd. 64, 1896, h. 1, u. 2, S. 234.

( 309 )

foremost condition to obtain correct notions concerning the pigmen-
tation of the skin is to understand properly the morphological
framework whereon the design is founded, and the manner in
which it is physiologically determined. In endeavouring to elucidate
these questions, it becomes evident that the simple distinction between
“design”? in a narrower sense and ‘“prime-colour’, is not sufficient
for a rational description of the manifold pigmentations of the skin.

According to my opinion, we ought at least to distinguish three
elements, constituting in their complete or partial combination the
“design” in its widest acceptation. In order to obtain a proper
distinction between these three elements, it is necessary to introduce
a quantitative criterion into the problem. Besides the respective plus
and minus in the pigmentation, we shall therefore have to make still
another distinction, by opposing to the prime-colour respectively an
excedent and a defect contrast in the production of pigment.

A few instances may suffice to elucidate this. In a white dog
with black ears black represents the contrasting colour just as white
does in a black horse with a white mark on its forehead. But in
the first case it may be called an excedent contrast, in the second
case a defect contrast. In an animal where the prevailing colour is
brown showing both black and white marks, we find combined the
three elements that ought to be distinguished: the brown prime-
colour, the excedent and the defect-contrasts. Starting from these
simple instances, we shall be able to compose a complete terminology,
by means of which the most important elements of the pigmentation
of the skin may be defined with tolerable exactness as to their shape,
extension and distribution. For instance the white mark on the fore-
head of the black horse we will call an isolated defect contrast.
Thus the dark stripes on neck and trunk of the zebra may be called
regular serial diagonal excedent contrasts, whilst the stripes on Galidictis
are regular serial longitudinal excedent contrasts. The morphological
and physiological basis for distinguishing between excedent and
defect-contrast, consists in the following points:

1. In a large series of cases excedent-contrasts are found in such
places where the innervation of the skin is likewise strongest, whilst
on the contrary defect-contrasts are found in such places where this
innervation is feeblest. 2. We may observe that the excedent-contrasts
often correspond as to their shape and distribution with the carica-
tures of the dermatoma’), whilst the defect contrasts often correspond

1) C. Wiyxter and G. van Risnperx, Structure and function of the trunk-derma-

toma Ill. These Proc. 1V, p. 509. Verslagen der Kon. Akademie vy. Wetenschappen,
22 Febr. 1902.

( 310 )

With the analgetie areas called into existence by the destruction of
sensibility in one or more segments.

A few instances may serve to illustrate this. The sensibility of
the skin under normal circumstances is strongest within a system
of lines and zones, corresponding with the average limits of the
dermatoma (of more precise imits we cannot speak because of the
overlapses). This has been proved clinically for man by LANGELAAN *),
experimentally for the dog by Winker?) and myself. Now if we
observe the dark stripes of the zebra, we find that beyond any
doubt these stripes, at least on neck and trunk, show a marked
accordance in their distribution and direction with the average limits
of the dermatoma, as these latter may be imagined to be, relying
on the data procured by Prysgr *), SHrrrincton *), TiRcK °), WINKLER °)
and myself respectively for the rabbit, the monkey and the dog.

Their number corresponds nearly with that of the segments of
neck and trunk; on the neck and on the trunk they are somewhat
wider apart than at the points of insertion of the extremities, this
being in perfect accordance with the fact demonstrated by WuINKLER
and myself that at those points the ranks of the dermatoma are more
thickly set. The distribution of the stripes on the extremities is not
so easily explained. On _ superficial view they resemble rings,
running around the extremity. In reality however each of these rings
consists of two symmetrical semicircles, passing by pairs into one
another by a definite angle on the outside and on the inside of the
extremity. Connecting together the different points at which the semi-
circles meet, we obtain two lines corresponding with the dorsal and the
ventral axis-lines of the extremity. The direction in which the stripes
run (in a caudal direction from the axis-lines), corresponds with

1) J. W. Laneetaay, On the determination of sensory spinal. skinfields in healthy
individuals. These Proc. Ill, p. 251.

2) See note 1 on the preceding page.

8) J. Preyer, Ueber die peripheren Endigungen der motorischen und sensibelen
Fasern des Plexus bracchialis. Zeitschrift f. rat. Medizin. N. 7, Bd. IV, 1854, 8. 52.

4) C. S. Suerrineton, loco citato, and: Idem If Ibidem vol. 190, B. 1898, p. 45—186.

5) L. Tircx, Vorliufige Ergebnisse von Experimental Untersuchungen zur Ermit-
telung der Hautsensibilitatsbezirke der cinzelnen Riickenmarksnervenpaare. Sit-
zungsber. der K. K. Akad. der Wissensch. zu Wien 1856, and: Die Hautsensibili-
tiitsbezirke der einzelne Riickenmarksnervenpaare. Aus dem literarischen Nachlasse
von weil. Prof. Dr. L. Tick zusammengestellt von Prof. Dr. G. Wept. Denk-
schriften der Math. Naturw. Classe der K, Akad. der Wissensch. zu Wien. Bd
XXIX, 1869.

6) CG. Winker and G. van Runperx, On function and structure of the trunk-
dermatoma 1. These Proc. Vol. IV, p. 266; II. 1. c. p. 308; III. 1. c. p. 509; IV.
I. cf: Vol. Vij ip. 347:

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that followed by the limitlines of the segments of the skin. The
number of the stripes, however, is greater than that of the segments
can possibly be. But for this difficulty too a solution can be found.
On considering the curve of sensibility of a normal trunk-skin, as it
has been constructed by Winker and myself on the basis of our
experiments, we find that at the dorsal median line, where the
central areas overlap one another on an average for one third and the
dermatomata for one half, a top of the curve i. e. a zone of
summation corresponds with every average limit-line between two
dermatomata. If now the overlapses amount to more than one half,
as they do on the extremities, the curve of sensibility will be much
more complicated and the zones of summation therefore more nume-
rous. Accordingly the dark stripes on the extremities correspond
likewise with the average lines of demarcation of the dermatomata.
In the zebra the excedent of pigment apparently is distributed in
accordance with the scheme of the intersegmental zones of summation,
and the design resulting from this distribution may therefore be
defined as consisting of intersegmental excedent-contrasts. Although
this instance may not be entirely isolated, still it is a rather rare
one. In many other cases we find that the excedent of pigment is
not distributed in accordance with the uniform scheme of inter-
segmental demarcation, but arbitrarily accumulated in certain
points or portions of the segments themselves. A large number of
white domestic animals for instance present black spots, showing a
marked similitude in their shape, distribution and extension with the
figures, denominated by WINKLER caricatures of the dermatomata.
The way in which the pigment is distributed, offers even an indication
of that peculiar significance which the point of entrance of the skin-
nerve apparently possesses for the innervation (maximum and
ultimum moriens of the central areas, of the dermatomata and of the
sensible skin-areas in general*)). Thus the series of black dots in many
species of sharks, amphibians, serpents and saurias, apparently corre-
spond nearly with the serial ranging of the points of entrance of the
dorsal and lateral nerve branches.

We will now turn to the defect contrasts. In deepcoloured speci-
mina of our domestic animals white-tipped ears or tail, a white belly,
or a white mark in the frontal median line of the head, or else
white toes, are frequently to be met with. It needs not being demon-

1) On function and ete. These Proc. Vol. VI, p. 347. G. van Rueerk: On the
fact of sensible skin areas dying away in a centripetal direction. These Proc,
Vol. VI, p. 346.

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strated that such marks represent either absolute or relative eccentrical
areas. Consequently I consider these marks to be eccentrical defect-
contrasts. The white marks in rabbits, to which attention has been
drawn by Winkie, are of a very different nature, being expressions
of segmental variability; in the series of equivalent segments, pro-
ducing pigment, one or two have lost this faculty; thence results
the defect, corresponding in shape, distribution and extension with
the segmental analgetic areas. A further instance may be forwarded
by the so-called Lakenveld cows, whose white ‘cloth-covering around
the trunk corresponds evidently with a series of pigment-less seg-
ments, which have become hereditary by artificial selection in breeding.
The above-mentioned white feet may be reckoned likewise to these
instances. In black dogs, horses or rabbits, white forefeet and a white
mark on the breast are frequently to be met with. Evidently these
mean something more than a simple eccentrical defect. It cannot be
doubted that such cases represent phenomena of segmental omission.

It is known by the experiments of Winker *) and myself that
the most eccentric skin-segments of the fore-feet (7 and 8 cervical
roots), consist only of the lateral portions of the dermatoma, the
dorsal parts having entirely vanished whilst the ventral parts are
lying exceedingly reduced at the ventral median line near the manubrium
stern. Accordingly this relation corresponds perfectly with that of
the above described defect areas. For this reason I consider these
latter ones to be segmental defect contrasts; they are the expression
of a segmental defect-varibility in the 7 and 8" cervical segment.
Analogous cases are not rarely found. Frequently the white areas
are so extensive that eventually a defect of the 5 and 6 cervical
segment may be assumed besides that of the 7 and 8. Analogous
relations exist in the posterior extremity, though we know less about
its segmental innervation.

I cannot possibly in these pages enter into more minute details
concerning the question of the segmental distribution of the colour
marks in the skin. An extensive essay on this subject is shortly to
be published. The preceding explanations will however be of sufficient
aid to form a judgment concerning my fundamental views and to
understand the conclusions, stated in the following summary. Doubtless
these conclusions may have some importance for clinical work,
because they prove beyond any doubt the great significance of the
segmental innervation for the trophic condition of the skin, and add

1) C. Winkter and G. van Ruwperx, Something concerning the growth of the

areas of the trunk-dermatoma on the caudal portion of the upper extremity. These
Proc. VI) ip. 392:

( 313 )

a new support to the probability of the hypothesis that a segmental
basis lies at the root of many pathological states, as naevus
pigmentosus ete.

CONCLUSIONS.

1. The distribution of the pigmentation on the skin of vertebrated
animals is in a large series of cases the expression of peculiar rela-
tions in the segmental innervation of the skin.

2. In the “skin-design’ taken in its widest acceptation, three
elements ought to be distinguished: the prime-colour, the excedent-
contrast and the defect-contrast.

3. In animals, whose skin is nearly wholly of one colour, the
excedent-contrast may be zonal (dorsal) or isolated.

An isolated contrast frequently corresponds :

for the head

a. with a definite central nerve-area: (the excedent-contrast in
the NV. trigeminus), or else with definite portions of these areas (Point
of entrance of the nerve in the hypodermis; the excedent-contrast
ex introitu; the supraorbital mark).

for the remainder of the body:

6. with definite isolated skin-segments, more pigmented than the
other segments, or with definite sub-divisions of these segments
(caricatures of the dermatoma; segmental excedent variability ; seg-
mental excedent contrast).

e. with zones of intersegmental summation (intersegmental excedent
contrast; the cross on the back of the ass).

4. The defect contrast in animals that are nearly wholly of one
colour may appear as a lack of this colour either zonal (ventral) or
isolated. The isolated defect contrast frequently corresponds with:

a. definite nerve-areas, being situated very eccentrical, either in
absolute or in relative sense. (Tip of the tail, tips of the ears, ventral
median line, frontal median line of the head, toes; they are all
specimina of eecentrical defect-contrasts).

6. with definite non-pigmented skin-segments (phenomena of seg-
mental-omission, segmental defect variability, segmental defect-contrasts).

5. Eimer’s type of the transversally striped animals ought to be
divided into two sub-divisions :

a. animals with broad, dark transversal stripes, which are less
numerous tban the segments of the body (fishes, sauria’s, serpents).
These broad transversal stripes correspond probably with groups of
strongly pigmented segments, alternating with other groups that are

( 314 )

less pigmented. (A transversal serially ranged segmental excedent
contrast).

4. animals with narrow dark transversal stripes, more numerous
than the segments of the body (mammalia, e.g. zebra’s). These stripes
correspond with zones of intersegmental summation. (A transversal
serially ranged intersegmental excedent contrast).

6. Eimer’s type of the animals with longitudinal stripes includes :

a. fishes, in which the dark longitudinal stripes, or else the dark
dots and spots ranged in long rows, correspond apparently with the
points of entrance into the hypodermis of the skin-branches of the
peripherical nerves. (An exeedent contrast ea itroitu).

6. amphibians and reptiles. Probably the precedent hypothesis holds
likewise for these.

c. mammalia. In the viverridae the longitudinal stripes apparently
have been produced by the confluence of rows of spots, which were
originally distributed intersegmentally. (Pseudo-longitudinal stripes).

7. Erer’s spotted type in the mammalia includes:

a. Irregular spotting. This is caused by segmental excedent and
defect-variability.

6. Uniform dotting. We may imagine this to have been produced
by the fragmenting of stripes, that occur un-interrupted in kindred
species of animals (leopards).

Meteorology. — “On frequency curves of meteorological elements.”
Bij Dr. J. P. VAN DER STOK.

1. The application of the theory of probability to the results of
meteorological investigations has hitherto been more limited than the
nature of the data would lead us to expect.

It is not difficult to indicate the reason for this fact. Nearly all
applications of the theory of errors to physical and astronomical
problems are induced by the desire to determine a quantity with the
greatest attainable. precision; the remaining uncertainty affords a
criterion for the value of the different methods employed and leads
to experimental improvements, by means of which the errors, or
departures from the average value, may be minimized.

These reasons for the application of the theory of errors fail in
meteorology: for the greater part of meteorological quantities and
climatological regions it is impossible to calculate average values
within a reasonable time and with a moderate degree of precision
and, if this were at all possible (e.g. for tropical stations), an increase

( 315 )

of precision would scarcely afford any advantage as we are unable
to reduce the deviations by improving the observations. Moreover
the knowledge of the most probable value is of minor importance
as the frequency curves in general are very flat and we cannot
attach the common idea of errors to the deviations which, after all,
are more characteristic of meteorological conditions than absolute
values.

Meteorological constants in the true sense of the word and to
which the methods and terminology of the theory of errors are
applicable, are nearly exclusively Fourier constants, obtained by the
analysis of periodical phenomena such as daily and annual variations,
and to these it is certainly desirable to apply the criterion of the
theory of errors more extensively than has hitherto been the case:
the theory of errors in a plane can be immediately and advantage-
ously used to get a clear understanding of the value of the results
obtained.

If however we abandon this basis of the theory of errors and proceed
upon the lines which have of late been followed by the sociological
and biological sciences, the matter appears in a different light; in
these sciences the principal object to be obtained is not so much the
mean value as the occurrence of deviations, or rather the nature of
the frequency curves.

Monthly means e.g. of barometric heights may be identical for
January and July as far as the absolute values are concerned, but
we may confidently expect the frequency curves for these months
to bear a totally different character. It is also extremely probable
that the frequency curves will show a considerable difference for
places in different latitudes or differently situated in relation to the
main tracks of depressions.

The constants which occur in the analytical expressions for these
curves may then be considered as characteristics of the climate and,
as in meteorology we possess more data than in most other branches
of science, a more thorough study of details is possible.

The principal questions are:

a. In how far are monthly means in accordance with the common
law of probability.

6. What is the form of the frequency curves constructed from
daily means or from observations made at fixed hours in as far as
these curves may be considered symmetrical.

c. An investigation of the skewness of these curves.

In this communication only the first of these problems will be
considered.

( 316 )

2. The material chosen for this inquiry consists of :

Ist. monthly means of barometric pressure at Helder, calculated
for the 60 years period Aug. 1843 to July 1902, the total number
being 720.

2>¢, monthly means of barometric pressure at Batavia for 37 years,
1866-1902, altogether 444 data.

3’, monthly means of atmospheric temperature for the whole of
France during the 50 years period 1851—1900, altogether 600 data.

Up to 1873 the data for Helder have been taken from a meteor-
ological journal kept by Mr. Van brr Srerr and, after his death,
from the annals issued by the K. Met. Instituut.

A Newman standard-barometer at Helder, which is known to have
been in use as early as 1851, has recently been tested and does
not show any appreciable errors, so that it may safely be assumed
that also the records of the station-barometer are sufficiently accurate
for our purpose.

The monthly means for Batavia have been taken from the returns
published by the K. Magn. en Met. Observ., and those for France
from Ancot’s “Htudes sur le climat de la France, Température,”
published in the Ann. du Bureau Central Météor. de France, Année,
1900, I. Mémoires, Paris, 1902, p. 34—118.

Table I gives the results of the calculations for Helder.

Let ¢ be the deviations of the individual data from the cor-
responding general average value and n the number of data available,

then:
Ie it
i , mean deviation,
9 = —, average deviation,
nr
1
h = ——., factor of steadiness,
My2
h' : id
1,’ = ——, idem.
oY 2’
A= number of years required to obtain a general mean value

with a probable error of + 0.1 mm. for the barometric height and
of + 0°.1 C. for the atmospheric temperature.
This number, calculated from the formula:
0.6745 M

= ————_,

0.1
is given instead of the probable error of the result with a view of
showing how difficult, if not how impossible, it is to fix normal

( 317 )

values of meteorological elements, at least in high latitudes. The
application of this formula is justified by the consideration that
monthly means for a given month may, as far as our actual knowledge
goes, be regarded as independent of each other, whereas e.g. daily
means are certainly not so.

If the deviations are distributed according to the normal, exponential

law :
oh aa. (1)
Wa Las Aer eee ORR a5. ONG
the quantity A’ must be equal to 4. Another criterion to ascertain
whether the distribution of deviations is regulated by the normal
law, as advocated by Cornu '), is obtained by calculating 2 by means
of the formuia :
2M
Lee Peo OO Gio. 2)
it is equivalent to the criterion previously mentioned as it holds
only when h=/V'.
The quantities J/ and h may be regarded as a measure of the

TABLE I. Monthly means of barometric height, Helder.

M 3 h h! A T
3 i
January........ 5.01 mm.) 4.04 mm. 0.44 0.440 1149 3.083
February.... .. 4.85 3.82 | 0.146 0.148 1071 3.222
WETGlea Senoaore 4.24 3.45 0.168 0.164 805 2.971
A\}0 Gl ace oe eeOr 3.36 | 2.74 0.214 0.206 14 3.007
WER cso ApODOORe 2.34 | 4-94 | 0.302 0.296 250 3.022
Nit@se gSonen gor 2.22 | 1.76 0.318 0.321 225 3.190
Sui\jeoganebooue 2.09 1.65 0.339 0.343 198 3.212
PAU SUSD cls ce'sie)- 2.12 1.69 0.334 0.334 206 3.158
September..... 3.01 2.45 0.235 | 0.230 AA 3.079
Octabers. «.1-- 3.35 2.62 0.214 0.215 510 3.262
November.. ... 3.74 3.02 0.489 0.187 637 3.063
December...... 4.99 4.00 0.142 0.441 1132 3.106
MICH noo oir i a

1) Ann. de l’Observ. de Paris. XIII, 1876.

( 318 )

variability and steadiness of the climate from year to year in so
far as this is determined by the oscillation of the atmospheric pres-
sure. By analogy to the secular variation of the elements of terrestrial
magnetism this instability might also be called secular variability.

Assuming this criterion to be correct, it appears from Table I
that there is every reason to suppose that at Helder the deviations
follow the normal law, the average value of a not differing more
than 1.8°/, from the real value.

On comparing the climate at Helder, which is highly variable
from year to year, with the climate at Batavia (in so far as in this
ease also the variability of atmospheric pressure may be taken as a
measure), we find totally different conditions.

A period of about ten years for the Eastmonsoon, and of twenty
years for the Westmonsoon months is already sufficient to obtain
total monthly means of the barometric height with a probable error
of + 0.1 mm. and for the dry months the available series of
37 years is quite sufficient to obtain a degree of certitude twice as
great.

TABLE II. Monthly means of barometric height, Batavia.

| M s h h' A t

January........ 0.84 mm.) 0.71 mm.) 0 845 0.792 32 2.759
February.......|| 0.75 0.62 0.938 0.917 26 3.004
Marchtyaacitelte 0.63 0.52 4.415 1.085 18 2.974
iNpril. Setieee as 0.42 0.36 4.704 4.581 | 8 2.715
May prrsescrets 0.44 0.32 1.603 Aerio 9 3.752
ITN SG 6 onodeae 6 0.40 0.28 Ao 1.990 7 3.934
Unb aes e otaor .. || 0.44 0.34 1.604 1.640 9 3.282
INTERTOR ooodoet ¢ 0.47 | 0.33 41.492 1.689 10 4,028
September ..... 0.44 0.35 1.598 1.606 9 3.173
October........ 0.54 0M 1 375 1.370 412 3.418
November. ..... || 0.65 0.53 1.088 1.063 19 | 22999
December...... 0.64 0.49 1.166 1.155 17 3.086

Meant. ccc: 3.235

The application of the criterion as to whether the deviations follow

( 319 )

the normal law leads to a far less satisfactory result for this place
than for Helder. The two values h and h’ of the factor of steadiness
show considerable and systematic discrepancies, the calculated values
of x for May to August being collectively too great, and those for
the other months too small. Although the total mean, 3.235, does
not differ more than 3 °/, from the real value, these differences
amount to + 15.7 °/, in the five dry months and to —6.5 °/,
in the seven months of the wet season.

Here, therefore, the secular variability cannot be regarded as a
purely accidental quantity unless another law, more complicated
than the normal one, applies and which is in some degree dependent
upon the monsoons. This might be the case if the atmospheric pressure
were dependent (and in a different manner in different seasons) upon
another factor, for instance the temperature, the variability of which
might still be according to the law of accidental quantities.

Similar systematic differences, varying with the season, between the
calculated and the real value of a are not apparent in the results
of the calculations for the atmospheric temperature in France, and
the general average value of 2 does not differ from the real value more
than 0.13 °/,.

TABLE III. Monthly means of atmospheric temperature, France.
M g h h' A T
ola

January........ 2.07 C. 4.73 C. | 0.34 0.326 495 2.869

February....... 2.03 1.70 0.356 0.382 188 2.855

Marchisn cc... ' ta) lp deze 0.446 0.452 4115 3.230

ENE) Foatec boen 1.20 | 0.92 0.588 0.616 66 3.444

WEN fog omen eee 1.32 1.07 0.536 0.529 79 3.067

UaliGioa seoaborise 1.14 0.94 0.629 0.623 59 3.450

itt be oy ope o oer 4.29 1.00 0.548 0.565 76 3.347

August ....-... 1.08 0.88 0.653 0.644 53 3.029

September..... 1.49 0.94 0.594 0.600 64 3.205

October. .\.,.)..:.- 4.25 4.02 0.565 0.551 71 2.994

November...... |} 4.50 4.22 0.472 0.464 102 3.043

December...... 2.44 1.84 0.294 0,306 264. 3.418

NRCG op Gooaee | 3.137

( 320 )

In the paper already quoted, Mr. Ancor assumed that the deviations
do not show systematic differences in different months, and he sub-
jects the deviations taken conjointly to the criterion of the law of errors.

This assumption is not justified by the results given in Table III,
from which it is evident that the values of 4 are subject to consider-
able and systematic variations and, if a satisfactory agreement
is still found between theory and observation, this can only be
accounted for by the fact that the probability of the occurrence of
deviations between fixed limits is expressed in a number of decimals
too restricted to indicate the differences which, as for Helder and
Batavia, must here exist between theory and _ practice.

No more can it be affirmed that, if a satisfactory accordance exists
between the calculated and the observed number of deviations
between given limits, the average value will also be the most
probable one. In applying this criterion, as well as in calculating
h' and a, a possible (and probable) skewness of the frequency curve
is not taken into account because, by treating the deviations without
regard to their sign, symmetry with respect to the ordinate of the
centre of gravity of the figure is tacitly assumed.

As the number of years over which the observations extend is
still far too small to allow frequency curves to be drawn for each
month separately, it is still worth while to consider the deviations
collectively, provided that at the same time the question be put,
what form the law of deviations will assume when they are com-
posed of groups which individually follow the normal law, the factor
of steadiness being different for different groups. Even then the
available data are insufficient to indicate with certainty a small
degree of skewness in the frequency curve, so that only the sym-
metrical form can be sought for.

3. If, as in our case, the different groups oceur with equal (sub)
frequency, it is not difficult to indicate im what respects such a
curve, the resultant of many elements, must differ from the normal
curve. The groups characterised by large factors of steadiness will
raise the number of small deviations above the number correspond-
ing with an average factor and contribute only in a small degree
to the number of large deviations, whereas, on the contrary, flat
curves with small factors will give rise to a greater number of large
deviations than is consistent with the normal law. Deviations of
average magnitude will then occur to a less degree than is required
by the common law ; consequently in drawing the two curves, they
will be seen to intersect at four points, as a minimum.

( 324 )

In a paper‘) published some years ago, Scots has drawn the at-
tention to the fact that differences of this description are almost always
found when sufficiently extensive series of errors are put to the test
of the normal law; in this paper he shows that these differences
cannot be explained by the omission of terms in Bussr1’s develop-
ment of the exponential law and suggests that their origin must be
sought for in the superposition of observations of different degrees
of precision.

In the observations alluded to by Scnots, it will in general not be
possible to estimate these degrees of precision any more than the
relative subfrequencies with which the different groups are represented
in the result; in the case of monthly means such as are being
discussed here, the factors of steadiness are approximately known
and the subfrequencies of the different groups are all identical.

If we arrange the 12 groups according to increasing values of h,
it appears that we may take its change to be uniform ; consequently
it is possible to find an approximate solution of the problem in
finite form.

We have then to consider 4 as a variable quantity z and to ask
what form the expression will assume for a sum of elementary surfaces:

ao

cf Ra ee eer Ee Ty Ge cl. (G))

—o

if ¢ varies in a continuous manner from / to H. If the subfrequency
of these elementary groups be also regarded as a function of z
(which occurs e.g. in the case of wind-frequencies), (3) must be
equated to g(z)dz, g(z) being subject to the condition :

H
fy@ar=1 Smee) On a) rol cae von (Ee)
h
The constant C' is determined by the expression
: 29(2)dz
Gq HORS Ts ee eae
Vn
and if, as in our case,
g (2) =e
1 edz
¢= —— rn
H—h (H—h)y/x

1) Vers]. Wis. Nat. Afd. K. Akad. Wet. I. 1893 (p. 194—209).
22
Proceedings Royal Acad. Amsterdam, Vol. VIII.

the resulting probability of a deviation being situated between « and
«+ dz is then:
H

da 2 on J
anya S* S wey

and the equation of the frequency curve:

& il eax? __ p— Hx
Swen |

Developing this expression we may put:
IPI

EEL, = Gast (2h)!
Y= sane é [2 -L 3 a -f- Sunn wv. | . (7)
If we put:
Un = af omy da
0
we find with the help of:
2 fo mid ti r=)
2
0
and
*e—Pt—e— 4 (
———— dz = log -,
. og 12

0
for the moments of different order with vespeet to the maximum
ordpnate:
|
uy = » Sad =
2h
1 H 1 GIS)
—— log , B=
(H—A)V/ a h 2V% Ah?
From a series of deviations following the law (6) the two character-
istie constants HT and A can be derived by computing the moments of
the second and third order. They are found to be equal to the roots

(8)

BL, = =

of the quadratic :
X—pX+q=—0

U,V 0 1
pee a, ee (9)
au,” 2u,

If we had put a similar series to the test of the normal law (4)
we should have found for the equation of the frequency curve:

( 323 )

y == Hh e—Hhx?
fy 4
or
Se G25 0
LEN Se (71—h)? : (2)
ae Ag oon. hag

nD

On comparing this expression with (7) it is at once seen that in
this manner too great a number of small deviations must be found,
as the module of the deviation zero, computated by (10)

“Th
Ee
is always smaller than that derived from (7):
H-+-h

The position of the four points where the two curves intersect
are found by equating the expressions (7) and (10); if the development
can be stopped at the third term they are given by the roots of the
biquadratie :

‘ (OO? GOO SGU Rol) GS ola Jom 9 - {(Iluh))
ay ae —- V h(H 1) ge 4 V ih

4(VH—yh)?
> VG:

With the help of the form. (8) for @, it can be shown that, if a
series of figures follows the law (6) the computation of a according
to (2) must necessarily lead to values which are somewhat too high :

2 Eh) fe Es

ie © ( log: ]

ue Fh ar h
Putting :

we find:
q eG) BCR:
loc ey, 1
og 2( ese Sg!
q’ q
9 I eee as
Uy P P 12
gala a 7 ie eee sme ecne sla)
eRe ie ete
tas 5 pt )

4. In the following applications of these reasonings to deviations
taken collectively for all months, the frequencies are reduced to a
total number of 1000: by exponential law is understood the simple,
normal law of errors (1).

22*

( 324 )

TABLE IV. Barometer, Helder.

Dev. mm. || Observ. | Exp. L Diff. Dev. mm. Observ. | Exp. L. | Diff.
a en ee
0.0°—0.45 104 | 100 + 4 5.95—6.45 214 25 lg 4
0.45—0.95 |} 199 | 108 +21 | 6.45—6.95 47 49, | —32
0.95—1.45 |] 121 | 406 | +45 | 6.95—7.45 Ae | 45y ee
1as—1.95 || 101 | 4100 | a= 7805 7 417 || es
1.95—2.45 97. | 92 | +5 | 7.95—8.45 18 8) | eho
2. 45—2..95 86. | 8%. |) Seo ems maton eelbere ako
2,953.45 || 68 7 | —7 | 8.95-9.45 || 40 Rate hats #5
3.45—3.95 50 65 | 45 9.45—9.95 7 | 3 | +4
3.95—4.45 43 | 56 As 9.95—10.45 Vie Bella 9) 0
4a =4.95. ||) “384 Sar Pls Saito ==10095)|| Meese 0
4.95—5 45 3 39 | —-8 [40.9511 45 OF 1 70 0
5ie-5 195 lna5 EO) alluesrye Mieateeall 3 9 eg

[ese CeO Oe es =) N28 Oil yy idle be
h (Exp. L) = ea = \/Hh = 0.1971,
Vu,
zx (form. 2) = 1.069 x 3.142.
Points of intersection observ. near dev. 2.95 and 7.95,
H = 0.2712 , h = 0.1433 (form. 9).
x (form 12) = 1.044 3.142.
Points of intersection (form. 1J) at dev. 2.60 and 9.19.
The sums of the differences between the limits of the observed
points of intersection are, as given in Table IV, +48, —70, +22.
If we also wish to compare these quantities with the result
of the theory, we have to integrate (6) between the limits deduced
from (11). For the limits @ and zero we find the frequency :
aH ah

1 2 :
3 [eA Hle—?dr—hle-"dr | (18
Hine etna —— if ri fe ‘| vee
0 0

By means of this formula we find between the limits calculated
by means of (11):

a Form. (6) Form.(10) Diff. Obs.
O— 2.60 558 509 4-44 +48
2.60—9.19 440 476 —3 —70

6
9.19—ete, 7 15 228 “1199

(325 3)

As the situation of the second point of intersection according to
the observations (7.95) shows a rather large discrepancy with that
given by theory (9.19), it is natural that only the sums of the
positive differences between the limits zero and the first point of
intersection agree closely.

Taken as a whole it may be stated that the secular variability of
barometric pressure at Helder is regulated by the law of accidental
events as completely as might have been expected considering the
scantiness of the material available.

A possible skewness of the curve is left out of consideration as
has been already remarked; it can, however, be but unimportant as
in 720 deviations 364 are positive and 356 negative.

The same cannot be ascertained of the secular variability of baro-
metric pressure at Batavia; the differences between the observed
frequencies and those calculated according to the exponential law
are not of such a well marked description as for Helder, so that a
determination of the points of intersection is out of the question ;
their situation can only be calculated as a result of theory.

TABLE V. Barometer, Batavia.

Dev. mm. Observ. | Exp. L. | Dilf. Dey. mm, || Observ. | Exp. L. | Diff.
|

0.000—0.02 || 446 | 435 | 441 | 0.995-1.095 || 34 % | +9
0.095—0.195 | 149 | 136 | 443 | 4.095-1.195]) 7 | 18 | 14
0.495—0.995 || 126 | 129 | —3 |4.195-1.995]] 7 | Ae ais
0.295—0 395 || 117 | 4118 | = 4} 4.995—1.395 0 | 8 JE 5)
0.395—0 495 101 | 104 | —3 1 .395—1 .495 5 | 5 0)
0.495—0.595 | 95 | 89 | +6 ]1.195-1.595]/ 7 | 3 | 44
0 595--0.605 || s0~| 74 | 24 | 1.595-4.695 | 2 a
0.695—0 795 || 52 | 59 | —7 |1 6051.79) 29 | 4 | 44
0.705—0.805 || 59 | 46 | +43 |1.795-1.80]) o | 4 | —4
0,895-0.995 | 29 5 a) ==6 | deeioectes TI somaemor temas

u, = 0.43805, pw, = 0.8156, p, = 0.2915,
h (Exp. L.) = 1.2586, = (form. (2) = 1.040 3.142,
7 = 1989, f= 10.006:
ge (form, 12)) = 1.0116 S€3:142,

( 326 )

Points of intersection (form 11) at dev. 0.399 and 1.620, For the
sums of deviations between these limits we find (form 13) :

a Form. (6) Form. (10) Diff. Obs.

0 — 0.399 559 522 + 37 + 17
0.399 — 1.620 431 474 — 43 — 19
1.620 — ete. 10 dq + 6 + 2

It appears from these results that the calculation of « cannot
always be regarded as a good criterion of the variability being
regulated by the law of accidental events. From a_ series of
numbers, composed, as the barometric departures for Batavia are, of
groups which follow neither the simple normal law nor the more
complicated law (6), still the calculation of a leads to a value which
is correct within 1°/,

TABLE VI. ‘Temperature, France.

Dev. C°. ones | Exp L.| Diff Dev. C°. |] Oserv. | Exp.L. pitt

| | |

i
0.00—0.15 73 78 | —5 | 2.45-2.35 || 27 36 *| Sexe
0.15—0.35 || 413 | 101 | 442 | 2.95—9.55 18 29 eal
0.35—0.55 || 108 98 | +410 | 9.55-9.75 8 24 | —16
0.55—0.75 || 87 9 | —8 | 2.75~9.95 || 95 19. || =E%6
0.75—0.95 |} 100 88 | +42 | 2.953.145 15 15 0
0.95-1.15 || 83 82 | +4 | 3.453 35 12 i ee
4.45—1.95 || 97 | 74 | 4-3 |) 3:35—3.55 8 es
1.35—1.55 |} 60 66 | —6 | 3.55—3.75 12 6 2 heats
4.55—1.75 || 70 59 | 441 | 3.75—3.95 5 5 0
1.75—1.95 |] 58 50 | +8 | 3.95—4.15 2 3°: aan
1,952.15 |] 30 43 | —13 | 4.45—ete. 9 9 0

2, = 1.207 9a 2.394 5 a 6.7103:
h (Exp. L.) = 0.4570, 2 (form. (2)) = 1.046 & 3.142,
FL 016275 hi O2739;
x (form. 12) = 0.140 & 3.142.
Position of points of intersection (form. (11) at dev. 1.09 and
4.87. Sums of deviations between these limits (form. 13) :

at Form. (6) Form. (10) Diff. Obs.
Oe 10S) 565 519 + 46 + 22
OS) a My 429 480 — 51 — 22

As a general result of this investigation it can be stated that,
aceording to theory, in all three series the number of small deviations
is greater than the simple exponential law would require, but to a
somewhat less degree than would follow from the law formulated
in (6).

The deviations of barometric pressure at Helder are in almost
perfect accordance with this frequency law and, therefore, for each
month separately with the normal law; the curve of deviations of
atmospheric temperature in France still shows many irregularities,
but, in general, it accords well with the law of form. (6); the
secular variability of atmospheric pressure at Batavia is not regulated
by the law of accidental events and its frequency curve shows
characteristic peculiarities in ‘different seasons.

Microbiology. — “Methan as carbon-food and source of energy
for bacteria’. By N. L. Sénncex. (Communicated by Prof.
M. W. Brwerinck).

Methan, which is incessantly produced from cellulose in the waters
and the soil, through the agency of microbes, and which, since
vegetable life became possible on our planet must have been formed
in prodigious quantities, yet occurs only in traces in our atmosphere.

As this gas is very resistant against chemical influences its dis-
appearance in this way is highly improbable. But the conversion of
methan into carbon dioxid and water produces a considerable quan-
tity of heat, and so it seemed worth investigating whether there should
exist any organic beings capable of feeding and living on it.

In the first place green plants were examined as to their power
of decomposing methan in the light. To this end some waterplants
were chosen, which seemed ito offer most chance of suecess, con-
sidering that the formation of methan, as an anerobie process, takes
especially place in stagnant waters.

In this way positive results were obtained with several species
of plants as Callitriche stagnalis, Potumogeton, [lodea canadensis,
Batrachium, Hottonia palustris, Spirogyra. So, tor example, in
one of the experiments in the light of a window to the North,
with Hottonio palustris, put in a flask containing 500 cc. of methan
and 500 ec. of oxygen, and inversely placed in a vessel filled with
water, all the methan disappeared from 7—21 May, so within
a fortnight.

( 328 )

In the dark, also, absorption of methan was with certainty observed.

However, the lapse of time preceding the first perceptibility of
the process in different experiments with the same species of plant,
varied very much, but when once set in it went on rapidly. When it
was moreover observed, that by carefully washing the plants the setting
in of the absorption was much slackened, whilst it seemed probable
that just then an acceleration would follow in case the plant itself
absorbed the methan, and especially when furthermore the absorption
was observed to take place only after a slimy film had covered the
water in the flask, it became evident that the oxidation was not
caused by the green plant itself, but by microbes living on it surface.

In order to study the process more exactly an apparatus was
constructed allowing us to pursue the absorption as well qualitatively
as quantitatively.

It consists, as shown in the figure, of two Ertenmryer-flasks of
+ 300 cc., each closed by an indiarubber stopper with two perfo-
rations and joined by a twice curved glass tube reaching to the
bottom of the flasks, which bears in the middle a glass cock. The
flask, destined for the cultivation of the bacteria, bears, in the second
perforation of the stopper, a tube with a glass cock to admit the
gasmixture; the other flask is fitted with a glass tube filled with
cotton wool.

The use of this apparatus is as follows: For the erude culture
the first mentioned flask is quite filled with the culture liquid t

( 329 )

Destilled water 100
k? HPO: 0,05
NH‘ Cl O01
Mg NH‘ PO‘ 0,05
Ca SO* 0,01

and inoculated with garden soil, sewage or canalwater, of whieh
the two last cause the quickest growth.

By the cock on the first flask a measured quantity of oxygen and
methan is admitted by means of a gas burette. The liquid is there-
by pressed into the other flask, and when it has lowered until a
layer of about 1 cM. remains in the first flask, then the middle-
cock is shut and at last the admission-cock.

The cultivation is effected at about 30°C. After a period, varying
from 2—4 days, a film is observed on the liquid, which rapidly
increases in thickness and then shows a distinet pink colour. Beneath
the film the liquid, clear at first, begins te display a considerable
turbidity caused by foreign microbes, which feed on the dead bae-
terial bodies of the floating film. Later on a great number of amoe-
bes and monads develop in the film and in the liquid, evidently at
the expense of the methan bacteria, no other material for food being
present. In the other flask no film appears on the liquid.

Transports to a same liquid in an apparatus like the former, easily
produce a new film, and, when garden soil is used for the infection,
it grows even faster than in the crude culture.

An analysis of the gas after about a week, shows that the methan
has quite or partly disappeared whilst a considerable quantity of carbonic
acid is formed. The film is found chiefly to consist of bacteria be-
longing to one single species, which has proved to be the microbe
which makes the methan disappear. It is a short, rather thick
rodlet, immobile in the film, mobile or immobile in the plate cultures.
Always the individuals are united by a layer of slime.

The length of this bacterium, which will provisorily be called
Bacillus methanicus, is 4—5 uw, its thickness 2—3 «.

It is not yet ascertained whether this species has already been
found under other conditions of life and described elsewhere without
the knowledge of its relation to methan. The question whether there
exists only one or more than one species possessing the faculty to
live on methan is also subjected to further investigation.

The methan bacterium is easily obtained in pure culture by eul-
tivation on washed agar, containing the necessary salts, at a tem-
perature of circa 30° C,, in an atmosphere of */, methan and */, air,

( 330 )

with which an exsiceator is filled and into which the plates are
introduced.

sy streaking a young film from a liquid culture on the said solid
medium already on the second day nearly pure slightly turbid
colonies are obtained, quite distinguishable by their size and their
slimy and lightly pink-coloured appearance. Such a colony, when
early inoculated into the above apparatus forms, after some days,
another bacterial film.

The methan, being in all the experiments the only source of carbon,
necessarily at the same time must serve as food and as source of
energy.

The quantity of carbonic acid in the culture flask indicates
the amount of methan which has served as source of energy.
The quantity of methan used for the formation of the bacterial
bodies may be measured by subtracting the quantity of produced
carbonic acid, expressed in ce., from the volume of disappeared methan.

So for example it was found that in an experiment in which were
added successively 225 ec. CH* and 320.7 ce. O? to 102 ce. ot
liquid, the flasks contained after a fortnight

78 cc. CO*
no», .CH*
A 2cCGx we Oe:

In the culture liquid 21 ce. of carbonic acid were solved, so that
126 ee. of methan had been assimilated for building up the bacterial
bodies, and 78 + 21 ce. CH* for the respiration, 148.7 ce. of oxygen
being assimilated.

Another experiment gave the following result.

Sueccessively added 200 ec. CH*
and 331 cc. O°.
to 108.5 ee. liquid.
After two weeks the gas contained
72.8 ee. CO
39 ec. CH*
138 ce. OF

In the culture liquid 18 ce. of carbonic acid were solved. Hence,
73.2- ce. CH* had been assimilated for the formation of the bacterial
bodies, whilst 90.8 ec. CH* were converted into CO’.

Some oxidation experiments were performed with permanganate
and sulphuric acid, in order to prove that a large quantity of organic
material had aceumulated. Thus, 100 ec. of the culture liquid,

described in the first experiment, consumed :

N. L. SOHNGEN. “Methan as carbon-food and source of energy for
bacteria ”

Bacillus methanicus (800).
Crude film on culture liquid in methan-oxygen atmosphere. Between the
bacteria mucus occurs.

iy
a

D5 a

Bacillus methanicus (1000).
Pure culture on agar with salts in methan-oxygen atmosphere.

Proceedings Royal Acad. Amsterdam. Vol. VIII.

(2305)

1 .
Before the cultivation 0 ce. 70 normal KMnO?,

After the cultivation 48.3 _,,

second experiment 100 cc. consumed :

” ”

At

[)

; 1
Before the cultivation 0 ce. 10 normal KMnO‘.

After the cultivation 26.5 __,, 93 3
Even this rough estimation gives the convincing result that much
organic matter is formed from the methan. Hence it follows that
methan is the starting point for the production of a relatively rich
flora of microbes, which as said above, may even at an early period
contain amoebes and monads living from the methan bacteria.

There can thus be no doubt but methan is, though indirectly, of
importance as a fish-food in the waters, as the said flora certainly
serves as such.

Further investigations concerning the natural history of the methan
bacteria and the relation between the assimilated methan and the
amount of organic matter produced are in execution.

H. Kaserer (Zeitschrift fiir das Versuchswesen in Oéesterreich,
Bd. 8 p. 789, 1905) seems also to have observed bacteria living
on methan, but he gives no particulars.

Microbiologie Laboratory
of the Technic High School at Delft.

Physics. — “Determination of the Tnomson-efiect in mercury.” By
C. Scuoutr. (Communicated by Prof. H. Haga.)

This determination has been executed as a sequel to that, undertaken
by Prof. H. HaGa, and published in the ‘Annales de I’Ecole Poly-
technique de Delft, I, 1885, p. 145; III, 1887, p. 43.”

A detailed account of the way, in which the experiments were
carried out, has been given in my ‘Dissertation’. The results
mentioned here were partly obtained afterwards.

The value of the THomson-constant was expressed by a relation,
got by integration of the differential equation, which Vurprer has
given for the points of an unequally heated homogeneous conductor,
when an electric current passes through it.

If the distribution of temperature is considered, after it has grown
constant, and in some portion of the conductor, confined by two parts
of a constant temperature, this equation is integrable, and the integral
is quite simple for the points halfway between these limits of constant

temperaiure, when all over the part between them the external

exchange of heat, by conduction, convection and radiation is small

enough to be disregarded with respect to the other thermal effects.
The THomson-constant 6 may then be expressed :

wherein 7 represents the strength of the current; 2 the resistance ; J the
mechanical equivalent of heat; q the section of the conductor; U the
difference of temperature between the two parts of constant temperature ;
/ the distance between those two parts; 2 47, the change of tempe-
rature which manifests itself in the middle-section when the current
is reversed; and 4 wu the rise of temperature in the same section
according to Joule’s law.

In order to be able to measure 4472 instead of 247;,u the
mercury was investigated in a U-shaped glass tube, put ina vertical
position, the curved part up. The upper part of this U-tube was
enclosed in a glass bulb, in which different fluids (acetone, water,
aniline, glycerin) could be kept boiling by an electric current. In
this way the upper part was kept at a constant temperature. For
the same purpose the bottom parts of the legs of the U-tube,
which were closed by small rods of platinum, were placed in
running tapwater.

In the parts of non-uniform temperature this temperature was
measured in sections halfway between the constant limits. If, after
the current has been sent through in one direction, there should
exist a certain difference of temperature between the two middle-
sections, this difference will suffer a change of 447; by reversing
the current, if the condition about the external exchange of heat is
fulfilled.

Therefore the parts of non-uniform temperature were enclosed in
a large vacuum-tube, for the greater part of glass, with a brass
bottom and, for the sake of practical advantages, the glass boiling
bulb and part of the condenser upon it were also enclosed in this tube.

In order to measure 4 wu, separate experiments were made, with

us nearly as possible the same current. By making the current
vo first through one leg and then through the other the diffe-
rence in temperature of the middle-sections was varied by 2 A

For measuring the temperature in the mercury the thermo-electric
difference between this metal and platinum was used. Different kinds

999
( O00. )

of platinum acted quite differently in this regard. The strongest
thermo-currents were obtained with Pt Ir of 10 to 20 °/,. A -wire
of this platinum was fused into each of the legs of the U-tube, as
aceurately as possible in the middle-section. These wires being con-
nected and a sensitive galvanometer being introduced into the cirenit,
the temperature-differences Arne and Au could be measured in

proportion. Should we have wished to measure each of those quanti-
ties separately, it would have been necessary to determine the thermo-
electric constants of this platinum with regard to mercury.

The unequality in temperature in the middle wires caused by an
inevitable lack of symmetry in the U-tube was compensated by
means of another thermo-couple. After each series of observations
the galvanometer deflection, given by this couple with a known
resistance and a known difference in temperature between the points
of contaet, was measured, in order to eliminate changes in the sensi-
bility of the galvanometer or in the distance of the scale.

+

The quotient oy eS determined indirectly. If the external exchange

of heat could be neglected, the temperature-gradient must be the
same all over the parts of non-uniform temperature, so long as the
eurrent did not pass through the mercury, apart from the distribution of
temperature near the limits. And in the middle-seetion the gradient
of temperature would remain very approximately the same, when the

=

: ; Rane
current did pass through it. Therefore ihe quantity i could be said to

be equal to the temperature-gradient in the middle-sections.

To measure this gradient in each of the legs of the U-tube on
both sides of the middle-section at a given short distance both above
and below it, another wire of platinum was fused in. The tempe-
rature-difference between these sets of wires divided by their distances

U
was put for ris

The wires last mentioned were of a kind of platinum of which
the thermo-electrical constants with regard to mercury had been
accurately determined beforehand. As the same thing cannot be said
about the wires in the middle-sections it is impossible to say any-
thing definite about the uniformity of the gradient resulting from the
experiments as they have been made. Preparatory experiments
however have shown, that when 7 does not exceed certain limits,
the gradient is sufficiently uniform.

Much trouble has been caused by wild thermo-electrie currents.

( 334 )

Especially in acommutator for the galvanometer-current these difficulties
arose. Contacts made by solid homogeneous copper have given the
greatest satisfaction. With this arrangement for measuring the
temperature the current through the U-tube, the chief current, had
to be cut off for a moment during the reading of the galvanometer.
Therefore the galvanometer commutator was combined with an inter-
rupter for the chief current.

Changes in the meridian during the experiments were eliminated
by noting, before the deflection, the position of the galvanometer-
mirror when at rest. This position was more or less affected by the
magnetic field of the chief current, but this obstacle was overcome
by systematically combining readings with reversed chief current
and galvanometer-current.

The galvanometer, made by CarprntierR, was of the THomson-type.
Provided with a sensitive set of magnets after PascuEn, suspended
by a quartz-fibre of + 7a, with electvomagnetical damping and
with coils of small resistance (2,76 £2), this instrument answered to
all the special requirements of the problem.

The strength of the current was determined by measuring the drop
of the potential at the ends ofa known resistance, and comparing this
with that at the poles of a Wesron-element. The potential differen-
ces were measured with a five-cell quadrant-electrometer (H. Haga,
These Proc. I p. 56).

The course of the experiments was the following :

A sufficiently Jong time beforehand the fluid in the boiling-reei-
pient was set boiling and the tapwater was allowed to run. Then the
current in the (tube was closed. When the distribution of the tem-
perature had erown constant, the positions of the galvanometer resp.
when at rest and detleeted were read. After five moiites these readings
Were repeated, but now the conmutator tor the galvanometer was closed
in the opposite direction. Then the current in the U-tube was
reversed and after LO or 15 minutes the galvanometer-readings were
resumed. In a corresponding way the measuring of the Joule-heat
was carried out.

In each series 8 deflections were read, as well for the determina-
tion of Ay,w as of Aw; first four of one quantity, then eight of
the other, and again four of the first. In the meanwhile during the time
necessary for the temperature to become constant, the current strength
was measured from time to time, and the temperature of the run-
ning water was read. In this way the following results have been

obtained ;

53°

58°

II (100°
154°

(335)

eo?

73

80

90
108°
124

The values I are averages of the results of four series each, which

have been given in my “Dissertation”.
The values II have been obtained with another similar instrument

it
500
OSS ;
°

300 = “ si :
200 ———
400 7

— 95 —h0 —75 —100 —125

+ Series I.
® Series IL.

ax 108

( 336 )

under about the same conditions. They represent the averages of
resp. 2, 2 and 1 series.

The meaning of those values for o is: When a current of one
ampere passes through a column of mereury, the THomsoy-effect will
cause a quantity of heat, equal to 6 (expressed in gram-calories) to
be developed in one second between two consecutive sections of the
temperatures ¢— 4° and ¢-+ 4°, if the current goes in the direction of
the increasing temperatures.

As the diagram added shows, the values I and II for 6 lie all but
in straight lines, passing through the origin, which means, that the
Tuomson-effect is proportional to the absolute temperature (7’).

’ Oo ee
The values II give fo S<10-") and the combination of

Oo
I and II give a 260 XA0=.

It is not clear what has caused the difference between I and II.
May be it is the effect of some difference in purity of the mercury
which is known by experiments on other substances to strongly
affect the THomson-constant.

Chemistry. — Prof. Francuimonr presents a communication from
Dr. D. Mon on an investigation commenced in 1903 as to
the “ester anhydrides of dibasic acids.”

Of the anhydrides of organic dibasie acids but very little is known ;
only the internal anbydrides, which cannot be formed except in
those cases where the position of the two carboxyl groups in the
molecule is stated to be favourable, have been investigated. But in
some cases at least we may expect others formed in the same manner
as those of the monobasic acids, namely by the co-operation of two
molecules instead of the exercise of the two functions of the same
molecule.

We may equally expect that when the dibasic acid has passed
into a monobasic one, for instance by changing one of the acid
functions into an ester or a salt, this will anyway yield an anhydride
in the same manner as other monobasic acids.

Of some mixed anhydrides which are also esters we know, for
instance, the ethyloxalylehloride but not the simple anhydrides.
One of the chief methods of preparing the simple anhydrides is the
one apphed by Geruarpr in 1853, namely, the action of acid
chlorides (mixed anhydrides) on salts. It is this method which, at
any vate with oxalic acid, has at once yielded the desired product,

( 387 )

Dr. Mot allowed ethyloxalylehloride to act with the usual pre-
cautions on the potassium salt of acid ethyloxalate covered with
ether and obtained a colourless liquid which distilled at 85°—90°
under a pressure of less than 1 millimetre, solidified on cooling and
then melted at 4°. The results of the elementary analysis and of the
determination of the molecular weight agree with what is required
by the desired anhydride

O O
ae Bs
C—— O——C
| |
eo) | eo
Oe ie
Nou, oc,H,
. ethylovalanhydride.

as does the decomposition by water. On being heated at the ordinary
pressure it is decomposed with evolution of gas.

Dr. Mo. obtained this substance in a still simpler manner by
acting with oxychloride of phosphorus on an excess of potassium ethyl]
oxalate. The investigation is being continued with other dibasic acids.

Chemistry. — ‘“TVhalictrum aquilegifolium, a hydrogen cyanide-

yielding plant.” By Dr. L. van Trani. (Communicated by
Prof. P. van Rompuren).

The communications from GuicNarp (Compt. rend. de Acad. des
Sciences du 24 Juillet 1905) as to the presence of a hydrogen cyanide-
yielding glucoside in the leaves of Sambucus nigra L. and other
varieties of elder have induced me to continue the experiments
previously made in the same direction. I have been able to confirm
the observations of GuiGNarD in every particular notwithstanding the
figures which I found for the HCN-content are lower than those
stated by him. This may, probably, be explained by the fact that I
did not test the elder leaves until the beginning of September whilst
GuieNarp made his experiments in June.

From 100 grams of fresh leaves of Sambucus nigra I obtained
8,3 milligrs. and from 100 grams of Sambucus nigra var. laciniata
7,7 milligrs. of HCN. No HCN was obtained from 100 grams of
Sambucus Ebulus.

The ornamental plant Thalictrum aquilegifoliam (which appears to

23

Proceedings Royal Acad. Amsterdam. Vol. VIII.

( 338 )

erow wild in the environs of Nijmegen) appears, however, to be
comparatively rich in HCN-yielding material.

If the leaves of this Ranunculaceus are crushed and digested with
36° a hydrogen cyanide-containing

water for 12 hours at 30°
distillate will be obtained on distillation.

The distillate from 100 grams of fresh leaves collected on Sept. 14
in the botanical garden of the Veterinary School yielded 248,8
milligrs. of AgCN = 50,2 milligrs. of HCN = 0,05 per cent. A
volumetric experiment which showed 53 milligrs. of HCN confirmed
this result.

A third experiment made with leaves, kindly forwarded to me
from the Botanical Gardens at Groningen, gave 0,06 per cent of
HCN in the distillate obtained from the same quantity of leaves.

I failed to obtain any HCN from the root of the plant and 142
erams of the fresh stem only yielded 4,4 milligrs. of HCN.

The leaves of Thalictrum aquilegifolium are therefore, compara-
tively rich in HCN-yielding material

No HOCN-containine distillate could be obtained from Thalictrum
flavum, Thalictrum minus and Thalictrum glaucum.

Hydrogen cyanide could not be detected in the leaves in the free
state. When fresh leaves were immersed in hot alcohol no HCN
could be detected in the alcoholic distillate.

The hydrogen cyanide is formed during the digestion and is, there-
fore, most probably liberated from a glucoside by the action of an
enzyme,

This enzyme is probably closely related to emulsin. I have obtained
it, in an impure condition, by extracting the fresh, crushed leaves
with water, and adding to the filtrate a large amount of alcohol.
The precipitate so obtained was carefully dried; it very readily
resolved amygdalin.

The glucoside present in Thalictrum aquilegifolium is not identical
with amygdalin but is probably so with phaseolunatin isolated from
Phaseolus lunatus by Dunstan and Henry (Proc. Royal Soc. LXXIT,
482, 1903), because in the hydrogen cyanide-containing distillate acetone
can be detected, but no benzaldehyde. The presence of the former was
shown from the iodoform-reaction with ammonia and tineture of
iodine and the solubility of freshly precipitated mercuric oxide in the
distillate.

Owing to the small quantity of leaves at my disposal it was
useless to attempt the isolation of the glucoside in the pure state. I
intend doing so next year, and also to watch the development of the
glucoside in the plant.

( 339 )

I may, however, state provisionally that this glucoside is either
insoluble or at most very slightly soluble in cold alcohol. When the
leaves, after being dried in an airbath at 80°, and then powdered,
were extracted with cold alcohol, no HCN and acetone could be
obtained by enzyme-action from the alcoholic residue.

When the extracted powder after being dried was mixed with
water, and then brought in contact with the enzyme, the aqueous
distillate showed abundant evidence of the presence of HCN and
acetone.

Utrecht, September 25, 1905.

Chemistry. — Prof. P. van Rompuren presents a communication :
“On the action of ammonia and amines on formic esters of
glycols and glycerol” (I).

As the action of ammonia and amines on ally! formate (Proc. June
24 °05) had yielded such good results to me, I have also included in
my research other formic esters, and I now communicate, briefly, the
results obtained with the formates of some polyhydric alcohols.

If gaseous ammonia is allowed to act on the diformate of glycol
it is first absorbed slowly with evolution of heat. If, when the action
is over, the liquid is distilled, nothing passes over at the boiling
point of the diformate (174°), but the temperature rises at once to the
boiling point of glycol, and then gradually to that of formamide. A
complete separation of the two substances, whose boiling point only
differs about 20°, does not succeed with small quantities, and although
it has been proved that the reaction takes place readily and almost
quantitatively, formamide cannot be obtained pure in this way.

One gram of the diformate when mixed with 2 grams of dipro-
pylamine gave a slow rise from 18° to 42°. The liquid being distilled
the formate again seemed to have disappeared, and a fraction could
be obtained at the boiling point of the glycol, and another at that
of dipropylformamide.

With 1.8 gram of benzylamine, 1 gram of glycol diformate gave
a slow rise from 18° to 80°. On distillation, the formate seemed to
have disappeared and the glycol being distilled off, nearly the theoretical
amount of benzylformamide was left in a pure condition.

If gaseous ammonia is allowed to act on the diformate of pro-
panediol (1. 2), which I prepared by heating this glycol with formie
acid, phenomena are noticed analogous to those in the ease of glycol

23*

( 340 )

diformate. After the action is over the ester has again disappeared,
and a mixture of propanediol (1. 2) and formamide has formed.

7 grams of this diformate being mixed with 10 grams of piperidine
the temperature rose from 20° to 120° and on fractionation it again
appeared that the ester had been completely converted into propanediol,
whilst the formylpiperidine, after a few distillations, could be separated
in a fairly pure condition. The boiling point was a little too low,
probably owing to traces of the glycol.

With 7 grams of benzylamine, the diformate of propanediol (1. 2.)
gave a rise from 20° to 110°. On distillation the formed glycol
passed over at about 190°. The residue which had been heated to
about 250° (thermometer immersed in the liquid) solidified on cooling,
and consisted of nearly pure benzylformamide. It may be distilled
at about 295° with only slight decomposition. The distilled produet
had a faint odour of carbylamine, and melted at 59°. By recrystal-
lisation the melting point rose to 61°.

If gaseous ammonia is passed into a mixture of formines of gly-
cerol, such as is obtained for instance by boiling glycerol with
formic acid, or heating with oxalic acid, and then removing the
free formie acid by distillation in vacuo, it is absorbed with great
evolution of heat. After expelling the excess of ammonia and distilling
in vacuo a rich yield of almost pure formamide is obtained.

In one of my experiments 66 grams of formine (yielding 65 °/,
of formic acid on saponification) was saturated with ammonia. In
the first distillation 22 grams of formamide m.p. 0° and 17 grams
dito m.p. — 2° were separated whilst 40 grams of glycerol re-
mained in the distilling flask. The yield was therefore practically
the theoretical one so that this method may be recommended for
the rapid preparation of formamide in large quantities.

With pure triformine *) the action of ammonia is slower than with
the above mentioned mixture. Triformine of glycerol eagerly absorbs
gaseous dimethylamine with strong evolution of heat, and on distil-
lation in vacuo a good yield of the dimethylformanide b. p. 153° is
obtaimed. Piperidine gives with triformine a considerable rise in
temperature (from 20° to 70 ).

Dipropylamine forms with triformine, at first, two layers. After a
little shaking (the temperature rose from 18° to 77°) the liquid becomes
homogeneous, and by distillation in vacuo a good yield of the dipropyl-
formamide formed could be readily obtained.

With chisobutylamine, triformine also gives two layers which do

1) I hope to communicate about this substance, shortly.

( 341 )

not disappear on shaking for a while, but if the liquid was allowed
to stand over night it became homogeneous, and on distillation in
vacuo yielded diisobutylformamide.

Formic esters of unsaturated glycols also seem to react readily
with amines, at least Mr. W. van Dorssrn, who is engaged in the
Utrecht laboratory upon the study of the 3.4-dihydroxy-1.5-hexadiene

CH, = CH — CH.OH

CH, = CH — en
obtained, on mixing 1 gram of the diformate of this glycol with 1.5
eram of benzylamine, a rise in temperature from 18° to 65°, and after
distilling off the glycol could readily isolate benzyiformamide m. p. 61°.

Mathematics. — “A local probability problem”. By Prof. J. C.
KKLUYVER.

The following problem was lately (Nature, July 27) proposed by
Prof. Prarson :

“A man starts from a point 0, and walks / yards in a straight
line; he then turns through any angie whatever, and walks another
/ yards in a second straight line. He repeats this process 7 times.
I require the probability that after these 7 stretches he is at a distance
between 7 and 7-+ dr from his starting point 0.’ ?)

I find that the general solution of this problem depends upon
the theory of Brssrn’s functions, especially that in some particular
cases it leads to the evaluation of certain definite integrals, involving
these functions.

Let OAA AA... Ay be
the broken line, the n stretches
of which need not be all of the
same length. Then the shape of
the figure, not its orientation in
the plane, is wholly determined
by the lengths @,a,,@,,.:, @n—1
of the stretches, and by the
magnitudes of the angles
PP1,-+>Pn—2, formed at the
origin of each stretch a, by
the stretch itself and by the
radius vector s,._;.

1) Recently (Nature, August 10) Prof. Pearson stated, that the solution for 2
very large was already virtually contained in a memoir on sound by Lord Rayterau,

( 342 )

In a turning point the rambler takes his new direction at random ;
hence for any angle g,, all values between O and 2a have an
equal chance, and the probability that those angles are respectively
included within the intervals, gz, pe + dyx, is equal to the product

@aypai? dg, .-- APn—o.

If we integrate this product over a region, determined by the
condition that the mt" radius vector s,—-; remains less than a given
distance c, the result will be the required probability W,,(¢;aa,a,...dn—1),
that the ending point of the path lies within the distance ¢ from
the starting point 0. °*)

The integration becomes less complicated, if we introduce in the
usual way a discontinuous factor. Choosing a function 7(9,9,,-, Pr—2)
such, that it vanishes when s,—;>c¢, and that it is equal to unity

Sp-1<. ¢, to each of the variables gj; we may give the whole
range from O to 22, and we have

Wile 34h, os a) ars f ae fos oe APn—2 T(P:Ps ee > Pn—2)-

For the function 7’ we may take Wesrr’s discontinuous integral,
that is, we may put

T(PsPys + + Prn—2) = of J, (uc) J, (usp—i)du,

ihe integral being equal to zero or to unity according to s,—; being
larger or smaller than c.
This choice of the factor 7’ makes a good deal of reduction possible.
If we consider the side ¢ of a triangle as a function of the sides
a and 6 and of the inclosed angle C;, the relation holds

Ute
J, (ua) J, (ub) = a (uc) dC,
A 4
0

and this formula can be repeatedly used in reducing the integral
Wr (Ge AA, +++ Gra):
So we get successively
1) In the case 7=2, we have, supposing @ +a, >¢>a—M, Wa(c;aq) =
a’? +a,?— :
arccos 5 . Of course for c>a-+a, We becomes equal to unity and
n 2aa,
it is zero for @a—& >.

( 343 )

1 Pais
J, (us, —2) J, (wan—1) = sf (Usn—1) IPn—2 y
0

2
1
J, (us,—3) J, (uay—2) = On J, (usp—9) dgn—1 ’
- 0
2x
1
J, (ua) J, (ua,) = ole (us,) dy ,
0

and consequently

WA GRC Co 1G) = fv, (uc) J, (ua) J, (ua,) «2. J, (wan —1) du.

0

From this result we infer, that the probability sought for is of a
rather intricate character. The n+ 1 functions / are oscillating
functions, and have their signs altering in an irregular manner as
the variable w increases. Hence even an approximation of the integral
is not easily found, and as a solution of Pearson’s problem it is
little apt to meet the requirements of the proposer.

From a mathematical point of view the integral presents some
interest. In fact, if we consider it as a function of c, it is readily
seen to be continuous and finite for all real values of c, and the
same holds for a certain number of derivatives with respect to c¢,
but a closer inspection shows, that this analytic expression, regularly
built up as it is, represents in different intervals different analytic
functions. To make good this assertion, we have only to remember
that the integral stands for the probability required in Pwarson’s
problem. Hence we know beforehand, that it always must be positive
and increasing with c, but that it never surpasses 1, this upper
limit being actually reached as soon as ¢ becomes greater than
a-+a,+...+-a,—1. Moreover, if we suppose a >a, +a, -+...+ an,
the inequality a >c-+ a, + a,...—+ a,_1 is possible for small values
of c. And if the latter inequality holds, the rambler of Prarson’s
problem necessarily arrives outside the circle with radius c, and the
probability is zero.

Thus, by solving the problem, we have found

ec >ata,.. + a1 yee Ll) *
BU eie Ne A me \ “fs, (uc) J, (ua) J, (ua,). «+ J, (wan—1) du,

0
quite independently of the number of the ./,-functions, showing

( 344 )

thereby at the same time, that the continuous analytic expression
cannot be regarded as a single analytic function.

The same still holds for values of c, not fulfilling one of the above
inequalities, though the integral is then continuously varying with c.

So for instance in the case n = 3, taking the stretches a > a, >a,
in such a manner, that a triangle is possible having these sides, I
am led to conelude from the discontinuities of the first derivative
that in each of the following intervals

I a,ta,—a>c>0 TRV; aa, a, >¢ >a ae

II a-—a,+a, >c>a,+a,—a V c >ata, +a,
Til a+a,—a,>¢>a—a,+a,

a distinct analytic function is defined by the integral.

Some further remarks may be made. On integrating by parts
we find
D

Wr (cad, . Gn =1) of, (ua) J, (uc) J, (way) +0. Jy (Man) du

0
— a, | J, (ua,) J, (ua) J, (uc)... J, (ani) du
0

. e . . . . . . ° . . . . . . .

or what is the same :

1 Wa (Gj aa, - Gro) = Wa (Gia, Gs) Was (Gignae cn) lee
Dividing both sides of the equation by 2-++ 1 we may interpret

the coming relation as follows: 7-1 equal or unequal stretches

being given, if of them, taken at random, are put together to a

broken line, according to the rules of Prarson’s problem, the proba-

1
bility is equal to Me that the distance between the extremities of
nr

this broken line is less than the stretch that was left out.

And from the same equation we deduce in the very particular case :
CG Ooi

1
n+] ;

or: the rambler of Prarson’s problem after walking along n equal

W,, (a3 a) =

stretches has the chance to find himself within a stretches’ length

n--
from his starting point.
In the most general case of the problem I cannot give a practical
solution; something however can be done, in the case: very large,
all stretches equal, treated already by Lord Ray.rien.

( 345 )

L
Putting na = L, e=—, we have
a

W,, (c; a") = Wi (e/L) = JJ, rates J, G a du.

0
Now by raising to the nt power the ordinary power series for

J, =) we get
n
aun k=0(—1)kF (au 2k S,(n)
>
Fe = |= Hee ec fal (E)- kl 2k?

where S; (7) stands for the sum hor squares of the coefficients of the
expansion (wv, -—- uw, +...w,)*, so that

Sree Sayin! 1 Si(m) 3 2

int en Dnt gee 4 2n )18ln® on? 2n4 + 3n>
Generally supposing » very large we may put approximately
Sz (n) 1

kl n2k nk’

and, substituting, we find that this approximation leads to the suppo-

sition
au? =
4n
de. C) —e 5
n

For small values of w the approximation is good enough. It is
true both functions behave quite differently when w becomes very
large, but as they are rather rapidly converging to zero, the actual
amount of their difference can be neglected. In particular | find that

the integral
fa OrAGs “Yi
n

n

is of an order of smallness certainly higher than that of the expression
n+1 n

2n Nea NE
n—2 \ox Pap)

while the order of smallness of the integral

oo oy?

[roe 4n du

n

is that of the expression

—— an

e (J,(~)—J,(9)) 5 q > Me

( 346 )

Hence if only @ be rather greater than unity, both integrals cannot
have an appreciable difference and we may put

ure n Cd

nan
= iss ee
W,(c/L) = [Jue 4” du=i—e ~ =1—e :
0
From this result it is evident, that W,(c/Z) for n very large is
always nearly unity. The rambler, walking along a very great
number of very short stretches will almost certainly arrive in the
neighbourhood of his starting point.

1 1 pet
Putting c= -—L, we find a =1—e " an ond result
n n t

nearly equal to the true value ——

at
Returning to the general expression for W,(c;aa, . . da—1) we observe
the possibility of differentiating the integral with respect to ¢ in the

ae

usual way a number of 2m times, provided 2m es
Suppossing ¢ >a-+a,-+...—-+ a,—1 and putting
J (ua) J,(ua,)... I ,(uan—1) =f (4),

we deduce by differentiation

1 =of/, (cu) f (u)du,

0

ud (cu) f (u)du , O= Ju'J,(cu) fu)du,
0

0 = JubJ,(cu) f (udu » O= JutJ(cu)f (udu,
0

@ oo
0 = fw—lJ (cu) f(ujdu , O= | w2J,(cu) f(u)du.
0 1)

These equations allow us to introduce into the integral a new BrssEL
function, the function J/on41 (uv). For Jon4i(w) is connected with
J, (u) and J, (uw) by the relation

Jomti (u) = Pogm (&) J, (&) = Piem—1 () J, (u),;

where

( 847 )

Po2m w= as + bw +... + bom uw”)

yom

and

1
P\2m—1 (uv) = aie (b,u + byw? + 22. + dop—1 u2m—l)

are a pair of ScHLAFLI’s polynomials.
Using this relation we obtain

ies)
= by = ct [vom Tanti (uc) f (u) du,
0
and as
b, = Lim w™ Pom (u) = 22" m!
‘u—0,
we have

22” m! = cont | uM Tomty (uc) J, (ua) Jy (ua,) ..« Jy (wan—1) du
0

with the conditions

n Ls
CSO Oy se fOr 5 om < #

Evidently the value of the integral would be zero, if instead of
the first of these conditions the condition

a>ecta+...+a-1
was satisfied.
In the same manner we might differentiate and also integrate with
respect to one or to several of the parameters a. This leads for
instance to the following results

nm even: 0 =| J, (we) J, (ua) J, (ua,)... J, (Uan—1) du

n odd : 0 =f. J, (uc) J, (ua) J, (ua,) .. «J, (uay—) du.

cata, ta,+....+ a1.
Still other results present themselves when Prarson’s problem is
slightly modified. Again putting

Ji, (ua) J, (ua,) .. 2 J, an) =f

and writing 9 for c, we get by differentiation with respect to @

( 348 )

aD

1 .
W,, (d2) = a o do dé fr J, (wo) f (u) du,

“ e

and here JW,,(d2) means the probability that the ending point of
the broken line falls on a given element d2 of the plane, the polar
coordinates of which are 9, 6.

By integrating over a given finite region we may deduce the
probability that the rambler reaches that region ’)..

First let the region be a rectangle /#, and let the rectangular
coordinates of its vertices be + p, + q, then we find for the corre-
sponding probability

ieee Ip teg
Wry (it) == Gs uf (u) im fas fay J, (u V&? + 77’).

—p —q
Now we have

2
Ji (uVs? + 77) = af (u § cos a) cos (u 4 sin a) da,
0

and therefore, effectuating the integrations with respect to § and to »,

9

nee sin (pu cos et) sin (qu st
W, (Rk) = =f Jw aw f cenpelon ayer aay) da.

u? sin COs a

0 0
A somewhat simpler expression is found, if changing the variables
we pass from w and a to
U == u COS a,
wusina.

Then the probability IV, (2) is expressed as follows:

(2)

om oO
4 sin pu sin qu es
W,, (2) = — [ue dw cease sata St (Ve? + w’).
Cd w
0 0

Again an evaluation of this double integral is generally not practi-
cable, but the problem itself gives the value of the integral, if both

1) If this region is a circle with radius c, the centre of which lies at a distance
b from the starling point O, we have at once

Wri (¢; baa, ... dn—1) = fv, (uc) J, (ub) J, (ua) J, way) ©. « Dy (Udn—1) Ue
0
for the probability, that the path ends inside the circle.

( 349 )

the coordinates p,q are surpassing the total length of the path. Then
the probability becomes a certainty and it follows that

a ={ fis dw ae SS ee St (Vo + + w w D)

with the condition

pandg >ata,+...+ a1.
In the general case of the rectangle the probability W,(R) is
independent of g, as soon as its length is superior to that of the path.
Assuming this to be the case, we remark that the value of the
slightly transformed integral

S Later sinpy sinw , ana
Wa) = a _[avar i if yt 7
o 0 ;

remains unaltered, when gq increases indefinitely, and we conclude that

wo ao

; x 4 pe ; ” sinw 2 (° sinpv
Lim W,(k) = = T(v)de a du = 15 J (v)de.
q=on 4 w Fy

0 0 0

Thus we have solved another modification of Prarson’s problem,

1
for half the result, added to Q” expresses the probability

ein Yt

W,(")=— ai File Fv) de,

that the rambler, starting on his a at a distance p of a straight
frontier /’, after walking along 7 stretches, will arrive at that side
of the frontier he came from‘).

As before we are enabled in a particular case by the problem
itself to assign the value of the integral. If we suppose that the
rambler cannot reach the frontier, that is, if we take

poatat...t+a—,

the probability becomes a certainty and we find

1) Obviously the probability Wn (") might have been derived from the proba-
bility Wnti (w+ p:eGd,...dn—1) by making w indefinitely large. Therefore we
may conclude that

sinup

pa winfs (v@+p) J (um) f (u)du = = a a I (v)dv.

( 350 )

an up
pele J, (va) J, (vay) J, (van—i) dv.
In the case n= 1, this is a known result to which another may
be added, if we take a>jp. When the single stretch @ is inclined
to the frontier under an angle less than
eae
are sil —,
the rambler remains at the same side and, all directions of the stretch
being equally possible, we have

Wf) = ==(4 -++ are sin 2),

oe
, 0 sin vp
are sin — = J, (va) dv.
a

0

hence

Mathematics. — “A definite integral of Kummer”. By Prof. W. Kaprryn.

In Cretie’s Journal, Vol. 17, Kummer has determined the value ot
the integral

supposing 4? to represent a positive quantity and p not an integer.
He finds:
Up = V(p+ hf (—p, &) + P(—p—l) br? f (p+2; 6");

where
2 v 3

ipt Spee t ppp ey +

Se — ey ze .
s=0 8/p(p+1).(p-+s—})

In the following pages we propose to study this integral for the
case that p represents a positive integer, and at the same time to
show that there is a simple connection between this integral and

the integral
oo v
a
a sk e © a? da;
b

where / is supposed to be positive.

i (pe) = 1+ —-

~

(351 )

It is rational to put in the integral of Kummer
: Sie &
assuming 2 to be an integer and ¢ an arbitrary infinitesimal, and
then to determine the limit for «= 0.

Let us therefore examine the limit of
U,-:= T'(n4+1—8) f(—n-+e, b?) + P(—n—1 +8) b2"+2—-® f(n+2—«, b?)
for «= 0.

Suppose
Vintl—s) = A, + A, e+ A, 8...
B,
f(—nte, d= 4B, +Be+...
C,
V(—n—1+8) =—+¢,4+ Cre+..
&
bent2—2§ — D+ Diet Diet...
f(a+2—¢8, ?) =F, + He+ Fe’ +,
then
A,B,+C DE,
= °+[A,B,+A,B,+C,D,E,+C,D,E,+C,D,E,)+ .

and the limit
U, = A,B, + A,B, + C,D,£, + C,D,E, + CDE,
for we shall see that
A,B, + C,D,£, = 09.
Let us now determine the various coefficients.
First we have

Mra+tl
(n+ 1—s8) = I'(n+1) — ¢ P(n+1) aaa
or if we put
P(e) _
Ta) w(x),

T'(n+1—8) =n! [lL—ew(n+1)+...],
thus
A, =)
A= — ioe (n+1).
To find B, and es we write

h2s

pp DS a a ee
IAS I ara UT RTP FTL
1 2 h2n-+-2+2s

eee
& =o (n+8-+1)(— n+2)(— n4+1+¢)...(—1+8)(1+8)...(6-+8)
If

( 352 )

1 21(0)
= = ——==(e)=2(0)| 1 ——-+...],
ae IPE, er ESTEE me if Tea@ 7 |

then we easily find

(—1)

nisl

4 (0) =
and

AO! 1 1s La val

TO) =a <= eg os os ay = (+ =F ear vee =)= y(1--n)—y(1-+-s),

therefore

B= (ae 2nt2 < bes bh (—1)"b2+2 > fer
om n! =p see. Teme ree )
1 2 (—1)s(n—s)! (—1)"b2n+2
= SS a oS et , 3\ oe
“ly s! e n!/(n+1)! el) ier eae)

H2n+42 S ay(1 + s)b2s
n! s—9 8/(n+s+1) r

For the evaluation of C, and C, we have
0 1

—(-1)"

1
r(- n—1-+ 8) = See I'(—n-+ 8) ’
1
I'(—n-+e) = apaee T'(—n+1-+6) ’

1
P2248) == Spee),

SAD GES
1
r(—1-+48) SST P(e),
so
r( 1+) (—1)r41 te
ph aeies €).
(n+1—e)(n—8) ... (2—«)(1—8)

Assuming

: = u(e) = (9) [pe + |
(ple a eae ee (0) otal

then

1
a0) = Gani”
ao) 1 , 1
u(o) m1 on

1
feet tlsuet2—w(),
whilst

P(e) = P(®) + Qe)

(353)

1 Tres ok pet Goll ie ear
a —_— eke = &
ce elas, 6 ae
= : OO 1 : : :
Se ee cara
If we no notice that
oo
dy
Q(0) = Oh) == — |) Sse = li(e—!)
J logy
]
and that out of the well known formula
ae x a“? a
ARE ae ooycais mea ko.
follows
ee 1 1 1
Cie an aide ani oy heer wh

then it is evident that

1
SG ee praants WAC) Ses 2

from which ensues

es (—1)rt!

i (n-+1)! ;
eu
O.= Gar Wet

Moreover we find
pnf2—2e — J242(elyd)—2= — H2n4271—_Qelgh + ...],
so
Dy == b2n-t2
D, = — 2b2"--2lgb
and finally

Co} b2s
‘(n+2—8,b?) = = oe
I Os eT f2—e)(n+38—6)... (rte Eb)

If again we put
1
(n+2-—«)(n+3—8) ...(n+s+1—8)

we find

v'(0)

»(0)

r(0) = - :
(n+2)(n+3)...(n+s+1)
iD) ees oe | 1 ee wie
MON eee nie =f PTS ep i) = (n+ s+ 2)—w(n+2),

so that

24

Proceedings Royal Acad. Amsterdam. Vol. VIII.

== 2)(e))/==10((0) Ee —-+..

|;

( 354 )

E, = f(n+2,0’) ,
co 9)),2s

E, = — w(n4+2)f (n+2,07) + = y(n + s4-2)b

s=0 sl(n+2).. .(n iL s+ 1)

With the aid of these values we find

A, B, =F C, IBY LE, = 0 9
—s)! ; 8) 2s
Eep umes eveanini ese ae)
s! s—0 s!(n+s s!(nts+)l’

CD B, ACD Ey Cy ye —
me > (7 i) Qs
= Ge wnt] 2070+ 2,0) — & GAG 2)
ea) ola +2)(n4 8) .. s+)
hence
au —s)!
Oe — SS (—1)s (n ) b2s ae
s—()

w(l+s)—p(n+s+2)]...(1)

— 1)" b2n+2 se 2lat
+ ) sos! [G@ien aay - g°

Let us now determine U, in another way to
To this end we differentiate the equation

another form.
LOR eNO
pee ed oe
U,, =| é Tar du 4
0)

give this result

we then get
- wal b2
1 dU, 5 ee oe is
—- ie. GUUS OHI TG =o a ac a
2b db (2)
0
Bae = is) U2
CRO Laie; a ae
= --- — = 2b fe FRU PAG De
2b db? 26? db
0
Out of the identity
v 32
aa Ss =>
, “2 dev + b* e v gn—2 dz ,

ve d ( é
deduce by integrating between the limits 0 and a

we moreover
i

wo Td ca b2 a)
> -—2 SS — 2 —— 7 — —
o “| e 2a dae = | e Pan de + v | é EA Fi (hia
0 0

0
hence we find for U» the differential equation

DP Uy 2 1 dU,
SiN see eG lt)
db

a 2)

db? b

( 355 )

U,
This differential equation we also find if we put # = 276 and v = ree
in Bwssen’s equation :

fe ee

dia? aw da Pi

therefore
U, =H [A I (210) + B+! (21 d)].
In order to determine the constants properly I notice that the
integral U, for 6=0O is equal to ‘”/ and vanishes for b=;
moreover we find

for 6:-=0 nl 7 n+ (2i b) iy
}
” ” or+l yr+l (2% b) ——— (2 jh ee
4
fi b ij Fl 7 fy 9% b bn+l pees
or == 60 bn rp 2% ee. ;
, aa 2V ab
; pnfl 204 aa
” ” n+l yr+l (2i pee 6 |
2V ab
thus
A) (— irri B

: in
0O=A+t Be = ;
and _ finally
ie ait? br+l [fr (2% b) a 7 Yut+i (27 b)] = mint? bn+l Hy (275) eo
That this value and the value (1) agree is easy to prove. For
according to definition!) we find:

} 2 1 \2+1 in Nip — AW
xe Yutt (2ib) = 2 TH (28) (u b + 5) = ia So) ee
a 40

s=0 s !

Q5

le yh = iG ene Inp(s > I) ap = 2 22);

from which ensues when we multiply by @+8 bert
wat? bel [Lr+1 (27 b) i Yrt+i (eh b)) = — 27n+3 Jn+1 ly 6 [n+ (2% b) i.

=1)s(u—
4 s | y & Us 5 —(—1)rbe S =

s=0 8! ae =o el(s ear
By

sts L)-+ pot n+2)].

Cee hy (a nyeeli st ean eS
s=08! (s+n-+1)!

2) NretseN, Handbuch der Cylinderf. page 16.

24%

( 356 )

the second member of the former equation becomes the second
member of the equation (1).
Let us now examine

A= fe Sic" Lt a ae oe ah eevee te ate (CE)
Here too we can find a differential equation satisfied by this integral.

By differentiating we find
rd

1d, [f-=- CeCe, ALS
onee =| a ae Bey inti eee (b)
b
rd? Ve, 1 dVn
26 db? OVS hey
x B
eo n—l
— — 2b fe v—2da— 2b"—1e—2b 4 Spee anaes eC)
b

whilst the integration of the identity

pe b2 U2
—“£—— —=<— — =r —
wid \e zu }— _e@ * wrdau + be T gn—2dax

between the limits 4 and o furnishes

wo ue b2 a) bo

nD
ee eal See Spee pee
— bre—2b — n| e * eda = — [- 2 wd +0" e = ae ie eee
3
b

b b
So we find for V7, the differential equation
a2, 2n+1 dV,

—4Vy = (n41)bn—le-2, 2. (5)

db? b db
If we now write the equations (a) and (0d)
Eee NSE 6
eee ee
and
dV
==) Vea = bn a2). rt cae (7)
db
it is easy to find out of (6) and (2)
ee 6 dU ny U
pe eo aa ta
and likewise out of (7) and (5)
b dVn—1 Z, bn 25
Vii nope remcinie Vari sige’ oak

Out of the last two equations we deduce the recurrent relation

( 357 )

hn
Va— $ Un= — —— [Vn 1— $ Ont 2 [Vn — § Una + = e--. (8)

by which we can reduce the evaluation of Van — + U;, to-that of
Vi.—i U,..

1 :
Let us now determine the value of ,—- U,. To this end we

start from the equation (c); this becomes for n= 0

Van Sle aN ee 1
aga eS ee pe -
1 V4 i Op Dh fe x Eg eRe = e—2b ,
ee

2b db? ON Gi
b

ob
By substituting in this integral — for 2 we find
2 6 b
Mea Gin - lea
f: mn) ak dz Se (Up V,) ’

hence the preceding equation becomes
UP 1G 1 dV, I
= 0 ES 4 —2b/ | a
OS ghte APRA he ( tS a
By subtracting from this according to (5)
gcc Veal ae a=?

= = —— sa
4 db? 4b db S Ab ’

we find

V Leas —2b
has Sa Nata Ce eS oO td ors on oO» (YD)

With the aid of equation (8) we get: .

: 1 5
we 5 h=(e4 se
ie ae 307
ee | ee Le
r 1 Tr D3 mp2 § 2)
Vp NEY + 5b? + 6b + 3)e—,
ae es 5
V,— os Ce aa bt + 106% +- 216? 4- 25h + 12 }e—22,
in which we can easily trace the following law :
1 1P(mtly, 14n+2)/ 2Nn48)!
(eee ayo ae [UI ie BL SF pn 24. tn! |e—28.. . . (10
ay Otibinn at n! 3/(n—1)! : 5 1(n-2)! aes: | oe

: : ap fel 1
Out of equation (8) and this one it is evident that Viga — > Uni

follows the same law; so the relation (10) is proved,

( 358 )

Mathematics. — “An article on the knowledge of the tetrahedral
complex.” By Dr. Z. P. Bouman. (Communicated by Prof. Jan
DE VRIES).

§ 1. When for an arbitrary ray out of a tetrahedral complex P;
represents the point of intersection with the face Aj; Ai Am of the
tetrahedron, then

Rip, Ps + PsP. — 9

where #& represents the given anharmonic ratio of the complex and
pi ((=1..6) are the Prickrr coordinates of lines.

By using the condition necessary for each ray of the complex,
namely

Pi Ps 1 Pa Ps + Ps Pe = 9
the equation of the complex becomes
Ap, Ps + Bp, eral C LL = 0,
where the anharmoni¢e ratio is given by
BA

eee

a

A given tetrahedral complex can always transform itself projectively
into another one with the same anharmonic ratio in regard to the
faces of the rectangular system of coordinates and the plane at
infinity.

§ 2. After having executed this transformation we can examine
whether a surface with two independent parameters can be found in
such a manner that the normals to be erected in an arbitrary point
on the oo’ number of surfaces passing through that point, are rays
of the given tetrahedral complex.

To this end we make the two determining points to lie infinitely
close to each other on each ray of the complex, so that each ray
is determined by one point (a, y,2) and the direction (dz, dy, dz) in
that point. The coordinates of lines now take the form:

Pr
Ps

So the equation for the complex becomes:

A (a dy — y dx) dz + B(y dz — z dy) dx + C (ede — a dz) dy = 0.

If now every ray of the complex is to be at right angles to a
surface z= f(x, y), then we have for each ray in each point of the

surface :

=ady—yd«a, p,=ydz—zdy, p,=—2de —2x dz,

pe dz, i da, = dy.

( 359 )

da:dy:dz=p:q:—1,
Oz dz
om oy
So the differential equation of the surface becomes :
— pgz (B — C) + yp (A — B) 4 ag (C — A)=0

where p=

or
av 1 y R re
By Reng hea Re eo

The complete integral with two parameters C and C, becomes :

On

=|/ Jovests | / vest
a eee | WAR Os y~—U,.
Rol Vu Tp Vy 1

It represents a surface of order four.

1
It is evident out of the equation that for R =F the surface re-
L

mains the same; only the X- and the J-axes have been interchanged.
(This is geometrically immediately made clear). So we have but to
examine the surface for, let us say, A> 1.

§ 3. It must be possible to find the equation of the cone of the
complex in a definite point out of the equation of the surface because
that cone is the locus of the normals to the oot number of surfaces,
passing through the point under consideration. If, 8, y represent the
cosines of direction of a ray of the complex in the point a,, y,, 2, then

a B
p= Al a

Substituting this in the differential equation and eliminating « and

B by means of the equations of the ray of the complex, namely

Ce Wie Ua eee
ais. Bo atiexy
we find for the cone of the complex :
(R—1) Za (w7—a,) (y—y,) hal Ry, (e—2,) (z— z,) = ay (y-—y,) (z—z,) = 0:
The planes of the coordinates forming the singular surface of the
complex, the cone of the complex must degenerate for each point
of one of these planes. For the point P(w,, y, = 0,2,) the cone
breaks up into y=O and into #,2+ (R—1)z,#=—Rz,2,, i.e. a
plane passing through P and parallel to the Y-axis. This plane is at
=
Pe

y

right angles to OP, if this line has for equation z= + a

(Comp. § 4).

( 360 )

§ 4. The drawing of the surfaces to be found offers no difficulties.
For kR>1 (§ 2) we must take C, positive and then we have to

distinguish the cases C 2 0,

So for C>>0 the surface consists of two separated parts connected
by points forming parts of a double conic in the XO Y-plane. The
planes «= + 47C' touch both parts according to equal ellipses and
no points lie between with z> 0.

The section with the XOZplane consists of two hyperbolae

Vile
with centres (: —+ kA uy
=

connected twice, and intersect each other in the points of intersection
of the double conie with the X-axis. The hyperbolae coincide in
the planes y= +WC\, where the common vertex of the double

) on the Zaxis. At infinity they are

conic is lying.

C’ becoming smaller, the two parts of the surface approach
each other and for C=O the conics meet in the planes 2 = +VC.
The surface becomes a ruled surface, so it breaks up into two
cylinders with axes in the XOZ-plane.

The axes have for equation <= + v ye a (Comp. § 3). The
ae

section perpendicular to these axes is a circle which is in accordance
with the signification of the axes as found in § 3.

§ 5. It is known that the normals of a system of similar, con-
centric ellipsoids form a tetrahedral complex '). So this system must
be a particular integral of the above-mentioned differential equation.

Let us put C=gC,+h (g and h being constants) and let us
operate in the ordinary way; we find C and C, as functions of the
variables out of:

PS 76) -
g : o(g+R)
gy? Eh - x?
2? — C= — g —_—__..
gas

Substitution in the complete integral furnishes:
ea Ct) —g’ +e=h.
gtk ©
Let us put in this equation g=—5. and let ¢ be the axis
along the Z-axis; we shall then find if we take a’ positively

1) Dr. J. pe Vries: On a special tetrahedal complex. Proceedings of Febr. 25
1905, Vol. XIII, pages 572—-577.

( 361 )

CF Yin nee en ge a — Cc
aia he? pias —- h', with ——— ana ok
ae
Likewise (y = a negative) the system of hyperboloids with
\ i)
two sheets
an be y" ; atc
SSS SS = h', with 1h SS
Ee Ger SO b?+te

a
and also (7= ise a positive ) the system of hyperboloids with one
52

sheet
BoE AN ae 2 e—a
—~+——— =H’, with R= ——.
tc Gi A ge c? +}?

§ 6. The “curves of a complex” are curves whose tangents are
rays of the complex. The coefficients of direction (a, B, y) in a
definite point (7, y, 2) must therefore be proportional to

Oz Oz '
Po SG ’
: Oa?! Oy

of one of those surfaces through that point. From this ensues that

a fi
—— andy Gr SS nish a, ob p and q must satisfy the
Y Y
equation :
we Al Q Ik
E arco —0,
p R=1. 9g IR
So the quantities x, 7, z, a, 8, y must satisfy :
z eels aw Al
Bi lah ea 0.

pee SUR RE Yo tena he
Let a curve of the complex be given by :
t=f,0, y=. -<=f@),
where s need not of necessity represent the length of the are, then:
i@) £0) £2 ©) 1
yan ALES Sain iar =
-A.@ | f'@1—RK " f'@)R—1
Amongst others all curves for all values of p to be represented by
e=Al +s, y=u(mtsy, z=v(atsp
satisfy this equation if only
l—n

?
m—n

which condition can be satisfied by putting 7= B, m=C, n= A.

( 362 )

For p= —1 these are twisted eubies. If we bring these through
a point (7, ¥;,2,) the o* curves all lie on the cone of the complex
of this point. This holding for each point, the bisecants (and not
only the tangents) are rays of the complex.

Indeed, all the twisted cubics pass through the vertices of our
tetrahedron and the four pianes passmg through a bisecant and these
four points have thus a constant anharmonic ratio. From this ensues
that the bisecants intersect the four planes of coordinates in the same
anharmonic ratio.

For p=1 we have the rays of the complex themselves.

For p=2 we have conics which can be nothing but conies of
the complex, e.g. for s = —/ the curve touches the plane Y OZ, ete.

For p=83 we have twisted cubics whose bisecants are not rays
of the complex, ete.

In general the tangents to the “curves of a complex” lie always
in linear congruences belonging to the tetrahedral complex. For such
a tangent namely we have

_ dv dy dz
a hee aus, =(n+s)—-.
From this ensues among others:
lz L+ s)du + k(m s) dx
ope at tide thm ty
z w+ ky
This is evidently always satisfied by rays of the complex, satisfying
at the same time:

. (& an arbitrary constant.)

xvdz — zedx =k (zdy — ydz) and kdy = — Radz,
for which we can write in coordinates of lines:
pe hp a) eNe —h) ee a
These satisfy the equations of the tetrahedral complex and lie in
congruences; the two linear complexes determining such a congruence,
are themselves special, and the position of their axes is evident from
their equation.

§ 7. Finally it proves to be simple to bring in equation the
curves which are drawn on an arbitrary surface in such a way that
the cone of the complex touches the surface in each point of the curve.

Let the surface be f(x, y,z)=90 and the ray of the complex
u—2, YU 2-2

Se = *, passing through the point «,,7,,2, of the
t ,

surface.
A ray of the complex in the tangential plane must satisfy

( 363 )
of of of
— 7 i 0,
4 0a, cm OY, a Oz,
and further according to the differential equation

(R —1)2, a8 — Ry, ay + «, By must be equal to 0.

The two rays of the complex in the tangential plane have but to

be made to coincide. The condition is:
= R(R ‘igs 1) 21 Yidsts == [= (k a 1) 21h; at Ryihy adi «,|’,

where /,, /;,,/, vepresent the differential quotients of / according to
x, y and z respectively, whilst analogous relations are easy to deduce.

From this ensues that the required curve is the intersection of

(aks 2) == 0
and
—4R(R—1)eyf,f,=(—A-)ef+ ky they.

Without entering into further details | only wish to observe that
when / (#7, y,2)=9 represents a plane, the curve can be nothing
but the conic of the complex. From the above mentioned equations
we therefore find a parabola (the conic of the complex touches the
tetrahedron plane at infinity) touching the three planes of coordinates
of the rectangular system of axes.

Physiology. — “On the excretion of creatinin in man. By C. A,
Prkeruarinc. Report of a research made by C. J. C. Van
Hoocrnnuyzn and H. Verrionen.

As the muscle tissue in herbivora as well as in carnivora always
contains a not unimportant amount of creatin, and creatinin is daily
excreted with the urine it may be concluded, that creatin is formed
as a product of metabolism in the muscles, and having entered
the blood is at least for a part excreted by the kidneys in the form
of the anhydride, creatinin.

But no agreement has been obtained about the question whether
the forming of creatin is bound to the labour, the contracting of
the muscles. To answer that question, researches have been made
whether the amount of creatinin excreted by the kidneys augments
after muscular labour. Different investigators have obtained different
results. Van Hoocennuyze and VureLorcH have resumed the research
anew, using a new method to determinate the amount of creatinin
in the urine, which was published some time ago by Fon). The

') Zeitschr. f. Physiol. Ghemie, Bd. XLI, S$. 223.

( 364 )

method of Forin is founded on the reaction of Jarré, which consists
in adding picric acid and an excess of caustic soda to a solution of
creatinin, whereat the liquid takes a brown colour, which cannot
be discerned from the colour of a solution of bichromate of potas-
sium. This reaction is employed in the following way: 5 cc. of
urine is mixed with 15 ce. picrie acid 1,2 °/, and 5 ce. of caustic
soda 10 °/,. After 5 minutes water is added to a volume of 250 ce.
This solution is compared by Forin in the colorimeter of Dusosce
with a */, normal solution of bichromate of potassium of which a
column 8 mm. high shows exactly the same intensity of colour as
a column 8,1 mm. high of a solution of 10 mgr. creatinin with
15 ce. pierie acid solution and 5 ce. caustic soda diluted to 500 ce.
Instead of the colorimeter of Dunosce, Van Hoocrnnvuyze and VErRPLOEGH
used a little instrument, constructed after their indication, which
answered completely to their demands. Immediately after each deter-
mination each of them performed 5 readings of the height of the
solution of creatinin at which its colour had just the same intensity
as a column 8 mm. high of the solution of bichromate of potassium.
The several readings of which the average was taken, never differed
more than 0,2, only very seldom more than 0,1 mm. —

It proved meanwhile that the temperature has influence on the
reaction in that sense that the colour of the creatinin solution be-
comes deeper by increase of temperature. Therefore the water used
for the diluting was always kept at a temperature scarcely differing
from 15° C. The relation found by Fontn was affirmed. A solution
of 10 mer. of pure creatinin in 500 c.c. treated in the indicated
way produced as the average of 10 readings 8.14 m.m. (max: 8.2,
min. 8.1) out of which a quantity of 9.951 instead of 10 mer.
would be deduced.

The results become less exact when the concentration of creatinin
is much larger or smaller than 10 mgr. in 500 e.c. Therefore the
determination was repeated when the readings became higher than 10.5
or lower than 5, with 10 ¢e.c. urine, in the first case diluted to 250
in the second to 1000 ¢.c. The method of Foun had great advantages
over the method of Nruspaurr used till now, in which the creatinin
is precipitated out of an alcoholic extract of the ure by means of
chloride of zine and after that weighed. Not only that the method of
Form takes much smaller quantities of urine, so that it renders it easy
to discern by the examination of different portions of urine the oseilla-
tions in the secretion in the course of the day, but it is also more
reliable. With the method of Nrupavgr there is always some danger
that under the influence of the alkaline reaction arising from the addi-

( 365 )

tion of milk of lime to separate the phosphates, a part of the crea-
tinin is changed into creatin. This danger may bé lessened but
not wholly avoided by acidifying the filtrate before evaporation by
means of hydrochloric acid, after which at the end the hydrochloric
acid must be eliminated by addition of sodium acetate in order not
to hinder the precipitation of creatinin zine chloride. But there are
other difficulties connected with the method of Nuusaver which can
never be totally removed. The urine is after the removal of the
phosphates concentrated till it obtains the consistency of syrup and
is than extracted with alcohol. In the mass of salts rendered hard
by the contact with alcohol, a part of the creatinin may be retained
undissolved. If in order to eliminate this difficulty the urine is not
very much evaporated, there arises another source of error. The
alcohol is diluted by the still resting water and the consequence
is that now the creatinin-zinechloride crystallises only partially. For
this compound is insoluble in absolute aleohol but not in alcohol
containing water. A too small quantity of creatinin is therefore
always found by the application of this method.

Van Hoocrnnuyznn and VireLorcn have investigated the solubility
of creatinin-zinccbloride in alcohol by putting dried crystals, prepared
from urine and purified as much as possible, in closed bottles under
alcohol of different strength at the temperature of the room under
repeated shaking and by determinating afterwards, by means of
Formn’s method, how much creatinin was dissolved in the aleohol.
They found:

in 100 C.C. alcohol 99 °/, trace of creatinin.

yt) aay Ma

” ” ” ” 93 He 0.6 mer. 56
76 99 -

” 2? ” ” 72 oh 32.1 ”? ”
A AS

Ee) » ”? ” 50 Hh 104.5 ” ”

In connection with this they obtained out of urine more ecreatinin-
zinechloride when the alcoholic extract before the addition of chlorid
of zine was again evaporated to almost dryness and then dissolved
by strong alcohol, than with the usual method. They could still show
ereatinin in the liquid filtered off from the creatinin-zine-chlorid as
well by the reaction of Wryt as by that of Jarre. So the method
of Nnusavrr always gives a loss of which the amount cannot be
estimated. One is therefore not entitled to attribute much value to
the little oscillations in the output of creatinin found by applying
this method.

By the method of Fouin on the contrary such a source of uncer-
tainty does not exist, when the time of the reaction — 5 minutes —
is rightly observed, the liquid is brought to the exact volume with

( 366 )

water of the temperature of the room and when the determination
is completed immediately afterwards.

Van Hoocrnnuyze and VurrLorcn have investigated by themselves
whether increase of the secretion of creatinin in consequence of
muscular labour eould be observed. For that purpose in every series
of experiments the urine was collected every day at appointed times
namely in the morning, in the first series of experiments at 9,
the following at 8 o’clock, in the afternoon at 12 o'clock and at
+'/, o'clock, at night at 11°/, o'clock. Every portion was measured
and divided into two equal halves. One of the halves was used for an
estimation of creatinin, the other halves were mixed, after which the
quantity of creatinin in the mixture was determined and moreover an
estimation of nitrogen was performed after the method of KyrLpAnL,

In this way the determination of creatinin was also controlled. In
all the series of experiments the conformity of the figure of the
total quantity of creatinin and the sum of the four portions was
very gratifying. The quantity of urine of one day was that collected
from 12 o’clock in the afternoon till the following morning 8 or
9 o'clock.

During each series of experiments a fixed amount of food was
taken, every day the same. Only in the first series coffee and tea
were still taken, in the later series only water.

I. From April the 8'*—24t 1904, seventeen days at a stretch, food
was taken which consisted of bread, butter, cheese, milk, oatmeal,
sugar, meat, eggs, potatoes and rice, daily an equal portion of each.

The food contained :

for v. H. 118 gr. proteid 146 er. fat, 326
” V. 115 ” ” Si ” ” 32

er. carbohydrat.; 40,8 Cal. p. Kg.
” »” 08.6 By Fy eet)

On working days moreover both consumed 50 gr. sugar.

The 141, the 16 and the 21 of April bicycle excursions were
undertaken at which they rode steadily on for 2*/, a3 hours without
resting. The other days were spent in the laboratory while the
evenings were passed peacefully.

The excretion of creatinin underwent no perceptible change in
consequence of the muscular labour. With both investigators it oseil-
lated not unimportantly during the whole experiment. It amounted
on an average to:

v. H. 14 days of rest 2.116 gr. daily (max. 2.401, min. 1. Aas or.)
NG be df fol OST b3.0e aCe, ae lee aes is Be 8550)
Vie EL: ; workingdays 2.147 ,, ONC RLS 2O 2ottes eee O2 oNaes)
Valine a Od tye) ayeeeee Oona oe -

( 367 )

The difference is so small that no value must be attached to it. On
the days which followed on the muscular exertion the figures of the
creatinin remained within the usual daily oscillations.

The secretion of nitrogen was rather irregular with both during
the whole experiment.

From June the 22°4 till July the 24 1904 (eleven days) the
experiment was repeated with less food which in particular was less
rich in proteid. It contained :

for v. H. 71.5 gr. proteid, 125 gr. fat, 351 gr. carbohydr.; 33.7 Cal. p. Kg.
NE. 80.5 ” Ly) 74.75 ” ” 358 ” +) 34.6 ” 7139)

>?

On July the 1st a bicycle excursion of three hours was undertaken
(50 KM).
The excretion of creatinin amounted on an average to:

10 days of rest Workingday
v. H. 1.983 (max. 2.042, min. 1.809 gr.) 1.997 gr.
Vi e203 9M Gs tao (tire Mem aOZ OE.) 2.049 _,,

On the days which followed the day of muscular labour the excretion
of creatinin did not increase either.

Ill. Whereas till now meat was still taken, in the series of
experiment II daily 50 gr., the experiment was now taken with
food which contained no creatinin at all, moreover it was made
poorer in proteid. The experiment lasted from July the 7 till the
29th 1904, 23 days at a stretch.

From July the 7 till 18 only bread, butter, cheese, rice and
sugar were taken containing :
for v. H. 50 gr. proteid, 115 gr. fat, 344 gr. carbohydr.; 31.2 Cal. p. Ke.
eee V0). 5, 3 74 ,, ,, 344 ,, D 30-8" ype

From July the 18 rice was partly replaced by potatoes and
the quantity of butter was decreased so that the ration became:
for v. H. 47 gr. proteid, 98 gr. fat, 337 gr. carbohydr.; 29,5 Cal. p. Ke.
Ne - OLE Ee tiaole by 3 BOL Meg :§ hatetes

On July the 28 and the 29" 5 egos were daily added to this food.

On July the 15, the 20% and the 23™ muscular labour was
again performed while the other days were passed in the laboratory
with occupations which exacted only little exertion of the muscles.
On July the 15 a bicyele excursion was undertaken in which
54 K.M. were covered in three hows. On July the 20% and 234
fatiguing indoors gymnastics were performed for 27/, hours at a
stretch with halters of 10 K.G. and with the chest-expander and

( 368 )

the combined developer of Sanpow; care was taken that all the
muscles of the body and the extremities were used.

When the first three days of the scanty diet, July the 7”, 8thand
9th in which the secretion of nitrogen fell with v. H. from 14,562
io 9,045 gr. and with V. from 15,721 to 10,234 er. are not counted,
as belonging to a transition-period, and neither the last two days,
July the 28 and 29" at which about 30 gr. more proteids daily
were taken, it appears that the excretion of creatinin has amounted
in 15 days of rest on an average every day to:

v. H. 1.836 gr. (max. 1.935, min. 1.693 gr.)
Vi. A296 2B (ey eee en CS come ong)

while on the working days was found :

v. H. July the 15 1.908, July the 20 1.921
and July the 23 41.974 er. creatinin.

V. July the 15 2.142, July the 20% 1.947,
and July the 23 1.937 er. creatinin.

Here then the figure with v. H. is always, with V. once above
the average on the working day. Meanwhile the deviations do not
surpass the oscillations, which are always found, also without im-
portant exertions of the muscles.

The figure found with V. on July the 15 does, it is true, surpass
the maximum in the period of the days of rest, but the difference
0,063 er. is so slight, that no value must be attached to that, in
connection to the lower figures of the two other workingdays.

On the two last days of the series on which no muscular labour
was performed, but on which more proteid was taken, the excretion
of creatinin was :

v. H. July the 28 1.955 er., and July the 29% 1.959 er.
Vial Re tate fates: 21055 Un ie me ae eam lea oS

while with both the secretion of nitrogen increased from about
8 er. to 11 gr. daily.

IV. In September 1905 a new experiment was taken, to examine
firstly whether preceding muscular exercise might perhaps bring
some change in the result, secondly to investigate the influence of
excessive labour and thirdly to see whether the excretion of creatinin
would be increased with excessive labour and totally insufficient food.

After performing daily for three weeks ata stretch indoor gymnas-
tics after the method of Sanpow, the experiment was begun Septem-
ber the 26 with food of the same composition as was used July the
18 till the 27%, hardly suflicient and poor in proteid. This food was

( 369 )

taken nine days at a stretch till Oct. the 4". On September the 29%
exercises were performed with Sanpow’s implements for 2'/, hours with
short intervals. On October the 2°¢ excessive labour was done,
consisting of a walk of 21 KM. in the morning from 9 till 12
o'clock, a walk of 10 K.M. in two hours in the afternoon and wor-
king with halters for ‘/, hour in the evening. On the six days of
rest between September the 27% and Oct. the 4%» (on the first day
Sept. the 26' the urine was not examined) there was exereted on
the average every day:
v. H. 1.859 (max. 1.977, min. 1.755) gr. creatinin
Neri O2on@ en 047 4." 1860). ,; -
while on Sept. the 29% there was found :
v. H. 2,001 V. 1.979, gr. creatinin
On®@October 2227. 15859) |) 16945-"

That not too much importance for the influence of muscular labour
on the secretion of creatinm must be attached to the somewhat high
figure of v. H. on September the 29 becomes clear when the sepa-
rate portions of that day are considered. In the first portion of
that day, that is in the urine excreted in the morning between 8 and
12 o'clock, so before muscular exertion was begun, 0,404 gr. creatinin
was already found to 0,331 gr. and 0,845 gr. in the corresponding
portions of the preceding and the following day.

After ordinary food had been taken for nine days, food was taken
in absolutely insufficient quantity for five days at a stretch, con-
sisting of bread, potatoes, butter and cheese. It contained :

”

for v. H. 36.6 gr. proteid 43 gr. fat 186 gr. carbohydrate; 15 Cal p. Kg.

Pee. 29.7 POEs 55 Loon sy 33 be Se a RO

On Oct. the 16% a bicycle ride of 42 K.M. in 2'/, hours was
undertaken in the morning. In the first hour 20 K.M. were done
but after that they could progress but slowly from hunger and fatigue.

In the afternoon a walk of 16 K.M. was taken from 2 till 5
o'clock and afterwards in the evening they worked with halters. The
result was that both felt still very tired the next day.

The calculation of the average has no value in this short experi-
ment. The course of the excretion of the creatinin was as followed:

ie dele Vie

Oct. the 14% 2.020 1.908

ee melo unmelia 0)2 1.934
8, LOLs ATS 1.899 workingday

so devthe ed 83a 1.938

Plot ea mlesod 1.868

25
Proceedings Royal Acad. Amsterdam. Vol, VIII.

( 370 )

Here too where the food was not sufficient for the organism to
defray the costs of the muscular labour, as appeared also from the
increase of the nitrogen secretion on the workingday, one can cer-
tainly not speak of distinct influence of muscular labour on the
excretion of creatinin.

It is however different when no food is taken at all for days.

Van Hoocrnnuyze and VerpeLorcn had an opportunity to make
observations about this too on the “Hungerkiinstlerin” FrLora Tosca
a strong, young woman, who lent herself for the investigation during
a starving period at the Hague, in a room which was opened to the
public night and day. The urine was collected every day in three
portions, in the morning from 10 o’clock till 4 o’clock in the after-
noon, from 4 o'clock in the afternoon till 10 o’clock in the evening
and from 10 o'clock in the evening till 10 o’clock the next mor-
ning; it was sent every day at a fixed time to the Physiological Labo-
ratory in Utrecht and was there examined at once.

In the morning of June the 10% 4905 the last food was taken;
after that nothing but mineral water (Drachenquelle) till June the
25%, Besides creatinin several other constituents of the urine were
determinated daily ; about this it will be sufficient to mention that from
the course of the secretion of nitrogen, urea, uric acid and phosphoric
acid it appeared sufficiently that no food was taken.

During the whole hunger-period of fourteen days complete bodily
rest was observed as much as possible save on June the 17 when
Tosca during two hours-with short rests, under direction of VERPLOEGH,
was occupied with gymnastic exercises with halters of 1 KGr. 13
different movements were made, the first ten 20 times each, the last
three 10 times each. The movements were so chosen that us many
muscles as possible were set to work.

The examination of the urine showed now that in hungering the
secretion of the creatinin as well as of the other products of meta-
bolism steadily decreased. But the muscular labour suddenly produced
an undeniable increase, not on the same day, but on the following.
Still on the third day the influence was to be perceived, which however
also was the case with connection to the total quantity of nitrogen.
On the first day, when food was still taken, the quantity of creatinin
amounted to 1,087 gr. Later on it decreased rapidly and rather regu-
larly till on the 8% day. On June the 17, the day of the muscular
labour, it amounted only to 0,469 to rise the following day to 0,689.
In the three days before the muscular labour 1,662 was secreted, in
the three following days 2,006 gr. creatinin. After that the secretion
decreased almost to 0,5 gr, daily, to remain rather constant then,

( 374 )

From the above mentioned it appears that even with perfectly
regular food and with avoiding of all excessive muscular labour the
daily secretion of creatinin, as was communicated already in 1869
by K. B. Hormann'), undergoes rather important oscillations. This
is not sufficiently taken into consideration by those authors who
as Morressizr?) and as Gruecor*) have deduced from their results
with series of experiments of three, four or five days, where the
creatinin was precipitated from the alcoholic extract of the rine as
a compound of zinechloride, that the excretion of creatinin increased
as a result of muscular labour. It seems therefore to me that more
value may be attached to the conclusion, which v. Hoogennuyze and
VerpLorcH drew from their observations, that in man only then
increase ot excretion of creatinin is caused by muscular labour when
the organism is forced, by abstaining from food, to live at its own costs.

If the creatinin which is found in the urine of normal and nor-
mally fed men and animals is not to be considered, even were it
for a small portion, as a product set free by the contraction of the
muscle fibre, the question arises what signification must be given to
this constituent of the urine.

Since Mnissner’s researches *) it is known that to make use of meat
as a food must lead to the excretion of creatinin, as creatin and
ereatinin, brought into the blood either by resorption out of the
intestinal canal or by injection under the skin completely or almost
completely is excreted as creatinin by the kidneys.

The quantity of creatin in meat is rather important. It is usually
mentioned as 0,2 a 0.3°/, of the fresh muscle substance °). With
the aid of Forr’s method v. H. and V. have determined the amount
of creatin in muscle. 500 gr. meat freed as carefully as possible of
fat and tendons and minced was mixed with chloroform water and
was pressed out after standing for some hours at the temperature
of the room. This was repeated twice. After that the pressed out
meat was boiled for two hours with water and after cooling pressed
out anew. The filtrates were mixed, boiled at weak acid reaction
io remove proteids, after cooling filled up to 4000 ¢.e, and then
filtered. 500 c.c. of the filtrate was concentrated to 100 ¢.c. and
filtered anew, 80 c.c. of this filtrate was boiled with 50 c.e. 2

1) Virchow’s Archiv. Bd. XLVIII S. 358,

2) These Montpellier 1891.

5) Zeitschrift f. Physiol Chemie. Bd. XXXI S. 98.

) Zeitschr. f. rat. Med. Bd. XXXI, 1868, S. 234,
5) Vorr. Zeitschr. f. Biol. Bd. [V, 1868. S. 77.
25*

(at2. )

H,SO, 48 hours in the waterbath, to change all creatin into creatinin.
After that the quantity of creatinin was determined colorimetrically.
Every time a determination of the same kind of meat of different
animals was made twice. So the following was found:

Beef I 3.688 gr. creatinin: 4.378 gr. creatin p. Kg. meat

3898 =. ne ASN. 3 ae ie PY
Mutton I 3.499 ,, e 4.059 _,, _ ra oh ss
Lie 3(608ee # ASD, 3 a ee a
Rove I By 7ilis) © ss AVS ohms > Bd a2 e
Nt ZEOROY 55 ANAL ‘3 tee x
Horse I 3.244 ,, - SHOOMES 2 5 ee i}
NE BB < 50 rt SO43i we, 5. SR Eats .

Even with an abundant use of meat or beef-tea the creatinin excreted
by the kidneys (1,5 a 2 gr. or still more in 24 hours) can but fora
part be derived from the food. It is moreover well-known and by
the above mentioned researches proved anew that the secretion of
creatinin sinks not or scarcely under the norm, when the food does
not at all contain creatin or creatinin. The organism itself forms
creatin as a product of metabolism from the proteids. It would be
possible that the nature of the proteid taken up as food was of signi-
fication for the forming of creatin. In that case it was possible that
especially such proteids would produce creatin, out of which by
hydrolysis much arginin, a more complicated derivative of guanidin
could be obtained. According to the researches of Kossren and his
disciples, from gelatin twice as much arginin can be obtained as
from casein; out of gelatin 9.3 °/,'), out of casein 4.8 °/, 7). VAN
Hoocenuuyze and VwureLtonen have therefore examined by a new
series of experiments whether the use of casein or gelatin increases
the secretion of creatinin and if such is the case in what measure.

V. On April the 7 1905 a beginning was made with the use
of the same food as in series IV.

v. H. 47 gr. proteid, 98 gr. fat, 3387 gr. carbohydr.; 29.5 Cal. p. Kg.
VAS ss G4 «5 4555 olaness 3 30 * eee
On April the 12%, 13, 14% 50 gram casein was taken prepared

after HAMMARSTEN from cow-milk, in the afternoon at 12 o’clock

25 gr. and in the evening at 6 o’clock once more 25 gr. To leave

the total chemical energy of the food unchanged, so much fewer petatoes

were taken on these days that the quantity of carbohydrates fell
from 337 to 287. After that the food was taken as on April the

1) Zeitschrift f. Physiol. Chem. Bd, XXXI, 8, 207,
2) Ibid. Bd. XX XIU, S$, 356,

( 373 )

7% till April the 19. April the 20% 24st and 22°¢ 50 eram com-
mercial gelatin, well washed in water, was taken every time in two
portions, each of 25 gr., just as the casein instead of 50 er. carbo-
hydrate. On April 23'¢ and 24 the first diet was again taken.

In 10 days in which the food daily taken contained 47 gr. proteid
(the first two days April the 7 and the 8", which were still under
the influence of the food taken the preceding days, the urine was
not examined) the secretion of creatinin amounted to:

v. H. on the average 1.813 gr. (max. 1.921 min. 1.706 gr.) daily
Vie iss See rSDOne Ge ee OUONs 8 A230 Raa
On the days on which casein or gelatin was taken the secretion
of nitrogen increased but the secretion of creatinin not or scarcely.

It amounted on the three casein-days to:

v. H. on an average 1.913 gr. (max. 2.009, min. 1.836 gr.) daily
Nera, a JP Soi Gere leo Aad Oode....\hr ys

and on the three gelatin-days:

v. H. on an average 1.800 gr. (max. 1.813, min. 1.783 gr.) daily
Mises * “Woe wentttel(. satpataed lito! Sep mans Iie 0} - eerie enter

Just as in the series of experiments III as was mentioned above,
where, after the daily addition of 5 eggs to food which contained
47 gr. proteid, only a too insignificant increase of the secretion of
creatinin was found to attach any value to it, it appeared now that
the addition of casein and gelatin had no important influence whatever,
although the added proteid was daily resorbed and desintegrated in
the body, as the determination of nitrogen taught.

A short time ago Foun has communicated ample researches about
the constituents of human urine and has come to conclusions *) with
which the observations of van Hoogrnnuyzk and VERPLOEGH are quite
in accordance.

In 1868 Meissner has drawn the conclusion from his obsers
vations that the origin of creatinin in the organism of mammals
must be quite different from that of the urea with which most of
the nitrogen is excreted from the body’). Fou draws this conelue
sion anew and, in connection with his observations about the secretion
of other nitrogen containing substances and sulphur-eompounds, starts
from this point in proposing a new theory about the desintegration of
proteid in the animal body, which he puts in the place of the wellknown
theories of Voir and of PriiicGer. In considering the desintegration
of proteids in the body, there has heen, argues Foray, generally laid

1) Amer. Journ, of Physiol. Vol. XIII, p. 45. p. 66 p, 117.
2) }. c. S, 295.

(374 )

stress almost only on the total quantity of nitrogen excreted, in
relation to the quantity taken up in the food, and not enough
attention has been paid to the quantities of each of the different
nitrogenous products of metabolism which are excreted with the urine.

When the quantity of proteid in the food is enlarged or diminished
then the secretion of nitrogen increases or decreases till after a short
time a condition of equilibrium has been again obtained when intake
and output of nitrogen are alike. The variability of the metabolism of
proteids does not manifest itself in connection with all nitrogenous
substances but for the greater part with connection to the urea.
The secretion of creatinin on the contrary and also in a less degree
that of urie acid is apparently independent of the richness of the
food in proteid. We must distinguish a desintegration of proteid
variable under the influence of the food, on which depends in the
first place the forming of urea and which according to Forty’s
conception takes place for the greater part if not wholly in the

digestive organs — in the cavity and in the mucosa of the intestine
and in the liver — and beside a much less variable desintegration

of proteid in the different organs which does not immediately depend
on the food but on the function of the tissues. In the tissues there
arise undoubtedly nitrogenous products of desintegrating of different
composition. To them belongs as has been stated by NeNcK1, SALASKIN
and their collaborators ammonia, which is changed into the harmless
urea by the liver. Moreover urea is formed in the organism in
other places than the liver. This product of metabolism proceeds thus
for a part, as Fouin expresses it “endogenously” in eonsequence of
the rather regular metabolism of proteid in the tissues and for another
part “exogenously” in larger or smaller quantities, as more or less
proteid is taken up in the digestive canal. It is however not possible
to distinguish these two parts from each other in the urine.

But on the contrary the secretion of creatinin, on which the digestion
of the food when it contains no creatin has no direct influence,
gives an indication about the intensity of the desintegration of
proteid in the tissues. In this respect the muscular tissue, must be
thought of in the first place, but not exclusively, as creatin is formed
undoubtedly in other tissues too.

It does not seem necessary to accept that all the creatin which
is formed in the tissues is excreted as creatinin. The observations
of Murssner give already rise to the supposition that creatin must
be considered as an ‘intermediate’ product of metabolism, as has been
stated by Burian and Scaur for the uric acid. Metssnur at least
could not quite retrace in the urine the creatinin brought into the

( 375 )

circulation. He did find, it is true, that after injection of creatin
under the skin, not only the whole injected quantity was excreted
again with the urine, but also 20 mgr. creatinin with it, but it
remained uncertain how much of it proceeded from the metabolism
of the animal itself.

To obtain an insight into this v. H. and V. have made anew
an experiment in which the same food was taken, with 47 gr. proteid
daily, as in the preceding experiment.

VI. The experiment lasted from Aug. the 17" till the 28 1905.
On the first day the urine was not examined. The oscillations in
the secretion of creatinin were very insignificant. In five days
from Aug. the 18" till the 224 there was secreted :

v. H. average 2.023 er. (max. 2.029, min. 2.017 er.) daily
Vi: . 2028 syne cae o2rO298, = ue lO Onee.p)

2)

s 92rd ae p Mae ae Breve im ha :

On Aug. the 23" each of them took in one portion 500 mgr. pure

ereatinin dissolved in water. On the same day there was excreted :
v. H. 2.420 gr. and V. 2.508 gr. The next day:

Wie W2.030F 5 ay eee ONS

On Aug. the 26 each of them tock again 500 mer. creatinin but

divided into 10 portions, 50 mgr. every hour. Now also the creatinin

was found back the same day for the greater part in the urine.

The excretion amounted to:

vy. H. Aug. the 25 1.998 Aug. the 26 2.425 Aug. the 27% 1.940
Ang. the 28t 1.951 er.
V. Aug. the 25% 2.045 Aug. the 26 2.467 Aug. the 27%
2.035 Aug. the 28 1.968 ¢

At least in three of the four determinations a part of the creatinin
brought into the blood was not found back in the urine.

From this experiment, which has still to be completed with others,
in which creatin will be taken instead of creatinin, it appears how
distinctly every change of some importance in the excretion of crea-
tinin can be shown with the aid of Fo.in’s method. So it gives the
more reason to trust the results of the above mentioned series of

9 ” >

Tr

experiments, and the conclusion derived from them, that creatin is
i product of metabolism which is not formed at the contraction of
the muscle-fibre, but proceeds in muscles and other organs by the desin-
tegration of proteid to which is bound the life of the cells, without
regard to the developing of energy to which they are able in
performing their peculiar functions. Only then when the organism
is deprived of food and must therefore seek the power of performing

labour in itself, the material which the muscles want for contraction
is taken from the proteids of the tissue; for this the tissues are forced
to more vigorous life, of which an increased formation of creatin is
the result.

Quite in accordance with the investigations and arguments of
Foun, v. Hoogrnnuyze and Verrpioren also found that though the
excretion of urea increases and decreases with the resorption of pro-
teids, the excretion of creatinin is not directly dependent on it. There
is dependence in so far that with total privation of food, the activity
of the organs becomes as small as possible and that then with the
intensity of the symptoms of life the secretion of creatinin becomes
extraordinarily small. In connection with this a statement made on
the last day of the hunger-period of Tosca is worth mentioning.
June the 25‘ she took milk and eggs in the evening after ten o'clock.
The urine which was collected the following morning at 10 ’clock
contained 0.375 gr. creatinin, more than double the quantity which
was excreted by her in the last days in that same period. This
sudden increase can certainly not be put to the account of the food
as such, as is shown by the very slight increase of the excretion of
nitrogen in the same period, but must be attributed to the stimulation
which the whole organism suffered by the putting into action of
the digestive organs after such a long vest.

Nok. Paton investigated a short time ago with the aid of
Fourn’s method the excretion of creatinin of a dog which was fed
with oatmeal and milk and moreover on one day with 5 eggs and
which got no food at all on other days‘). According to the author
the results seem to indicate that in the dog there is a relationship
between the production of creatinin avd the intake of nitrogen.

The secretion of creatinin shows a somewhat too large irregularity
in the communicated series to admit the making of conclusions.
But if the impression of the author is right, there may be thought
here also of a stimulating effect of the food on the whole organism.

Just as Forms, van HoocgrnnuyzE and VerreLorcH have observed
not unimportant individual varieties in the excretion of creatinin
with mixed food. Without doubt the quantity of meat which one
is used to take, influences it. But with persons living pretty well
under the same circumstances the difference seems to be less great
when the weight of the body is considered. In 5 students a secre-
tion of 26, 26.9, 27.4, 29,4 and 31,5 mer. ecreatinin pro bodily
weight of one Kgr. was found in 24 hours,

1) Journal of Physiol. Vol. XXXII, p. 1.

(377)

Van Hoogenntyze and Verpetoren have also examined the urine
of some sucklings. Always creatinin could be shown, more distinetly
with the reaction of Jarré than with that of Weynt. On account of
the small concentration and the trifling quantities of urine which
could be collected an accurate colorimetric determination was not
possible. In four cases however a_ sufficient quantity of urine
(145—60 ee.) was obtained, to admit at least of a somewhat reliable
determination. In 10 ce. urine which was diluted to 50 ce. after
having been mixed with picric acid and caustic soda, there was
found :

I child 8 days old, 1.141 mgr. creatinine

Haase. 120 5 NUCH “5
NE 2months ,, 0.41 ,, E
IV ” 2 ” ”» A. ” ”

It is remarkable that in case III which coneerned a weak child
which was fed exclusively on cowmilk, the quantity of creatinin was
so much smaller than in the three other children who were all
strong and brought up by human-milk.

The above mentioned proves, as it appears to me, that the method
of Foun is an acquisition of importance of which may be expected
that it will aid in penetrating deeper than before into the knowledge
of metabolism.

Physics. — “On the theory of rejlection of light by imperfectly
transparent bodies.” By Prof. R. Sissinan. (Communicated by
Prot. H. A. Lorentz).

1. The laws of metallic reflection have been derived first by
Caucny'), later by Kerrener’) and Voier*), while Lorentz *) has
developed them from the electromagnetic theory of light. By different

1) Cavcny, Compt. Rend. 2, 427, 1836; 8, 553, 658, 1839; 9, 726, 1839; 26,
86, 1848; Journ. de Liouv, (1), 7, 338, 1839. Caucny gives only general remarks
on the way followed by him. Derivations of the results have been given, inter
alia by Beer, Pogg. Ann. 92, 402, 1854; Evrinestausen, Sitzungs-Ber. Akad,
Wien, 4, 369, 1855; Eisentour, Poge. Ann., 104, 368, 1858: Lunpaquist, Pogg,
Ann., 152, 398, 1874.

2) Pogg, Ann., 160, 466, 1877; Wied. Ann., 1, 225. 1877; 3, 95, 1878; 22,
204, 1884, Kerteter has, also in consequence of Vorer’s observations, modified
his developments, and given a final form to them in the “Theoretische Optik”, 1885.

3) Wied. Ann., 28, 104, 554, 1884; 81, 233, 1887; 48, 410, 1891.

4) On the theory of reflection and refraction of light, 1875; Scutémitcu’s Zeitschr.
f. Math. u. Physik, 23, 196, 1878.

ways these investigators arrive at exactly the same results. The
relation inter se of the mechanic theories has been elucidated by
Drupe'). In 1892 Lorenrz*) derived the laws of the refraction of
light by metal prisms, which had already been given by Voice *)
and Drupe*), from a few simple principles. Concerning the nature
of the vibrations of light no special hypothesis is introduced. This
investigation of Lorentz enables us to develon the theory of metallic
reflection in a simple way.

2. The simplest disturbance in a metal is that represented by:
HO (Ga eige i of oa co co (Ll)
In this w is the distance from the bounding plane of the metal.
This disturbance is caused when light falls perpendicularly on the
metal. Here we meet with the particularity, that the planes of equal
phase determined by the goniometric factor of (1) coincide with
these of equal amplitude which follow from the exponential factor.
From the assumption that the metal is isotropic and the deviation
from the condition of equilibrium in the light disturbance is a vector
determined by homogeneous linear differential equations, LORENTZ
derives, what other disturbances are possible in the metal. Assume
that the bounding plane of the metal is the V/-plane, and that the
plane wave-fronts are perpendicular to the /-plane. Then a distur-
bance is possible, represented by :

Aes Eihisin(¢t —(Ola——i8)) i 5 el eee)

if
Hee Oy pie ee een nore ao Go (2)
THONG Gz) — ACP) =n A 5 oo (4)

are satisfied.

The planes of equal amplitude and phase are given by /, = const.,
,»= const. In this /, is the distance to the plane in which the
amplitude is A, and /, that to the plane in which the phase has the
value s. a, and a, are the angles of the normals of the planes of
equal amplitude and pbase with the X-axis,

3. From (38) and (4) the principal equations for the propagation
of light in metals may be immediately obtained. If light penetrates
from the surroundings into the metal, then the planes of equal
amplitude are parallel to the bounding plane. The exponential factor

1) Goltinger Nachrichten 1892, 366, 393.
2) Wied. Ann., 46, 244, 1892.
8) Wicd. Ann., 24, 144, 1885.
4) Wied. Ann., 42, 666, 1891.

(2379 )

in (2) passes into e~?* and «,—=0. In this case @, may be called
the angle of refraction in agreement with what takes place for
perfectly transparent bodies. Denote it by @, then (4) passes into :
IOUS (Silo. “Ob G0. 6. oa o (@))
Let us now put P=2ak:4, where 2 is the wave length in the
air, and / the coefficient of absorption. In (2) we put Q=2a:42,,
where 4, represents the wave length in the metal. Be 4:2, =n, then
we may call the index of refraction of the metal m, in agreement
with what happens in transparent bodies. In the same way Q = 2an: 2.
Let us call the values of 7 and n, when the light propagates in the
metal perpendicularly to the bounding plane /:, and x,. Then in (1)
abe A, = 27 124°: 2.
Introducing these values into (3) and (5), we get:
[Pe G5) as [NT te Ge eG ip aol cage i (s))
NSCOSYOr—h cg Waren Rolin sy eRcAr gio, “<A Wi!)
In order to bring our formulae for the disturbance in the metal
at the bounding plane in harmony with those for the disturbance in
the air, we must put sind: sina = 4:2, =norsini=nsina, when
7 is the angle of incidence. It follows from (6) and (7) that both
the index of refraction and the coefficient of absorption depend on
the direction of propagation, i.e. on the direction of the normal
of the planes with equal phase.
If (7) is written in the form:
PROP (2 = (PS) a ie = 6 te, ee, (3h)
it follows from (6) and (8) that:
2n? = —k,? + n,? + sin? i 4- V(k,? — n,* + sin’ i)? + 472k? « « (9)
2 = =k? —n,? + sin? i t+ Y(k,? — ny? + sin® i)? + 4n,7k,?. . (10)
They denote in what way / and n depend on the angle which
the direction of the propagation of the disturbance falling on the
metal forms with the normal to the bounding plane ’).
For an opaque mirror of silver deposited on glass by a chemical

1) Kerretern was the first to derive these equations, (see inter alia Pogg. Ann.,
160, 408, 1877) which, of course, also occur in Voter's theory. Voter puts the
quantity corresponding to P equal to 27k: A,, so that Voter's mk corresponds
to the coefficient of absorption & introduced here. It is not correct that Gaucuy
already gave these equations, as Drupe observes (Wied. Ann. 35, 515, 1888),
They have not been given explicitly in this theory. This appears, indeed, from
the fact that Beer (Pogg. Ann., 92, 412, 1854) substitutes other relations for
them, which are not correct. Derivations of the principal equations were given by
Wenrnsicke (Pogg. Ann. 159, 226, 1876) and Kerrener, Pogg. Ann. 160, 468, 1877,
See also Kerrerern, Wied. Ann., 49, 512, 1893 and Theoretische Optik, p. 198,
§ 85, Zur Geschichte der Hauptgleichungen,

( 380 )

way, and very firmly attached to the glass, the values J = 72°345,
/] = 42 21'.7 were found for principal angle of incidence / and principal
azimuth /7') (H being the angle which the plane of polarization of
the reflected light, being restored by compensation to plane polarization
makes with the plane of incidence) from which follows for:

j= OF 20° 40° 60° 80° 90°
n= O295 0,450 0,800 0,928 0,990 1,03
= 2788 2,90 2,95 3,01 3,04 3,05

In the same way I found for a steel mirror’), / = 77°23'5,
H = 26°34', so that for:
HOF baal F902
n = 2,684 2,794 2,799
hk: = 3,404 3,491 3,496
As follows from (9) and (10), & and 7 increase with the angle
of incidence ¢ From (8) follows, that always 1? > sim?t. Media
which absorb the light, can never reflect the light totally.

4. Normal to the planes of equal amplitude the amplitude decreases
in ratio 1:e—! over a distance 2: 22%. In the planes of equal phase
the points whose amplitudes stand in the same ratio, lie at a dis-
tance 2: 2a k sin (a,—a,).

According to (6) and (7) depends on &. The velocity of propa-
gation depends therefore on the way, in which the amplitude in a
plane of equal phase varies. If ¢ =O, it follows from (6) and (7)
that f=, ,2—=n,. The planes of equal phase and amplitude can
therefore only coincide with a propagation normal to the bounding
plane. If this took place in every direction, 4 would be zero according
to (8), so the substance would have to be perfectly transparent.

When the planes of equal phase and amplitude are normal to
each other, «= 90°. For light that penetrates into the metal from
outside, the planes of equal amplitude are parallel to the bounding
plane, so for a= 90° those of equal phase are perpendicular to it.
The propagation then takes place parallel to the bounding plane.
This is in harmony with what follows from (7) and (8). According
to (7) k,n, = 0 for ¢= 90° and so according to (8) either 4 = 0 or
i — sini. The first case leads us back to perfectly transparent media.
For == sini there is total reflection. This however, can only be
the case with light absorbing media, if 4,7, =O or, as n, > 0, if

1) Sissineu, Thesis for the doctorate, p. 88, 1885, Arch. Néerl., 20, 207, 1886.
2) Stssinau, Verh. Akad. vy. Wetensch., Amsterdam, deel 28, 1890; Wied. Ann.,
42 132, 1891.

(3st)

k, = 0. So the coefficient of absorption of the medium normal to the
bounding plane had to be 0. For metals this is not the case, so that
no total reflection can occur there, as has been observed above.

It is well known that with total reflection on perfectly transparent
media the planes of equal phase and amplitude are normal to each
other for the disturbance in the second medium which is propagated
parallel to the bounding plane. Voier showed, that this case also
occurs for a disturbance, which leaves a prism of a substance which
absorbs light, when plane waves fall on it and the dimensions of
the prism are large with respect to the wave length ’).

From (6) and (7) we may derive (4,7 — n,’) cosa = n, k, (= — =),
From this follows, that according as /::7 increases, a differs more
from x: 2, with which we have got back a result of Vorer’s ’).

5. Ersentour*) showed, that by the introduction of a complex
index of refraction, we arrive at Caucuy’s results for metallic reflection.

In the followmg way it may be shown that for metals a complex
quantity corresponds to the index of refraction of transparent bodies.
With observance of the conditions (38) and (4), (2) is a possible distur-
bance. In this /, and /, are the distances from the point for which
(2) holds, to the plane of equal amplitude, in which the amplitude
is A and the plane of equal phase, in which the phase is s. We
may also write for (2):

Ae -Pit—P2? sin (cb—q, --q,e—8) . . « . . (Il)
because the planes of equal phase and amplitude are normal to the XZ-
plane. The normals from the point 2,2 on the two above mentioned of these
planes are respectively (p,7-+p,z): Vp,°+p,” and (q,7+9,2):V 9.7 +41"
Boy that P= /p,*+p;, Q= V4a,7+4,"-

In the same way as (11) a possible disturbance is also:

Ae?" Px cos (eb — q,®— 4.2 —8)+

The differential equations, which are supposed homogeneous and
linear, are therefore also satisfied by :

AePit12? feos (ct— g,a—9,2 —8) + sin (ek—q,e—q,2z—s)}
or by
Ae i (ct—ainFy)—elgetoel—st , fw. (12)

For a perfectly transparent medium p, = p, = 0. The velocity
of propagation is then v=c:Vq,’+9,”, or c being c=2z: 7,

1) Wied. Ann., 24, 153, 1885.

2) Wied. Ann, 24, 150, 1885.
8) Pogg. Ann., 104, 368, 1858:

( 382 )

v= 2a:7Vq,'+ 9,7. Let the velocity of light in the air be V, the
index of refraction of the perfectly transparent medium 7, then:
n* = VAI (g7-q.s)\ Acca
From (12) follows, that for a metal g, + ¢p, occurs instead of q,
and the quantity g, + q, for q,. Let 7, be the quantity, which for
a metal corresponds to tie index of refraction n of the perfectly
transparent medium, then
2 yal’ rial 2 a9
Rm = dx? IP + Pa? + 9a" F 2 P19 + P2Ga)}-

The cosines of the angles formed by the normals of the planes
of equal amplitude and phase with the Vand Z-axis, are respectively :
Pit VPY+Ps > Ps? VP +p," and 1° V+ 9s" Dan Fc V9q,?-+937

With observance of the above given values of P and Q and in-
troduction of the angle @ between the planes of equal phase and
amplitude p,q, +p.q2 = PQ cos a. Thus:

72 '°'2
r= — (—P? + Q* = 2PQ cos a),
4x?
or according to (3) and (5):
i Vee , . 2 :
Ueetopeey aol pel ee GUL)

Hence the so-called complex index of refraction of a metal is

cag al

i (0) a= Let 4, be the wave length in the metal for light

entering normally, then according to (1) q=2m:4,=22n,:4 and
Ped 7 Re ONO = = Ur

6. It follows from what precedes, that in accordance with EIseNLOnr *)
we can deduce the expressions determining the amplitudes for the
metallic reflection from those for the reflection on transparent bodies,
if we replace n by n, = ek,. Let the incident beam of light have
the intensity 1 and let it be polarized in the plane of incidence.
The reflected disturbance may be represented by the real part of
ae we = etelct+z, Here sinr=sini:n. Putn=n, = th,, then as
sin (t + 7) sin(t + 7)
passes into Ae+. The disturbance reflected by metals is the real
part of Ae tet+z—*), in which A is the amplitude and & the dif-
ference of phase with the incident ray. In this way we arrive at
the well known expressions for the metallic reflection. I may be

1) Cf. also Lorentz, Theorie der Terugkaatsing en Breking, p. 163,
Scuiémitcn’s Zeilschr., 28, 206, 1878.

( 383 )

allowed to place them here side by side, after which I shall give
some expressions which enable us to determine the optical constants of
a metal from the quantities measured, and also some approximative
formulae for the calculation of the principal angle of incidence /
and principal azimuth /7 from n, and 4/,.
Light polarized // plane of incidence.
reflection by

transparent bodies metals
Intensity
Incident light Reflected light
1 sin*(t—rr) — (cos i—y ni—sin*i)?
sin?(i+7) e~ (cos i+ Vn? —sin?i)? +k?
Difference of phase with incident beam
180° tp) = — si :
1—n?— hk?
Light polarized 4 plane of incidence
Intensity
; tg*(i—r) ae n*cos*(i—a)-+ k*cos*i s
tg?*(i—r) n'cos'(i+a)+keosi ?

Difference of phase with incident beam
Opator 0 <a F,
SUM seen <a tien 902

oe 2k(k? + n2cos?a— sin’) cos t

(hk — n°cos" — sin*i)cos*t — n’cos’a cos 2i-k?

tug = (Ci

From this follows also:

2k sin i tg i

tg (pi — $)) = —

n? cos? a — tg? i sin? i SLES

n and k apply to the disturbance in the metal, arising from plane
waves, which fall on the metal with an angle of incidence 7. The
angle of refraction @ is determined by Sina@—= Sind: n.

7. The expressions obtained are in perfect harmony with those of
Caucuy. First the relations of § 6 may be brought to the same form
as these. For this purpose we put in accordance with Brrr’):

MUOse— | n — smn t= Ucosu, k= U Sma.) 5) (is)

Substituting this we get

cos? t+ U? — 2U cosicosu

32

VS : Oy op Se
: cos?t + U? + 2U cosicos u

Put

1) Beer, Pogg. Ann. 92, 413, 1854.

( 384 )

then

From the value of 7 follows also

, ’ cos t
cot f = cos u sin | 2 Ba tg i Peete: Go) (15)

; cos t
td Pp = sinutg 2 Ba tg a}

Be Ri: R, = tgh, then

Further we get:

cos? ((—a) a K? cos? 2
n® cos? (i +e) + k cos? i

In the corresponding expression of Caucuy the value of Cos 2h
is given. From the value of fg? follows, as

tq? i—

2n cos a sin? i cost

(n2cos?a+- k*)cos*i+ Sinti ©

cos 2h = (1l—tg 2h): (1-+-ta2h) , cos2h =

Ace ording to (13) this passes into
: sinitgt a
cos 2h = cos u sin | 2 Ba tg U —— ee EDC (15)
In the same way becomes

2 2U sinitgi ; sintitgt
tg (Pl — Gp) = sin u ——__——_ = sin uttg | 2 Bg tg — a . (16)

U? — sin® to* i

The expressions (14), (15), (16) have the same form as the corre-
sponding ones of Cavcny, only according to LorENtTz’s notation 69
stands for U’, the angle t+ © for w.*)

Just as from 7, U and wu the quantities R,, R), gp, and g may be
derived, which determine the reflected beam of light, U7 and w may
be calculated from 7 and two of these four quantities. U and u
depend therefore in exactly the same way on the angle of incidence
and the optical properties of the meial as Caucuy’s corresponding
quantities cg and t+ @. At the same time it appears that two constant
quantities suffice for the determination of the optical behaviour of metals.
They are here n, and &,, which have a definite physical meaning,
with Cavcny o and t, whose meaning is not so obvious. Whatever

1) Lorentz, Theorie der terugkaatsing en breking. p. 166. According to E1seNLoHR’s

. . BY oQ- a ~— = « Pave
notation (loc cit. p. 369, 370) U=cH,uw=e-+u. As Rp and Rr: Rp, gy and
yi—g, have the same form as Cavcuy’s equation, this holds also for Rjand g ,

( 385 )

system of two determining quantities is chosen, however, the ampli-
tudes and phases of the two components of the reflected light pola-
rized in and normal to the plane of incidence, whether caleulated
in one way or in the other, will have the same values. The two
systems of formulae are therefore identical.

Cavucny *) calls the so-called complex index of refraction oe*. This
quantity being represented here by x, + th,, 6cost = n,, Osint = f,.
The auxiliary quantities 9 and @ have been introduced by Cavcay
for the determination of the so-called imaginary angle of refraction r,
determined by sin 7 = sini: (n, + ,)*). In order to express 9 and
® in the quantities used above, it may be observed that:

cos*-r sin® i: sin® r = cos* x(n, + th.) = (n, + th,)? — sin? i
or with the aid of (6) and (7):
cot r sint = neosa + th.

As Cavcuy gives:

cos r = ge” *) and n, + tk, = oe*, neosa + tk is equal to erect)
or

neoosa== Ucosu=gocos(t+om) . . . . (17)
k= Usnu=oosn(t+o). - . . . . (18)

The equations (17) and (18) allow us to deduce our auxiliary
quantities from those of Caucny and reversely *).

8. According to § 7:

2U sinitgi
cos 2h = cos u Th sin' i igi stg (PI—G)) =

These two equations may serve to determine U and wu, and from
this the optical constants x, and 4, with the aid of (13), (6) and (7).

From the values of cos 21 and ty (yi —g,) follows :

; 2U sinitgt
sin u

U?*—sin? i ti

= U? — sin* it? i
stn 2h cos (PP) Ta sini tg? iz
or 2

2 sin® i tg’ 2

U* + sin® itg i
From (19) and the value of cos 2/ follows :

1 — sin 2h cos (i—¢p) = (19)

7 sin i tg i cos 2h (20)
cos wu == ——————_-_ . ... .
1 — cos (~l—Gp) sin 2h

1) Lorentz, 1.c. Theorie der terugkaatsing en breking, p. 164, Scutésicu’s Zeitschr.,
28, pg. 206, Etsentonr, p. 369.

2) See note 1 on the preceding page.

8) Cf. also Kerrever, Wied. Ann. 1, 242, 1877; 22, 212, 1884. Formulae for
the calculation of ; and « are given by Lorentz, Theorie der Terugkaatsing en
Breking, p. 164, 165, Eisentonr, Pogg. Ann., 104, 370, 1858.

26

Proceedings Royal Acad. Amsterdam. Vol. VIII.

( 386 )

Further
2U sin u sini tg i

VU? + sinritg i
From this and from the value of tg (gi—g,) follows

)

sin 2h sin (pi— Pp) =

; sin itgisin(~, —@,) sin 2h
0 snu= “ (f= 9») A

(21) °)
So from the restored azimuth 4 and the difference of phase

yi—g, at an arbitrary angle, Ucosu and Usinu or neosa and k

”

1 — cos (~i—Gp) sin 2h

are to be derived for that angle. As n cos a= (n> — sin’ i, we get
afterwards n, and *, with the aid of (6) and (7). By means of them
we can calculate gj—gp and h for every angle.

9. As a rule we introduce the principal angle of incidence J,
for which gj;—g,=2a:2. The restored azimuth at this angle is
called the principal azimuth H. As well from (20) and (21), as from
15) and (16) we may derive, when we add the index J to the
values of all the quantities for the principal angle of incidence :

O-= sin Litg Dl <_, cost; == 1008 2-5) eee
According to (13)
p= Urpsnuz=sm ligt sn 2... . = 9 ae)

(n* cos? a), = ny? — sin? I = sin? I ty I cos” 2H
or
ny = t9) I (V—sin Lain 2) ee eee)
We may also write (24):
np kp sat F*)y Se ee ee)
The optical constants n, and &, are obtained from :
n, —k,° =n? — ns = ty’ I (1—2 sin? I sin? 2 A)
or
ny k= (n cos\a)7 ky = 4 sin® Tito? Lan 4 H -. . . (26)

10. When x, and &, have been given, we find by elimination
of n; and k; from the two first members of the two equations (26)
and (25) an equation of the sixth degree for the determination of
ty I. There may be given also approximating formulae for the deter-
mination of / and // from n, and /,. From nj—k,=n,’—k,* and
k Via si’ T=n,k, follows

Qn = sin ltni—khe+y (sinl—n,—k,2 4+ 4h, 2807 LD

Substituting this in n+ 4/= tg’ I, we get:
') This equation was already given by Kerrerer, Wied. Ann,. 1, 241, 1877.

2) Kerrerer calls this equation an analogon of the law of Brewster, Wied.
Ann 1, 242, 1877.

( 387 )

sin'T + Qsin?I(k,? — n,°) + (k,2 + 2.) = sintTtg’T . . (27)
With metals »,?-+-4,° is comparatively large compared to the two
first terms of the first member of (27). By approximation we get
therefore :
sin’ Lig? L=k, + 2,5
from which follows with the same degree of approximation
1
(ee
Introducing this in (27), we get:
sinltqI = Veen, ete E Cs io
eee Ea ky?
In the following way we get an approximate value for H. From
(23) and (24) follows :
nt — kf =n, — kh = sin? 1 + sin’? Lito L cos 4 A,

sel = 1 —

(28)

so
24° —k, —sin’L
cos 4. H = ——___—_—_
sin’L tg I
1—cos4 H

From this follows, as ty? 2H = , after substitution of the

1+-cos4 1
approximate value

oh Tye + 9 ky —ny”
sin? Ttg’ I = (n,* + &,?) 41 + sue snr
Cy No

which follows from (27),

k
tg 2H = = 1 + sin? I 836 oh to ot (AD) 9)
ny

ny ER

11. Finally it may be observed that the relations hold for any
value of &. The reflection on perfectly transparent bodies is therefore
a limiting case for the metallic reflection. *)

Chemistry. — “On the chlorides of maleic acid and of fumaric
acid and on some of their derivatives,’ By Dr. W. A. van
Dorp and Dr. G. C. A. van Dorp.

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

1) Corresponding approximate formulae were given by Drupe in WINKELMANN,
Physik II. 1, p. 823, 824.

2) Cf. Vorar, Wied. Ann., 24, 146, 147, 1885.

(October 25, 1905).

KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM.

PROCEEDINGS OF THE MEETING
of Saturday October 28, 1905.

(Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige
Afdeeling van Zaterdag 28 October 1905, Dl. XIV).

Gjoyaypunaaanp Buss

H. pe Vries: “Central Projection in the space of Loparscnersky” {Ist Part). (Communicated
by Prof. J. Carpryaar), p. 389.

H. J. Hampurcer: “A method for determining the osmotic pressure of very small quantities
of liquid”, p. 394.

Evcen Fiscuer: “On the primordial cranium of Tarsius spectrum”. (Communicated by Prof.
A. A. W. Husrecut), p. 397.

H. A. Lorentz: “On the radiation of heat in a system of bodies having a uniform tem-
perature’, p. 401.

H. ZwaaRvDEMAKER: “On the ability of distinguishing intensities of tones. Report ofa research
made by A. DEENIK”, p. 421.

Erratum, p. 426.

The following papers were read:

Mathematics. — “Central Projection in the space of LOBATSCHEFSKY”’.
(ist part). By Prof. H. pe Vares. (Communicated by Prof. J.
CARDINAAL).

(Communicated in the meeting of September 30, 1905).

1. Let an arbitrary plane r be given in hyperbolic space; let
the perpendicular be erected in an arbitrary point O, of r, and let
finally an arbitrary point O be taken on this perpendicular. We can
now ask what we can notice if we project the figures of space
out of U as a centre of projection on tr as a plane of projection or
picture plane, and inversely, how the exact position and situation of
the figures in space can be determined by means of their projections,

27

Proceedings Royal Acad. Amsterdam. Vol. VIII.

( 390 )

In the following a few observations will be given on these two
questions.

Let us suppose an arbitrary plane € through the line OO,, standing
therefore at right angles to +; in that plane we can draw through
O two straight lines p,, p, parallel to the line of intersection e¢ of €
and rt passing through Q,, therefore also parallel to r itself. The
angles formed by p, and p, with OO, are equal; they are both
acute, and their amount is a funetion of the distanee OO, = d.
Lopatscurrsky has called each of these two angles the parallel angle’)
belonging to the distance d, and has indicated them by aq); if d is
given, the parallel angle is found out of the relation

ig */, Ma =e-%,

in -which for the number e the basis of the natural logarithmic
system may be taken, if only the unity of length by which OO, is
measured be taken accordingly *). As far as the range of values
of Ma) is concerned, I only observe that the parallel angle = 7/, x
for d@=0, decreasing and tending to O if d increases and tends
to o.

If the plane ¢ rotates round OO,, then p, and p, will describe a
cone of revolution round OQ, as axis; this cone is the locus of all
straight lines through O parallel to 7, and distinguishes itself in many
respects, in form and properties, from the cone of revolution of
Euclidian Geometry; the plane 7 is an asymptotic plane, to whieh
its surface tends unlimited, and from the symmetry with respeet to
OQ tollows easily that another plane t* like this exists, also placed
perpendicularly on OO,, but in the point O,* situated symmetrically
to O, with respect to O. So the cone is entirely included between the two
planes rt and +*, and these two planes having not a single point in
common (neither at finite nov at infinite distance), are divergent ;
however, they possess the common perpendicular O,0,*, and their
shortest distance is 2d. The cone discussed here will be called for
convenience, sake the parallel cone x belonging to the point 0.

2. The parallel cone divides the space into three separate parts;
let us eall those two parts, inside which is the axis OO,, the interior
of the cone, the remaining part the exterior; it is then easy to see
that the points of space behave differently with respeet to their
projectability according to their lying inside or outside the cone;
for a point P inside the cone the projecting ray OP forms with

1) F. Eyeet: ,N. I. Loparscnersky. Zwei geometrische Abhandlungen”. Leipzig,
Teusner, 1899, p. 167.
2) F. Enaez, 1. c. p. 214.

(391 )

OO,*) an acute angle smaller than the parallel angle, from which
ensnes that OP (or perhaps the prolongation of PO over () must
meet the picture plane; the point of intersection P’ is the central
projection of P. However, for P outside the cone the acute angle
between OP and OO, is greater than the parallel angle; so now
OP is divergent with respect to 1, from which ensues that points
outside the cone possess no projections at all; points on the cone on
the contrary do, but these projections lie at infinity.

From the fact that points outside the parallel cone are not projectible
we need not infer that these points cannot be determined by Central
Projection; if such a point is regarded let us say as the;point of
intersection of two straight lines, and if these can be projected by
Central Projection, their point of intersection will also be determined
in this indirect way.

3. Let a straight line / be at right angles to r in a point D
of tr. As the line OQ, is also perpendicular to rt, it is possible to
bring a plane through / and OO,, the trace of which connects 0,
with JD. This plane will intersect the cone x in two generatrices,
Py» Pas We now assume that / cuts these two lines in two points
P, and P,, and — to fix our thoughts — that P, lies between
D and P,. The line / possesses two points at infinity, V,

Geena
which both lie inside the parallel cone; let us suppose that V,,,
lies under the picture plane and J’,, above it, then the succession
Gathespomtson,-/ 1s this: Va. Ds 2) PioVas:

The projecting ray OV,, cuts rt in a point V", of e lying between
D and O,; we shall call it the first vanishing point of 1. In like
manner the ray OV,, prolonged over V will cut the line e in a
point V’, lying in such a way that (, lies between V’, and V’,;
we shall call V’, the second vanishing point of 7. The point O,
does not lie in the middle between V’, and V’,, on the contrary
it is closer to V’,; if namely we let down the perpendicular OS out
of 0 to 7 a quadrangle is formed with three right angles, namely
at O,, D and S, and from this ensues that the fourth 7 SOQ, is

acute. Now OS is the bisectrix of 4“ V,,OV,,, and therefore the
perpendicular in O on OS the bisectrix of 7 V’,OV’,; this perpendicular
must be placed, as “ SOO, is acute, between OV,, and OV’,, and
from this ensues “ V’,OO, <7 V’',OO,. If we now let the
rectangular triangle V’,OO, rotate about the side OO, till it lies on
the triangle V’,O0,, then we can immediately find that O, V7,< 0, V’,.

1) By OO, we understand the straight line prolonged at both ends unlimitedly,
by OP however a semi-ray starting from 0.

27*

( 392 )

It is clear that the central projection /’ of 7 coincides with the
trace e of the projecting plane O/=e, and at the same time that
/ is determined by its point of intersection and by one of the two
vanishing points; the second will be found by letting down the
perpendicular OS out of O to 7, and by setting off at the other side
of OS an angle equal to the parallel angle, formed by OS and
the only existing ray parallel to 7. But further can be remarked
that / is also determined by its two vanishing points, or what comes
to the same by its two points at infinity ; to find 7 we should but
have to bisect the angle formed by the two projecting parallel rays
of /, to mark on the bisecting line a segment OS corresponding
to '/, 7V,,0V,,, as parallel angle, and to erect the perpendicular
in S on OS.

The line / is divided into four segments by its two points at infinity,
its point of intersection and its two points P,, P, (whose projections
lie at infinity), and /’ in like manner by its points at infinity P’,_,
?”,,,, the point D, and the two vanishing points V’,, V’, of 7; the
connection between these different segments of / and /’ is as follows.
To the infinite segment ’,, D corresponds the finite segment V’,D,
and to the finite segment D/P, the infinite segment D?P’,,; to the
points. between P, and P, no projections correspond, because the
projecting rays of these points are divergent with respect to 1;
to the infinite segment P,V,, on the contrary a segment of /’
again corresponds, namely the infinite segment 1’,, V’,. There now
remain on /’ only the points between the two vanishing points, to
which also belongs 0,; to these no points of / correspond, their
projecting rays being divergent with respect to /.

§ 4. If a line /17 is to cut the surface of the parallel cone
in two points, the length of DO, may not exceed a certain upper
limit, so that the results just found do not hold for a// lines Jr.
Let us again suppose through OO, an arbitrary plane «, and let us
now first regard OO, itself. If we let down out of O, on to p,
the perpendicular O,7, then because p, is parallel to e, the angle
TO,P’,, is the parallel angle belonging to 0,7, and therefore angle
TO,O is smaller than this parallel angle, because O,O cuts the
line p, (namely in “Q); and OO,P’, being equal to 90°, the paral-
lel angle 7'O0,P’,,, > 45°, and angle 7'0,0 < 45°. Ifin ¢ we move /,
first coinciding with OO,, in such a way that it remains in D per-
pendicular to e, namely towards the side of P’,, (therefore from
P’,.)» then the perpendicular DZ’ on p, becomes continually greater,
and so (see N°. 1) the parallel angle 7DP’,,, continually smaller;
as soon as the perpendicular D7’ has attained such a length that the

{ 393 )

parallel angle corresponding to it is precisely 45°, the complement
becomes 45° too, and therefore / parallel to p,, but on the other
side of D7’ compared to e; / will still intersect p, in a finite point
P,, for as it enters the triangle OO,P,, at D, does of course
not contain the point 7?”,,, and is divergent with reference to OO,,
it can leave the triangle only ina finite point of p,; but it will eut p, in
an point at infinity ?,,, being at the same time J’,,,. So its projectio
consists of the segment of the line e of V’, over D to P,, and
the isolated point P’,, is equal to V’,, ;
by two of the three points D, V’,, V’,,.-

Be now too it is determinec

The point D lies at a certain distance 7 from O,; if we describe
a circle in tr about ©, as centre and with 7 as radius, and if we
erect in all points of that circle the perpendiculars on 1, a surface
appears which may be called a cylinder of revolution, of which the
circle just mentioned is the gorge line; the lines / (41) lying inside
that cylinder have two different vanishing points (with the exception
of OO,, whose projection is a single point), the lines /on the cylinder
have a finite and an infinite vanishing point, and the lines / outside
the eylinder miss the second vanishing point.

As for the shape of the cylinder it is easy to see, that the plane
1 (see N°. 1) is an asymptotic plane; and + itself being evidently «
plane of orthogonal symmetry, the plane 1** normal to OO, in the
point O,** symmetrical to O,* with respect to + will be a second
asymptotic plane; so the distance of these two planes is 4d.

5. In Euclidean Geometry the lines Jt are at the same time
those which are parallel to OO,, but in Hyperbolic Geometry this
is different; here we have to regard the lines having in common with
OO, the point I’,, lying under the picture plane at infinity, and
those having with OQ, in common the point V’,, lying above r.
A line 7 of the former kind lying in the vicinity of OO, has a
picture point D, two points P,, P,, and a second point at infinity
lying inside the cone x; its first vanishing point coincides with O,,
whilst the second lies on YO, in such a way, that O, lies between
D and that point.

If the perpendicular OS let down out of O to 7 becomes conti-
nuaily larger, the first particularity appearing here is that 7 becomes
parallel to the generatrix p, of cone % lying in the plane O/; then
it is at right angles to the bisectrix of the obtuse angle formed by
p, and OO,. All lines having this property form an asymptotic cone
of revolution ') with vertex V,,, whilst r* is an asymptoti¢ plane;

!) Hf. Lrepwany, “Nichteuklidische Geometrie”’, Collection Schubert XLIX, page 63,

( 394 )

as base cirele we can obtain a circle with finite radius in x. For
the generatrices of this cone the second point P?, coincides with the
second point at infinity; so the projection consists of the infinite segment
O,DP’,, and the isolated second point at infinity of this line.
For lines / outside this cone this isolated point vanishes, and on
account of this the second vanishing point; for its determination remain
however J, and the first vanishing point O,. Now however, the
perpendicular QS. still inereasing, / can become parallel to p,, and
hence parallel to e or to r; it is then at right angles to the bisectrix
of the acute angle between p, and OO,, as well as to that of the

=

right angle between ¢ and O,V,,,

which bisectrices are respectively
divergent. All lines showing this property form a second asymptotic
cone of revolution, for which however 1 is now the asymptotic plane ;
they have a picture point at infinity, but are no less determined by
this point and the first vanishing point 0,.

If 7 also lies outside this second cone, it becomes divergent with
respect to rt, so it loses its picture point D; but now its second point
at infinity lies again inside the cone x, which makes it projectible, so
that in this case / has two vanishing points but no picture point ;
however, the two vanishing points are sufficient for its determination
(see N°’. 3). The originals at infinity corresponding to it are both
under the pieture plane; in connection with the preceding it would
be preferable, in order to avoid confusion, to say that / has in this
case two “‘first points at infinity’ and therefore also two “first
vanishing points”.

The lines containing the second point at infinity V,, of OO,
behave in like manner; we again find two asymptotic cones of
revolution, one with the asymptotic plane 7, a second with the
asymptotic plane r*, and we terminate with lines with two “second
vanishing points” and without picture point.

Delft, September 1905.

Physiology. — “A method for determining the osmotic pressure
of very small quantities of quid.” By Prof. H. J. Hampureer.

lt not unfrequently happens that one wishes to know the osmotic
pressure of normal or pathological somatic fluids of which no more
than ‘/, or ‘/, ee. are available. I recently had such a case when an
oculist asked me what should be the concentration of liquids used
for the treatment of the eye. It seemed to me to be rational — and

the investigations of Massarr ') justified this opinion — to prescribe

1) Massart, Archives de Biologie 9 1889, p. 335.

(395 )

concentrations of the same osmotic pressure as the natural medium
of the cornea and conjunctiva, namely the lachrymal fluid.

Until now, however, this pressure had not been measured, at any
rate by a direct method, probably on account of the difficulty of
obtaining a quantity of that fluid, sufficient for the customary methods,
viz. the freezing-point and the blood corpuscles method.

So I tried to find a method with which '/, cc., if necessary */, ce.
of liquid should be sufficient. | succeeded in finding such a method.

It is based on the already known principle that the volume of
blood corpuscles is greatly dependent on the osmotic pressure of the
solution containing them. ‘)

This principle has been applied here in the following manner.
The fluid to be examined is put into a small, funnel-shaped glass
tube, the cylindrical neck of which is formed by a calibrated capil-
lary, closed below. *) Let this quantity be */, ce. Into other similar
funnel-shaped tubes of the same size are put solutions of Na Cl of
different concentrations (OlSae/ OL9 bon eee ele ee ee eye,
1-4 °/,, 1.5°/,, 1.6°/,) and to each of these 0.02 cc. of blood is

0? a> 0>

added. After half an hour — during which time the corpuscles are
sure to have found osmotic equilibrium with their surroundings —
the tubes are centrifuged until the sediments no longer alter their
volumes. It is obvious that the osmotic pressure of the fluid under
examination will be equal to that of the NaCl solution in which
the sediment of blood corpuscles is the same as in the fluid examined.
We passingly remark that this solution of NaCl in the ease of
lachrymal fluid contained 1.4 °/,.

A few remarks must be added.

Firstly it may be asked whether the blood which is added to the
fluid to be examined does not appreciably alter the osmotic pressure
of that fluid,

Assuming that the blood used contains 60 pCt. of serum, 0.02 ee.
of blood will contain 0.012 cc. of serum. If the quantity of fluid
was 1/2cc., the total quantity of fluid will now be 0.012 + 0.5 =
0.512 ce. If the fluid to be examined had an osmotic pressure of
a 1.2 pCt. Na Cl solution and the serum that of a 0.9 pCt, Na Cl
solution, dilution of the fluid with the serum will have produced a
auth ; .9.012 * 0.9 + 0.5 & 1.2
liquid with an osmotic pressure of TOI SS Sl TON

012 + 0.5
NaCl. The osmotic pressure of the fluid is consequently reduced

1) Hampurcer, Centralblatt f. Physiologie, 17 duni 1893.
*) Hampurcer Journal de Physiol. norm. et pathologique 1900 p. 889,

( 396 )

by 0.0L pCt Na Cl by the addition of 0.02 ce. of serum, a difference
which cannot even be detected with the BrckMANNn apparatus.

If instead of 1/2 cc. of fluid only 1/4 ce. has been used, a similar
calculation shows that the osmotic pressure of the fluid under exami-
nation is diminished by a 0.014 pCt. Na Cl solution, corresponding
to a depression of scarcely 0.00840, a difference of depression,
lying near the limit of accuracy of Brckmann’s determination of the
freezing point.

However, if the difference were greater, this could be no objec-
tion to the method, since also the Na Cl solutions are mixed with
the same quantity of blood.

The second remark concerns the pipette and the tubes.

In order to measure accurately, the bore of the pipette must be
narrow and accordingly the instrument itself long. The column of
0.02 ce. of blood has a length of 143 mm. The same remark applies
to the funnel-shaped tubes. The calibrated capillary part has a length
of 57 millimetres and a volume of 0.01 cc. and is divided into
LOO equal parts, which can easily be observed with the naked eye,
fractions being estimated with a magnifying glass.

The use of funnel-shaped tubes of the same length, but with a
still smaller volume of the capillary part than 0.01 e¢e., which
would enable us to make determinations of the osmotic pressure
of much smaller quantities of liquid than */, ce., would give rise
to technical difficulties, on which IT will not dwell here. No more
shall I mention here the special precautions in experimenting, neces-
sary for obtaining accurate figures. This subject will be dealt with
elsewhere.

In order to give an idea of the reliability of the method, a table
follows, containing two series of parallel experiments. (See p. 397).

The agreement between the figures (each division represents
0.01 : o- 76
ao = 0.0001 ce.) is seen to be very satisfactory.

A third remark concerns the possibility of making one or more
additional determinations with the same fluid, for checking the
result obtained. All one has to do is to drain off the liquid above
the sediment by means of a finely drawn tube or pipette and to
convey it into another funnel-shaped tube, to add again 0.02 ce. of
blood and to centrifuge in the same way. The liquids in the NaCl-
tubes are treated in the same way. Undoubtedly one changes the
osmotic pressure of the liquids a little by again introducing 0.02 ce.
of blood, but this is done with all the liquids and so the alteration

-

| Volumes of the sediments after centrifuging for.

Salt solutions.

4 hour. 4 hour. | 4 bour. | 4 hour. 45 min.

| |
Na Cl 0.9 %, 74 69 68h ye | 768 68
» > 73 69 68 68 68
NaCl 4.1 9%, 71 65 64 64 64
» » 68 64 644 o44 644
NaCl 1.2% 6s |. 6 63 613 614
» » 69 } 654 63 62 62
NaCl 1.3 % 67 | 62 60 59 59
» » 67 62 60 | 59 59
|
NaCl 1.4 9/) 69 62 5&4 57 57
» » 67 624 58 574 574
NaCl 1.5 % 62 | 58 55 | 55 55
» » 64 | 59 56 56 56
lachrymal fluid TASiral awe ye pu oleate 57 57
the same lachr. fl. 764 | 64 60 | 57 | 57
|

has no influence on the result, as would be the ease if fresh solutions
of NaCl were taken each time.

A last remark concerns the applicability of the method. It cannot
be used so generally as the freezing point method: It cannot be
applied with gall since this fluid contains substances causing haemolysis ;
it also fails for urine, since this fluid contains a relatively large
quantity of urea which contributes considerably to the osmotie
pressure but has no influence on the volume of the blood corpuscles.

For a number of other fluids, as blood serum, lymph, cerebro-
spinal fluid, saliva, lachrymal fluid, ete., the method can be success-
fully applied. It does not matter whether the fluids are coloured, for
the determination only depends on the volume occupied in the fluid
by the blood corpuscles.

Zoology. — “On the primordial cranium of Tarsius spectrum’.
By Prof. Dr. Eveen Fiscurr of Freiburg i. B. (Preliminary
paper). Communicated by Prof. A. A. W. Husrrcut.

An investigation of the primordial cranium of Tarsius speetrum
seemed particularly interesting to me as it might fill up a gap I had
found when making a comparative study of the cartilaginous skull
of apes and man on one hand, and of lower mammals (the mole)
on the other *). So I was exceedingly happy when Prof. Husrucut,

1) Fiscuer E. Das Primordialeranium yon Talpa europae. Anat. Hefte Bd. 17,
1901 and: zur Entwickhingsgeschichte des Affenschiidels. Zeitschr. f. Morph. und
Anthrop. Bd. 5, 1903.

(398 )

with generous kindness, placed at my disposal, out of his rich and
valuable collection of embryos, such stages as were proper for this
investigation.

In what follows a brief description will be given of the form and
development of the chondrocranium as it appears at the height of
development; this description is based on the reconstructed waxmodel
which I made of the skull of an embryo of 34 mm. length. Other
details of this embryo are shown in Kupen’s Normentafel ').

Since an extensive and illustrated description will follow elsewhere,
I shall be very brief here and give no detailed information as to
literature and comparisons. For the first and also for the nomen-
clature used and the meaning of many only shortly mentioned
details [| refer to Gaupp’s brilliant comparative of the development
history of the vertebrate cranium in Herrwie’s Handbook 7’).

The basal plate is broad behind and well developed; anteriorly
it delimits the foramen magnum. It is perforated by the hypo-
elossus. Laterally is has a fixed connection with the ear-capsule.
This connection, however, is pierced by the narrow and long, almost
slit-shaped foramen jugulare. Behind if, starting from the junction
of the basal plate and the ear-eapsule the cartilaginous plate de-
velops which upwards represents the parietal plate, backwards and
inwards the tectum synoticum. This tectum is a very narrow strap.
So in this respect Tarsius resembles the young foetus of monkeys
and man (ef. Bonk, Petr. Camp. IL) and differs from the other mam-
mals, where a broad plate is found.

Further forwards the basal plate itself becomes very remarkably
narrow, so that here it consists only of a thin, round projection.
At the same time it is separated by long slits from the two ear-capsules,
with the anterior parts of which it only coalesces again in the region
of the sella, This thin projection rises rather steeply, and in the
sella region it becomes quite considerable with its two processus
clinoidei posterivres. The two slits terminate close by, after having
erown very narrow. Their existence seems to be very rare in mam-
mals; they are defects which may be compared with the fenestre
basieranialis posterior of Reptiles (Gaupp).

The ear-capsules themselves showed no peculiarities; they are

‘) Husrecur und Kerser, Normentafeln zur Entwicklung von Tarsius spectrum
und Nyclicebus tardigradus. Jena 1906. Tabelle N’. 36. Fig. 20 a.—e.

I am also greatly indebted to Prof. Ketset for enabling me to use the splendid
series of sections of Husrecur’s Tarsius embryos on which his own investigations
were effectuated.

2) Gavpr. Die Entwicklung des Kopfskelettes. Hertwig’s Handbuch 1905. Cap. 6 p.573,

( 399 )

moderately erect, the fossa subarenata is only indicated. The fora-
men acusticum and higher upwards the foramen Nervi facialis
mark the border of the vestibular and cochlear parts; to the former
are attached above the parietal plates; they are very small and
insignificant. A foramen jugulare spurium perforates its base. Fron-
tally they send out a very short processus marginalis posterior,
exactly as in the monkey skull. On the exterior of the ear-cap-
sules lie, in exactly the same way as I described for the mole and
embryos of apes, the cartilaginous stirrup, anvil and hammer, pas-
sing into Mrckk1’s cartilage.

The orbito-temporal part is characterised by its relatively pheno-
menal length. The continuation of the cranial trabecle from the
saddle groeve, where it had much broadened, is a narrow high
ridge, a true septum interorbitale (Gaupp) still more extended than
I found it with apes, although not so high as there. So the cranium
is clearly tropibasical. By this jong septum, which in front of course
passes into the nasal septum, the nasal capsule is far separated
from the brain capsule; it lies far in front of it, exactly as with
Reptiles. The relatively large eyes of Tarsius are probably the cause
of the survival of this extremely primitive formation.

Where the deseribed cartilaginous beam broadens into the hypo-
physis groove it sends out, fairly deep towards. the base, a round
stalk at each side, bearing the small ala temporalis, which tapers
in the same way as with the foetus of man and ape. It does not
serve as a cranial wall yet, and has no Foramina rotunda and ovalia
yet. Above it starts with two roots the large ala orbitalis. Between
the roots the two foramina optica, the right and left one, are very
close together, so that only the thin septum mentioned separates
them. The orbital wings, themselves large plates, are neither bent
upwards so strongly as with the lower mammals (the mole), nor
do they extend laterally in such a perfect plane as with ape and
man, but their shape is exactly between the two extremes, they
slant sideways and upwards. Also the circumstance that their pos-
terior end lengthens out into a real, although very thin taenia
marginalis, which nearly reaches the parietal plate (there remains a
very small gap), shows a similar transitional stage between the
Primates and the other mammals. The anterior parts of the alae
orbitales are not connected with the nasal capsule as usual (also in
the sheep e. g. this connection is wanting according to Duckrr).
Below the sphenoid beam are, isolated from it, the roundish ptery-
goid cartilages, quite independent.

Proximally the septum interorbitale, as has been stated, passes

( 400

into the nasal septum. The nasal capsule has a remarkable resem-
blanece with that of the Primates; there is no trace of the double
tube form of other mammals.

The two apertures for the olfactory nerves are both simple, with-
out any formation of cribrosa.

This part of the future nasal root is relatively broad, whieh is
especially conspicuous with regard to the completed cranium.

Basally the whole nasal capsule has a slit-shaped opening, i. e.
the bottom (lamina transversalis post. and ant.) is lacking, this being
also characteristic for man and partly for apes, with whom J still
found an indication of the bottom (Semnopithecus). About the yet
slightly developed conchae, the cartilages of Jacopson, the alar ear-
tilage enclosing the nasal entrance, nothing particular can be mentioned.

Mecke’s cartilage proceeds well developed as far as the point of
the chin and here has a continuous connection, without any trace
of a suture, with that of the other side. ReicuErt’s cartilage proceeds
continuously to the tongue bone.

On the dermal bones I will not dwell here; besides the upper
squama of the occipital, resp. interparietal, all membrane-bones are
present; the annulus tympanicus is only ?/, of a ring; frontal and
parietal extend as yet to such a small height that the top of the
skull is mostly covered with skin only.

When we now survey the whole cranium, as sketched above, we
find two important characteristics. On one hand appears the exceed-
ingly close relationship of the developing cranium of Tarsius and
that of ape and man. In spite of clear specific peculiarities, it
evidently stands much nearer to these than to the other known
mammalian crania. This affords a new proof for the correctness of
Huprecut’s opinion as to the position of Tarsius in the system.
At the same time an investigation of the primordial cranium of
true Lemurs becomes necessary and promises important results. This
investigation will shortly follow.

Secondly the resemblance between this type of skull and that of
Reptiles is striking; like the skull of monkeys, so that of Tarsius in
its cartilaginous stage, pleads unmistakably for unity of plan and
origin of the Reptilian and Mammalian skull (ef. Gavpp’s various
articles). In our case the position of the nasal capsule, the septum
interorbitale, a series of details in the arrangement of the foramina,
the cartilaginous straps, ete. point clearly in that direction.

The study of each single form may in this way contribute to the
solution of the problem of phylogenesis.

Freiburg t. B., October 1905.

( 401 )

Physics. — “On the radiation of heat in a system of bodies having
a uniform temperature’. By Prof. H. A. Lorentz.

(Communicated in the meetings of September and October 1905).

§ 1. A system of bodies surrounded by a perfectly black enclosure
which is kept at a definite temperature, or by perfectly reflecting
walls, will, in a longer or a shorter time, attain a state of equi-
librium, in which each body loses as much heat by radiation as
it gains by absorption, the intervening transparent media being the
seat of an energy of radiation, whose amount per unit of volume
is wholly determinate for every wave-length. The object of the
following considerations is to examine somewhat more closely this
state of things and to assign to each element of volume its part in
the emission and the absorption. Of course, the most satisfactory
way of doing this would be to develop a complete theory of the
motions of electrons to which the phenomena may in all probability
be ascribed. Unfortunately however, it seems very difficult to go as
far as that. I have therefore thought it advisable to take another
course, based on the conception of certain periodic electromotive
forces acting in the elements of volume of ponderable bodies and
producing the radiation that is emitted by these elements. If, without
speaking of electrons, or even of molecules, we suppose such forces
to exist in a matter continuously distributed in space, and if we
suppose the emissivity of a black body to be known as a function
of the temperature and the wave-length, we shall be able to calculate
the intensity that must be assigned to the electromotive forces in
question. The result will be a knowledge, not of the real mechanism
of radiation, but of an imaginary one by which the same effects
could be produced.

§ 2. For the sake of generality we shall consider a system of
aeolotropic bodies. As to the notations used in our equations and
the units in which the electromagnetic quantities are expressed,
these will be the same that I have used in my articles in the
Mathematical Encyclopedia. We may therefore start from the following
general relations between the electric force ©, the current @, the
magnetic force § and the magnetic induction ¥

ie
HUG) Ny seme Noy Nise Poul ok, vay iL)

( 402 )

In these formulae ¢ denotes the velocity of light in the aether.

In the greater part of what follows, we shall confine ourselves to
cases, in which the components of the above vectors and of others
we shall have occasion to consider, are harmonic functions of the
time with the frequency n. Then, the mathematical calculations can
be much simplified if, instead of the real values of these components,
we introduce certain complex quantities, all of which contain the
time in the factor e¢ and whose real parts are the values of the
components with which we are concerned. If U,, %,, %. are com-
plex quantities of this kind, relating in one way or another to the three
axes of coordinates and in which the quantity ¢?! may be multiplied
by complex quantities, the combination (2, %,, %-) may be ealled a
complex vector % and %,, %y,, %. its components.

By the real part of such a vector we shall understand a vector
whose components are the real parts of ,, A,, U.. It will lead to
no confusion, if the same symbol is used alternately to denote a
complex vector and its real part. It will also be found convenient to
speak of the rotation and the divergence of a complex vector, and
of the sealar product (2, 3) and the vector product [u. B] of two
complex vectors % and %, all these quantities being detined in the
same way as ithe corresponding ones in the case of real vectors.
EK. g., we shall mean by the sealar product (2. 3) the expression
WB + W/By+ Az Bz.

It is easily seen that, if €, , © and B are complex vectors,
satisfying the equations (1) and (2), their real parts will do so
likewise. The denominations electric force, ete. will be applied to
these complex vectors as well as to the real ones.

One advantage that is gained by the use of complex quantities
lies in the fact that now, owing to the factor e”, a differentiation
with respect to the time amounts to the same thing as a multiplication
by zn; in virtue of this the relation between € and © and that
between © and 8 may be expressed in a simple form. Indeed, we
inay safely assume that, whatever be the peculiar properties of a
ponderable body, the components of © are connected to those of € by
three linear equations with constant coefficients, contaiming the com-
ponents and their differential coefficients with respect to the time.
In the ease of the complex vectors, these equations may be written
as linear relations between the components themselves; in other
terms, one complex vector becomes a linear vector function of the
other. A relation of this kind between two vectors % and B can
always be expressed by three equations of the form

( 403 )

Mia U, a Vig q, ale Pis U.,
Dy = v,, Uz + v,, Uy + v,, We
BS, =»,, Az + 7,, Uy, + »,, Uz,
which we shall condense into the formula
Hi (p)) ale
According to this notation we may put © =(jp’) £, or, as is more
convenient for our purpose,

¢
tf

\

Sa Grn Speke. Ser eek St. ts he a(S
the symbol (p) containing a certain number of coefficients p which
are determined by the properties of the body considered. As a rule,
these coefficients are complex quantities, whose values depend on
the frequency 7.

As to the relation between 8 and , we shall put

B= (u) $,
or

Scat G)F Oiaseeanr ee 2 ass = A)

We have further to introduce an electromotive force which will
be represented by a vector ©, or by the real part of a complex
vector €,. The meaning of this is simply that the current © is sup-
posed to depend on the vector €-+ ©, in the same way in which it
depends on © alone in ordinary cases, so that

Ge iCe— (pO te a te een)
Similarly, we may assume a magnetomotive force .9., replacing
(4) by ’
Dy Wnts awe eae ee)

This new vector , however, does not correspond to any really
existing quantity; it is only introduced for the purpose of simplifying
the demonstration of a certain theorem we shall have to use.

As to the coefficients we have taken tegether in the symbols (p)
and (qg),- we shall suppose them to be connected with each other in
the way expressed by
Pris = Pa17 Pos — Psa P31 — Piss? 9 ee (7)
and

Gia Fars) Yas —— F399 Gor —— Gis 3 ed ten ers (3)

The only case excluded by this assumption is that of a body
placed in a magnetic field. ,

For isotropic bodies we may write, instead of (5) and (6),

(Gafni Cheers re Sota ce Wee (9)
ite Oita tee et ee) CO)
with only one complex coefficient p and one coefficient q:

( 404 )

y 3. Before coming to the problem we have in view, it is necessary
to treat some preliminary questions. In the first place, we shall
examine the vibrations that are set up in an unlimited homogeneous
and isotropic body subjected to given electromotive and magnetomotive
forces, changing with the frequency . This problem is best treated
by using the complex vectors.

We may deduce from (1)

rot rot § = — rot ©,
;
or
alee 1 <
grad. dw —A f=—rotC . . « = 2s ay
c
and similarly from (2)

1 :
grad div C— AE=——retd. - . .=. . (2)
€
Again, always using the equations (1), (2), (9) and (10), we find

dw S=0 , dy ©€=0,

dir = — div J, div € = — div &,,
S 1 nS : 1

rot © = —(rot & + rot ©.) = — — ¥ 4+ — rot G,
Ww pe p

TE 1 =
= — —(b + Be) + = rot Ge,
1O0he ie
1 : : = il :
rot B = —(rot H + rot He) = —€ + — rot De
q 08 q

d serie eg ere
= Fe + Ed + ae

pg
so that (11) and (12) become
: ere: 1
AH— = § = — grad div De + awe = ——rot Sey
PE pg pe
= 1 = hare = 1 S
A&— Seay € = — grad div & + > We + — rot De.
Pqe pe ge

The solution of these equations may be put in a convenient form
by means of two auxiliary vectors % and 9. If these are determined

by
1

[Ee eGR coo ye ce (IS)
py
ieee
Ag——~-5=—S,- ° . . . . (14)
pqe

we shall have
: Ne 1 e

H = grad div Y = —— D = rot N, )

S 5 Pde pe

( 405 )

z ; | ay 1 : a
€ = grad div A — —~— A — — rot See ees (16)
PIE qe
Finally, putting
Cait NOM NO. Po ha, OMG ety scum. (EL)

we get instead of (13) and (14)

Beer:
AXI——A=—E,,
y?
A) = — Dz,
v

4 — —— — E(t) a5, oe reer he)

1 1
[hee SENOS) nae toy (19
vee Ben: = (19)

Here dS denotes an element of volume situated at a distance 7
from the point for which we wish to calculate % and 9, and the

. r . . . =
index (« — “) means that, in the expressions representing €, and .,
2 3

? ite
for that element of volume, ¢ is to be replaced by t ——.
vD

The algebraic sign of v is left indeterminate by (17). We shall
choose it in such a way that our formulae represent a propagation
of vibrations ‘sswing from the elements of volume in which &, and 9,
are applied.

1
For aether we have g=1 and, as may easily be shown — =in, v=ce.
P

§4. We have next to establish the equation of energy. The calcu-
lations required for this purpose, as well as those we shall have
to perform later on, may be much simplified, if we replace all
discontinuities at the limit of two bodies by a gradual transition
from one to the other; this may be done without Joss of generality,
because, in our final results, the thickness of the boundary layers
may be made to become infinitely small. A further simplification is
obtained by leaving out of consideration the imaginary magnetomotive
forces, and by supposing the coefficients « and q to be real. The
coefficients p,,,P,, ete. however will always be considered as complex
quantities. We shall decompose them into their real parts, which
we shall denote by «,,,¢,,, ete., and their imaginary parts, for which
we shall write —7?,,,—78,,, ete., so that p,, =«,, —7ifj,, ete.

28

Proceedings Royal Acad. Amsterdam, Vol. VIII.

( 406 )

The equation (5) now becomes
CC, = (a) C— 2 (ByiCy ee eee)
or, if we define a new vector D by means of the equation
C= De eee as is (i!
C+ €, = (ai@ =F (6) Oo os cae
In the deduction of the equation of energy we have to understand
by &,&,, 6 and D the real vectors. For these we have the formulae
(1), (2) and (21), and besides, since q, @ and £ are real, the relations
(4) and (22).
From (1) and (2) we may draw immediately
c(H. rot €) — (€. rot H)} = — (H.B) — (E.G),
the left-hand member of which is
div &,
if we define the vector S by the equation
G=c[(€.H],... 2 4 0s ees
i.e., if we understand by it the vector product of € and 9, multiplied
by ¢.
In the right-hand member we have in the first place

(9.8) = ae B
J) + Rap ce )y
as may be seen from (4), if (8) is taken into account, and further,
in virtue of (7), (21) and (22),
3 eG if
(€. 6) = ((a) ©. @) + 5 ap (0) SAS . GB).

Our equation therefore takes the following form, in which the
meaning of the different terms is at once apparent,

aeO 10: ee sine

(€..Q)=((@)€.6) 4+ =n a, (#) De Date aE () . B) + div S.
The first member represents the work done by the electromotive
force per unit of volume and unit of time; in the second member
11 (i(@) CO) cece ied al tenes (EE)
is the expression for the quantity of heat that is developed per unit

1 : ;
of space and unit of time. Further, 5 (o: B) is the magnetic and

1 ' 2
—n((8)D. D) the electric energy, both reckoned per unit of volume.
9 «

The vector © denotes the flow of energy, so that the amount of
energy an element of volume dS loses by this flow is given by
dw S ds.

( 407 )

§ 5. We may now pass to a theorem which I have formerly
proved in a somewhat more cumbrous and less general way. In
order to arrive at it, we have to use the complex vectors, supposing
at the same time the existence of magnetomotive forces; we have
therefore to apply the formulae (5) and (6).

We shall consider tivo different states with the same frequency n,
both of which can exist in the system of bodies. The symbols &, 6,
ete. will be used for one state and the corresponding symbols,
distinguished by accents, for the other. We shall proceed in a way
much like the operations of the last paragraph, with this difference
however, that we shall now combine quantities relating to one state
with quantities belonging to the other.

We shall start from the relation

e{ (8'. rot &) — (&. rot ')} = — (9. B) — (€. ©’).
Here the expression on the left is equal to
cediv[¥. 9']

and on the other side we may put

(9'. B) = in (45! B) = in ( (gq) BB) — (He. B),
(§.6) = (( €. €) — (&. ©,
so that we find
div [S. $'} = — in((Q) B- B) — ((HE- ©) + (He. B) + (G. ©).

The theorem in question is a consequence of this formula and the
corresponding one that is got by interchanging the quantities belonging
to the two states; we have only to subtract one equation from the
other. Since, by (8) and (7)

((q) B. B) = ((q B- B) and ((p) ©. C) =((—) €.
we find in this way

eps.6'| —div [C,H] = (o', B) — (0, 9) + &. ©) —E2 ©.

We shall finally multiply this by an element of volume dS, and
take the integral of both sides over the space within a closed surface
o. If we denote by n the normal to the latter, drawn outward, the
result will be

fi [E. H'n — [E. Hn} do = | f (9's.B)—(De.d!) +( E..€) —(C'e. ©} dS (25)

§ 6. There are a number of cases in which the first member of
this equation is zero.

a. HK. g. we may suppose the system to be limited on all sides
in such a way that it cannot exchange rays with surrounding bodies;
we can realize this by enclosing the system in an envelop that is

28*

( 408 )

perfectly reflecting on the outside. If, under these circumstances, the
surface 6 surrounds that envelop, we may put in every point of it
C= 105 C05 10 a0

b. If the envelop is made of a perfectly conducting material,
both the electric foree © and the force €' will be normally directed
in every point of its inner surface. Consequently, if the latter is
chosen for the surface o, we shall have

LS 3D]a—OsandaiC oie = 0:

c. Finally we may conceive a system lying in a finite part of
space and surrounded by aether, into which it emits rays travelling
outwards to infinite distance. Taking in this case for 6 a sphere of
infinite radius, we shall show that for each element do the factor
by which it is multiplied in the equation (25) vanishes. The direction
of the axes of coordinates being indeterminate, it will suffice to
prove this proposition for the point P in which the sphere is cut
by a line drawn from the centre O in the direction of the axis of 2.

Now, if we confine ourselves to those parts of €, $, €' and 5’
which are inversely proportional to the first power of OP, as may
obviously be done, we may consider the state of things near the
point P as a propagation of vibrations in the direction OP, the
electric and magnetic force being perpendicular to that direction and
to each other. Denoting by @ and 6, a’ and 0! certain complex
quantities, we may write

G10; g, = aeit, E. = bent,
De = 0; Df, = — bem, Hz. = aes,
Ce == ()); SF — aleint, om = bieint,
ee ea ena
ea 0' Hy = — dein, ae,

and we have at the point P, since in it the normal to the spherical

surface is parallel to the axis of 2,

[ED], —[c. oh, = €y D'. — €, )— ( ey H, — CLS Dy, )= 0.
These considerations show that in many cases the equation (25)

reduces to

fies 9- G8 ji dS =(tc. C')— (He - B)dS . (26)

§7. It is particularly interesting to examine the effects produced
by an electromotive or a magnetomotive force which is confined
to an infinitely small space 8. Let P be any point of this region,
a a real vector having everywhere the same direction 4 and the
same magnitude |«|, and let us apply in all points of 5S an electro-
motive force a¢®!, Then we shall say that there is an ‘electromotive

( 409 )

action” at the point P in the direction 4. We may represent it by
the symbol

a8 eint
and we may consider its intensity and its phase to be determined
by the real part of |a| Se.

In a similar sense we can also conceive a “magnetomotive action”
existing in some point of the system.

These definitions being agreed upon, equation (26) leads to the
following remarkable conclusions.

a. Let there be, in the first of the two cases we have distinguished
in the preceding paragraph, an electromotive action q Se at the
point P in the direction h, and in the second case an electromotive
action a’ S’e* at the point 7” in the direction 4’, there being in
neither case a magnetomotive force. Then the integrals in (26) are
to be extended to the infinitely small spaces S’ and S$ and the result
may be written in the form

(a’.€p)S'=(a.@'p)S,
if we represent by @p the current produced in /’ in the first case
and by @’p the current existing in ? in the second.

Hence, assuming the equality

Ja]s

|

| a’ | Ss! 7
we conclude that
(Spi SS (Ola ate Gu ebe IL ae nee (27)
The full meaning of this appears, if we write the two quantities
in the form
Cop pert and Cyp =p eter).
Indeed, (27) requires that
el Dy
and we have the theorem :

If an electromotive action applied at a point P in the direction A
produces in a point ?’ a current whose component in an arbitrarily
chosen direction /’ has the amplitude w and the phase », an equal
electromotive action taking place at the point 7’ in the direction A’
will produce a current in ?, whose component in the direction
has exactly the same amplitude « and the same phase yr,

6. Without changing anything in the circumstances of the first
case, we shall now assume, that in the second the vibrations are
excited not by electromotive forces, but by a magnetomotive action
a’ S' et, at the point /' in the direction 4’. We then find

—(a'. Sp) S' = (a: @'p)S8,
and, if we put

( 410 )
fu SSS ia st

— By p = Cap, Brest ca cem Ghee ae (28)

a theorem similar to the former.

§ 8. The absorption of rays being measured by the amount of
heat developed, the expression (24), in which © is the real current,
will be often used in what follows. It may be replaced by

w= (§- 8),
if we write § for the vector (a) ©, so that
Sc = a,, ©, a,, Sy pe, Gz ete. = = oe ag)

Now, by a well known theorem, the axes of coordinates may
always be chosen in such a way that the coefficients @,,, @,,,@,, in
these equations become zero. Denoting the remaining coefficients by
@t,, @,, @, We have for the relation between § and ©@

Se =a, 2, by = 4, Cy, F2—-4, 2,
and for the development of heat
Wis Oy Cx aC a CaF, 5) Cees See

The directions we must give to the axes in order to obtain these
simplifications, may properly be ealled the principal directions ; in
general, they will not be the same for different frequencies. This is
due to the fact that the coefficients in (29) depend on the value of 7.

It is also to be noticed that by this choice of the axes of coor-
dinates, the coefficients Bigs BasscPs1, 200: Dis, Days Dax owillemetain
general, be made to become zero.

In the case of an isotropic body we may take as principal diree-
tions any three directions perpendicular to each other.

§ 9. Thus far we have only prepared ourselves for our main
problem. In the next paragraphs we shall first consider the absorption
by a very thin plate surrounded by aether on both sides, and receiving
in the normal direction a beam of rays. Combining the result with
the ratio between the emissivity and the coefficient of absorption of
a body, we shall be able to determine the amount of energy, radiated
by the plate in a normal direction, and our next object will be to
calculate the intensity we must ascribe to electromotive forces acting
in the plate (§ 1), in order to account for that radiation. This will
lead us to a general hypothesis concerning the electromotive forces
acting in the elements of volume of a ponderable body and we shall
conclude by showing that, if these electromotive forces were applied,
the condition required for the equilibrium of radiation would always
be fulfilled.

anil js.

§ 10. Let the plate be homogeneous, with its faces parallel to the
first and the second principal direction. We shall take these for the
axes of wv and y, placing the origin Q in the front surface of the
plate, i.e. in the surface exposed to the rays, and drawing the axis of
< toward the outside. As has already been said, the absorption will
be caleulated by means of the formula (80); it will therefore be deter-
mined by the components of © and by those of &, on which they
depend. Now, our problem is greatly simplified, if we suppose the
thickness 4 of the plate to be infinitely small and if, in caleulating
the absorption; we confine ourselves to quantities of the first order
of magnitude with respect to 4. The quantity aw relating to unit
volume, we may then neglect all infinitely small terms in © and &;
consequently, we need not attend to the changes of these vectors in
the plate along a line perpendicular to its faces. Moreover, in virtue
of the well known conditions of continuity, the values of ©, and &y
within the plate will be equal to those existing in the aether imme-
diately before it; also, ©- will be 0, because it is so in the aether.
For ©, and, &, we may. even take the values, existing in the inei-
dent beam, the reason for this being that the values belonging to the
reflected rays, (the vibrations reflected at the two sides being taken
together) are proportional to the thickness, if the plate is infinitely thin.

It is seen by these considerations that in the case of a given
incident motion, ©, ©), €: are the only unknown quantities in the
three equations connecting the components of © and ©. We need not,
however, work out the solution of these equations.

Finally, it must be kept in mind that, in the case of harmonie
vibrations, the mean value of w for a lapse of time comprising many
periods is given by

1 i '
w= —{ a, (©)? + a, (G))? + o,(G)}, . « » (Bl)

a

if (Cz), (€,), (€z) are the amplitudes of the components of the current.

§ 11. We shall in the first place assume that in the incident rays
the electric force is parallel to the axis of w. Let its amplitude be a,
Then, an element w of the front surface will receive an amount
of energy

== CO: 0) alone anets So 9-6 (6)
; (82)

per unit of time.
Within the plate, there will be electric currents in the directions

of # and y. These will have amplitudes proportional to a, and for
which we may therefore write:

( 412)

Cy=sa , 6)=ga
denoting by j/ and g two factors, which it will be unnecessary to
calculate. From (31) we deduce for the heat developed in the part
w 4 of the plate,

1
(qf tageoh

and, dividing this by (82), for the coefficient of absorption
1
3H Ee ORG) IN eo. ore nS. ay ((853')
c

Our next step must be to obtain a formula for the emission. For
this purpose we fix our attention on a surface-element w' parallel to
the plate and situated at a large distance r from it, at a point of
OZ. The electric vibrations issuing from the plate may be decom-
posed in the first place into vibrations of different frequencies and
in the second place into components parallel to OX and OY.

After having effected this decomposition, we may attend to the
amount of energy travelling across w' per unit of time, in so far
as it belongs to vibrations having the first of the two directions and
to frequencies lying between the limits » and n+ dn. Now, if the
plate were removed, and if instead of it a perfectly black body of
the same temperature were placed behind an opaque screen with an
opening coinciding with the element @, the radiation might be repre-

sented by
kw! dn

3 ee eee
an expression which may also be regarded as indicating the ratio
between the emissivity of a body of any kind under the said cir-
cumstances and its coefficient of absorption. The experimental inves-
tigations of these last years have led to a knowledge of the coeffi-
cient & for a wide range of temperatures and frequencies.
By Krrcunorr’s law, the flow of energy across the element o’,
originated by the part
ols
of the plate, in so far as it is due to vibrations of the said direction
and frequency, is found by multiplying (84) by (83). Its amount is
therefore
kS (a, f? + a@, 9°) w! dn
’

cr?

Fes Sc, TES)

and we have now to account for this radiation by means of suitable

electromotive forces applied to the plate.

( 413 )

§ 12. We shall first put the question what must be the amplitude
a, of an electromotive force acting in the direction of OX with the
frequency mn, if this force is to produce, on account of the electric
vibrations parallel to OX, a flow of energy

kSa,f? w' dn

Se ee ois ce cl (GO)
across the element w' at the point P. Since this flow may be
represented by

cha’,

ho| —

if 6 is the amplitude of ©, at the point ?, we must have
b= a) DES aan,
cr
The amplitude of the current @, = &, must therefore be
i ——$—<—<—
SCR QV oes oo oF (Gz)
At this stage of our reasoning we may avail ourselves of the
theorem of § 7, a. Indeed, if the electromotive force €,,; in the part S
of the plate must have the amplitude a, in order to call forth at
the point P a current €; whose amplitude has the value (37), a,
will also be the amplitude we must give to an electromotive force
€,,, acting in an element of volume 8 of the aether near P, if we
wish to bring about by its action a current with the amplitude (37)
in the plate. This is the condition by which we shall determine the
value of a,.

§ 13. The solution is readily obtained by means of the formulae
(18) and (16). If, in an element of volume S of the aether, €.. = a, e",
€., —0, €.,=0, we shall have

-
aS intt— F
Ta ———e ( ) 2) == 0) 0 == 0

howe
and
2 3 ne
— uae. "Uy,
Ue ad 0 ve ea bat! aa]
as may be easily seen, if the equations
1 merit
p=—, Gg), Bema,
umn

are taken into aecount.
In the differential coefficients of 2, we may omit all terms con-

= 1
taining the square and higher powers of —. Hence, in a point of the
a

( 414)

oY,

axis of 2, which passes through the point P, = (i);

Ow
In this way, the electric force in the aether immediately before
the plate is found to be

Cr — ,
; 4 ac" :
Its amplitude is
a,n?s 38
pa ee (38)

and that of the current ©, within the plate
avs
Axe

This must be equal to the expression (37). The solution of our

47 ¢ 2ha, dn
“= WA SSUES pe eee ee
n 5

In the preceding formulae S means the volume of the portion of

problem is therefore

the plate we have considered. Now, after having decomposed this
portion into a large number of elements of volume s, we may bring
about just the same radiation by applying in each of these an
electromotive force in the direction of ON with the amplitude

4ne 2ka,dn
¢, = ae \A el ae Parra a (0)
n Ss

provided only we suppose the electromotive forees in all these
elements $ to be independent of each other, so that their phases are

distributed at random over the elements.

Indeed, from the fact that the force whose amplitude is (39),
acting in the space S, gives rise to a radiation represented by (86),
we may conclude that an electromotive force with the amplitude
(40), when applied to the element s, will produce a flow of energy

ksa, fo! dn

a
across the element w’. A similar expression holds for each elements
and, on account of the circumstance that the vibrations due to the
separate elements have all possible phases, we may add to each other
all these expressions. We are thus led back to the result contained
in (36).

§ 14. Whatever be the nature of the processes in the interior of
an element of volume, by which the radiation is caused, they can

( 415 )

undoubtedly be considered as determined by the state of the matter
contained within the element; for this reason an electromotive force
equivalent to those processes can only depend on quantities deter-
mined by that state; it cannot be altered by changing the state of
the system outside the element considered, or the form and magnitude
of the whole body. The formula (40), which indeed is determined
by the state of things within the element s, must therefore be applied
to an element of volume of all ponderable matter. It will be clear
also that we have to add the following formulae for the amplitudes
of the electromotive forces in the directions of y and ,

4c 2ka,dn Ane 2ha,dn
= —- , a4=—| f/f —-— 65 (E54)
n s nr s

As to the phases of the three electromotive forces, we shall suppose

them not only to change irregularly from one element to another,
but also to be mutually independent in one and the same element,
so that the phase-differences between the three forces have very
different values in neighbouring infinitely small spaces. In virtue of
this assumption the intensities of the radiation due to the different
causes may be added to each other.

Till now we have only accounted for the flow of energy (86), a
part of the total flow represented by (85). We shall show in the
next paragraph that the remaining part

kSa,g w' dn

ee Qe ese 4 (ES)

cr

is precisely the radiation brought about by the electromotive forces
we have supposed to exist in the direction of O Y, and that the forces
acting in that ot OZ cannot give rise to a radiation across the
element '. After having proved these propositions, we may be sure
that, as far as the electric vibrations parallel to ON are concerned,
the plate has exactly the emissivity that is required by Kircunorr’s
law. Of course, the same will be true for the vibrations in the
direction of OY,

§15. It may be immediately inferred from the theorem of § 7, a
that the electromotive forces applied to the plate in the direction of
OZ, i. e. perpendicularly to the surfaces, cannot contribute anything
to the radiation we have considered. Indeed, we know already that
an electromotive force ©&,, existing in the aether at the point P can
produce no current ©. in the plate; consequently, an electromotive
force €,. in the plate cannot cause a current ©, at the point P.

As to the effect produced by the electromotive force with the

( 416 )

amplitude @, acting in the direction of OY, this may be found by a
reasoning similar to that we have used in §§ 12 and 13. Let us
suppose for a moment an electromotive force of the same direction
and intensity to exist in an element of volume s of the aether near
the point 7. The amplitude of the electric force €, in the aether
immediately before the plate will then be (efr. 38)

ans

dae?’

and that of the current ©, in the plate

If follows from this that, if the element s in the plate is the seat
of an electromotive force Can with amplitude a,, the current
©, = €, at the point P will have this same value. The amplitude
of the electrie force ©, will be

; a,nsq g > ats
b' = —_— =- y2k sa, dn

2p cr

and the corresponding radiation across the element o'

ee ee ee eS O> Cuan
—cb? w == — Tapas ae ae
2 cr

This leads immediately to the expression (42).

§ 16. We are now in a position to form an idea of the state of
radiation in a system of bodies of any kind. After having divided
them into elements of volume s, and after having determined the
principal directions at every point, we conceive in each element the
electromotive forces whose amplitudes are determined by (40) and
41), the phases of all these forces being wholly independent of each
other. In representing to ourselves the state of things obtained in this
Way, we must keep in mind:

Ist. that the principal directions and the coefficients @,, @,, 4,
will, in general, change from point to point and will depend on the
frequency 2.

2ndly that for each frequency m or rather for each interval dn of
frequencies, we must assume electromotive forces of the intensity we
have defined in what precedes, all these forces existing simultaneously.

We shall now show that, if the temperature is uniform throughout
the system, the condition for the equilibrium of radiation will be
fulfilled in virtue of our assumptions. Of course, it will suffice to
prove this proposition for a single interval of frequencies d 1.

( 417 )

Let s and s' be two elements of volume, arbitrarily chosen, /
one of the principal directions of the first element, /’ one of the
principal directions of the other, @, and ey the coefficients relating
to these directions.

In virtue of the electromotive force ©, acting ins in the direction
h, there will be in s’ in the direction /’ a current ©, with a certain
amplitude (Gy); by (81) the development of heat corresponding to
this current will be per unit of time

1

= Can (GDP ES om. kc 0 6 oa (ER)
Similarly, we may write

1 ,

aon (KOs eH teeounce wh wich 6) 6 olavo (Ee)

for the heat developed in s on account of the current @', produced
in this element in the direction / by the electromotive force acting
ins’ in the direction /’.

Since each of the three electromotive forces in § calls forth a
current in the element s' in each of its principal directions, there
will be in all nine expressions of the form (43). These must be
added to each other, as may be seen by observing that the total
development of heat, represented by (81), is the sum of three parts,
each belonging to one of the components of the current and that
the three electromotive forces in § are mutually independent. The
sum of the nine quantifies will be the total amount of heat s' receives
from gs, and in the same way we must take together nine quantities
of the form (44), if we wish to determine the amount of heat
transferred from s' to s. We shall have proved the equality of
the mutual radiations between the two elements, if we can show
that for any two principal directions, the expressions (43) and (44)
have the same value.

Let us eall a, and a’, the amplitudes of the electromotive forces
originating the currents whose thermal effects have been represented
by (48) and (44). Then, in accordance with (40) and (41),

4m¢ 2k ay, dn 4c 2ha'ydn .
= 0 — eve (40)
n Ss n Ss

Now, by the general theorem of § 7, a the amplitudes (G,) and
(@',) in (43) and (44) are proportional to as and ays. Taking
into account the formula (45), we infer from this

(Cp)hs (Cpe = ai 84. a Ss — a) Sens,

an equation, which leads directly to the equality of (48) and (44).

( 418 )

If the system of bodies is entirely shut off from its surroundings,
the equality of the mutual radiation between any two elements
implies that the state is stationary.

In order to show this, we fix our attention on one particular
element 8, denoting all other elements by s'’. By what has been
said, the sum w, of all quantities of heat which s receives from
the elements s' will be equal to the sum w, of the quantities of
heat it gives up to them. But, if the system is isolated from other
bodies, each quantity of energy lost by s will be found back in
one of the elements s'; w, is therefore the total amount of energy
radiating from s and the equality w,= ww, means that s gains as
much heat as it loses.

§ 17. We shall finally assume that the system contains a certain
space which is occupied by an isotropic and homogeneous body Z,
perfectly transparent to the rays; we shall examine the electro-
magnetic state existing in this medium, if all bodies are kept at the
same temperature. To this effect, we must begin by a discussion of
the radiation that would take place, if the body Z extended to
infinity, and if it were subjected to an electromotive or magneto-
motive action (§ 7) at a certain point 0.

A perfectly transparent body is characterized by the absence of
all thermal effects. This means that the coefficient @ is zero, as
appears by (30). We have therefore

Di 18s aa st aoe ee ae
the coefficient g being real and positive, and the equation (17) becomes
vic One wo a ee (47)

I shall take here the positive value.

Let us first apply to an element of volume S$ at the point O,
which I shall take as origin of coordinates, an electromotive force
©, = ae’, but no magnetomotive force. Then

as inf t— —
feast SR eee ahyijen'y
4ar :

What we want to know, is the amount of energy radiating from
O, i. e. the flow of energy through a closed surface surrounding
this point. In caleulating this flow, the form and dimensions of the
surface are indifferent; we shall therefore consider a sphere with O
as centre and with an infinite radius 7.

Then we may omit all terms in © and . containing the square

1
and higher powers of —, and we find from (15) and (16), attending
a

( 419 )

to (46) and (47) and taking the real parts

2 asn r— a r
ee = 6 cosn| t — — ],
4nrv r v

¢ asn® ay ? * aSn? we r
= -— cosn| t — — 5 2 = — —— .— cosn| t——],
Ss Anxry? 7 v dary? 7? v
asn 2 r E aSn y r
Nd, =), Dy = SS HOB (6 lin SS COST alee
AarBev r Vv darpevr v

The electric and magnetic foree being known, the flow of energy
through the sphere may be calculated by means of (23). Its value is
a? S? né
12278 ©
If we perform a similar calculation in the assumption of a magne-
tomotive force with amplitude a, acting in the space §, the result is
a? 8? nt

122 q0° ;

bat 0 5 Reena (a8)

Be ae Slee el (49)

§ 18. Let P be a point of the body Z mentioned at the beginning
of the preceding paragraph, / an arbitrarily chosen direction and let
us seek the amplitude (€/) of the electric current, or rather the
square of the amplitude, produced by the radiating bodies, confining
ourselves to the interval of frequencies dn.

We shall divide the bodies into elements of volume s and we shall
denote, for one of these elements lying at the point Q, by h one of
the principal directions, by «, the coefficient relating to it, and by
ay, (cfr. (45)) the amplitude of the electromotive force acting in that
direction.

The amplitude (€)) produced by this force at the point P is equal
to the amplitude of the current @;, existing in the element s, if an

: ns : : ans , :
electromotive force €,,, having the amplitude —g (is applied to an
LN

element of volume S of the aether near P. In order to express myself
more briefly, I shall understand by A the radiation that would be
excited by an electromotive action at the point P in the direction
/ of such intensity that the product (&,;) S has the value 1. The
amplitude (@)) in P, of which we have just spoken, will be found
if we multiply by a,s the value which, in that state, (€,) would
have in the element s. Hence
A

WA 32 x? e kh a;,s dn (frQ)’
(Cup)? = ay? 8? (C4)? = — epee
v

6407 kdn 4 u
= —————— ,, (50)

n?

( 420 )
if we write wh for the development of heat in the element 8, which,

in the state A, is due to the current in the principal direction /h.
Now, starting from the expression (50), we shall obtain the total

value of (€;p)? by an addition, in which all elements 8, each with

its three principal directions, must be taken into account. In a

system, completely shut off from surrounding bodies, = w* will be

the total amount of energy, emitted by P in the state A; we can
therefore determine it by the formula (48), putting aS = 1. This
leads to the result
16 ake? ndn

3 Bu af

In the same way, using the theorem of § 7, 4 and the expression
(49), I find

(Cip)? =

(Sip = 16 ake? n* dn

3qv

These results being independent of the place of the point P and
the choice of the direction 7, we come to the conclusion that the
state of things is the same in al parts of the medium Z and that
both the electric and the magnetic vibrations take place with equal
intensities in all directions. The amount of the electric and magnetic
energy per unit of volume is now easily found. According to § 4
the first is

1
7 PBI)? + (Oy)? + (2):

for the value of which one finds
4akeidn

v :
by remembering that for every direction /,
Ci 1 ¢ 2
(2)* = —, Cy’:
n
The magnetic energy may likewise be determined. Referred to unit
volume it has the value

1 A : .
aad [(Bx)? + (By)? + (S37),
and this is easily calculated, since for every direction /,
2 Do aw
(39? = = yy.
nr

The result is that the two kinds of energy are distributed over
the body ¢ with equal densities. This has been known for a long

( 421 )

time, as has also been the rule implied in our formulae, that these
densities are inversely proportional to the cube of the velocity of
propagation v. It must further be noticed that, if the medium ZL is
aether, the density of the energy of the radiation becomes

8 akdn

c

This agrees with the meaning we have originally attached to the
coeflicient & (§ 11).

§ 19. There is one point in the foregoing considerations that may
at first sight seem strange, viz. that the intensity of the electromotive
forces we have imagined should depend on the magnitude of the
elements of volume s. It must be kept in mind however, that these
forces have no real existence, and that we do not pretend to have
found something concerning the causes by which the phenomena are
produced. That the magnitude of the electromotive forces must be
taken inversely proportional to the square root of the volume of 8
is simply a consequence of our assumption that the force has the
same phase in all points of such an element. For a given amplitude
of the electromotive force, the radiation would therefore be propor-
tional to s*, and we had to make such assumptions concerning that
amplitude, that the radiation became proportional to s_ itself.

In connection with these remarks it must be observed that we
have no reasons for ascribing to the dimensions of the elements
of volume some particular value. These dimensions are indifferent as
long as we consider only the radiation at finite distances and the
transfer of energy between neighbouring molecules lies outside the
theory I have here developed,

Physiology. — “On the ability of distingriishing intensities of tones”.
By Prof. H. Zwaarpemaker, (Report of a research made by
A. DBENIK.)

The “Unterschiedsschwelle” for impulsive sounds (dropping bullets
and hammers) has been studied frequently and many-sidedly, but
regarding the “Unterschiedsschwelle” for intensities of tone we have
had at our disposal till now only some information communicated
by M. Wien in his thesis.

M. Wien found the value of the “Unterschiedsschwelle”’ for the
three tones, to which he limited his investigation to be as follows:
for a average 22.5°/, (with 18.2 and 27 for extremes) for e’ 17.6°/,

29

Proceedings Royal Acad. Amsterdam. Vol. VIII.

( 422 )

(one determination) for a’ average 14.4°/, (with 10.8 and 22.5 for
extremes). It appeared desirable to perform such an investigation
through the whole scale and to establish it in other regards also on
larger foundations. At my request Mr. A. Deentk has executed a
very great number of observations of this kind, and I take the
liberty to communicate his results here in short, and refer the reader
to an ample description in a thesis on this subject by Mr. Drentk
which will soon be published.

Experiments with the tuning-fork.

A tuning-fork kept vibrating by electro-magnetism is started in a
room at the side of the sound-free cabinet of the physiological
laboratory and is kept vibrating at a fixed amplitude. This amplitude
may be measured microscopically by means of the triangle of GRADENIGO.
Normal to the axis of this tuning-fork a circle divided into grades
is placed, to which two hearing-tubes are attached in such a way,
that their radial prolongations cut the axis of the tuning-fork in the
tuning-centre. These hearing-tubes can be moved along the whole
circumference of the seale, and can be brought at pleasure into the
interference-planes of Kanssiinc, in the planes of maximum-sound
or between.

The hearing-tubes are led into the interior of the sound-free
cabinet by means of thick-walled caoutchoue tubes which were
still further acoustically isolated. There by means of a ~T—tap
alternately the one or the other of the tubes may be listened at or
perfect acoustic rest can be obtained by bringing the tap into a
closed position.

An assistant now displaces one of the hearing-tubes, while the
other hearing-tube is fixed in the plane of maximum sound, every
time through some grades at a time into the direction to the inter-
ference plane of Kirssnine till a distinct difference has been signa-
lised by the investigator (descending method). After the position of
the tube has been read this is pushed on and then brought back
in the same way till the investigator observes that the existing
difference in intensity becomes indistinct (ascending method). Again
ihe position of the tube is read off and the average is taken.

The observations take place in the above mentioned way “unwis-
sentlich” and at five sueceeding times. From the ten figures obtained
in this way the average is taken at last, which indicates in grades
of the seale a lowest ‘Unterschiedsschwelle” for the concerned
amplitude,

To be able to transpose these angle-values into absolute values,

( 423 )

in the sound free cabinet, which has been internally covered with
trichopiese, the greatest distance at which sound is still perceptible is
determined for the intensity of sound in the maximumplane and for
that in the discovered “Unterschiedsschwelle” plane. If we accept that
in case of absence of reverberations, as we may suppose here, the
sound intensities decrease proportionate to the quadrates of the dis-
tances, the sound intensities stand mutually in the same proportion as
the quadrates of those distances. If we eall the distance at which
the tone sound is perceptible in the plane of maximum sound 7 and

r—r?

that for the somewhat weaker sound 7,, then the quotient —j,—
i

u
represents evidently the ‘“Unterschiedsschwelle’, which in this case
may be indicated as “‘untere Unterschiedsschwelle” because the stimula-
tion distinguished from the chief is taken weaker than the chief
stimulation.

TABLE I. Experiments with the tuning-fork.

Tone level. ce niude Ar ‘ Unterschiedsschwelle”
microns if (average).
al 640 0.29589
800 0.34429 33.2 %,
1040 0.35657
c 20 0, 22698
40 0.26932
70 0.29825
100 0.30835 29.3 9/,
150 0.31003
200 0.31540
300 0.32006
ce 2 0.23435
2 0.20243 19.5 9%,
2 | 0.44865

Kuperiments with organ-pipes.

An accurately tuned, wide, covered, wooden organ-pipe is placed
in a felt tent in a room at the side of the soundfree cabinet in such

( 424 )

a way that the sound may be listened to through a caoutchoue tube
in the cabinet. This organ-pipe is permanently blown by air which
was supplied by a presspump driven by water and afterwards dried
with chloride of calcium. The supply of this air takes place along
a long system of leaden tubes, which shows inside the cabinet a
division into two parts and afterwards a reunion. To this two sepa-
rate branches by micrometer screws removable diaphragm openings
of Aubert are attached, which may be widened or narrowed at
pleasure. The reunion takes place in a ‘T-tap, which may also be
directed by the investigator, and down the current are placed the
necessary measuring apparatus for determining the pressure and
volume of the air passing to the organ-pipe. These measuring appa-
ratus are placed within the reach of the investigator, so that he
himself can do the reading off.

The investigator arranges in the first place the width of the two
diaphragm-openings in such a way that the sound may be called
equal in the two positions of the tap. Then he enlarges one of the
diaphragmata (the other remains constant) till a distinct difference is
perceived (ascending method). This he does five times. After this the
difference between the two tone intensities, which were alternately
listened to, was enlarged and the diaphragm position was ascertained
by descending at which the difference became indistinct (deseending
method). This again was done five times. The same takes place con-
formally in narrowing the diaphragm-openings. So the first series
leads to a ‘“obere’” the second to an “untere Unterschiedsschwelle’.
The determinations which were made for each tone with two chief
intensities have evidently taken place ‘“wissentlich” in this way. At
last a pressure and volume determination of the supplied air is made
for the found diaphragm widths. The first takes place by means of
a watermanometer, which for inereasing sensibility has been put
sloping; the second with an aerodromometer'). The energy offered to
the organ-pipe could be caleulated with the usual formula e = air-
volume > pressure x 981. This number, multiplied by a constant
factor, different for each pipe, indicates the acoustic energy.

As in the expression of the ‘prozentische Unterschiedsschwelle”

AR : ‘
ae the constant factor oceurs both in the numerator and the deno-
v

minator, the constant factor of the organ-pipe falls away from
the further calculation and we can also come to a trustworthy result
of the ‘“prozentische Unterschiedsschwelle” without its preceding

1) Arch. f. (Anat. u.) Physiologie 1902 supplement. p. 417.

( 425 )

determination. The final result for each tone is in this way the
average from 40 determinations.

Tone level

Relative intensity |

of the

| chief stimulation ')

1392 408
1120.55
1560.168
1243.35
1213 63
861.798
1412 200
788 .97
107.412
86.129
132.414
104 725
140.800
114.444
139.96
101.152
135.976
101.764
134.552
98.424
951 O41
139.438
332.072
230.888
424.636
280.908
295. 68
218.621
260.100
183.272
580.190

PA BL He i.

Unterschiedsschwelle.

oo
ww
WW
orn |

oo
ree
ox
=

oc
to
bo
=

oe SS oS 9S SS OS SS SO SS SO OG:
m 2 Gee y 2 . —= . < . . ‘< .

=)

(0°)

>
Ae

oo

oc oo

oo

oo

1) For the calculation of the absolute intensity
must still be multiplied by a constant factor which nowever falls away in the

calculation of the “Unterschiedsschwelle” and is of no consequence.

eooos

219
.237

Ses)

210
201
297

179

184
.158

168

152

166
108
134
442
105

132
138
108

108
082

101
122

121

114
107

155

145
200
178
194
.229

0.232

0.218

0.192

0.188

20.4

the number of the second column

( 426 )

Differences
of intensities.

LE Gs 3 (GK miele BIDS 5 (Or 6 (Ge h(E s
SS 4:9: 5% 9, 45 FIES;
Smallest perceptible difference of intensity by the scale.

CONCLUSION.

|. From the results of the experiments with the tuning-fork
proceeds that the law of Weer is valuable, when taken in a general
way, but not exactly for the investigated middle-strong and weak
intensities.

2. From the results of the organ-pipes proceeds that the most
favourable “Unterschiedsschwelle” is found with ct and that from
there to the ends the power of distinguishing differences in intensities
decreases rather regularly.

BRR AST USM:

p. 380 line 7 for 0,990 1,03 read 1,02 1,04.

(November 22, 1905).

KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM.

PROCEEDINGS OF THE MEETING
of Saturday November 25, 1905.

—D OCo—_—_—__—_\—

(Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige
Afdeeling van Zaterdag 25 November 1905, Dl. XIV).

QiOEN Ean ae Ss

Eve. Dusois: “The geographical and geological signification of the Hondsrug, and the
examination of the erraties in the Ne shern Diluvium of Holland”, (Communicated by Prof.
K. Martin), p. 427. Z

D. J. Korrewee: “Huycens’ sympathic clocks and related phenomena in connection with
the principal and the compound oscillations presenting themselves when two pendulums are
suspended to a mechanism with one degree of freedom”, p. 436.

H. W. Baxuvis Roozrsoom: “The different branches of the three-phaselines for solid
liquid, vapour in binary systems in which a compound occurs”, p. 455.

F. M. Jarcer: “On Diphenylhydrazine, Hydrazobenzene and Benzylaniline, and on the
miscibility of the last two with Azobenzene, Stilbene and Dibenzyl in the solid aggregate con-
dition”, (Communicated by Prof. H. W. Bakuvis RoozEgBoom), p. 466.

A. P. N. Francurmmonr and H. Friepmann: “The amides of z- and S-aminopropionic acid”, p. 475,

J. D. vAN DER Waats Jr.: “Remarks concerning the dynamics of the electron”. (Communi-
cated by Prof. J. D. van DER Waats), p. 477.

R. Sissincu: “Derivation of the fundamental equations of metallic reflection from Caucuy’s
theory”. (Communicated by Prof. H. A. Lorentz), p. 486.

P. H. Scuourr: “A tortuous surface of order six and of genus zero in space Sp, of four
dimensions”, p. 489.

W. Verstuys: “The PLucker equivalents of a cyclic point of a twisted curve” (Communicated
by Prof. P. H. Scuoure), p. 498.

“Preliminary Report on the Dutch expedition to Burgos for the observation of the total solar
eclipse of August 80, 1905,” communicated by Prof. H. G. vay pr Sanpe Bakuuyzen, in behalf
of the Eclipse Committee, p. 501.

’

Geology. — “The geographical and geoloyical signification of the
Hondsrug, and the examination of the erratics in the Northern
Diluwium of Holland.” By Prof. Eve. Duzors. (Communicated
by Prof. K. Martin).

(Communicated in the meeiing of September 30, 1905).

To those who do not know the Hondsrug from a personal visit
the name generally suggests an imposing hilly ridge, or perhaps
even a small mountain range. Visiting it for the first time, one is
disappointed in finding it to be no more than a nearly imper-

30

Proceedings Royal Acad. Amsterdam. Vol. VIII.

(80)

ceptible undulation of the ground, which only in some parts scarcely
deserves the name of hill. Before one is aware of it, its “summit” has
been reached, and it is probably only owing to the rather steep
slope of the Drenthe plateau towards the valley of the Hunze and the
extensive Bourtanger marsh, that this part of the country has received
its peculiar name. Without these the “ridge” would possibly be passed
unnoticed. However, the fact remains that there is a slight, irregular
elevation of the ground, rising at the most but a few meters above
the country on its western borderline, which, running from the
South-East to the North-West, is almost entirely confined to the
Province of Drenthe and has its Northern end a short distance
beyond the town of Groningen.

From a geological point of view, the Hondsrug is interesting on
account of the numerous erratics found there, several shiploads of
which are yearly collected. This, however, is a peculiarity not limited
to the Hondsrug: until recently similar boulders were also met with,
in as large numbers, in other parts of Drenthe and Friesland, but in
the more inhabited districts of these provinces they have, for the greater
part, already been dug out. Another point which until lately lent a
certain importance to the Hondsrug, was the generally accepted notion
of it being a terminal moraine. This interpretation, first started by
Prof. van Caiker, and especially based on his exploration of the
northern termination of the Hondsrug, in the town of Groningen
and in its vicinity, has successively been adopted. By a number of
papers, dealing with the Hondsrug in Groningen, published during
the last twenty years Prof. van Canker has not a little contributed
to give to this insignificant ridge a rather prominent geological
importance. Almost from the outset of his investigations, VAN CALKER
expressed his positive conviction that the Hondsrug is a terminal
moraine. As early as 1889, he writes’): “Seit meinen ersten ein-
schligigen Untersuchungen stand meine Ansicht fest, dass der Honds-
rug eine Endmoriine reprasentire, eine Moranenablagerung, welche
einem laingeren Stagniren im Riickzuge des Gletschers, vielleicht bei
einer gleich gerichteten Bodenwelle entspricht. Und mein Vermuthen,
dass diese eine weitere siidéstliche Erstreckung habe, wurde bestatigt,
als ungefahr 38 K.M. siidéstlich von hier bei Buinen in Drenthe
beim Aufgraben von Geschieben auch solche mit abgeschliffener und
geschrammter Oberfliche zum Vorschein kamen und noch etwa
26 K.M. weiter siidéstlich von dort, bei Nieuw-Amsterdam solche
von mir selbst gesammelt wurden, und ich an letzterer Localitat die
Grundmordane constatiren konnte.”’

) Zeilschr. der Deutschen Geologischen Gesellschaft. 1889, p. 3ol.

( 429 )

But when we compare the descriptions of Prof. van CaLker with
those of the terminal- and bottom moraines of other countries, it appears
doubtful whether even that part of Groningen examined by vaN
CaLker, notwithstanding its “tremendous accumulation of stones and
large boulders”, deserves the name of terminal moraine, and may not
in fact rather be considered as a bottom-moraine’). It can only have
been the shape and direction of the Hondsrug and the presence of
the numerous erratics found at its surface, which induced Prof. van
CaLker and others to regard this steep ridge of the Drenthe plateau
as a terminal moraine. Of its internal structure, except for the portion
which terminates in Groningen, no notice had been taken.

However in 1891, Loris, after his exploration of the high peat-
moss of Schoonoord, already expressed the opinion that those who
had really visited and explored the Hondsrug were not justified in
calling it a terminal moraine. He considers it to be the border of the
Drenthe plateau, slightly folded back by the moving ice-sheet’).

A few years ago I had several opportunities of visiting those parts
and making the exploration alluded to by Lori. To me it became
quite evident that the Hondsrug in Drenthe is not a terminal moraine.
Its geological structure, which I investigated more closely over the
Southern half of its length in Drenthe and but partially over its
Northern half, entirely refutes this interpretation. I found its nucleus
not composed of morainic material, but to be of fluviatile origin and
to consist of Rhenish Diluvium*) At the same time I also observed
that this fluviatile nucleus — although but slightly — was distinctly
vaulted. The second problem therefore to be solved, was to find the
cause of this vaulting, about which I could not agree with Lori,
who ascribes it to the motion of land-ice from the North-East. I
could not admit the possibility of the ice-sheet folding the soil
without perceptibly disturbing the nucleus of the fold, for the
contortions do not enter deeply into this nucleus; its stratification
has, in general, been well preserved. Basing my deductions on the
phenomena observed in the ice-sheet of Greenland, to which the
diluvial land-ice may be most aptly compared, I proposed several
possibilities which might account for this peculiarity. I suggested
the possibility of the ice having moved in the longitudinal direction

1) F. J. P. van Catxer, De ontwikkeling onzer kennis van den Groninger Honds-
rug gedurende de laatste eeuw. Bijdragen tot de kennis van de provincie Groningen,
etc. p. 217. Groningen. 1901.

2) Handelingen van het Derde Nederl. Natuur- en Geneeskundig Congres, 1891,
pp. 347 and 349.

3) These Proceedings, V, p. 93—114.
30*

( 430 )

over the Hondsrug, a certain elevation of the soil underneath it at
the same time having taken place from some cause or other. I
supposed, as one possibility, the mean pressure of the ice to have
been somewhat lessened, or its progress to have been easier just
above the present ridge. I imagined the change in the direction of
the ice-stream to have been occasioned by the Northern ice-sheet
being pushed back by the British ice-sheet in the German Ocean.

Recently, Dr. H. G. Jonker, one of Prof. van Caukrr’s youngest
pupils, has refuted my views in these Proceedings’). Though, from
a few lines at the end of his paper, it appears that he agrees with
ine on the main point, — namely, that, as to its geological compo-
sition, the Hondsrug in Drenthe consists of a fluviatile nucleus,
covered with a glacial deposit, and that on this account it cannot
possibly be considered a terminal moraine. But on the other hand,
he advances numerous arguments to disprove the probability of a
change in the direction of the ice-stream, and one of the ways in
which I conceived the raising of that ridge could have been effected.

In the first place I wish to refute, as briefly as possible, the
objections raised by Dr. Jonker to the explanation of the last-named
point suggested by me.

As mentioned before, the results of my investigation principally
related to the portion of the Hondsrug situated in the Southern part
of Drenthe, about half of its entire length in that province. I myself
mentioned several spots where the glacial covering does not consist
of sand but of loam. This circumstance however is not inconsistent
with the statement that the ridge, i general, is less rich in clay
than its Western borderland. Neither does it exclude a freer move-
ment of the ice-sheet over the Hondsrug, which I suggested as a
probable agency in the formation of this ridge.

As far as I am able to judge from the few excavations I visited
in the Northern portion of the Hondsrug, it seems to me that, in
general, its structure does not differ from that of the Southern part.

Dr. Jonker further mentions a few spots in the North of Drenthe
where the glacial cover of the Hondsrug consists of boulder-clay,
viz. in the neighbourhood of Gasselte and Zuidlaren. Of the latter
locality and also of some places near the town of Groningen, where
much boulder-clay is found, Dr. Jonker himself says that the hilly
character of the Hondsrug is less distinctly to be recognized, and
that the Hondsrug is hardly noticed there. These spots therefore may
be left out of account.

With regard to Dr. Jonknr’s reference to the borings of the Dutch

1) Vol. VIII (1905), p. 96—104.

( 481 )

Society for the Reclaiming of Heaths, I make the following remark.
Through the kindness of the Direction I was enabled to consult the
original registers together with the maps, relating to the borings, and
afterwards controlled these im sw. I ascertained, by many controlling
borings, that the borings of the Society are lying too far apart to
give an approximately exact idea of the presence and distribution
of the loam; besides that it is decidedly incorrect to state that ‘red
clay especially occurs on the Hondsrug and chiefly in its highest parts”.

The other solution, which I suggested, in a second paper dealing
with the Hondsrug, as a possible explanation for the origin of the
longitudinal vaulting of the ridge (an explanation which is independent
of the distribution of the boulder-clay and boulder-sand and which
at the same time throws some light on the origin of the strange,
round hill “Brammershoop’’), which Dr. Jonkur leaves unnoticed.

I believe to I have given already sufficient reasons for the opinion
I hold, that, generally speaking, boulder-clay and boulder-sand have
been, from the first, two distinct kinds of deposits, and that the
latter has not proceeded from the former. I will only add, that to
Dr. Jonkrr’s statements “that the percentage of stones in the boulder-
clay creases very much towards the surface’, I can oppose the
results of other and, I believe, more extensive statements, where
either the reverse was the case, or the stones were uniformly distri-
buted. This disparity is easily explained from the great local diffe-
rence in that quantity, justly observed by Dr. Jonker.

The vanishing of limestone-boulders does not prove the washing-out
of the loam, for it may have been occasioned by solution alone, without
washing; calcarious pebbles, originally present in clay or sand, may
disappear when the underground-water is not saturated with bicarbonate
of lime, and they may be preserved when this is indeed the ease. I wil-
lingly allow that the reason why, for instance, the clay of the Mirdum-
Cliff is especially rich in absolutely unmodified calcarious stones (the
finest scratchings lave been preserved), and that on the contrary, in
other parts, not a single calcarious pebble is found in similar clay,
need not be attributed to local differences in the original composition
of the ground-moraine. But this cannot be said with regard to the
flints, and especially not in respect of the clay itself. Clay of the
tough kind, called boulder-clay, is a very resistant substance. Expe-
rience in the field teaches that there can be no question of a wash-
out of particles of clay from a similar mass. The motion of the water
through the clay is far too slow for it. If Dr. Jonker had more
frequent opportunities of studying boulder-clay and sand abroad,
especially in England, he would, undoubtedly, have modified his

opinion on this head. I do not in the least question Dr. JonkEr’s
assertion, that in some places he ‘“‘can decidedly conclude from the
relief whether we have to do with boulder-sand or with clay”,
because there is such a large difference in the resistance which
boulder-clay and sand offer to erosion; but in most parts of Drenthe
it is dmpossible to judge from the appearance of the surface, whether
the ground underneath is sand or clay: this I learnt from consulting
the mentioned register of borings and also from numerous small borings
on my own account. The parts of the bottom-moraine from which, in
the opinion of Dr. Jonxnr, the boulder-clay has “disappeared”, and
the “intermediate stages between original boulder-clay, and altogether
washed-out boulder-clay”, therefore undoubtedly, have been varieties
existing from the beginning.

With regard to the occurrence of flint, which among our erraties
has been rather disregarded, there really exists an important differ-
ence between boulder-clay and boulder-sand. This I learned especially
too through the “comparative mechanical analysis’, recommended by
Dr. Jonker. Further I recollect that among the stones found in
the sand on the Hondsrug, I did not come across a single flint;
on the other hand I met with flint in all the clay-pits in the
neighbourhood, and, taking also into account the small fragments,
| found it even largely represented. The single exception which
Dr. Jonker observed in a pit of loamy sand near Groningen, is no
proof against this general experience. Besides, the bed was only loamy
sand, not boulder-clay. He too found flint in several clay pits on
the Hondsrug in Drenthe. In the fact, that on the whole (for it is
necessary to compare places lying outside the Drenthe Hondsrug as
well, because our Northern diluvium is generally considered as
belonging to one and the same glacial epoch) there is, with regard
to the presence of flint, an evident difference between boulder-clay and
boulder-sand, I find another proof in favour of my opinion that,
generally speaking, the one has not proceeded from the other by
a wash-out. Neither the occasional absence of flints from boulder-
clay nor the occasional presence of these stones in boulder-sand, are
proofs against the general tendency of my argument.

In the preceding I have endeavoured to give a succinct refutation
of the objections raised by Dr. Jonker against one of the solutions I
proposed to account for the vaulting of the Hondsrug, —a question
which is only of secondary importance.

But L gladly avail myself of the opportunity to discuss a point
of far greater importance, on which Dr. Jonkrr has expressed an

(433 )

,

opinion, namely, the direction of motion of that part of the ice-sheet
which reached our country.

From an examination of sedimentary-rock erratics from the Gro-
ningen part of the Hondsrug, the results of which he stated in his
dissertation, which appeared last year, Dr. Jonknr came to the same
conclusion as ScHropprr vAN per Kouk had arrived at from the
examination of igneous-rock erraties, especially from the Eastern
parts of the country, and as others too, namely “that the glacial
flow which has produced the glacial diluvium in the North of the
Netherlands was a Baltic one.” He even thinks it possible to trace exactly
the course taken by the glacial flow which “has created the Groningen-
diluvium”’. To these statements I have to make serious objections.

For long years, neglecting the available direct means of tracing
the direction of the glacial flow, such as the examination im situ of
the Quetschsteine — a study already recommended fourteen years
ago by our ever-lamented ScHrogpER VAN DER Konk — it has been
a custom in the Netherlands to be guided, in the determination of
the direction of the glacial flow, exclusively by the solid rocks from
which the stones carried towards us by the ice were derived. It
was not taken into account, and indeed was not at all known in
former time, that the great Ice Age, during which the Northern
Diluvium of our country was deposited, was preceded by another
glacial epoch, of lesser importance, it is true, for the Northern ice-
sheet did not reach our country then, but which was notwithstanding
the first real glacial epoch, by which the Pleistocene period was
introduced. In that first glacial epoch, the Scanian Epoch of Prof.
James Gurikiz, there lived in the North Sea the arctic fauna of the
Weybourn Crag, and, during the melting period of the Alpine ice,
our country received the Rhenish Diluvium.

In that same epoch, in Seandinavia and in the uplands to the
East of the Baltic, on the plateau called Fennoscandia, an ice-sheet
was formed which, following the slope of the land, terminated in
the North Sea as drift ice, and, on the other side, descended into the
basin of the Baltic, as the first Baltic glacier. It is well known that
the sculpture of the Scandinavian peninsula and of Finland has
been accomplished almost entirely during the Tertiary period, ata time of
a much higher level of those countries. The ice, which afterwards repeat-
edly passed over these parts, removed principally only the loose
material, smoothing the surface. Thus the /irst ice-sheet found all
the superficial deposits, accumulated on the rocky land-surface in the
preceding long period of erosion, both on that highland and in the
basin of the Baltic with its other environments,

( 434 )

Doubtless, already in the first glacial period, a transport of stones,
on a large scale and over considerable distances from the solid rocks,
has taken place, to the North Sea and especially in the basin of the
Baltic. The earliest Baltie glacier has been traced as far as Schleswig.
When at the later, much more considerable accumulation of ice, the
North Sea also was filled up with inland ice '), it may be reasonably
inferred that the British portion of it has carried along with it the
erratics which at that earlier glacial epoch dropped from the drift-ice
to the bottom of the sea.

In this manner we account for the finding of erraties of Scandi-
navian origin on the coast of Hast-Anglia. They are however
not so plentiful in those parts as Dr. Jonker supposes. Among
thousands of stones of British origin, occasionally one Scandinavian
stone is met with. I believe that indeed not a single geologist in
England is of opinion that the Scandinavian inland ice ever reached
the shores of Britain.

Undoubtedly the inland ice of that second or great glacial epoch,
which brought to our country the Northern diluvium, largely swept
up and transported the morainie débris deposited in the basin of the
Baltic, especially in its Western parts, during the preceding glacial
epoch. A large percentage, perhaps even the majority, of the erratics
thus again taken up and carried much further by the ice, must origi-
nally have come from a direction entirely different from that which
would answer to the glacial flow, by which they were then carried
along. Consequently, the presence of numerous stones of Baltic origin
in the bottom-moraine at the town of Groningen and in its neigh-
bourhood, is no reason why we should assume that the course of
the glacial flow has been from the Northern and Eastern parts of
the Baltic towards Groningen.

The abundance of flints in our Northern diluvium and the direction
of the glacial striae in the southern parts of Sweden, indicated on
the well-known map of Navnorst, rather suggest a more westerly
origin. Moreover it appears questionable if on more extensive study of our
erratics — those found in the bottom-moraine of Texel and Wieringen
have been almost entirely neglected — the Baltic character of those
stones found in the Diluvium of the northern parts of our country
can be maintained. Cousidering the great local differences existing in
the composition of the ground-moraines the erratics of such a small
spot as the Hondsrug in Groningen, prove but little.

I may here be allowed to mention a few other facts distinctly

1) There are good reasons for not admitting here pack-ice, as does the well-
known American geologist SatisBury.

( 435 )

supporting the conception that the coalescence in the North Sea of the
Northern ice-stream with another coming from Britain, may have
caused a deflection in its course over the northern parts of our
country, and changed its direction into one from North-West to South-
East. Owing to this meeting of Scandinavian and British glacial flows,
an enormous ice mass filled up the North Sea, in connection with the
ice-sheet extending over Holland, North Germany and the British
islands, the edge of which, as a high wall, faced the South. Accord-
ing to KiockmMann, WannscHarrn, Ruror and others, between this
wall of ice and the mountains of Middle Germany, Belgium, France
and the southern parts of England, the melting water rose several
hundreds of meters high, and in this water the deposition of the léss
took place. With regard to our country, [ entirely agree with this
view. The structure of the léss in the South of Limbourg decidedly
shows, in several places, its origin as a sediment deposited by
very slowly running water containing a large amount of drift-
ice, an opinion formerly advocated by Dr. A. Erens. In several loca-
lities of the Limbourg chalk-plateau (in the adjacent parts of Belgium
even as high as 300 M. above sea-level) erratics of Southern origin
are found in or were excavated from the l6ss, especially veined
quartzites from the Ardennes, sometimes measuring 2 M. and even more.

If thus we have to admit such an extensive and powerful ice-sheet
with considerable accumulation in the North Sea, — and at the same
time infer from the direction of the glacial striae on the rocky sub-
soil in North Germany, that one and the same glacial flow, owing
to local conditions, has taken at the same or contingent points very
different directions, deflecting even more than 90°, — I do not consi-
der it impossible that in its course over the Hondsrug, and in general
over the northern parts of our country, the direction of the glacial
flow may have deviated entirely from that of the flow passing
over the North of Germany.

Taking into consideration the still very limited knowledge we possess
of our erratics, and in view of the arguments in favour of a secondary
transport of perhaps the greater part of these stones, I consider the
suppositions which I advanced before, and which I have now somewhat.
more developed, as to a possible modification of the direction of motion
of the Norihern ice-sheet over our country, not only warranted but
necessary as a working-hypothesis for further investigation. I doubt
whether Dr. Jonker himself will now still adhere to his belief that,
“in ease this conception is the right one a great number of researches
into our ““Seandinavian diluvium’”” would become doubtful and
it would be advisable at once to begin a revision’. That Diluvium

( 436 )

will in every case remain Scandinavian, or rather Northern. At the
same time I would recommend a closer study of the Diluvium in
Texel and Wieringen, in order to ascertain, whether it contains
erratics, the origin of which may be traced to other parts than
of those which are found in the eastern parts of our country.
What I consider to be very “doubtful” indeed, is the right to
trace the direction which the Northern glacial flow is supposed to
have taken, solely from the examination of erratics, found at such a
large distance from the rocks of their origin. In reference to this
matter, I would strongly recommend “revision” and would especially
suggest a wider field of investigation than the Hondsrug in Groningen.

Mathematics. —- “Huycuns’ sympathic clocks and related phenomena
in connection with the principal and the compound oscillations
presenting themselves when two pendulums are suspended to
a mechanism with one degree of freedom.” By Prof. D. J.
KORTEWEG.

(Communicated in the meeting of October 28, 1905).

Introduction.

1. When in February 1665 Curistiaan Huyerns was obliged to
keep his room for some days on account of a slight indisposition he
remarked that two clocks made recently by him, and placed at a
distance of one or two feet, had so exactly the same rate that every
time when one pendulum moved farthest to the left the other deviated
at that very moment farthest to the right’). Yet when the clocks
were removed from each other one of them proved to gain daily
five seconds upon the other.

At first Huyerns ascribed this “sympathy” to the influence of the
motion of the air called forth by their pendulums; but he soon
discovered the real cause — the slight movability of the two chairs

?

1) Ce qu’ayant fort admiré quelque temps’; he writes: ,j’ay enfin trouvé
»que cela arrivoit par une espéce de sympathie: en sorte que faisant batlre les
»pendules par des coups entremeslez; jay trouvé que dans une demieheure de
ytemps, elles se remettoient tousiours a la consonance, et la gardoient par apres
,constamment, aussi longtemps que je les laissois aller. Je les ay ensuite eloignées
ylune de l'autre, en pendant l’une & un bout de la chambre et l’autre 4 quinze
»pieds de la: et alors j'ay vu qu’en un jour il y avoit 5 secondes de difference
,et que par consequent leur accord n’estoit venu auparavant, que de quelque
»sympathie”’. Journal des Sgavans du Lundy 16 Mars 1665. Oewvres de GurisTIsAN
Huygens, Tome V. p. 244.

( 437 )

over the backs of which rails had been placed with the clocks
suspended to them *).

‘) ,J’ay ainsi trouvé que la cause de la sympathie.. . ne provient pas du
»mouvement de l’air mais du petit branslement, du quel estant tout a fait insen-
sible je ne m’estois par apperceu alors. Vous scaurez done que nos 2 horologes
»chacune attachée a un baston de 3 pouces en quarré, et long de 4 pieds estoient
,appuiées sur les 2 mesmes chaises, distantes de 3 pieds. Ce qu’estant, et les
»chaises estant capables du moindre mouvement, je demonstre que necessairement
,les pendules doivent arriver bientost & la consonance et ne s’en departir apres,
yet que les coups doivent aller en se rencontrant et non pas paralleles, comme
»l’experience desia l’avoit fait veoir. Mstant venu a la dite consonance les chaises
she se meuvent plus mais empeschent seulement les horologes de s’écarter par ce
»qu’aussi tost quiils tachent a le faire ce pelit mouvement les remet comme au-
»paravant”. Letter to Moray of March 6th 1665. Oeuvres, T. V. p. 256.

Compare Journal des Scavans du Lundy 23 Mars 1665, Guvres T. V. p. 301,
note (4), where Huvyerys withdraws his first explanation to replace it by the
correct one and likewise his *Horologiwm Oscillatorium” where his experiments
and his explanation are developed on one of the last pages of ‘Pars prima”.

A somewhat more detailed account of those observations is moreover found

Fig. la in one of his manuscripts, from which we
derive the diagrams found here and the expla-
nation Huycens deemed he could give of the
phenomenon :

»Utrique horologio pro fulcro erant sedes duae
»quarum exiguus ac plane invisibilis motus pen-
,dulorum agitatione exitatus sympathiae praedictae
ycausa fuit, coegitque illa ut adversis ictibus sem-
»per consonarent. Unumquodque enim pendulum
ytune cum per cathetum transit maxima vi fulera
ysecum trahit, unde si pendulum B sit in BD
ycatheto cum A tantum est in AC, moveatur
,autem 6 sinistram versus et A dextram versus,

,punclum suspensionis A sinistram versus im-
D Cr 9S K »pellitur, unde acceleratur vibratio penduli A, Et
,rursus B transiit ad BE quando A est in catheto
,AF, unde tune dextrorsum impellitur suspensio 6, ideoque retardatur vibratio
»penduli 6B. Rursus B pervenit ad cathetum BD quando A est in AG, unde
»dextrersum trahitur suspensio A, ideoque acceleratur vibratio penduli A. Rursus
,»B est in BK, quando A rediit ad cathetum AF, unde sinistrorsum trahitur sus-
»pensio 6, ac proinde retardatur vibratio penduli B. Atque ita cum retardetur
ssemper vibratio penduli 6, acceleretur autem A, necesse est ut brevi adversis ictibus
,consonent, hoc est ut simul ferantur A dextrorsum et B sinistrorsum, et contra.
yNeque tune ab ea consonantio recedere possunt quia conlinuo eadem de causa
,eodum rediguntur. Et tune quidem absque ullo fere motu manere fulcro mani-
,festum est, sed si turbari vel minimum incipiat concordia, tune minimo motu ful-
ycrorum restlituitur, qui quidem motus sensibus percipi nequit, ideoque errori
,causam dedisse mirandum non est”.
We give this explanation for what it is. Huyeens, who never published it, will
probably himself, at all events later on, not have been entirely satisfied by it.

( 438 )

2. Although Huyens’ observations were published in the Jowrnal
des Scavans of 1665, and are moreover mentioned in his ‘‘Horologium
oscillatorium”’, they seem to have been forgotten when in 1739
correlated phenomena were discovered by Joun Exiicorr '). What he
observed at first was this: of two clocks N°. 1 and N°. 2 placed in
such a way that their backs rested against the same rail *), one, always
N°. 2, took over the motion of the other, so that after a time N°.1
stopped even if at first N°. 2 had been in rest and N°. 1 exclusively
was set in motion. Later on he found that the mutual influence was
greatly increased by connecting the backs of the clocks by a piece
of wood’). He also made both clocks go on indefinitely by giving
their pendulums the greatest possible motion, when alternately they
took overa part of the motion from each other, a¢cording to a period
becoming longer as the clocks being placed without connection with
each other had a more equal rate *). At the same time he observed
that both clocks when connected with each other in the way described
above assumed a perfectly equal rate lying between those which
they had each separately.

3. Sinee then different mechanisms where suchlike phenomena

Indeed, it is nothing but the friction which can finally cause that of the three
possible. principal osc'llations only one remains. Every explanation in which friction
does not play a part must thus from the outset be regarded as insufficient.

1) Phil. Trans. Vol. 51, p. 126—128: “An Account of the Influence which two
“Pendulum Clocks were observed to have upon each other,” p. 128—135:
“further Observations and Experiments concerning the two Clocks above mentioned.”

2) “The two Clocks were in separate Cases, and... the Backs of them rested
“against the same Rail.”

’) “I put Wedges under the Bottoms of both the Cases, to prevent their bearing
“against the Rail; and stuck a Piece of Wood between them, just tight enough
“to support its own Weight.”

4) “Finding them to act thus mutually and alternately upon each other, I set
“them both a going a second time, and made the Pendulums describe as large
“Arches as the Cases would permit. During this Experiment, as in the former, |
“sometimes found the one, and at other times the contrary Pendulum to make the
“largest. Vibrations. But as they had so large a Quantity of Motion given them
“at first, neither of them jiost so much during the period it was acted upon by
“the other as to have its Work stopped, but both continued going for several
“Days without varying one Second from each other”... “Upon altering the Lengths
“of the Pendulums, I found the Period in which their Motions increased and
“decreased, by their mutual Action upon each other, was changed; and would be
“pyrolonged as the Pendulums came nearer to an Equality, which from the Nature
“of the Action it was reasonable to expect it would.” Later on we shall see that
there was probably an error in these observations. The continual transmissions
of energy and the perfectly equal rate of the clocks exclude each other to my
opinion.

( 439 )

of sympathy may appear have been investigated theoretically and experi-
mentally; among others by KuLer *) the case of two scales of a balance
of which Danirt Brrnoviii’) had observed that they in turns took over
each other’s oscillations; by Porsson*), by Savart*) and by Résa1*)
the case of two pendulums fastened with Porsson to the extremities
of a horizontal elastic rod, or with Savartr and Résar to the horizontal
arms of a [T-shaped elastic spring; by W. Dumas") the case of a
pendulum, beating seconds, with movable horizontal cross rails, on
which other pendulums were hung; by Lucien pg La Rive’) and
Everett *) the case of two pendulums joined by an elastic string ;
whilst finally CeLuérimr, FurtTwAne ier and others developed the theory
of the motion of two pendulums of about equal length of pendulum,
placed on a common elastie stand, in order to determine experimen-
tally, and to take into account in this way the influence exercised
by the small motions of such a stand on the period of the oscillations °).

However, we see that the more recent investigations, with the
exception of the work of W. Dumas, who does not purposely mention
the phenomena of sympathy, relate to mechanisms where elasticity
plays a part; whilst it seems probable that this was not the case
or at least in only a slight degree in the experiments of Huye@Ens
and E.uicorr.

1) Novi commentarii Ac. Sc. Imp. Petropolitanae, T. 19, 1774, p. 325—339.
Routn, Dynamics of a system of riyid bodies, Advanced part, Chapt. Il, Art. 94,
giving the right solution, has justly pointed out an error in Euter’s solution and
likewise in the one signed D. G. S. appearing in The Cambridge math. Journ. of
May 1840, Vol. 2, p. 120—128. Euter’s treatment of the phenomenon of the trans-
mission of energy is also defective, as he does not lay stress upon the necessity
of the two almost equal periods, in this case of his quadratic equation admitting
a root nearly equal to the length of the mathematical pendulum by which he
replaces the scales.

*) Nov. Comm. |.c. preceding note, p. 281.

3) Connaissance des tems pour lan 1833, Additions, p. 3—40. Theoretical.
This memoir was indicated to me after the publication of the Dutch version of
this paper.

4) D’Institut, 1° Section, 7° Année, 1839, p. 462—464. Experimental.

5) Compt. Rend. T. 76, 1873, p. 75—76; Ann. Ec. Norm. (2), II, p. 455--460.
Theoretical.

6) “Ueker Schwingungen verbundener Pendel”, Festschrift zur dritten Sdcular-
feier des Berlinischen Gymnasiums zum grauen Kloster. Berlin, Wrmmann’sche
Buchhandlung. 1874. The investigations themselves are according to this paper
from the year 1867. Theoretical and experimental.

7) Compt. Rend. T. 118, 1894, p. 401—404; 522—525; Journ. de phys. (3),
III, p. 537—565, Experimental and theoretical.

8) Phil. Mag. Vol. 46, 1898, p. 236—238. Theoretical.

%) See for this the Hncyclopddie der mathematischen Wissenschaften, Leipzig,
Teubner, Band IV, I};, Heft 1, § 7, p. 20—22.

( 440 )

So it seemed worth while looking at the question from another side, and
studying the behaviour of a very generally chosen mechanism *) with
one degree of freedom, and with two compound pendulums attached
to it; noting particularly the case that both pendulums have about
equal periods of oscillation, whilst at the same time for the applica-
tion of the phenomena of sympathy of clocks the intluence of the
motive works will have to be paid attention to.

Moreover it is worth noticing that the results obtained in this way
will also be applicable to the case that the connection between the
two pendulums is brought about by means of an elastic mechanism,
every time when practically speaking only one of the infinite number
of manners of motion is operating which such a mechanism can
have. Such a manner of motion will have a definite time of oscilla-
tion for itself, which will play the same part in the results as if it
belonged to a non-elastic mechanism with one degree of freedom.

Deduction of the equations of motion.

4. Let § represent for any point of the mechanism with one
degree of freedom, to be named in future the “frame”, the linear
displacement out of the position of equilibrium common to frame and
pendulums; let $” be its maximum value fora definite oscillation to
be regarded as equal on both sides for small oscillations ; let $, and
¢, be its values for the suspension points O, and QO, of the pendulums;
let MW be the mass of the frame; let m, and m, be that of the
pendulums; a, an a, the radii of gyration of the pendulums about
their suspension points; gy, and gy, their angles of deviation from
the vertical position of equilibrium; v,, y, and 2,, y, the horizontal and
the vertical coordinates of O, and of O,, h the vertical coordinate of
the centre of gravity of the frame; taking all these vertical coordinates
opposite to the direction of gravitation.

So we begin by introducing for the frame a suitable general coor-
dinate u, for which we choose the quantity determined by the relation

Mir = | Odo, {ae

where the integration extends to all the moving parts of the frame;
this quantity might therefore be called the mean displacement of
the particles of the frame.

1) We assume with respect to this mechanism no other restriction than that the
motions of each of its material parts just as those of the two pendulums take
place in mutually parallel vertical planes, i.o.w. we restrict ourselves to a problem
in two dimensions.

( 441 )

For small oscillations of the frame we can put: wu=nw™, S=nh ,
where x is a function of time, but the same for all the points of
the frame.

So we have for such vibrations:

Mu? = M (num ))? =| (nS)? dm =| a dm ;

so that 4)fu? proves to represent the vis viva of the frame.

For the vis viva of the first pendulum we find, if 4, denotes the
distance between its suspension point QO, and its centre of gravity,
and if g, is reckoned (like g,) in such a way that a positive value
of g, increases the horizontal coordinate of the centre of gravity :

4 [m, $,? + 2m, kh, 2, 7, + m,a,? 9,7] =

a, ore . dz, . .
=m, w+ 2k, g,—uta,’g,7]3
du du

therefore for the entire vis viva of the whole system:

’ dS, : dS, ; “9 2 3 2 Z
Py) Mm, \ ms (5 | fet £m, 2,7," £m, 0,79,"

Uu
da, a0 dx, 676
Se LOA RO LL et so also co (CO)
and further for the potential energy *)

= d*h Dy, ay, 3 1 3 2
Va—=—'5.9 Ue +m, du? +m, du ws 3m, gk, py? + 4m, 9k, gy, (3)

5. To simplify further we introduce the new variable wu’ determined by:

| dee (ae, \?
M'u? =| M+ m,| — }+m, (| —) = Mv? +m,5,2+m,5,7; (4)
du du
where
Miele Nag eat eta Men twis) | Sy one ea rar) tor ()

represents the entire mass of the whole system; this variable w’
dé, dé,
and B
du du
indeed all such derivatives appearing in the formulae, may be regarded

as constant.

is proportional to wu, because for small vibrations

as

1) Indeed that potential energy amounts to Mgh ++ mgy, + magyg — my gk, cos g, —
— Mz Jky COS 7 + a constant. By developing according to w, taking note that on
dy, dy,

a + mg mS equal to 0 and by proper

choice of constant, we can easily deduce (3) from it,

nae ae dh
account of the equilibrium JM ae +m

( 442 )

Out of this proportionality follows ae

(ds, ? ds,
Mu? = ire Lm, +m, {| —
du du

which proves that } J/'w' represents the vis viva of what we shall
call the reduced system, which system consists of the frame and of the
masses of the pendulums each transferred to the corresponding

7m, c vtm

—
(=r)
—

Hee

suspension point O, or O,.
If now likewise we introduce the vertical coordinate /’ of the centre
of gravity of the reduced system, so that M/'h'= Mh + m, y, + m, ¥,,

271

€
the first term of (3) transforms itself into 4 ¢ ee u*, for which,
au

however, on account of the mutual proportionality of w and wu’ we
Bh

may write: $9 J’ aia w?.
du

So for the reduced system it holds that 7’ = }M'w? and Vi=

ah! : :
,w*; ifnow we write for this system the equations of motion,

and if we then introduce the length /' of the simple pendulum which
is synchrone to this system‘) we shall easily find:
dh! nH

Thus we finally may write for (2) and ok

i)

. : dx, :
T=} M'w? + im,a,79,7 + Lins dy? Pg 7+ mk, ai xy! L, saree Fi 2! G33 - (8)
Vtg M'Q)-1u? + 1m, gk, gy? +imgh gp, . . . @)
Application of the equations of Lagrange and substitution of the
expressions :

i (an)! g Y Sed g F . g 0
oS ul Sip Vaeeate Dire oath Pe aT een ae (0)

leads further easily to the eee

(m) ha da

MAN (Veh rans yf > 2, + mak! ty = 10) 55 tea)

lee m) q

—? me ai (a) we
du ky

ry t 4 :

din ym) +(4 —2)x,=0. . . (8)
du! ky

1) Should the reduced system be in indifferent equilibrium as was probably the
case in Enucorr’s experiments /' is infinite; if it were in unstable equilibrium this
would correspond to a negative value of /'. We shall again refer to these cases
in the notes. In the text we shall always consider 7’ positive, hence the reduced
system stable.

( 443 )

where x, and x, denote the maximum deviations of the pendulums
and 2 the length of the pendulum synchrone to one of the principal
vibrations.

6. In order to put these equations still more simply, we

3 2

a a,
first introduce the lengths of pendulum J, aie and 1, = | ofthe
4 V9
two suspended pendulums, secondly the maximum deviations in hori-
zontal direction of their suspension points :
dx ( dx
ee) = 3 iG) stati aay) ae : Py ,
du, du’,
It is then easy to find the following system of equations equivalent
to the equations (11), (12) and (13), namely :

F (a) = (—4) 4 —4) ¢,—4) — 6,2 U1, (,—A) — 6,7 81, (—4)=0; (14)

x (m) = (m)
Si Ss
= sit eo Be lee ces tbat aamee ae (Les)
or ST aR ae (ie)
where : is
»} 3
,_ 7m & (6)? Sys Sh a ke () (16)
Semin ae (ay | so Ue TE (etm.

We must notice here that c, and c, are numerical coefficients,
the first of which depends only on the first pendulum and its
manner of suspension, the second on the second pendulum.

Taking note ‘of the signification of wv’ and §,, and observing that

‘ ; (m)
for instance &,°” : wu’ =&,:1u' on account of the supposed small-
ness of the vibrations, we can write for the above after some reducing :
; m,§," a m,§.° fees noe
— 53 = hy)

a ¥ 1
m,$,?7+m,$,? +f dm *

holding at any moment of the oscillation, where § denotes the hori-
zontal, § the linear deviation out of the position of equilibrium of
an arbitrary poit of the frame, and where the indices relate to the
suspension points O, and Q,, whilst the integrations must be extended
over the whole frame.

If we finally remark that the relation between every § and every
S$ is the same as that of the fluxions, we can give the significa-
tion of ¢,* and c,* also in the following words:

c,? is equal to the proportion, remaining constant during the motion,
between on one side the vis viva of the horizontal motion of the
suspension point O, in which the mass of the first pendulun is con-

31

Proceedings Royal Acad. Amsterdam. Vol VIII.

( 444 )

centrated and on the other side the entire vis viva of the reduced
system multiplied by the distance between suspension point and
centre of gravity of the jirst pendulum and divided by its length of
pendulum ; and in the same way c,°.

Discussion of the general case.

7. Passing to the discussion of equation (14) we notice that
in the supposition 7, > 7, we have: F' (+ o) neg.; F(/,) pos.; F'(d)
neg.; (0) = /1,/, (4 —c,? —c,), and therefore with reference to
(17) where &,:/, and k,:7,< 1, F (0) always positive.

So there are three principal oscillations. The slowest, which we
shall call the slow principal one has a synchrone length of pendulum
ereater than the greatest length of pendulum of both suspended
pendulums ; of the intermediate principal one the length of pendulum lies
between that of these two pendulums; of the rapid principal one it
is shorter than the shorter of the two'). Further we can note that
when / >/,>/, the length of pendulum of the slow principal

one is greater than / and that for /,>>/,>>7 the rapid principal
one has a smaller length of pendulum than /.

The following graphic representation gives these results *) for the
ease I’ > 1, > /,, practically the most important.

1) This is the case for 7’ positive and this proves that when the reduced system
is stable. this must also be the case for the original system with the two suspended
pendulums. If /' is infinite, thus the reduced system at first approximation in
indifferent equilibrium, then the slow principal escillation has vanished or rather
has passed into an at first approximation uniform motion of the entire system, which
would soon be extinguished by the friction. The two other principal ones remain
and their lengths of pendulum are found out of the quadratic equation:

(j—A) (lg—A) — G2 G (lg—A) — C2? lg (h—A) = 0.

For /' negative /' (0) becomes negative too, but F' (— ce) positive, so then always
one of the principal lengths of pendulum is negative. From this ensues that when
the reduced system is unstable, this is also the case for the original one.

2) Of course these results are in perfect harmony with and partly reducible from the
well-known theorem according to which when removing one or more degrees of
freedom by the introduction of new connections the new periods must lie between the
former ones. To show this we can 1. fix the frame, 2. bring about two connections
in such a way that the pendulums are compelled to make a translation in a vertical
direction when the frame is moved. In the latter case it is easy to see that the
time of oscillation of the reduced system must appear.

For the rest these same results are found back in the main, extended in a way easy
to understand for more than two suspended pendulums, in the work of W. Dumas,
quoted in note 6, page 439 which I did not get until I had finished my investi-
gations. By him also the length of pendulum of the reduced system is introduced.
However, he has not taken so general as we have done the mechanism of one
degree of freedom, on which the pendulums were suspended.

( 445 )

Fig. 2. 8. With respect to the manner of oscillating
of the two suspended pendulums we shall
eall it the antiparallel mode when the simul-
taneous greatest deviations are on different

rapid principal sides as was the case in the observations of
oscillation Huyeuns, in the reverse case we shall call it
rapid pendulum 7, the parallel mode.

It is easy to see then from (15) that the follow-
ing three possible combinations will always
interm. principal appear, namely: for one of the three principal
oscillation oscillations the mode of oscillating of the pendu-
lums is the antiparallel one, for the two other
ones the parallel one, but in such a way
that for a definite greatest deviation of the
pendulums in a given sense the frame takes
for each of these two other principal oscilla-

tions an opposite extreme position *).

If thus for instance &,™ and §,™ have
equal signs as was certainly the case in the
mechanism used by Hvyerns (see fig. Ia)
slow principal _ and also in that of Exticorr, the antiparallel

oscillation : : te 5 ;
mode of oscillation observed by HuyeEns

slow pendulum /,

reduced system

belongs to the intermediate principal one.

9. For the application to the behaviour of two clocks connected
in the manner described we first consider /, and /, as very different
from each other, and that neither c, nor c, is small. In that case
it is evident from the values of /(/,) and F(Z) differing greatly
from naught that neither of the principal lengths of pendulum nearly
corresponds to 7, or /,; however from (15) then ensues that the
oscillations of the frame are of the same order as those of the pen-
dulums at every possible mode of oscillating.

Now it is of course not at all impossible that the principal oscillations
or certain combinations of them once set moving, might remain sustained
by the action of one or of both motive works under favourable
circumstances with sufficiently powerful works and when means have

1) Dumas has: ,dass, wenn.... die Aufhiingepunkte der Nebenpendel tiefer als
,die Drehungsaxe des Hauptpendels liegen, alle Nebenpendel von kiirzerer als der
,zu erzielenden [principalen] Schwingungsdauer in gleichen Sinne mit dem Haupt-
,pendel Schwingen miissen, alle anderen im entgegengesetzen Sinne”. This too
follows immediately from the formulae (15) which, indeed, correspond essentially
to those of Dumas. ;

31%

( 446 )

been taken to decrease sufficiently the frictions in the frame. However
in such a case the behaviour of the two clocks would differ greatly from
what was observed concerning the phenomena of sympathy; and in
the more probable supposition that the motive works will prove to
be unable to sustain a considerable motion of the frame, which
motion would absorb a great part of the energy, each of the principal
oscillations as well as each combination of them will after a certain
time have to come to a stop.

So we shall leave this general case, and pass to the discussion of
three special cases, which are more important for the consideration
of the phenomena of sympathy, namely A the case that /, and 7,
differ rather much, but where c, and c, are small numbers, B the
case, that /, and /, differ but little, but c, and c, are not small,
C the case where /, and /, differ but little and c, and e, are
both very small. In all these discussions we shall suppose /’ > 7, > 7,
and /' differing considerably from /, and /,. The treatment of other
special cases, e.g. c, small but c, not, will not furnish any more
difficulties if such a mechanism were to present itself ’).

A. Discussion of the case that l, and 1, differ rather much but

where c, and c, are small’).

In this case F'(l’), F(l,) and F(/,) are all very small, from
which is evident that each of the three roots of equation (44) is
closely corresponding to one of these three quantities, so that the
graphic representation of Fig. 2 looks as is indicated in Fig. 3.

From this then ensues according to (15) that for the rapid principal
oscillation the oscillations of the rapid pendulum are much wider
than of the slow one *), and that for the intermediate principal oscillation

1) Also the case /'=o differs in nothing, as far as the results are concerned,
from the eases treated here but by the vanishing of the slow principal oscillation.

2) The smallness of each of these coefficients may according to (16) be due to
three different causes, namely 1. to the smallness of A, :/, which will not easily
appear in clocks, 2. to the fact that the masses of the pendulums are small
with respect to that of the frame, 3. to the fact that the pendulums are suspended
to points of the frame whose horizontal motion is a slight one compared to that
of other points of that frame. It is remarkable that this difference of cause has
hardly any influence on the considerations following here, and therefore on the

phenomera which will present themselves.

3) Then still when in (15) £‘") might prove to be very small compared to £,™) ;
for as a first approximation for 7,—A we find: ¢)7/'l,: (//—/2), and therefore z, =
— Mi (l'—1y) (W™)2 712g Keg L' EQ. So the motion of the frame determined by 2'™) is slight
compared to that of the rapid pendulum and consequently ; is small compared to x.

( 447 )

Fig. 3. the opposite is the case. For the slow prin-

A cipal oscillation the oscillations of both pen-

dulums are either of the same order as those
of the frame or smaller still; the latter is
the case when the third cause mentioned in

rapid principal osc. note 2 of page 446 is at work.
rapid pendulum 7; Suppose now a’, 7, and 2, to be small oscilla-
tions belonging respectively to each of the
three types of the principal oscillations, namely
the slow one, the intermediate one and the rapid
one, each having the same small quantity
interm. principal osc. of total energy «= 7'+ V; then every
slow pendulum 4 ~ e9mpound oscillation can be represented by
w=K'a'+Kk,a,4K,a, and its total energy will

be equal to (KX? + K,* + K,’) «.
Let us then start from an arbitrary com-
pound oscillation for which A', A, and K,
reduced system have moderate and mutually comparable
< slow principal osc. Vojues; it is then clear that the motion of
one clock, namely the one with the rapid
pendulum will be dependent almost exclusively on the rapid prin-
cipal oscillation, that of the other clock on the intermediate one.
It is true, that slight periodical deviations in the amplitudes will
present themselves, which are due to the two other principal oscil-
lations, but these can have no influence of any importance on the
periods according to which the motive works regulate their action ;
so that therefore one of the motive works will be able to contribute
to the sustenance of the motion A,z,, the other to the motion A,z,,
but neither of them to the sustenance of the motion K'z’. So this

will vanish first.

What takes place furthermore will depend on the power of the
motive works, and on the frictions presenting themselves during
the motion of the frame. If those powers are great enough to
conquer the frictions when the pendulums deviate sufficiently to keep
the motive works in movement, a motion A, x, + A, x, will remain,
where the values of A, and 4,, thus also of their proportion, will finally
depend exclusively on the power of those motive works and on the
frictions. A theorem the proof of which we shall put off to § 44, to
be able to give it at once for all cases, shows that in general such
a motion can be sustained rather easily; it is the theorem that for
principal oscillations whose / differs but slightly from /, or 7, whatever
may be the cause, the kinetic energy of the motion of the frame

( 448 )
will be small compared to that of the corresponding pendulum. For
such a motion A, 2, + A, 2, remaining in the end, the two clocks
will each have their own rate*) whilst however slight periodic
variations in their amplitudes are noticed, caused by the cooperation
of the two remaining principal oscillations whose periods differ con-
siderably if /, and /, are sufficiently unequal.

11. Let us now however suppose that /, and /,, differing at first
considerably, are made to correspond more and more, for instance
by displacement of the pendulum weights. The chief consequence will
have to be that, according to equation (15), the amplitudes of both
pendulums will become more and more comparable to each other,
for Aya, as well as tor Ayx,, in consequence of which to obtain
their motion for the compound oscillation A\a, + Aya, we shall
finally have to compose for each of them two oscillations with com-
parable amplitudes, and whose periods of oscillation differ but shghtly.
As is known this leads for both pendulums alternately, to periods
of relatively greater and smaller activity, 1.0. w. to the phenomenon
of transference of energy of motion from one pendulum to another
and back again; the period in which this alternation of activity
takes place will be the longer according as /, and /, differ
less 7).

Now however a suchlike behaviour of the two pendulums accord-
ing as it gets more and more upon the foreground when /, and /,
approach each other, becomes less and less compatible with the regular
action of the two clockworks. For, during the period of smaller activity
of one of the pendulums the motive work corresponding to it will
finally, when the remaining activity has become much smaller than
the normal, come to a stop. Then one of the two will take place :
either the principal oscillation which is sustained particularly by this
work is powerful enough to keep on till the period of greater acti-
vity has been entered upon, and this will be deferred the longer
according as /, and /, differ less, ov it is not so. In the first case
the clock can keep goimg with alternate periods in which it ticks and
in which it does not tick, which phenomenon may of course present

1) Both rates however a little more rapid than for independent position.

*) These phenomena remind us of what Etuicorr observed later on (see note (4)
p. 438). However the correspondence is not complete, as in the case treated here
both clocks retain their different rate, whilst Etticorr mentions emphatically that
the two clocks did not differ a second for many days. We shall therefore have

to again refer to these observations at case C.

( 449 )

itself in both clocks’). In the second case the clockwork stops
entirely ; the corresponding principal oscillation vanishes, and the
pendulum performs only passively the slight motion which is its
due in that principal oscillation, which can now be sustained indefi-
nitely by the other motive work.

This is the phenomenon remarked by Etnicorr in his first expe-
riment when the clock n° 2 regularly made n° 1 stop.

We have now gradually reached case C where c, and c, are small
and where /, and /, differ but slightly ; this case demands, however,
separate treatment, for which reason we shall discuss it later on.

B. Discussion of the case that 1, and 1, differ but very Little,

but where c, and c, are not small’).

Before passing to the case C we shall treat the simpler case now
mentioned which will lead us to phenomena corresponding to those
found by Huyeens.

To this end we put /, —/,-+ A, and substitute this in the cubie
equation (14). Then by writing for one of the roots of that equation
/, +d and by treating 4 and J as small quantities we shall easily
find for the length of pendulum of the intermediate principal

oscillation the value
2

¢
Got a iy Die coy sca te yi tat f ay ES
eit gt (18)
from which is evident that this length of pendulum divides the
2

distance between /, and /, in ratio of ¢,?:c¢,?.
The two other roots satisfy approximately the quadratic equation:

CC =) SSO Ue Pee. eG)

1) This was really observed by Kxuicorr (l.c. p. 132 and 133) for both clocks,
however only temporarily, for at last the work of the first clock came entirely
to a stop. Compare for the rest the experiment of Danie, Bernouti with the two
scales mentioned in § 3.

2) If l, is perfectly equal to /;=/, then of course (14) has a root ~=J for
whose principal oscillation according to (15) the frame remains in rest. The remaining
roots are found by means of the quadratic equation (/’—A) (J—a)—(e2+-¢22) V0.
One of them will nearly correspond to / if ¢, and cz are both small fractions. All
this in accordance with Rovurn’s solution (l.c. note (1) page 439) which refers
exclusively to this case and also to that of Euter (barring what is remarked in
that note).

( 450 )

Fig. 4.

They correspond to the slow and_ the
rapid principal oscillation differimg considera-
bly in general in length of pendulum from /
and /,') and therefore by reason of (415)

skrapid principal giving rise to oscillations of the frame which
oscillation are of the same order of magnitude as those
of the pendulums.

ee ; So unless special measures are taken with
slow pendulum , respect to the decrease of the friction of the
frame, these oscillations will have to stop,
the more so as they are not sustained by

the action of the motive works.

So the only oscillation which will be able
to continue for some time is the intermediate
principal one whose length of pendulum is lying
between /, and /,; entirely in accordance with
the observations of HuyGrns*) and also with

meauced yy aica) those of Exticorr described in note (4) p.4388
when for the latter we overlook for a moment

mr the observed periodic transference of energy.
slow princip. osc. :

C. Discussion of the case that 1, and 1, differ but very little and

that at the same time c, and c, are small numbers.

13. The remarkable thing in this case is that now the remaining
quadratic equation (19) is also satisfied by a root differing but little
from /,. So there are now fwo roots of the original cubic equation
situated in the vicinity of /,, one found just now and expressed by (18)
and the other which is likewise easily found by approximation and
represented by the expression

2 3) ]'
j ROR SEOD Ei oe oa
’—l,

This root is, at first approximation, independent of A = /, —1,;
so when the lengths of the pendulums approach each other sufti-
ciently, it is, though small, yet many times larger than A. These

1) See the graphic representation of Fig. 4.

2) See however note (3) p. 452; from which is evident that the case which
really presented itself in Huyaens’ experiments is probably not the one discussed
here, but the more complicated case C.

(451 )

Fig. 5. conditions are represented by Fig. 5, where
A - we have moreover to notice that the third
root belonging to the slow principal oscillation
differs but little from /’.
We can now show that for the rapid prin-
cipal oscillation as well as for the intermediate
sk rapid principal ose. One, although not in the same measure, the
rapid pendulum t, oscillations of the frame remain small com-
eine Ua Soren pared with those of the pendulums.
Generally this is already directly evident
from the equations (15); this is however
not the case when the pendulums are suspend-
ed to points of the frame whose horizontal
motion is an exceptionally slight one*). In
that case we refer to the general theorem to
be proved in the following paragraph, and
from which what was assumed ensues im-
veduced system mediately.
slow principal ose. Let us note before continuing that now
for the rapid as well as for the intermediate
principal oscillation the two pendulums possess amplitudes which
are mutually of the same order of magnitude.

14. The indicated theorem can be formulated as follows: when
the length of pendulum of a principal oscillation approaches closely to
l, of 1, then the vis viva of the reduced system, thus a fortiori of
the frame alone, is continually small with respect to that of the pen-
dulum corresponding to 1, or (,.

To prove this we compare in formula (&) the three terms:

Ak

4 M'u'?; m, k, 5

'
au

‘uy, and }m,a,’?y,*. For the proportion of the

: Thier” Saale
second to the third can be written ae u':4g,, or on account
au

da (m) sim)
F =i lec be ony as 2m i, = 2(4—L):1,. The
au

second is therefore, when 2 approaches /, closely, small with respect
to the third, which can thus be regarded in such a ease to represent
at first approximation the vis viva of the first pendulum.

of equation (10), 2

1) That is to say, when the third cause mentioned in note (2) p. 446 has given

rise to the smallness of ¢, and cy.

For the proportion of the vis viva of the reduced system to that of

the pendulum referred to we can write *) :
M'u?: m, a,” y ==! (u”)? Min Gh t=
= Miu — mn a En eae

If now c¢, is not small, as in case ZB, then we have in this manner
already proved what was put. In case A we substitute 2—=/,—d
in the cubie equation (14) after which we find easily at first appro-
ximation, c, being likewise small *), d=/,—A=c,?/7,:(—1,), by which
what was put is likewise proved.

In case C finally, which occupies our attention at present, ensues
from (20) for the rapid principal oscillation 7, —%—=(c,?-++c,”)/'7, : (’—L,);
from which is evident after substitution of /, and c, for /, and e¢, in
(21) the correctness of the theorem also for this principal oscillation,
hence a fortiort for the intermediate one; unless c, be small but yet
much larger than c,, which restriction does not exist for the inter-
mediate principal oscillation.

15. From these results must be inferred that in the ease C under
consideration the rapid principal oscillation as well as the intermediate
one when once set in motion will each be able to maintain them-
selves under the influence of the motive works, when the condi-
tions of friction in the frame are not too unfavourable. However, the
intermediate principal oscillation will have, if the difference in rate
between the two clocks was originally very slight, a considerable
advantage on the rapid one, the motion of the frame being much
slighter still in the former case than in the latter. And this will
probably be the reason that in the experiments of HuyeEns as well
as in the later ones of Exiicorr evidently the intermediate principal
oscillation exclusively *) or at least chiefly *) presented itself.

1) According to (10), (15) and (16) taking at the same time note of the sig-
nification of 7,, @ and fy.

*) For c, small and ¢, not, the proof runs in the same way, although the
expression for 5 becomes a little less simple.

8) With Huyerns. In his experiments the masses of the pendulums were certainly
slight with respect to those of the frame, so that without doubt c, and ¢ were
small and the case C was present.

!) With Exuicorr, where at least at first according to the observed transferences
of energy also the rapid principal oscillation must have been present. Although
Enuicorr used according to his statement very heavy pendulums, we have probably
also the case CO with him. [f we do not assume this then it is more difficult
still to make the perfectly equal rate of his clocks tally with the observed trans-
ferences of energy. The presence of two principal oscillations evident from these
would have been continued indefinitely in case B, so the clocks would have
retained an unequal rate.

( 453 )

Savart on the contrary has effected with the aid of his T-shaped
spring at whose ends almost equal pendulums were attached both
principal oscillations *).

But besides these two principal oscillations which deviate in their
periods of oscillation, and moreover by the circumstance that the pendu-
lums will move in a parallel mode for one and in an antiparallel
mode for another, there is still a third manner of motion which
must be able to continue indefinitely.

16. To prove this let us again start from an arbitrary compound
oscillation w = K'a' + Kya, + K,x,; then unless the friction in the
frame be extremely slight the oscillation A's’ will soon disappear.
When however in the remaining motion A, is much smaller than Ay,
it is clear that as the intermediate principal oscillation is then the chief
one for the motion of the two pendulums, the motive works
of both clocks will regulate themselves according to it, so that they
will not be able to contribute to the sustenance of the principal
oscillation Aya, which will thus likewise have to die away, so that
finally only a pure oscillation A,a, will be left, for which both
clocks will follow the rate of the intermediate principal oscillation.

If on the contrary after the disappearance of the slow principal
oscillation A, is much smaller than A,, it will have to be the inter-
mediate principal oscillation, which dies away, whilst the rate of the
clocks will finally regulate itself entirely according to the rapid one.

But in the intermediate case, when the proportion of A, to A, lies
within certain limits, also a manner of motion will be able to
appear under favourable cirenmstances where both principal oscillations
are sustained for indefinite time, whilst each of them will govern
the behaviour of one of the two clocks; for from the equations (15)
it is easy to deduce that in general the proportion between the
amplitudes xz, and x, is different for both principal oscillations *).
Then the values of A, and A, and so also their proportion will in the
long run be entirely governed by the power of the motive works,

1) Lc. note (4) page 439. Savarr had however /’</,=/,; therefore with him
it is the slow principal oscillation which plays the part given here in the supposition
>t, > ly to the rapid one.

*) By substitution of the value (18) for A we find for the intermediate principal
oscillation %1 3 % = €;—? £10"): ¢.—2 £m); whilst the substitution of (20) furnishes
for the rapid principal oscillation

e.? pe HE —] (mn) , 3 »2)1'1 (n)
seme) a ete MT A Pletbe M] . ,
ae Peay

so for very small values of A we have for this one xy : %g = £,(m): E,(m),

ore

( 454 )

connected with the frictions presenting themselves, i.e. these values
will be independent of the initial condition. At the same time the
two clocks will show a different rate '), of which clocks one therefore
will have to sustain the rapid principal oscillation, the other the
intermediate one. Periodic transference of energy will then take place.

Probably it will not be easy to realize this condition, character-
izing itself particularly by the fact, that one of the clocks goes consi-
derably faster than would be the case when placed independently *).
The initial conditions will then have to be chosen in such a manner
that from the very beginning one oscillation will predominate for one
clock, the other for the other clock. And this will become all the
more difficult as c, and c, become more and more equal, therefore
according as the two clocks become more and more alike and
are suspended in a more symmetric way. For, so much smaller
will, according to what was mentioned in note (2) p. 453 be the
difference in proportion of the amplitudes x, and z, at each of the
oscillations. *)

17. Finally we wish to point out how we must represent to our-

1) So this differs again from what Exticorr observed in his last experiments,
so that these cannot be regarded as the realisation of this case, though they have
the transferences of energy in common with it. However, between the fact of
those transferences and the assurance that both clocks have entirely the same
rate exists a contradiction, as we have already seen, which is not to be solved.
Indeed, those transferences can be explained by interference only, so they require
the cooperation of two oscillations of different periods; but these oscillations must
both be sustained if the state is really to continue indefinitely, and then each
of them by one of the motive works where the oscillation referred to will predominate
the other one. See also the last note.

To me itseems most probable that with Exzicorr the transferences of energy existed
only at first indicating the temporary presence of the rapid principal oscillation.
Etuicort’s wording is not emphatically against this conviction.

2) The difference from case A is of course only quantitative. In both cases the clocks
go faster than when placed independently, but in case C the acceleration of the quickest
clock becomes much greater than that of the less rapid one (see § 13). A gradual
transition presents itself then, and the case of Exiicorr was probably situated on
that transition-line.

3) The idea that perhaps each of the motive works might be able to take over
one principal oscillation and the other in turns had to be set aside after a closer
investigation. If we compose in the well-known graphical way two oscillations of
unequal amplitudes and of periods of oscillation differing but little, it is evident
that the motive work will go alternately somewhat quicker and somewhat slower
than will correspond to the period of oscillation of the greatest amplitude, but this
can never go so far that the rate of the smaller amplitude is taken over, not even
for a short time.

( 455 )

selves the transition of case A into case C. In case A in which the
rate of the clocks differs greatly, the manner of motion which is most
difficult to realize in case C, namely the one, where the clocks have
each their own rates, is the normal one. Yet the two other manners
of motion also are possible, i.e. those where exclusively one of the
principal oscillations appears ; however in these cases, the pendulum of
the least active of the two clocks will still perform a shght oscillation
though not sufficient to set its motive work in motion.

If now starting from case A we reach case C) i.e. if the rate of
the clocks is taken more and more equal, the state of motion with
mutually different rate of the clocks becomes continually more diffi-
cult to realize, finally perhaps impossible ; whilst for the two other
possible imanners of motion the pendulum of the second clock too
keeps performing greater and greater deviations till these deviations
are finally sufficient to set its motive work also in motion, so that
both clocks go quite alike, either with the rate belonging to the rapid
principal oscillation or, what is more easily realized, with that or
the intermediate one.

Chemistry. — “The dijferent branches of the three-phase lines for
solid, liquid, vapour in binary systems in which a compound
occurs.” By Prof. H. W. Bakuuis RoozmBoom.

(Communicated in the Meeting of October 28, 1905)

A chemical compound, formed from two components, need not to
be regarded as a third component, when this compound is somewhat
dissociated, at least when it passes into the liquid or gaseous state.
Instead of the triple point we then get a series of triple points, the
three-phase line, indicating the co-related values of temperature and
pressure at which the compound can exist in presence of liquid and
vapour of varying compositions’) This was advanced for the first
time in 1885 by van per Waats. The equation for that line was
deduced by him’) and shortly afterwards*) applied by me ina few
instances where it was always admitted that the vapour tension of
the liquid mixtures gradually diminished from the side of the most
volatile (A) towards that of the least volatile component (2).

In the first considerations as to the course of the three-phase line

1) There exist several other three-phase lines which are not considered here.
2) Verslag Kon. Akad. 28 Febr. 1885.
3) Rec. Tr. Chim. 5, 334 (1886)

( 456 )

and the parts which could be realised in different binary systems,
the line was generally divided by me into two branches according
as the coexisting liquid contained more A or more & than the
compound,

Fig. |

(ag

In figure 1 branch 1: C7RF represents liquids with more A and
branch 2: /'D liquids with more B.

At the commencement, special attention was called to the iumpor-
tant fact that in the first branch a maximum pressure occurs at 7’
where the heat of transformation of the three phases passes through
zero. Less attention was paid to the fact that the maximum tempe-
rature FR does not completely coincide with the point /’, where the
composition of the liquid becomes the same as that of the compound,
but is situated either on branch J, if the compound expands when
melting, or on branch 2 if the reverse is the case; this may be best
understood if one remembers that the melting point line of the com-
pound #’A meets the three-phase line in the point /’. Although indi-
cated in the first publication of van ppr Waats and in my more
extended paper‘) this point remained in the background hecause,
practically, the difference in temperature between /’ and F is very
small. Afterwards *), Van per Waats worked it out more carefully
and only recently Smrrs*) has fully considered the peculiarities of the
p.«-tigures between / and f, after these had become important

1) Rec. 5, 339, 340, 356, 1886.
2) Verslag Kon. Akad. April 1897.
8) These Proc. June 1905.

( 457 )

from the point of view of the hidden equilibria which continuously
connect with each other the lines of the liquids and vapours coexis-
ting with the solid phase.

In the systems which formerly came most to the front, the diffe-
rence in volatility between the two components was so large —
such as with water and salts — that on the whole three-phase line
no vapour occurred which had the same composition as the compound.

If however, the difference in volatility is less pronounced, a case
may occur where the equality in composition between vapour and
compound is attained somewhere. Van per Waans foresaw that pos-
sibility in 1885, bat not until 1897 did he point out how such a
point, occurring on the three-phase line below the point /’, indi-
cates the maximum temperature at which the compound may’ still
evaporate in its entirety, and how in that point the subliming line
of the compound meets the three-phase line. Such a point is indi-
cated in fig. 1 by G, the subliming line by GZ.

It was, however, thought very desirable to elucidate the manner
in which, in such a case, the equilibria solid-vapour, solid-liquid and
liquid-vapour join each other on the three-phase line by a repre-
sentation in which is also shown the change of the concentrations
of liquid and vapour along the three-phase line of the compound
with increasing temperature.

Dr. Smits *) recently gave a representation of this by working out
a connected series p, v-sections of a spacial figure, which in the case
of a binary compound takes the place of my spacial tigure, where
only the components occur as solid phases.

A good example may be found in StoRTENBEKER’s *) research on the
system chlorine + iodine. There it is found that both the compounds
JC] and JCI, yield at their melting point a vapour containing more
Cl, but at a lower temperature they have a point on their three-
phase line where the vapour becomes the same as the compound.
STORTENBEKER had noticed this fact during his research, but had not
followed the matter up. After I had completed in 1896 my jp, ¢, x-
figure for binary mixtures, I also projected the spacial representation
for this case, and I had then already come to the view, by graphical
methods, that the point G is the highest temperature at which a
compound can exist near vapour of equal composition.

Bancrort *), in consequence of VAN DER WAALS’ publication, tried
to elucidate the case of JCI by a representation of partial pressures,

1) These Proc. June 1905.

2) Rec. Trav. Chim. 7. 183. 1888.
3) Journ. Phys. Chemistry 3, 72. 1899,

( 458 )

which appears to me less suitable, to survey the connection of the
phase-equilibria. The representation now worked out by Smits (see
his communicaiion fig. +) is a p, projection of my own spacial
figure with p,¢, as coordinates which, however, had not yet been
published.

This representation is well suited to explain at once which are
the transformations which take place on the different parts of the
three-phase line, owing to change in pressure or temperature, and
finally lead to the disappearance of one of the three phases.

Those transformations are dominated first of all by the connection
of the compositions of the three-phases.

From the figure it will be seen at once that, if we indicate the
solid compound by S, the coexisting liquid by Z, and the vapour by
G, the order of the compositions of the phases commencing with
one richest in the volatile component A, is as follows:

on branch CTRF : GLS
pe os! FG : GSL
5 3 GD: SGE:

The only transformation which can take place between three
phases is such that one is converted into two others, or reversely.
That one must then necessarily be the middlemost in composition,
consequently successively L, S, G.

The most rational division of the three-phase line is obtained when
this takes place according to the transformation which occurs between
the phases, and we will, therefore, call in future the branches on
which ZL, S or G are the middle-bodies, the branches 1, 2, 3.

The transformation of 1 into 2, therefore, takes place in the point
F where S= JL, that of 2 into 3 in the point G where S= G.

If now we observe in what direction that transformation takes
place, for instance on applying heat, we have

on branch 2: SoG+L
a5 33 3 : S+ Ll G
on the other hand on branch 1 we have:
on the part TRF: S+G—L_ branch la
Mur bea hve CT: Le=S+G ae (i)
whilst in the point 7’ itself, both transformations are without heat
effect. The reversal of the direction of the transformation causes
retrograde phenomena, on increasing or lowering the temperature.

A reversal of the direction of the transformation caused by a
change in pressure also takes place on either side of the point /
on branch 1, or on branch 2 if the compound melts with contraction,

( 459 )

and in this way retrograde phenomena by variation in pressure
become possible.

On the branches 2 and 3 a reversal of the direction of the trans-
formation caused by heat supply is as a rule not probable, as this
always consists in the evaporation of the solid matter, coupled with
melting of the same, or evaporation of the liquid, processes which
generally want a supply of heat"). The readiness of the reversal on
branch 1 is, therefore, closely connected with the fact that the liquid
phase is here the middle body.

If we consider in an analogous manner the character of the three-
phase line on which the most volatile component A occurs as solid
phase, the order of the phases is here SG'L, therefore the line AC
represents branch 3; in the point A, Gand L become simultaneously
equal to S; consequently, there exists no branch corresponding with
branch 2 of the compound. On the three-phase line D2 where the
least volatile compound £ is the solid phase, the order is GLS,
therefore DF corresponds with branch 1.

In the previously studied binary compounds the volatility of the
one component was so much smaller than that of the other, that on
the three-phase line only the branches 1 and 2 were noticed; if
the second constituent is sufficiently volatile branch 8 may be met’)
with as in the case of JC] and JCI,.

Such is the state of affairs in the case that the vapour tension of
the liquid mixtures gradually decreases from 100°/, A to 100°/, B.

If now, however, a minimum or a maximum occurs in the vapour
tensions the possibility may arise that, somewhere on the three-phase
lime of a compound, the liquid and vapour phases, wich coexist
with the solid phase, become equal in composition; and the question
arises what significance this fact possesses for the division of the
three-phase line.

In his communication cited Dr. Sairs has for the first time given
the three-phase lines for both cases and also the p ,v-projections of
the appertaining spacial figure but has not further investigated the
character of the different parts of the three-phase line.

Let us first take the case that a minimum oceurs in the p-.-lines
for liquid-vapour. If the compound in liquid and gaseous state was

1) The special cases where reversal might take place will not be considered here.

*) If branch 3 is wanting because on branch 2, S nowhere becomes equal to G,
there is still a possibility that this occurs somewhere on the three-phase line
which the compound with the least volatile component as solid phase and vapour
gives below the point D. This we cannot further enter into.

32

Proceedings Royal Acad. Amsterdam. Vol, VIII.

( 460 )

not at all dissociated, that minimum would coincide with the coni-
position of the compound.

Fig. 2

t

The three-phase lines would then appear about as shown in Fig. 2.
Instead of one continuous line for the compound, there would be
two branches sharply meeting in /’, CY and DF, both exhibiting
the character of branch 1, and therefore the order GZS of the three
phases, and both becoming tangent in /’, to the melting point line.

The sharp meeting in / is caused by the fact that there is no
continuity between liquids or vapours containing an excess of A or
of 4, if the compound itself on its transformation into liquid or
vapour, that is in /’, remains totally undissociated and therefore con-
fains no trace of A or 4 in the free state. In this case /’ is a
triple poimt for the compound.

In case of the least trace of dissociation we, however, get continuity
and the branches C7’ and Ds’ unite to one three-phase line of the
compound, which therefore assumes the general form deduced by
Smits, and is represented in fig..3. The minimum in the vapour and
liquid) line. now, however, shifts towards a composition differing
from that of the compound, generally all the more as the volatility
of A and £F differs more and the dissociation is greater. Unless
special influences") decidedly modify the partial pressures of the
components in the liquid phase, the minimum will generally be
situated at the side of 2. From the p, a-representation deduced for
this case by Sirs, it will be easily seen that, proceeding along

1) Such as the existence of several compounds.

( 461 )
branch CF of the three-phase line and continuing over FD, the

order in which two of the three phases become equal in composition
is as follows:

point /: i)
point G: C5)
point H: == 6%

From this it follows firstly that, if somewhere on the three-phase
line of the compound liquid and vapour become identical (point //),
there is certainly also a point G where vapour and solid become
equal, as G is situated between // and F’.

Fig.

Let us now consider the character of the different parts of the
three-phase line. From ( to HH, the state of affairs is just the
same as in Fig. 1. C/’ is, therefore, again branch 1 with the order
GLS for the composition of the phases, /G branch 2 with the
order GSL and GH branch 3 with the order SQL.

Whilst however in Fig. 1 the character of branch 3 continued up
to D, a change oceurs at H because L = G. It is easy to deduce
from Dr. Suirs’s p, v-figure that the continuation HD of the three-phase

line again exhibits the character of branch 1, the order of the pha-
ses is just as on C7’F': GLS, with this difference that G is now
the richest in the component 4 whilst on branch C7'F’ the vapour
was richest in A. Because in H the compositions of 4 and G be-
come equal, a transformation in that point of the three-phase line

32*

( 462 )

only oecurs between those two and, therefore, the tangent HW to
the three-phase line must be the line indicating the p,f values for
the series of liquids and vapours having equal composition.

Just as in #’ oceurs as tangent to the three phase line the mel-
ting point line FA, which is the extreme limitation of the equilibria
between solid and liquid, and in G the subliming line GZ, which
is the extreme limitation for the equilibria solid and vapour, the
tangent in #7 is the line HAM, which is the boiling point line of the
liquids with a constant boiling point, and also the extreme limitation
for the equilibria liquid-vapour *).

The points # G and H are, therefore, points of strictly related
significance; they are the points where the order of the phases sud-
denly changes.

Let us now further consider branch /7). In fig. 3 ocenrs a point
of maximum pressure 7’, and of minimum pressure 7). The first
point is quite comparable with the maximum 7’ in the branch C7,
the part D7, is again // on which, on heating, the transformation -
L—+>S+G takes place, the part 7, 7, is branch /a, to which
belongs the reverse transformation, whilst in 7’, itself the heat of
transformation passes through zero.

Owing to the continuous connection of D7, 7, to HG, we
necessarily get a small rising part 7 #7 of branch 1, after the line
has passed through a minimum 7” The possibility of this minimum
may be explained as follows:

Just beyond 7, the amount of heat necessary to convert S + G
into 4 can at first increase, because 4 and G both approach in
composition to 9S, so that the quantity of G concerned in the said
transformation diminishes with regard to S. But as we approach on
the three-phase line the point #7, / and G approach each other
more than they approach S (for point /7, where = G‘, is reached
sooner than G, where S—= G); consequently the ratio of the phases
G/S, which transform themselves in 4. becomes again larger and
the heat required for this again smaller until it finally becomes zero
at 7 and beyond this point negative, in other words the trans-
formation again becomes = S+ G; the smail part 7’ /7 again
represents /> and keeps on doing so up to the point // where the
transformation in branch 3 takes place.

As the minimum 7” does not coincide with the point H where

L=G, a small modification must be made in the p,.-projection of

') In the figure the lines AM and ZG intersect. In the spacial figure this is

howeyer, a crossing,

( 463 )

the spacial figure given by Dr. Surts in his fig. 5. His three-phase strip,
which I will rather call two-phase strip because it is formed by the

Fig. 4

lines indicating the liquid) and vapour existing by the side of the
compound, assumes the form of fig. 4, in which the particular points
of the threephase line fig. 8 with which this figure corresponds are
indicated by the same letters. The line is extended so far that italso
includes the maxima 7’ and 7, and so shows in which respects it
differs from the case corresponding with fig. 2, and of which the
strips have been indicated by Smits in his fig. 2.

If the minimum in the liquid-gas surfaces should be very little
pronounced, another type of the three-phase line may be expected,
which is represented in tig. 5 in which both minimum and maximum
have disappeared in branch //D, the whole line having the character
of branch Id.

In fig. 4 this would result that beyond the point /7 vapour and liquid
lines keep on a downward course, which may be the case if the
composition of L and G, which coexist with the compound, shifts
but little with the temperature so that the increase in pressure which
would occur owing to the shifting towards the side of 6 is more
than compensated by the decrease in pressure caused by the fall
in temperature.

Up to the present not a single example has been studied where
a three-phase line of the type fig. 38 or 5 made its appearance. Still
it is not difficult to see that both must frequently exist in the case
of dissociable compounds with sufficient volatility of the two com-

( 464 )
ponents. Examples will be found in the compounds of NH, or amines

with volatile acids as HCl, HBr, H,S, HCN or organic acids as formic
acid, acetic acid, in chloral hydrate or alcoholate, ete.

Fig.9

f

When the compound becomes less dissociated, fig. 3 will assume
more the character of fig. 2. To this belongs, perhaps methylamine
hydrochloride. As the dissociation becomes greater and the volatility
of £# differs more from A, the point /7, where L = G, will be fur-
ther removed from /. In the case of amine salts of organic acids
it is already known that the liquid with a constant boiling point
lies much closer to the acid-side than the compound.

If the volatility of 4 decreases very much, fig. 5 may form.

If the line AZM lies strongly to the side of 4 the case might happen
that the point /7 did not occur on the three-phase line of the com-
pound, but on that of the component 5. In fig. 3 and 5 branch 3
is represented by the three-phase line ALG as well as by BLG.
In the case mentioned, the line 4G starting from 4 would at
first represent branch 3 but after passing the point 1 = G@ it would

represent branch 1 either 14 or later even ta. These branches then
join on branch 8, the three-phase line of the compound. Of this, no
instance is as yet known. In the systems HCl, HBr, HJ and H,O
the ice line runs to very low temperatures, and therefore to very
high concentration of HCl ete., but the line /7J/ when ruining to lower
temperatures also runs toa higher acid concentration, so that according
to PickurRING’s data on the coexisting liquids the minimum in HCl— H,O
would fall on the three-phase line of the third hydrate, in HJ — H,O
on that of the fourth hydrate (both on the side of the solutions
richer in water) in HBr— H,O even just before the melting point

( 465 )

of the fourth hydrate at the side of the solutions richer in HBr —
in no case, therefore, on the ice line.

Let us further consider the case where liquid and vapour become
equal at a maximum pressure. Here, this point will lie generally on
the side of the most volatile component and as the compound becomes
more dissociated and the difference in volatility of its components
greater, the chances are that the composition of liquid and vapour
at which they become equal, differs more from the compound.

From this originates a form of the three-phase line which is in
general indicated by Fig. 6. The point // is now shifted to the top
branch at the left side of the maximum 7’ in branch 1. The part
HC now exhibits the character of branch 38. The line HAZ, which
indicates the maximum pressures of the series of liquids and vapours
having an equal composition, is tangent in 7 to the three-phase line
and forms the extreme limitation of the equilibria between liquid
and vapour. The three-phase lines for solid A and solid 4 both
exhibit the character of branch 1.

Owing to the non-coincidence of the points 7 and 7 a similar
correction must be applied to the p,.-projection of the two-phase
strip given by Dr. Sirs as has been done by me in Fig. 5 in the
case of the minimum.

The type fig. 6 will, presumably, not frequently occur, as a
combination between two bodies is as a rule accompanied by a
reduction in pressure and therefore, the occurrence of a maximum

( 466 )

pressure in the series of the liquid-vapour equilibria is but little
probable. At the moment there only seems an indication that the
case occurs with PH, Cl.

If the line H/J/ is situated much more towards the side of A it
might then also happen that the point AZ did not occur on the three-
phase line of the compound, but on that of the compound A, so
that branch 3 on this line follows on branch 1 and disappears
from the three-phase lines of the compound.

In a future communication I will discuss the boiling phenomena
of the saturated solutions corresponding with the said branches of
the three-phase lines.

Crystallography. — “On Diphenylhydrazine, Hydrazobenzene
and Benzylaniline, and on the miscibility of the last two
with Azobenzene, Stilbene and Dibenzyl in ‘the solid state.”
By Dr. F. M. Janerr. (Communicated by Prof. Bakunuts

RoozmBoom).

(Communicated in the meeting of October 28, 1905).

The following research was undertaken to furnish a new contri-
bution to the knowledge of the relation of the crystal-symmetry of
organic compounds and their power of yielding crystallised mixed
phases with each other’). Originally, it only aimed at the investigation
of Hydrazobenzene and Benzylaniline in their connection with the
series, investigated by Bruni, Garwiii, CaLzouart and Gorni, of
Azobenzene, Stilbene, Tolane, Dibenzyl and Benzylideneaniline, but
afterwards, Diphenylhydvazine, which is isomeric with Hydrazobenzene
was also included.

Diphenylhydrazine.
(C, H,), N—NH,; melting point: 44° C.

This compound, which I obtained through the kindness of Prof,
5S. Hoocnwerrr of Delft, crystallises from ligroine in the form of
colourless, large, lustrous crystals, which exhibit a rather varying aspect.
On exposure to light they rapidly assume a brown colour.

') Compare F. M. Jagger, These Proc. VIL. p. 658.

( 467 )

Triclino-pinacoidal.
4 Zao \ a:6:¢=0,7698 : 1: 0.5986.

| Ar 89 ak” == feisyuy
/ B = 137°28’ 8 = 137°283’
C= 89529; y= 90° 43’

very plain.

Forms observed: 6 ={010', broad and
ed lustrous; i= {110}, somewhat narrower and
reflecting less sharply; p = {110}, very lustrous
, and broad; c= {O01}, well developed and

1
|

|

!

|
The approach to monoclinic symmetry is
|

|

I

|

1

yielding fairly sharp reflexes; o-={111}, very
lustrous and well developed. The crystals are
mostly flattened towards p, or they may be

developed isometrically with a slight elongation
along the c-axis. It is peculiar that in the
vertical zone the co-related parallel planes of
the forms m, p and 4 are generally very
unevenly developed. Perhaps we may have
here a new example of the presence of an

Ee acentric crystal; the nature of the surface of
ee i the parallel planes is also often different on
Fig. 1. a plane and its corresponding contreplane.
(Diphenylhydrazine). Etched figures could not be obtained.

No distinct plane of cleavage.
Measured : Calculated :

Bios == (O10): 10) = 62°6! —

Cin (ONO (AO) =" 62) o4/, =
OO = (O10 (OO), ==" 89 13"/, —
p:¢ = (410): (001) =* 48 53 =
o:m = (111): T10) =* 73 38 —
m:¢ == (140) (001) — 49 24 49°28’
o:c =(111):(001)= 56 54 56 54
p:o =(110):(411)— 78 42 78 36
0:6 = (414): (010) = 58 457/, 58 43
pim = (110): (110)= 54 597/, 54 592/,

In the vertical zone the situation of the optical elasticity axes was
almost parallel to the direction of the c-axis; but on 6 the angle of
inclination amounted to about 10°, on m only about 12. An axial
image could not be observed.

( 468 )

The sp. gr. of the erystals is 1,190 at 16°; the equivalent volume
154,62. Topical axes ¥: w: wo = 6,0956 : 7,9182 : 4,7399.

Hydrazo-Benzene.
C, H, .NH—NH.C, H,; melting point: 125° C.

(Hydrazobenzene).

When recrystallised from a mixture of alcohol and ether, the com-
pound forms small, thin, colourless, square plates.

Rhombic-bipyramidal.
a:6:¢= 0,9787 :1 :1,2497.

Forms observed: c= {O01}, strongly predominant and very lustrous ;
o = $111}, sharply reflecting; ¢ = {021}, lustrous, always very small
developed; @ = $221}, narrow; m= ;}110} very narrow and often
wanting altogether. Thin-tabled towards c.

Measured : Calculated :

c:0 = (001): 411) =* 60°46’ a

0:0 == (441): (411) =* 75 14 =

G2 g, = (O01) 021) o8a 68°12!
aby =i bys (Oyu) ey 24s) 13 36
ey CPL SG (0) =" aksy AY 15 38
c:m= (001): (410, = 89 56 90 O
OO (221) (221) — 84 41
m:m = (110): (410) = 88 36 88 46
w:@ = (221): (221) = 3058 31 16

Very completely cleavable along {OOL{.

On c¢ the situation of the directions of extinction is orientated
towards the side ¢:q. An axial image could not be observed.

Sp. gr. = 1,158 at 16° C.; the equivalent volume is 158,89.

Topical axes: 4: yy: @ = 4,9567 : 5,0645 : 63291.

( 469 )

Benzylaniline.
C, H, .CH,—NH .C, H,; melting point: 364° C.

Vig. 3.
(Benzyl-Aniline).

From ether or alcohol the compound ecrystallises in large colourless
crystals flattened towards @ which, however, never exhibit measurable
end planes. The best crystals are obtained from methyl alcohol. They
are then mostly twins towards {100} or sometimes parallel-crystalli-
sations. The end planes are generally curved and unsuitable for
measurement. With some of the better developed crystals more
accurate measurements could be executed.

Monochno-prismatic.

a:6:c=2,1076:1:1,6422:
8 = 76°363’.

Forms observed: a= {100}, most broadly developed of all sind
strongly lustrous; ¢ = {OOL}, somewhat narrower and strongly lustrous;
s = {021}, bent and curved, sometimes less opaque and flat; 7 = {203},
well developed and lustrous; @ = $2421}, indicated as extremely
narrow vicinal form, mostly wanting.

Measured: — Calculated :
a:¢ = (100) : (001) =* 76°36"/,) —_
Oe TP SAVE 203) =—=oyoy! baie —
)

€:s = (001) : (021) —* 72 377/, =
s:$ == (021): (021) = 34 45'/, 34545 ye
a:s =(100):(021)— 9358 93 58
pes (203): (021) = 75 74 59'/,
c:r == (001): (203) = 29 53 29 52

Very completely cleavable towards {001} and {100!. Twins
towards {LOO}.

In the zone of the 4-axis orientated extinction everywhere; the
optical axial plane is {OLO{. On a and ¢ a black hyperbola is visible
in convergent light; one axis forms with the normal on @ an angle

( 470 )

Fig. 4

Huydrazobenzene.
120° Stilbene.

Azobenzene®

Oo 10 2 30 0 60 VO 80 90 400

Binary meltingpoint lines of Azobenzene + Stilbene

and of Azobenzene + Hydrazobenzene.
of about 12°. The apparent axial angle in a-monobromonaphtalene
amounts to about 90°. Strong, inclined dispersion with @ > v.
The Sp. Gr. = 1,149 at 14° C.; equivalent: vol. = 159,25.
Topical axis: 4: wy: w = 7,6220 : 3,6164 : 5,9389.

As regards Hydrazobenzene and Benzylaniline the following obser-
vations must be made.

(441)

Some time ago, Brunrand Cramician!), and Garett and CaLzo.art’)
coneluded, on account of cryoscopic abnormalities, to a formation
of mixed phases in the solid state between, Dibenzyl, Stilbene, Tolane
and Azobenzene, and to the ‘somorphogenous substitution in aromatic
molecules of the atomic-combinations:

— CH, — CH, —
— CH = CH —
= —=C——

MUN aE

According to Brunt and Gornt*), Benzylideneaniline: C,H, . CH =
N.C,H, may also form mixed crystals with St//bene and Azobenzene

so that according to them the atomic combination: — CH = N—
ought to be ineluded in the above series. The question now arises
whether the combining forms : NH — NH — and —CH, —NH —
which find in) Hydrazobenzene and Benzylamine their most simple
representatives, analogous with the above derivatives, belong to this
isomorphogenous series or no.

The important question, however, arose whether we have really
the right to speak here of an isomorphism, as we are not allowed to
conclude at once that an isomorphism exists merely on account of
the power of mixing in the solid state only.

Borris’), however, demonstrated that the four firstnamed sub-
stances exhibit such a close form relationship that this is practically
indistinguishable from true isomorphism.

Dibenzyl : C,H, . CH,—CH, . C,H,.

Monoclino-prismatic a:b: ¢ = 2,0806 : 1: 1,2522; 8 = 64°6'

siilioene= C,H... CH= CH .C,H,.-

Monoclino-prismatie » : 6 :¢ = 2,1701 : 1: 1,4008 ; 8 = 65°54!

Tolane : C,H, . C=C .C,H,.

Monoclino-prismatic a: :c = 2,2108 :1:1,3599 ; 8 = 64°59

Azobenzene: C,H;.N=N.C,H,.

Monoclino-prismatic a: 6:¢ = 2,1076 : 1: 1,3312 ; 8 = 65°34

Here, however, we meet with differences in aspect, optical orien-

}) Brunt and Cramicran, Soluzioni solide e miscele isomorfe fra i compostia catena
aperta saturi e non saturi; Rendic. Lincei (1899). 8. [. 575; Gazz. Chimic. Ital.
(1899). 29. 149.

2) Garetu and Catzorart, Sul comporiamento crioscopico di sostanze aventi i
costituzione simile a quella del solvente ; Rendic. Lincei (1899). 8. 1. 585; Gazz.
Chim. Ital. (1899). 29. (2). 258; Rendic. Lincei R. Accad. (1900). 9. (1). 382:

3) Brunt and Gorni, Gazz. Chim. Ital. (1899). 1. 55.

3) Borris, Atti Societa Ital. di Se. Natur, Milano. (1900). 39. 111—123. Abstract
Z. f. Kryst. 34, 298.

( 472 )

ation eic. which are greater than is allowed in strictly isomorphous
substances, so that it is better to speak of isomorphotropism mstead
of isomorphism.

Now according to Garenit and Canzonart. Dibenzyl and Benzyl
aniline form mixed erystals; also Azobenzene and Benzylaniline*).
This in connection with the results previously obtained by MutTHMANN”)
according to which the Terephthalic-methyl-ether is isomorphotropous
with Ajys—, and A,3— Dihydroterephthalic-dimethyl-ether and the
Ays— and 4,;— Dihydroterephthalic-Niethyl-ethers ave isomorpho-
tropous with the A,— Tetrahydroterephthalic-diethyl-ether, whilst, in
addition the p-Diovyterephthalc-ethers behave in an analogous manner
to the p-Dioxydihydroterephthalic-ethers and are capable of forming
with these mixed phases in the solid state, the Italian investigators
believe they are justified in coming to the conclusion that if two
aromatic substances can form mixed crystals, their hydro-products
can do the same.

The universal application of this rule is at once upset by Hydrazo-
benzene and Dibenzyl, which, eryoscopically, behave quite normally
but differ in their crystalline form as shown above.

It was, therefore, to be expected that Azobenzene and Hydrazo-
benzene would form no mixed crystals. Experiments taught me
indeed that from their mixed solution in ether Hydrazobenzene is
deposited first in colourless, perfectly pure crystals. Afterwards these
are accompanied by pure red crystals of Azohbenzene; they were
verified by the melting point.

I have also determined the melting point line of mixtures of the
two substances. This line has two branches and an ordinary eutec-
ticum. situated at 59°,25 and corresponding with a concentration in

oy

Azo-compound of 76.2 mol

Hie
Here are a few data:

Azobenzene melts at 67°.8 C.

- + 9.6°/, Hydrazobenzene ,, ey 2h CL

Be Seen 7 a As COSC

7 + 23.8 °/, x yes EDC.

" + 47.0°), . ss Oh Oa es ae Gls

“sf + 70.5 °/, i eater (OP UO)

Hydrazobenzene J Sel SOM

1) Bruni, Ueber feste Lésungen. Samml. chem. techn. Vortrige. Bd. VI. (1901).
p. 48.
®) Murumann, Z. f. kryst, 15. 60; 17. 460: 19. 357,

According to my research there is here no question of an isomor-
phism with an appearing hiatus.

All this is quite in accord with the deviating crystalline form of
Hydrazobenzene.

It deserves attention that Brunows") has investigated p-Azotoluene
and p-Hydrazotoluene. He finds;

p-Azotoluene (143° C.)-Monoclino-prismatic
O20 3605687 31 edd B= 897-44

p-Hydrazotoluene (128° C.

)
Gave 0627

-Monoclino-prismatic
Ja OZOms a == 69.49),

Notwithstanding these deviations, also that one where {O01} of the
first substance plays the role of {100! at the second he declares
these compounds to be “isomorphous”! Of more than a mere morpho-
tropic relation there can be no question here, and the so called
isomorphogenous replacement of —-N—N— by —-NH—NH— does
not help us here.

As Dibenzyl and Benzylaniline can yield mixed erystals and as
according to Bruni the latter yields mixed crystals with Azobenzene
an analogy in form is to be suspected here. This may indeed be
brought to light by assigning to Borris’ forms in Azobenzene: S100%,

S001}, {140}, (2013, ‘403! respectively the symbols: }LOO}, LOL, §4108,
101%, {103}, that is to say by calling the form whieh should be
{701} with Bonrts, {O01}.

Then we have :
Azohbenzene : G0 6241076); 1 71,4220). 3 == 762.39)
Benzylaniline : a:b:¢=2,1076 :1:1,6422: »— 76°.362'.

Therefore, a relation which, having the same ratio a:b and an
equal angle $, looks as if it ought to be considered as a ease of
isomorphotropism bordering on isomorphism.

It must, however, be pointed out immediately that this explanation
is not a rational one as the other forms of Azobenzene observed by
SonrIs Obtain in this way very complicated symbols.

It must also be observed that the meltingpoint of Benzylaniline
(363° C.) is lowered by addition of small quantities of Azohenzene.

Whilst the Benzylaniline used melted at 361° C. and the Azobenzene
at 68° C., the following melting points ¢ were found for mixtures -

') Birrows, Rivista di Mineral. e Cristall. Ital. (1903). 30. 34—48. Abstract
Z. f. Kryst. 41. 273.

934 °/, Benzylaniline + 64°/, <Azobenzene, t = 35° C.
ssa), 35 aie bs ¢ = 324°C:
48,6", : 51,4°/, is f= 49" e.
23,7°/, a 76,3°/, ‘3 f= 614°C:

On crystallisation of the two compounds from a joint solution in
alcohol + chloroform, small aggregated orange-coloured needles were
obtained which were unsuitable for further investigation. But it seems
that solid solutions are formed here.

As regards Azobenzene and Stilbene with which Dibenzyl forms
solid solutions, I have tried to determine, in mixtures of the two
substanees, the points of initial and final solidification in the usual
manner. It appeared that these two substances whose isomorphotro-
pism is almost indistinguishable from real isomorphism, also agree
with real isomorphous substances in this respect that they ean form
a continuous series of mixed phases from 0°/, to 100 °/,. The lower
branch of the continuously-rising melting point curve is situated so
close to the higher one that a sharp determination was quite
impossible.

The following melting points were found:

Azobenzene melts at 68° C.
4 + 22,9 °/, Stilbene melts at 82°.4C.
a + 66,94°/, P, OMe:
Stilbene melts at 120° C.

The melting point of Azoehencene is therefore raised by addition of
Stilbene. The lowering of the melting point of S#2/bene is not propor-
tional to the number of molecules of added Azohenzene (incorrect
rule of Kisrrr) but takes place more slowly.

The mixed erystals obtained from mixed solutions were homogeneous
and of a bright red colour. They erystallise beautifully. The mixture
being sublimed the vapour deposits mixed crystals as is also observed
by Brunt.

Finally, we wish to observe that /Hydrazobenzene, notwithstanding
the difference in symmetry, shows in its parameters some degree of
analogy with Azobenzene if we take 0 = {211}, o = {421} and
m = {210}.

Azobenzene : G66 210162 aka? 2008 = oOwoue
Hydrazobenzene: a:b:e= 19574: 172 12249735 — 907

In the case of Diphenylhydrazine, notwithstanding its great simi-

larity to the monoclinic system, there is no question of such a

distant analogy.

( 475 )

»

Chemistry. — “The amides of « and B-aminopropionic acid.”. By
Prof. A. P. N. Francnimont and Dr. H. FrimpMann.

(Communicated in the meeting of October 28, 1905).

Some time ago, I was asked for information as to a substance
isolated in 1873 by Baumsrark from some urines. He describes it
as white prisms some millimetres long and resembling hippuric acid
not only in form but also im lustre. If is fairly readily soluble in
boiling water, with difficulty in cold water and spirit of wine, in-
soluble in absolute alcohol and ether, and suffers no decomposition
when heated to 250°. It is a neutral substance which, however,
yields with acids, hygroscopic compounds difficult to crystallise, and
gives a precipitate with mercuric nitrate. The analysis led to the
formula C,H, O N,. By means of nitrous acid he obtained from. it
a liquid acid and from this he prepared a zine salt, which in zine
and water content corresponds with the zime salt of sarcolactic acid.
This zine salt was very soluble in water and in spirit of wine, and
from this he concluded that it really consisted of zinc sarcolactate.
He also showed that under the influence of alkalis, one nitrogen
atom is readily converted into ammonia and the other into ethylamine,
carbon dioxide being also formed.

From this he concluded that his substance was the amide of
e-aminopropionic acid or, as he called it, the diamide of lacue acid.

Briustuin in his text-book, mentions BaumMsTark’s substance as the
amide of @-aminopropionic acid but adds a point of interrogation,
and rightly so, for Baumstark has tried to control the conclusion
drawn from his analytical research by synthetically preparing the
diamides of the lactic acids, or amides of the amino-propionic acids,
and comparing them with his substance from the urine; from those
experiments he drew the opposite conclusion, and declared the previous
idea to be incorrect.

But if we look at the synthetical methods applied by Baumsrark,
it is at once evident that he could obtain nothing else but mixtures,
which he has not separated but seems to have regarded as pure
substances -whose properties were totally different from that of his
substance from urine.

Moreover, many, particularly physical properties such as melting
point, solubility, neutrality ete. of the substance from urine, are not
those which we might reasonably expect of the amides of the amino-
propionic acids.

None of the amides of the aminopropionic acids being known, I

39

Proceedings Royal Acad. Amsterdam. Vol. VIII.

( 476 )

instructed Dr. FrimpMANNn to prepare both by a method leading to
pure products, in order to end the present uncertainty.

The a@aminopropionic acid (commercial alanine) was converted in
the usual manner into the hydrochloric methyl ester, which melted
at 158°. By the action of silver oxide or aqueous caustic soda, in
presence of ether, the free methyl ester was prepared, which is very
volatile, and under a pressure of 15 m.m. passes over between 38°
and 41° as a colourless liquid (spec. gr. 13,5° = 1,0309), which
however, after some days changes into a solid mass (alanine anhy-
dride), presumably owing to the interaction of the two functions —
amine and organic ester. A nitrogen determination in the liquid
gave figures corresponding with those required by the amino ester
CH,.CHNH,.CO,CH,. The amino ester was mixed with a saturated
methylalcoholic ammonia and left to itself for afew days. A distilla-
tion at 85° at a pressure of 20 mm. removed the ammonia and alcohol
and left as residue a colourless oily liquid of a strongly alkaline
reaction which is but litthe soluble in ether and benzene and_ solidi-
fies in a dessiceator. After being recrystallised from alcohol, the
substance on analysis appeared to be pure. It crystallises in needles,
is very soluble in alcohol, very hygroscopic and melts at 62°. On
prolonged heating ammonia is set free and alanine anhydride is
formed, as was indeed to be expected. The e«-aiminopropionic amide
CH, CH NH, .CO.NH, gives also a well-crystallised compound with
hydrogen chloride, a fine crystallised orange-red chloroplatinate, which
is readily soluble in water, but litthe so in alcohol, and a bright
yellow picrate, little soluble in water, which on being recrystallised
from alcohol melts at 199°. Thus we have sufficiently characterized
this amide for the present. It seems to be decomposed already at
ordinary temperature in an exsiccator.

The p-aminopropionic acid was prepared but witha slight modification,
according to HooGrwerrr and Van Dorp from succinimide *). Like the
e-compound it was converted into the hydrochloric methyl ester,
which melts at 95°.

From this was prepared in the manner described above the free
B-aminopropionic methyl ester, which under 18 m.m. pressure distills
at 57° —59° as a colourless liquid and is pure as follows from analysis.
After a few hours it is decomposed with formation of crystals. By
direct treatment with methylalcoholic ammonia the amide was at
first obtained as an oily liquid, which was purified by repeated

') This was simply prepared like many other amides (when they and their
acids can resist a fairly high temperature) by heating the acid in a current of
ammonia until no more water is expelled.

( 477))

solution in methyl aleohol and precipitation with ether, when it gave
good analytical results. On being cooled, it beeame solid, and on
inoculation with a trace of the solid material, it yielded beautiful
erystals, melting at 41°. The B-aménopropionic amide NH,.CH,.CH,.
CO.NH, is very hygroscopic, very soluble in alcohol, with difficulty
in ether, has a strongly alkaline reaction, absorbs carbon dioxide
from the atmosphere and yields a well-crystallised hydrochloride.
Both aminopropionic amides are, therefore, now known and not
identical with the substance obtained by Baumsrark from urine.

Physics. — “Remarks concerning the dynamics of the electron.”
By Prof. J. D. van per Waats Jr. (Communicated by Prof.
J. D. VAN DER WaAats).

The theory of electrons is usually deduced from the following

equations :
dialled 4 uate ts eA)
rot {) = _ rises) an Heal en Og ust Regsele 1 0102)
i — : Bee esse tre crak nes (LEED
CUS eet Oe ee Mm Rane er AEE POSTS (01/9)
f= = [v6] sees east (id A

The units and notations used are those of Lorenrz’s article on the
“Elektronentheorie” in the ‘“HKneyelopadie der Mathematischen Wis-
senschatien” V 14.

The equations (I)...([V) determine the field, the motions of the
electrons being given. Equation (V) will be independent of the for-
mer four, and determine the motion of the electron in the electro-
magnetic field, ef representing the force exercised by the field on the
electric charge. The application of equation (V) is however attended
by peculiar difficulties. If in mechanics a body is given its mass is also
supposed to he known; then if we know the force, the acceleration,
and the law according to which it will move may be calculated.
Now if in forming f we take into account all the forces also those
excited by the electron itself, then the case that we ascribe a “real”
or “material” mass to the electron offers no fundamental difficulty.
For the case, however, in which the electron has no real mass the
equation (V) assumes the form:

force = O

( 478 )

dv -
without a term with mt the righthand member which enables
at
us to determine the acceleration. If on the other hand in forming f
we take into account only the external forces, we may introduce the
electromagnetic mass mm’ and write :
/ dv

force = m' —
dt

The mass mm’, however, is not known, and we can only calculate
it if we know already the law according to which the electron
moves. But then of course we need not make use of equation
(V) any more.

ABRAHAM‘), accordingly, does not make any use of equation (V)
for determining the motion of the electron. He does use an equation
which in Lorenrz’s notation, the integrals being taken throughout
infinite space, may be written as follows:

{foo Snr {fps

But this equation is deduced from the equations (/)...(/V) and
so it is not equivalent to equation (J7) which must be independent
of them. The only use which is made of equation (J’) is the

: : ; 1
introduction of the name force for the quantity o(s + —[vb] }, of
-

1
the name momentum for — [db], and of the name electric mass
;

for the quotient of the so defined force and the acceleration. But
the real problem: how will an electron with given shape and
charge move in a given field, must be solved beforehand indepen-
dently of this nomenclature.

Yet it is evident that the equations (/)...(/J’) will in general
be insufficient for the determination of the motion of an eleetricai
system. In the case that we ascribe a real mass to the electron, it
is obvious that we must know the force acting on it. But also on
the supposition that the electron has no real mass — and in what
follows we will confine ourselves to this case — another set of
equations is required for the determination of the motion. For
equation (//) enables us to determine » if » is always known, and
inversely to determine » if we know >. but it does not enable us
io determine both these quantities. The assumption that the motion
1S coo stationary is equivalent to a relation between y and b.

Apranam, “Dynamik des Elektrons.” Ann. der Physik IV. B. 13, 1904, bl. 105.

For such a motion the equations (/)...(/ 1’) are therefore sufficient
to determine the motion. If the motion is not quasi stationary then
the equations (/)...(/V) are not sufficient, and we must make use

of equation (7), which may be written:

; 1
[[fels+—to + ternal dS=0 . . (Va)

the integral being taken”throughout the electron.

If we wish to state the meaning of this formula with the aid of
the conceptions force and mass, we may say: the real mass of the
electron being zero, it is impossible that a force should aet on it.
We may, however, set these conceptions aside, and simply state: the
electron places itself and moves in the electric field in such a way,
that the relation (WV) is permanently satisfied.

dy

It is true, this equation has the form: foree = 0 without m—
in the righthand member. Yet it may serve to determine the motion.
This is owing to the fact that the expression of the force itself
contains the velocity » and the angular velocity g. In general we
may choose such values for these quantities that the equation (V)
is satisfied. To some extent therefore we return with the dynamics
of an electron to the standpoint of mechanies before Ganimer: the
forces do not determine the acceleration but the velocity. If we
might assume > and to be given throughout all space and at. all
times, > and g would be determined by the place of the electron, and
we should get a differential equation of the first order for the
determination of the motion of the electron.

The question is in reality less simple, because > and § depend on
the former motion of the electron. This causes a time-integral of a
function of » and g to occur in the equation of motion of the electron.
So we get integral equations as Sommmrreip has used in his treatises
“Zur EKlektronentheorie I, I] and III’’.") In some eases the integrations
may be effected, and then we get functional equations.

If the electron moves rectilinearly without rotation, and if it
moreover has an axis of symmetry the direction of which coincides
with the direction of the translation, then the terms of equation
(Va) which contain » or § disappear and the equation reduces to :

[[ferws=o ‘

In this case it is no longer possible to satisfy the equation by

1) Géttinger Nachrichten 1904, p. 99 and 363 and 1905 p. 201.

( 480 )

means of a suitable choice of the value of » and 4, and now it is
the place of the electron which must be such that the equation is
satisfied. If the electron stood still the equation would cease to be
satisfied in a following moment because of the propagation of the field-
forces, it must therefore suffer a displacement in such a way that
the relation continues to be satisfied. So the equation determines the
velocity, though the velocity itself does not occur in it.

This remark may perhaps serve to elucidate the results of Som-
MERFELD concerning the motion with a velocity greater than that
of light, and this is principally my aim with this communication.
In the following | shall denote a velocity, greater than that of light,
with B and one smaller with ».

We see at once that the supposition of SommeErrerp that the velo-
city of an electron moving with B will suddenly decrease to » when
the external force is suddenly suppressed, cannot be accurate. For
if we take d to be the sum of two parts 5, the external field and
d, the field of the electron itself, then we have at the moment ¢
before the suppression of the external field:

{fe (d, d,) OS == (5

aad

But as » requires an external force, {| J
e/ee

0d, dS is not zero, so

.

neither can {{J o>, dS be zero. This last quantity is independent
a/e/se

of the velocity at the moment 7 itself, and so it cannot be made to
disappear by any choice of the velocity, and there is no possible
way in which equation (J7@) can be satisfied.

If we imagine the velocity of an eleetron moving with 8 momen-
tarily to decrease to y, then the required external force will not
suddenly beeome zero, but at the first instant it remains unchanged,
and only gradually if varies in accordance with the new mode of
motion. This thesis applies to every discontinuity in the velocity
provided the motion be rectilinear and the electron have the required
symmetry. For the ease that the initial velocity is zero it follows
from SoMMERPELD’s complete calculation of the force. We see again
the conformableness of the dynamics of an electron with a theory
of mechanics in which no inertia is assumed: the force required for
a discontinuous change in the velocity is not only not infinite, but
even zero; the foree, which acts before the discontinuity, remains
unchanged at the moment of the discontinuity.

We cannot be astonished at the fact that we do not find a possible

( 481 )

Way of motion for an electron moving with B, when the external
force is suddenly suppressed. The same applies to an electron moving
with »; if the motion is aecelerated, and so if a force acts on the
electron, and if this force is suddenly suppressed, the equation (Va)
cannot be satisfied in any way. This is because the momentary
disappearance of the external force is an impossible supposition.
Even an infinite acceleration would not satisfy equation (Va). The
internal force namely depends only on the former motion of the
electron, and not on the velocity or the acceleration at the moment
ey }
itself. SOMMERFELD’s conclusion that a motion with aS does not
require an external force holds only if the initial velocity is », and
is nothing else but a statement in other words of the fact that the
force acting on an electron whose velocity is at the moment f mo-
mentarily — i.e, with infinite acceleration — brought from » to &, is
zero at the moment f. If however we begin with a constant velocity
B,, and change the velocity at the moment ¢ suddenly to ¥B,, then
the force is not zero at the moment ¢, though the acceleration be
infinite, but it has that value which corresponds with a constant
velocity B,.

It may however be asked what will happen, if the force acting
on an electron with B does not suddenly decrease to zero, but
gradually. SOMMEREELD says about this case only that the sudden fall
to yr, which be expects from a sudden suppression of the force, will
make room for a gradual fall. But as his expectation concerning
the case of a sudden suppression of the force appeared to be inac-
curate, we might suppose that also this expectation will appear not
to be satisfied. The more so because Sommerrenp found a negative
value for the electric mass of an electron moving with B. We might
therefore expect that a decrease of the force would cause an
acceleration. This, however, is not the case, and here we see how risky
it is to introduce the conception of mass in the theory of the motion
of electrons, to which it is essentially strange.

The negative mass, which SommMprernnp ascribes to the electron
means nothing else, but that in order to move with a given %, the
electron requires a greater force when in the active interval the
velocity was on an average greater than B,, a smaller force when
it was less. By active interval is meant the time during which the
electron emitted the fieldforees, which at the moment ¢ act on the
electron. The greater the velocity during the active interval, the
greater the force, and inversely the smaller the velocity the smaller

( 482 )

the foree'). But it does not follow from this that also a greater
force is required if the retardation exists only im the future. On the
contrary when the velocity has decreased to B, < ¥, the velocity
during the active interval has been smaller than ¥, on an average, and
so also the force required will be smaller than that which corresponds
to a constant velocity B,. So with a gradual decrease of the velocity
corresponds a gradual decrease of the foree. The reverse of this
ds.
thesis is not always true: if 5 is a continuous function of ¢ then the
dx
velocity will also vary continuously. If on the other hand r is
discontinuous though & be continuous then y will vary discontinuously.

A diminution of the force is therefore accompanied by a diminution
of the velocity, and inversely. The behaviour of an electron moving
with B corresponds in this respeet with that of a body with a posi-
tive mass. If the force acting on the electron decreases gradually to
zero, the velocity will fall to v.

Though it seems to me that there is no reason to doubt whether
the behaviour of an electron has been deseribed here accurately
though only in general outlines, and though a complete caleulation
of the motion is not practicable in consequence of the great intricacy
of the formulae, I will show in one simple case that the force
required for a given motion agrees with the above description. 1
imagine to that purpose an eleetron which for some time moves
with a constant velocity ¥. At the instant ¢ the motion is suddenly
accelerated with a constant acceleration p. In order to render the
calculation possible we will assume that we may apply the formulae
for quasi stationary motion. We will ealeulate the force at an instant
/ in the first interval *), so ¢>> 7. The calculation does not present
any difficulties, and can be carried out in the way indicated by
Sommperrenp. After introduction of the approximation for the quasi
stationary motion we may everywhere separate the terms as they
would be for a constant velocity X, (we call the sum of these
terms §,) and the supplementary terms which depend on the
acceleration, and whose sum will be denoted by &,. In this way
we find:

1) This rule is given by Somaerrerp though his calculations show that it does
nol hold good with perfect generality. In most cases and also in the present one
it will give a true idea in general ouUimes of the value which the force must
assume,

2) SomMERFELD II] p. 206.

o2ma* _ S2ma* _ 8 ac® pet?
Se ES Ce ae age . as
Say 32 5 oo? 4
xy shy
(r : ie. dy | Cais ]
= a(p d= tak — p — LAL
Tree 2a v(vte), J (2 om da I v* f
0
ry
‘ lp ih v— 2c
— pt — -- 2p + a se dx +- p A 20) pda
vy? du he
0 0
2a 2a
efvte) dg \ 1 na (G3) 9 (em ell
— pt? as r9) 29 + a y diaz +- pt? es a Y gp — dx
2u? da uv vy x av
Ly) Ly
+ six other integrals which are obtained by substituting — ¢ for ¢

and «, for x, in the above.

The signification of the symbols is as follows : @ is the radius of the
spherical electron which is supposed. to be charged with homogeneous
cubic density; ¢ is the charge of the electron, ¢ the velocity of
] ae
20 «

a, is (v- c)t-+-7/, pi and «,—(—c)t-+*/, pt?. In the expres-

sions for «, and wv, the term '/, pl’ may, however, be neglected.

light, 7 the numeric value of B; g the function 2¢—.a +

Without performing the integrations completely we may draw the
following conclusions :

dst. All the terms of 8, contain /¢ as a factor. So we have 5} = 4,
if ¢ vanishes. No sudden increase of the force is therefore required
if the motion is suddenly accelerated, as is the case witha body with
positive mass; neither a sudden diminution of the force as would
be the case with a body with negative mass. The foree remains
unchanged.

2nd, No terms with the first power of ¢ occur in 4,, therefore
ds ds
= = 0. Even the derivative is therefore continuous at the point
dt; =0) dt
=O. This agrees with our remark that a discontinuity in the
derivative of & only occurs if the velocity changes discontinuously.

3", Kor establishing the sign of 4, for ¢= very small, we have
only to take into account the terms with #. There are also terms
with #//¢) but the sum of their coefficients is zero. If we perform
the integrations as far as is required we find :

(Let | eetre) vote

eo —}2apt?
| 3 vu v—ce

( 484 )

+ * representing the force of the field of the electron itself, — %
is the external field required for the motion. So we see that the
sign of the external force X, agrees with that of p, and that
therefore acceleration requires increase, retardation decrease of the
external force.

We conclude that the behaviour of an electron moving with B&B,
though in many respects it differs considerably from that of an
ordinary body, does not show at all that paradoxal character, to which
we should conclude from the expression negative mass. Nothing
prevents us from assuming that electrons really can behave in such
a way. Accordingly I de not see any reason for assuming with
Wien ') that a moving eleetron must suffer a deformation in order
that the possibility of a motion with B, as it requires an infinite amount
of energy, will be precluded.

Finally a remark concerning the series of the emission spec-
tra of elements. The equations of motion of the electron are
integral or functional equations, and may be developed into dilfe-
rential equations of an infinitely high order. An infinite number of
constants oecur accordingly in the solution. If the equations are
linear, these constants represent the amplitudes and phases of har-
monic vibrations; the system may therefore vibrate with an infinite
number of periods *). We are inclined to think that the periods of
the lines of a speetral series are the solutions of sueh an equation.
We have then the ereat advantage that we need not aseribe to the
electron a degree of freedom for each line in the spectrum. A degree
of freedom in the atom is then not required for each line, but only
for each series of lines.

SOMMERFELD tries fo account for the spectral series by means of
the vibrations an electron performs when it is not subjected to
external forces. The periods which he finds, do not agree with those
of light. It seems to me that we might have expected this a priori.
For the vibrations of light are not emitted by isolated electrons but
they are characteristic for atoms or positive ions, and are influenced
by the forces by which the electron is connected to the other parts
of the atoms or ions. But also with the aid of these forces we cannot
account for the spectral series without a much better insight into

}) W. Wien. Uber Elektronen. Vortrag gehalten auf der 77. Versammlung Deut-
scher Naturforscher und Arzte in Meran p. 20.

2) Comp. also these Proceedings March 1900 p. 534. Then however, I thought
erroneously that the solution obtained in this way was different from that, which
I had first developed with the aid of integrals of Fourier.

(485 )

the way in which these forces act, and of the properties of the elec-
tron, than we have as yet obtained. If e.g. we introduce the so
called quasi-elastic force into the equations of motion of the electron,
then this does not bring us any nearer to our aim. In order to show this
we may write the equations for translation of an electron in the
form of a differential equation as Lorentz has done in equation 73
p. 190 of his article “Elektronentheorie” in the Eneyel. der Math.
Wiss. V 14. If we introduce the quasi-elastic foree — fe we may
write the equation as follows:

‘ 1 Ma .

Je ar aol Aa |

da , dita :
AR de + A, Te = |)

As it is only my aim to determine the order of magnitude I have not
determined the coefficients (A, and A, have been determined by Lorentz).
The only thing we have to know is that the order of magnitude of

- I ; ey é Anti a = x
the ratios of two successive coefficients 1s es . The solution
In ¢c
of this equation is «= ce where s is a root of the equation:
fa ans] | Ay 8 A, 8. = 0

This equation has two kinds of roots, namely Is! two roots for
which the other terms are small compared with /—+ A, s?; these
will represent the light vibrations; 2°¢ an infinite number of roots
for which s is so large that f may be neglected compared with the

é

other terms. For these s must be of the order —, and the period of
a

a
the order —. The appearance of the term / has little influence on
a
the value of these roots, the periods of these vibrations are there-
fore nearly independent of the quasi-elastic force, and an isolated
electron might have executed vibrations with nearly the same periods,
We might have expected a priori that we should find periods of the

2a

order : it represents the time required for the propagation of an
¢

electri¢ force over the diameter of the electron. The periods of these

vibrations are of the same order as those of the rotatory vibrations

the periods of which have been accurately calculated in the interesting

treatises of HurrGnorz') and SoMMERFELD.

The lines of the spectral series are not accounted for in this way.
Yet the periods of the rotation and translation vibrations of the
isolated electron must have a physical interpretation. Perhaps we
should see them appear if we sueceeded in forming the spectrum of
RONTGEN radiation.

a) Herciorz, Gott. Nachr., 1903.

( 486 )

Physics. — “Derivation of the fundamental equations of metallic
reflection from Caveny’s theory”. By Prof. R. Stssinen. (Com-
municated by Prof. H. A. Lorentz).

1. It has been pointed out in a previous paper’) that the theories
of metallic reflection drawn up by Caucry, KerreLer and Vorer and that
by Lorentz lead to identical results. It must therefore also be possible in
the theory of Cavcuy to derive the two relations which the three last
theories furnish between index of refraction and coefficient of absorp-
tion for normal and oblique incidence of the light that penetrates
into a metal, the so-called fundamental equations. These fundamental
equations may first be obtained by paying regard to the connection
of the quantities which the theory of Cavucny and the other theories
introduce for the description of the phenomenon. Cavucny determines
the so-called complex angle of refraction 7 by sim 7 —=sini: oe? and

aaa
sin?
cos = oe"). From this follows 1— ——| = oe, so that:
O-e-
67 cos: 2 == OG? cos 2i(e=\\ep)) sins apes eee tO)
G7 Sin 2 T= (07 Grsiniai(Gi-t=1@)| Ist ee et (2)

If we pay regard to the relations between o, + and n, and 4,,

index of vefraction and coefficient of absorption for normal incidence,

iu

and to the equations (17) and (18) of the preceding paper’), the
equations (1) and (2) appear to be nothing but the fundamental
equations, given in equation (6) and (7) of the previous paper.

2. On account of the close connection between the theories of
metallic reflection if must, however, be also possible, to derive these
fundamental equations from Cavcny’s theory without paying attention
to the connection with the others. The fundamental idea of Cavcny’s
theory is the introduction of a complex index of refraction. Denote

this again by », + oh, = cet, so that
PESO CI Wy Le oo 6 6 oc. ((®))
and
SUP == SUB ROE ore 8 ee le op, (Kt)

while we put

0 oe 3 a é e ‘ rs 5 2 2 (5)

Let the V/-plane of a rectangular system of coordinates be the
jane of incidence of the light penetrating into the metal, and the
| |
Zane ine the bounding plane of the metal, the X-axis being directed

1) Sissinau, Riinese Proc. VII p. 377.
2) Loc. cit. p. 385,

( 487 )

from the surrounding medium to the metal. Assume plane waves to
fall on the metal. The vector of light in the refracted ray is then
determined by :

z@ cosr + z-sinr

ao ES GeO es Wek een (G)

A sin 2x

In this 4 is the wave-length in the air’). The phase is determined
with respect to a point in the bounding plane.
With the aid of (4) and (5) equation (6) passes into

to) _}

; t wye + zsimie-F:0
A sin 2% ie ~ Tae (x, + ) Peat | ()
(7) satisfies also the differential equations for the vector of light
in the metal which are supposed homogeneous and linear, if the sine
is replaced by a cosine.
If the are occurring in (7) is called g, also
Acos p —tA sing *)
satisfies.
The light-vector in the metal ean therefore be represented by
—2n1( 7)
Ae—2n4 & ¢ FO ai ete oe oe (eh)
In this:

; ST sinaNn Bacosi Wes ,
a =| Ow sin wW — z sine ~J—+{ ov cosw + z sini —}|—. (9)
Oy) 7) o J:

5 UEBEN Dr ; PSO Ce

b =| oa cos w + 2 sin 1 —— ]}— —| 0 sin w — z sini — —. (10)
Ss ‘ -
Ona OFA

3. From (8) follows, that the planes of equal amplitude are
represented by :
Cla en Oe ate Wee as tei ay (LU)
In this is, according to (9)

5 Re De Mn COSt Talc’
q ——'— S27 2 = $7. ———— as
2
As from (3) follows
Twine COET 35 We, Mave gi —— 0:

1) Lorentz showed that, also when a complex index of refraction is introduced,
at the bounding plane the values of the light vector in the two media harmonize.

Cf. Theorie der terugkaatsing en breking, 1876, p. 160.

*) It appears from § 5 of the previous paper (loc. cit. p. 381), that if the index
of refraction is put my +k, this expression is A cos g £1 A sin 9.

( 488 )

and the planes of equal amplitude run parallel to the bounding
| | | g
plane. This is necessary as it is assumed that the light enters the
metal from the outside.
The planes of equal phase are represented by:
Meena (Cbs 5 A cy 5 0 o (ll)
If we introduce again n,:/, = cotr, then according to (10)
0 cos(t-+-a) :
Dy =~ ky ———— « . ws. s. (18)
2 Sin T
k

—————-
> 64 sint

9 Sime

reer ace en 3.25)

4. Let « be the angle between the normals of the planes of
equal amplitude and phase. The former running parallel to the
bounding plane or the )’Z-plane, @ is the angle of the normal of the
planes of equal phase with the Y-axis. Thus cos @ = p,:Vp.2+4q,"
from (15) and (14):

or if we introduce the values p, and q,

sina
cos @ = Q cos (t - w): [Ye cos* (t + ) te ia owas (15)

From this follows:
sine sin?t
SU recat le cos? (t +- w) + | arent. « ((1k5)
0) + Omen
« being the angle of refraction corresponding to plane waves with
an angle of incidence ¢ (see § 2 of the preceding paper), we get:
1? = sin? 4%) sina = 070° cos (Mt) = sin. |e la)
Let the coefficient of absorption belonging to 2 be 4. Normal to
the planes of equal amplitude the amplitude decreases over a distance
2 in ratio 1 to e—**:4. As q,=0, we get according to (8) and (9) :

2ake 270
s cl !
—-— (n, sin w + &k, cos w)

Bie ae
from which again follows, when cof + is substituted for 7, : /,:
k=k, 9 sin (t 4- w) : sint
or on account of (3):
KONO Pa) weeeo wc to oo o (lls)
5. The fundamental equations follow immediately from the values
found for the index of refraction and the coefficient of absorption.
The equations (17) and (18) lead immediately to:
n? — k? = 6? 9? cos 2 (t + @) + sin? i.
According to (1) the second member of this equation is equal to
*__f,?. In this way the first

o cos 2t or according to (3) to 2,

fundamental equation is obtained.

( +89 )
Further follows from (15), (17) and (18) :

Ly yan = Cy Aa te Se (9)
nk cos & = 5 6* O* sin 2 (t -+- w).
According to (2) the second member is equal to 6? si 27 and so

according to (3) to ,4,, and thus the second equation has also
been derived.

To conclude we may remark, that here the reversed course has
been taken from that by which in the preceding paper the oeeur-
rence of the so-called complex index of refraction was derived from

the two fundamental equations ').

Mathematics. — “A tortuous sur face of order sia and of Jernus

zero in space Sp, of four dimensions.” By Prof. P. HL. Scouts.

1. We begin by putting the following question :

“In space Sp, are given three planes ¢,,@,,@, and in these are
“assumed three projectively related pencils of rays. We demand the
“locus of the common transversal of the triplets of rays corresponding
“to each other.”

Notation. We indicate the vertices of the rays of pencils by
O,,O,, O,, three corresponding rays and their transversal by /,, /,, /,
and /, the points of intersection of / and /,,/,,2, by S,,S,,.
the pencils of rays by (/,), (4), (4). Let further P,,, P,,, P,, indicate
the points of intersection of the planes a,,@,,@, two by two, and «

S, and

the plane P,, P,, P,, which has a line in common with each of
the planes «,, a,, a,, namely with e, the line P,, P,,=a,, with a, the
iheneen= ie —— a, with a, the: line <P.) 'P\. =a,. | We. take ‘for
granted that not one of the three vertices (,, O,, 0, coimeides with
one of the points P.,, P.,, P.,:

2. The answering of the given question offers no more difficulties,
as soon as the locus of poimt S, in @, is known; so we shall first
find this. Hach ray /, of pencil (/,) furnishing a single point JS,, it
is a rational curve, whose degree surpasses the number of times a
transversal / passes through QO, with unity. Now two transversals
/ pass through O,. For the pencil of planes ((, /,) with (0, @,) as
bearing space and Q, VU, as axis marks on the line of intersection
m of (QO, a@,) with a@, a series of points (7) projectively related to
the pencil of rays (/,), from which ensues that there are two rays /,
passing through their corresponding point 7? and that therefore there

1) See loc. cit. § 5.

( 490 )

are two lines O, P? cutting the corresponding line /,, i.e. that two
transversals / pass through @,. So the locus of JS, is a cubic curve
and 4S,
with

s,° having QO, as node, and so we find for the loci of JS,
in a, and «a, in the same way rational curves s,°

3

and s,°

OY, and (V, as nodes.

3. To determine the degree of the scroll of the lines / we first
investigate what this scroll has in common with an arbitrary space
through «@,. Each point @Q lying outside @, which this space has in
common with the scroll gives a line / having two points in common
with that space, therefore lying entirely in that space. So that space
can contain besides s,* only a certain number of generatrices / of
the scroll. As the line of intersection of @, with the assumed space
through «, has three points im common with s,* the number of
eeneratrices to be found is three and the scroll, having a system of
lines of order six in common with the assumed space, must be a
tortuous surface O" of order six. So it is cut by an arbitrary space
according to a twisted curve of order six; this section in general
not degenerating is rational, its points corresponding one by one
to the lines / and therefore to the rays of each of the pencils
(1), (4), (4,). So the surface is of genus zero.

We call the locus just found — however not yet what was meant
in the tithe —- a surface, to show by this that the number of poimts
is twofold infimite; by the predicate “tortuous” we express that it
is not situated in a threedimensional space.

+. By considering the three projective series of points (4,), (4,), (A,)
marked by the three projective pencils of rays (/,), (/,), (/;) On ay, dy, A,
we easily prove that the plane @ contains three generatrices of O°.
For it happens, we know, three times that three corresponding points
), (A,) lie in a
same right line, which then becomes a generatrix / of O°; for, the

A,, A,,A, of the projective series of points (4,), (A,
conics enveloped by the lines A, A, and A, A, connecting each point
A, with the corresponding poimts A, and A, have besides qa, still
three common tangents.

To the rule that the tangents in a point of O° drawn to O* are
situated in a plane, the points of intersection of two non-successive
generatrices / form an exception. In such a point, through which
the surface passes twice, a tangential plane will belong to each of
the two lines /; so it can be called a ‘biplanar node”. From the
above is evident that © possesses six biplanar nodes, the three
points O,, O,, Y, and the three points of intersection of the genera-

(491 )

trices lying in @; moreover we shall see directly that the number
of those nodes is in general six, becoming infinite when it surpasses
six, as takes place in the surface to be considered presently and
which is indicated in the title.

5. We point out the fact, that the found surface O° is determined
by the projective correspondence of the curves s,* and s,* ine, and
a,, and we now show that this correspondence, characterised by the
particularity of the corresponding triplets lying on a, and a,,'is not
the most general one can think of. To that end we take two rational
curves s,* and s,* in two planes @, and «,, which planes for convenience’
sake we assume for the present to be lying in our space, and which
curves with the nodes O, and O, we suppose to be brought into
projective correspondence in the most general manner. Are there
then — we ask — to be found on s,° three collinear points to which
on s,° three likewise collinear points correspond? The answer runs
affirmatively; what is more: each point of s,° forms one time a
part of such a triplet and the bearing lines form a pencil of rays.
If namely the point A, of s,* corresponds to the point A, taken
arbitrarily on s,°, and if the central involution of the points B,, C,
of s,* collinear with A, is represented by (B, C,), the non-central
involution of the corresponding points 6,, C, of s,* by (B, C,) and
the central involution of the points B,', C,' of s,* collinear with A,
by (B,'C;,'), then the two involutions (B, C,), (B,' C,') have a pair
of points in common. If 6,°, C,° is this pair and B,°, C,° on-s,* the
pair corresponding to it, then A,, B,°, C,° and A,, B,°, C,° are two
corresponding collinear triplets. If now Q, is the point of intersection
of two such like lines /,',7," in @, and Q, the point of intersection
of the corresponding lines /,', 7," in «@,,, then the triple involution
(A, B, C,) marked by the lines through Q, in s,° must correspond
to the triple involution (A, 5, C,) marked by the lines through
Q, in s,*, with which we have proved what was asserted above.

With the aid of the preceding it is easy to show in how far the
particularity of the corresponding triplets lying on a, and a, is a
real one or an apparent one. With respect to the planes @, and a,
placed in our space it is evidently an apparent one; for not one time
but an infinite number of times it happens that three collinear points
of s,* correspond to three likewise collinear points of s,*. If the
planes @, and a, are placed in Sp, in such a way that an arbitrary
point P, of «@, coincides with an arbitrary point P, of «,, then
however the three points in which s,* is cut by the line P, Q, will

correspond to three collinear points of s,*, but the line through Q
34

3

2

Proceedings Royal Acad. Amsterdam. Vol. VIII.

( 492 )

bearing the last three points will in general not pass through
P,= P,. So then there are no lines a, and a, to be drawn through
the point of intersection P,, of the planes @, and «, cutting s,° and
s,° in corresponding triplets.

3

6. We shall now consider the more general case of two projectively
related curves s,° and s,* lying in such a way in a, and a, that
through the point of intersection P,, no triplets of corresponding
points bearing lines a, and a, are to be drawn. The argument leading
to the order of the seroll, which is the locus of the line connecting
the corresponding points of those curves, retains here its force. So
we have but to determine the number of nodes. Of course O, and
VY, ave nodes. If farthermore A,A, and 6,4, are two generatrices
cutting each other outside «@, and «,, then A,b, and A,2, pass
through P,,, as they must cut each other. So we consider the central
triple involution (A, 6, C,) marked by the pencil of rays with P,,
as vertex in s,* and the non-central triple involution (A, B, C,) of
the corresponding triplets of s,°; then the latter furnishes as envelope
of the sides of the triangles A, B, C, a definite curve of involution
which makes us acquainted by the number of its tangents through
P,, with the number of nodes not lying in @, and a, of the new
surface O°. Now the class of the indicated curve of involution is
four; for evidently four tangents pass through the node O, of s,*.
If to the two points of s,* coinciding in O, the points J/,, VN, on
s,° correspond, and if P,, MW, and P,, N, cut the curve s,* still in
the point J/,', M," and N,', N,", then the lines connecting P,, with
the corresponding points J/,', J7,", N,', NV," are the only tangents of
the curve of involution passing through P,,. So O° has here also
six nodes.

12

7. It is now easy to see that the first surface O° of the three
projective pencils of rays is found back, if the correspondence of
and s,* is given in such a way that through the point
of intersection P,, of a, and a, lines a, and a, pass bearing two
triplets of corresponding points. The plane a,a, is then again a plane
a through three generatrices of O° and the line a, represents three
of the four tangents to be drawn through P,, to the above found
curve of involution, whilst the fourth tangent causes us to find a
node not lying in «,,@, or «. If we now eut the surface by the
space determined by « and this node, the section will consist of the

3

the curves s,

three generatrices in @ and a curve of order three with a node,
i.e. a rational plane cubic curve. The plane of that curve is then

( 493 )

the plane «,, the node of that curve the point O, of the first generation.

8. We now ask what arises when the planes «¢, and @, are
placed in such a way in Sp,, that the points Q, and Q, coincide and
therefore each line drawn through the point of coincidence P,, in
a, is to be regarded as line a,. We then find that to every line a,
through P,, in «, a definite line a, through P,, in @, corresponds,
so that there is an infinite number of planes e. The locus of these
planes @ is a quadratic conic space with P,, as vertex; for the
pencils of rays of the lines @,,a@, through P,,
other in @,,a@, are evidently projectively related. This quadratic conic
space must contain, as it contains all generatrices of O°, this tortuous
seroll itself.

Now that the generatrices of this particular surface O°, being the
surface indicated in the title, group themselves into triplets lying in
a plane, there must be a locus of nodes. This is of order four. If
namely we project the surface O° by means of the just found qua-
dratic conic space out of P,, on to an arbitrary space not containing
P,,, the projection is a quadratic scroll having the projections of the
planes « as a system of generatrices. Of this surface 0? the projec-
tions of @, and a, form thus two lines of the other system ; for each
of those two planes has a line in common with each of those planes
@ and from this ensues that the sections of a, and @, with the space
of projection must have a point in common with the sections of the

corresponding to each

planes «@ with that space of projection. So in that space of projec-
tion each plane through one of the two lines contains a line of the
system corresponding to the planes « and therefore the projections
of four nodes, namely one on the first line and three on the second.
So the projection of the nodal curve out of P,, on to the assumed
space of projection is a curve of order four lying on Q?, which has
one point in common with each of the generatrices of one system,
and three points with each of the generatrices of the other system.
So the nodal curve itself is a tortuous curve of order four; it is
rational as its projection is.

Considering the surface O* we see at the same time that the sur-
face O° admits of an infinite number of planes cutting it according to
a rational cubic curve, namely each plane through P,, and one of
the lines of the system to which the projections of «, and @, belong.

So we find the following theorem :

“If we assume in two planes «, and «, two projectively related
“rational cubic curves, if in these planes we determine the vertices
“(),, Q, of the corresponding central triple involutions on those

34*

( 494 )

“curves and if now we place those planes in S, in such a way that
“(Q, and (, coincide in the points of intersection ?,, of the planes,
“the locus of the line connecting the pairs of corresponding points of
“the cubic curves forms a tortuous surface with the following properties :

a. “It is projected out of P,, by a quadratic conic space, on
“which two systems of planes are lying;

b. “It is cut by each plane of one system according to a cubie
“eurve with a node, by each plane of the other system according
“to three generatrices ;

ce. “The cubic curves in two planes of the first system have no
“point in common, neither have the triplets of lines in two planes of
“the second systems; each cubic curve, however, is cut by each
“oeneratrix ;

d. “The generatrices cause a mutual projective correspondence
“among all cubic curves and the cubic curves among all the gene-
ratrices.”

9, From the preceding ensues immediately that the tortuous scroll
with a nodal curve 4* can be represented on a plane. If in a plane
6 we assume arbitrarily two pencils of rays with different vertices
T,, T,, and if we allow three arbitrary rays a,, 6,, c, of the former

to correspond to three rational cubic curves so Shy a of O°, three
} )) )

arbitrary rays a,, 6,, c, of the second to three generatrices Ua), (5),
i.) of OF, then to each rational cubie curve Sy) corresponds a definite

ray p, of the first pencil, to each generatrix /,) corresponds a definite
ray g, of the second; so we can assign the point of intersection of

Sb) and /,) to the point of intersection of p, and q,. The elements

of exception of that representation are immediately found. If to the
line connecting the vertices of the pencils of rays counted with the
first pencil the curve s* corresponds and counted with the second pencil
the generatrix /, and if S is the point of intersection of s* and 7,
then to point 7’, corresponds the line /, to point 7, the curve s* and
reversely to point S the line 7, 7,. To each point P of the nodal
curve k correspond two points P', P" of 6 collinear to 7’,, because
in the correspondence of ° to i) the node of Sy) represents two

Dy
different points and two points of J) correspond to this point. As
T, forms part of two representing pairs, the pairs belonging to the
nodes of the generatrix / belonging to 7’, this point is node and
the curve of order four. This is also evident when we consider the rays
of the other pencil. On each ray q lie two points of the curve forming

( 495 )

part of pairs corresponding to the points of the nodal curve lying
on /,, whilst 7, corresponding to the node of the curve s° is
likewise node of the curve. So in 6 to the nodal curve /* corresponds
a curve k!*(7\?, 7,2), having 7, and 7, as nodes and being of genus
unity. And to the rational space sections /° of O° correspond in o
eurves k"(7,, 7,5) through 7, with 7, as triple point, which is
found immediately when we remember that an arbitrary space has
three points in common with each of the rational cubic curves of O°
and one point with each generatrix of O°. As is proper each of
those rational curves 2" (7',, 7’,°) has with the representation /'*(7',*, 7.)
of the nodal curve 4#* hesides 7 and 7’, four pairs of points in
common, corresponding to the four points of the nodal curve lying
in the selected space, whilst two curves 4'*(7’, 7,°) cut each other
besides in 7, and 7’, in six points corresponding to the six points
of intersection of O° with the plane of section of the two spaces.

10. The locus of the bisecants of a tortuous curve of order four
is a curved space of order three having the imdicated curve as nodal
curve. For the twofold infinite number of bisecants furnishes a triple
infinite number of points and three of these lie on an arbitrary right
line /, because the curve projects itself out of 7 on to a plane not
intersecting / as a rational curve of order four and this plane curve
possesses three double points. If we apply this to the nodal curve
kt of O°, taking into consideration that the generatrices of this scroll
are all bisecants of 4‘, we find:

“The tortuous scroll O° with the nodal curve 4* is the complete
section of a quadratic conic space with a curved space of order three,
of which the first passes once, the second twice through 4.”

Whilst the cubic space is the locus of the bisecants of £*, the
quadratic conic space with P,, as vertex is the locus of the planes
containing three points of &* and passing through P,,.

The tortuous surface O° with a nodal curve ** is determined by
this curve and the point P,,. As P,, lies arbitrarily with respect to
k* each tortuous curve /* in Sp, is nodal curve of a fourfold infinite
number of surfaces O°.

11. We observe that the case just considered of the correspon-
dence of the curves s,* and s,*, where a tortuous scroll with a
double curve /* is formed, is not the most particular one that one
ean think of. If for instance — instead of starting from two rational
curves s,° and s,*° taken arbitrarily in «, and «4, — we start by making
the pointfields @, and a, to be in projective correspondence and then

( 496 )

continue to assume two rational curves s,* and s,* corresponding
to each other in this manner, then of course to every three col-
linear points of s,° correspond three likewise collinear points of
s,*, and therefore we can take for the above determined pair of
points Q,, Q, any corresponding pair of points of @,,@,. In this
special case a plane @ through three generatrices will present itself
already for arbitrary position in Sp, and the position, that an infinite
nuunber of those planes present themselves, will be able to be brought
about in a twofold infinite number of different ways; in the last
case however the three generatrices lying in a plane @ pass through
a point, as the series of points lying on the lines of intersection
a,,d, Of this plane with @,,@, are perspectively related, so that the
locus of the nodes becomes a conic instead of a £*. In both cases
surfaces O° are formed differing from the above also in this respect
that they admit not only of a single but of a twofold infinite
number of spaces through three generatrices.

12. Also when we start from two projective rational curves
s,*, s,° in not projectively related fields a great number of special
cases are left for consideration. So the point of intersection P,, of
the planes a@,, @, can lie

a. on one of the curves s’,

6. on both curves s*,

* and correspond to itself,

c. on the two curves s

d. it can be the node of one of the curves s*,

e. it can be the node of one of the curves and lying on the other,

f. it can be the node of one of the curves and forming on the
other part of the two points corresponding to this node,

g. it can be the node of both curves,

h. it can be the node of both curves and in such a way that one
pair of points coinciding in this node has a point in common with
the other,

i. it can be the node of both curves and in such a way that the
pairs of points coinciding in this point correspond to each other.

Of course the number is still inereased if we further permit
the pointfields «, ,@, to be projectively related. We do not wish to
-investigate more closely all these special cases. Neither do we intend
to investigate here the scrolls presenting themselves in both cases
of projective or non-projective pointfields a, ,«, as the locus of the
line P,P, connecting corresponding points P,, P, of other curves
of the same genus and of the same order, which are projectively
related. We only wish to observe that these scrolls will lie in the

( 497 )

case of the projectively related pointfields «,,¢@, on the locus of
the line P,P, connecting corresponding points P,, P, of the planes
which is a quadratic or a cubic space according to the point

Gigs

of intersection ?,, of «, and e, corresponding to itself or not.

13. We conelnde with the deduction of the equations of the above
found cubic and quadratic spaces which have in common the
surface O° with the nodal curve £* and to this end we start from
this curve. If the curve /* is given by the system of equations:

Oe h(E Oa 8) SAN) An Sey (Uk fos AT (1)
— and in this way the simplex of coordinates can always be taken —,
and if the poimt which is the vertex of the quadratic conie space
with respect to that same simplex has the coordinates (¥,, 7, Yas Yas Ya)s
then the equations

1 | hs Ba, Ge Be |
| By BEG. | |
| | &, &, Uy XL,
; ,
vay uy |= OF == 00 5) au (2)
| Yo Yr Ya Ys
Mace cl |

Gays isa Ya
represent those two spaces. We see namely immediately that the
first determinant by insertion of the relations (1° shows three equal
rows, i.e. that the cubie space represented by the first equation must
have the points of the curve £* as nodes, and must thus contain each
bisecant of 4*. Further it is equally clear that the second deter-
minant by insertion of the relations (1) shows two equal rows and
that, when substituting y; for 2;, two pairs of equal rows appear, from
which ensues that the quadratic space represented by the second
equation passes through £* and has a node in y.

A more direct deduction of the equation of the locus of the.
bisecants of the curve 4* was communicated formerly (Proceedings
of the February meeting of 1899 vol. I, page 313). It is founded
on the wellknown lemma, according to which the product of two
matrices 1," and M,”* with + rows and / columns, taken according
to the rows, vanishes identically for 7 >. This same lemma leads
to the deduction of the equation of the locus of the planes con-
taining three points of 4*, and passing through (y,, Y¥,, Y., Ys Ys)» An
arbitrary point P of the plane P, P, P, through the points P,, P,, P,
of £* corresponding to the parametervalues 2,, 2,,2, is represented by

Op —w oy RU pe Atte, Aat (10d 28s A eae et sory (3)

If the plane P, P, P, passes moreover through the given point

Yor Yrr Yoo Yao Ya), Also the relations

( 498 )

oyi=q, Ar tag + 94%, @=—0,1, 2,3, 4) 4% - 27 (4)
hold, and now the equation sought for is found by eliminating the
nine quantities 2,,4,,43, Pi, Ps Pa Vis Jar J, Out of the ten equations
(5) and (4). This takes place by inserting the values given by (3) and (4)
in the left hand member of the second equation (2). For by this we find

’

Ba Coy tease ae | | Le oly pel | jd, Ox) 7
ae iid Uae a | | 4, 4, A, | PA, Pod, PsA,
0° OF | so al == I).
| Yo Yi Ya Ys | | Ae Age Ags | q V Pak)
| |
|
| Fo Ya Ys | a A ase Piacch one |

We considered in the above cited communication equations forming
the extension of the first of the equations (2) to the curve 42" of
the space Spo2,. In connection with this we shall notice that the second
of the equations (2) admits of corresponding extensions, in which
those of the first are included. However, these will be developed
elsewhere.

Mathematics. — “The Pricer equivalents of a cyclic point of a
twisted curve.’ By W. A. Verstuys. (Communicated by Prof.
P. H. ScHourTE.

If a twisted curve C' admits of a higher singularity (eyelie point)
of order n, of rank 7 and of class m, it is to be represented accord-
ing to HatpHen') in the vicinity of this singular point M by the
following developments in series:

C=,
eel tals
2a etrtn {¢],
where [t| represents an arbitrary power series of ¢, starting with a

constant term.
If n, r and m satisfy the conditions that

9 n and r,

2° r and m, Zz
3): nm and r+m, ey)
4° nt+r and m

are mutually prime, then this higher singularity MM (n,7,m) for

1) Bull. d.l. Soc. Mat. d. France t. VI p. 10.

( 499 )

the formulae of Cayiey-PLiicknr and for the genus is equivalent to
the following numbers of ordinary singularities :

n—I cusps ~,

(n—1) (rn+-r—3)

9

a

nodes H,

m—1 stationary planes a,

ae —3 4
(ce logins double planes G, (B)

r—I1 stationary tangents 6,

(r—1) (r+-m—3)
9

double generatrices w,,

(r—1) (r+-n—3)

a

double tangents o,.

For a curve with only ordinary singularities we always have
W, = W,.

If the curve admits of higher singularities, then the tangents in
these singular points will not have to count for as many double
tangents to the curve as they must count for double generatrices of
the developable belonging to the curve. The number @ will then be
different for the formulae of CayLny-PLickrr, relating to a section
and for those formulae relating to a projection, i. 0. w. the singularity
w of a twisted curve appearing in a term (# + @) is not always the
same as the one appearing in the term (y + o).

So the formula

y—« = v—w')
‘is no longer correct as soon as the curve has higher singularities
for which order and class are unequal.

The above as well as the following results do not hold for a
common cusp 8(2, 1,1) and fora common stationary plane e@ (1, 1, 2),
the conditions (A) not being satisfied for these cyclic points.

Through the singular point Jf (n,7,m) pass

n (n+ 2r-++m—4)

9
a

branches of the nodal curve of the developable O belonging to the
curve C.

All these branches touch the curve C in M and have in MM with
the common tangent

(n+7) (n+ 27--m—4)

9
~

coinciding points in common.

1) SALMON. 3 Dim. § 327.

(500 )

These branches have in J the same osculating plane as ( and
with this osculating plane they have in Jf
(n-+-r-+-m) (n+ 27-+-m—A4)
; =

coinciding poimts in common.
From the conditions (A) ensues that (n-+-2r-+-m) is even, so that
the three above numbers are integers.
The second polar surface of O according to an arbitrary point
meets in the point MM (n,7,m) the cuspidal curve
(n+ r—2) (n-+-r-+-m)
times and the nodal curve
n+2r+m—4
“SST pee a (n-+-r—2) (n4+-r+m)

times.
Each point R, where the tangent in J still meets a sheet of

the surface O, counts for
r+ rm—m—r
points of intersection of the nodal curve with the second polar surface.
In the equation of Cremona ') serving to determine 2 (number of
cusps of the nodal curve) we must add for every singular point
M(n,r,m) in the second member of the equation a term

(n-++-r—2) (n+-r-+-m).
In the equation of Cremona’), serving to determine t (number of
triple points of the nodal curve) we must add for every singular
point to the second member of the equation a term

ee Orlane taae)

and for the corresponding points R a term
(n+-2r+-m—A) (r+-m) (r—!).

The decrease of 4 and ¢ arising from the presence of a point
M(n,7,m) is not equal to the decrease of 2 and rt caused by the
ordinary singularities necessary to form a singularity J/ (n,7, m). So
the equivalence of the values expressed in (4) does not extend to

numbers which are found by means of a second polar surface.

Delft, November 1905.

1) Gremona-Curtze, Oberflichen § 104.
2) loc. cit. § 109.

ae)

( 501 )

Astronomy. — “Preliminary Report on the Dutch expedition to
Burgos for the observation of the total solar echpse of
August 30, 1905,” communicated by Prof. H. G. vAN DE
SanpE Baknuyzen, in behalf of the Eclipse Committee.

In March 1904 the Eclipse Committee determined to fit out on
a small scale an expedition to Spain to observe the total solar
eclipse of August 30, 1905. The means for it were found from some
liberal gifts of private persons and of societies (Provinciaal Utrechtsch
Genootschap, Teyler’s Stichting, Utrechtsch Oud-Studentenfonds, Natuur
en Geneeskundig Congres). As observers the same persons were
appointed who had been sent to Sumatra in 1901: Messrs. W. H.
Junius, J. a. Winrrrpink and the undersigned. The observations were
to include the spectrography of the corona and of the sun’s limb
and, provided a fourth observer should offer himself to join as a
volunteer, the radiation of heat of the corona.

A volunteer was soon found in the person of Mr. Mott, assistant
for physics at Utrecht, and so the entire programme could be
worked out.

The outfit of the expedition consisted of:

a siderostat with a coelostat apparatus ;

two slit-spectrographs, to be directed on the coelostat mirror ;
a prismatic camera, to be directed on the northern polar mirror ;
a heat actinometer;

a pyrheliometer ;

a sextant with accessories ;

three chronometers and other auxiliary apparatus.

As the principal instruments were also used for the eclipse of
1901, I refer for the description of them to previous publications
(These Proc. HI p. 529).

The sextant and two of the three chronometers were kindly placed
at our disposal by His Excellency “de Minister van Marine” out of
the collection of instruments at Leyden.

Ox the 13 of August the party arrived at Burgos. This town had
been chosen for the observations not only on account of its favou-
rable situation and other outward advantages, but also because, as
far as was known at the time, it would not be visited by other
expeditions. These advantages were lost through the visit of H.M. the
King of Spain, on which occasion the town council of Burgos organised
a series of festivities which seriously interfered with the astronomical
work. For it is chiefly owing to those feasts that in spite of all

( 502 )

endeavours we could get no assistants from among the educated
inhabitants of Burgos. At last one volunteer was found for the
spectrographie observations, and fortunately on the day of the
eclipse some assistants offered their help; without this help the
measurements of the heat radiation especially would have been
entirely impossible.

The eclipse has been observed under very untoward circumstances.
The station of observation, the hill Lilaila, at 3 kilometers south east
of Burgos (some 18 kilometers north of the central line) was a true
desert of sand where clouds of dust and sand were blown up by
the usually very strong wind, from which the tents, kindly lent us
by the Spanish war administration could only partly protect the
instruments.

Especially the siderostat, which as a matter of course could not
be entirely covered, suffered very much from the sand-storm ; although
it had been cleaned on August 29, the wheelwork did not work
properly on August 30. The piers once being erected, it was impossible
to change the station of observation; moreover, Lilaila offered the
advantage that we could make use of the determinations of time
and geographical coordinates made by the Madrilenian astronomers
in whose camp our instruments were standing.

The weather on the eclipse day was very unfavourable. The 1%
contact could not be observed owing to clouds, and though there
were some bright moments between the 1st and 2°¢ contact, the
observation of totality seemed hopeless. One minute before the
2nd contact the rain ceased, the caps of the siderostat mirror could
be taken off, the clouds broke, and the corona was fairly visible
during 3'/, minutes, sometimes even clearly visible.

Unfortunately totality began 20 seconds earlier than the computa-
tion had predicted, —it seems that also in Algiers and at other places
in Spain a fairly large difference has been stated between observation
and computation — so that the observers were taken by surprise by
the phenomenon, much to the detriment of a smooth carrying out
of the programme.

For a detailed description of the observations I refer to the annexed
papers, which show that the results for some instruments, the very
unfavourable circumstances considered, may be called satisfactory.

At the end of my report I wish to acknowledge thanks to the
Madrilenian astronomers, who hospitably made room for us in their
camp and who were very obliging to us in all respects; to the
Spanish civil and military authorities who kindly allowed us exemp-

(503 )

tion from import-duties and placed some tents at our disposal; and
lastly to the Compania del Norte who forwarded the luggage of the
eclipse party by express at reduced rates.

The Secretary of the Committee
A. A. NIJLAND.
Utrecht, November 1905.

SuppLeMENT I. Measurement of the heat produced by the integral
radiation of the corona and of the solar disk, by
Prof. W. H. Juuius.

The object of our heat observations was, as in 1901, 1s*. to settle
the question of the order of magnitude of the coronal radiation, and
2d, to determine the curve of the total radiation from the first until
the fourth contact, with the aim of deriving from it the distribution
of the radiative power over the solar disk.

The investigation has been carried out with the same actinometer
that had been constructed for the Sumatra eclipse '); in it the rays
are caught directly on a thermopile, without the intervention of lenses
or mirrors. As long as the radiation was sufficiently intense, absolute
determinations with ANnesrrom’s pyrheliometer were also made at
intervals, in order to make sure whether the indications of our
sensitive actinometer might be considered proportional to the received
radiation. Such proving to be the case even for intense radiations,
we were quite justified in assuming proportionality also to exist for
the feeble radiation falling beyond the range of the pyrheliometer.

The astronomers of Madrid had a small house built in the obser-
vation camp; they kindly allowed us to dispose of one of the rooms
for setting up the galvanometer and performing the necessary
laboratory work.

Four persons were required for manipulating the apparatus, two
inside and two outside the room. Mr. W. J. H. Monu, who has also
had a prominent share in the preparation of the observations and
the setting up of the instruments, was in charge of the absolute
measurements and of noting down all the readings together with the
corresponding times. The operations of directing and exposing the
actinometer and the pyrheliometer at signals, given by the observers
inside the room, were performed very punctually by P. Enrurerio
Martinez S. J., phys. prof. at Valladolid, and P. Antonio bE

1) Total Eclipse of the Sun. Reports on the Dutch Expedition to Karang Sago,
Sumatra, N°. 4. Heat Radiation of the Sun during the Eclipse, by W. H. Junius,

( 504 )

Mapariaca, 8. J., theol. prof. at Burgos, to whom we express once
more our sincere thanks for their very valuable assistance. I myself
reeulated the resistance in the circuit of the thermopile, and read the
oalvanometer deflections.

The conditions for measuring radiation were much more favourable
now at Burgos than during the 1901 eclipse at Karang Sago; for
then the phenomenon was permanently veiled by rather thin clouds
of very variable transparency, covering the whole sky; this time
heavy clouds caused the Sun to be indeed absolutely invisible now
and then, but between the epochs of the first and the fourth contact
there were intervals in which the phenomenon showed itself in per-
fectly clear patches of the firmament. The favourable periods have
all been utilized; thus we were able to determine some parts of the
radiation curve very sharply. After the results of the 81 observations
had been plotted down on millimeter paper, we saw that the missing
parts of the curve could be inserted with a fair chance of exactness.

Fortunately the time between second contact and 11 minutes after
third contact was among the favourable periods. This period, however,
had been preceded by full half an hour during which no obser-
vations could be made; and as the rift in the clouds, through which
totality just became visible-in our camp, came quite suddenly, we
were not prepared and lost at least a minute after second contact
in arranging our apparatus for highest sensitivity. Nevertheless we
compared three times the radiation of the corona with that of a
portion of the sky at a distance of about four degrees from the Sun.
The observed deflections were 9, 13 and 33 scale divisions; then a
sudden increase showed that totality was over. The effect produced
by full sunshine corresponded to 1800000 divisions, when reduced
to the same resistance of the circuit. So the smallest effect observed
during the total eclipse was ae of the radiation of the uneclipsed
Sun, or about 7/, of the radiation of the full Moon. This value
must be considered as an upper limit to the radiation
emitted by those parts of the corona, which were not
screened by the Moon at the epoch of central eclipse.
Indeed, the radiation must pass through a minimum about the middle
of totality, and we are not sure that the first of the three obser-
vations above mentioned corresponded exactly to the central position.
Moreover, since a few thin clouds may have traversed the compared
fields, there is some uncertainty left.

An account of the observations made before and after totality and
a copy of the resulting radiation curve will be found in the com-

plete report shortly to be published. The shape of the unscreened
part of the solar disk being known for every moment, and the
corresponding radiation being given by the ordinates of the curve,
we have the data for calculating how the apparent emissive power
increases from the limb unto the centre of the disk. This method
avoids certain sources of error by which the results must be disturbed
when’ the distribution of the energy is measured in an dmage of the
Sun, viz.: the diffusion of rays by the Karth’s atmosphere and by
the optical train, as well as the consequences of variable radiation
emitted by the apparatus. We find a greater difference between the
heat from the limb and that from the central parts, than has been
obtained by the other method. According to our measurements the
decrease of the integral radiation from the centre to the limb follows
nearly the same law that was found by H. C. Vogrn, with the
spectrophotometer, to hold for rays of wave-lengths between 500
and 600 uu.

SuppLeMEnT II. The prismatic camera. By Prof. A. A, NisLanp.

The prismatic camera was mounted above the northern polar
mirror in such a manner that the dispersion direction is almost
perpendicular to the expected crescents of the first and the second
flash.

The programme was as follows:

1st flash: 5 exposures, each of *'/, second on one plate at intervals
of 3 seconds ;

totality: 2 corona exposures, each of one minute and a half, at two
different positions of the instrument so that the plate in the
second position might show a part of the spectrum which
did not occur on the first corona plate.

2°4 flash: 5 exposures in the same manner as those of the 1st flash.

Capett’s spectrum plates were used.

As soon as we found that we could not count upon assistance
of volunteers I had after some training acquired the necessary
skill to carry out this programme, yet I disliked the prospect of
having to do everything entirely by myself. Therefore I gladly
accepted the help of Dr. J. Kapan (from St. Petersburg) who, having
arrived at Burgos on August 29, immediately offered his assistance.
I wish to express here my cordial thanks to him for his skillful aid,

(506 )

The two corona negatives show traces of the corona rings 4 3987
and 4 53808. In consequence of the general cloudiness the plates are
veiled, to which it is undoubtedly owing that the very bright green
corona ring, which visually was so clearly visible, has produced such
a faint image.

The plate taken of the second flash failed entirely because the end
of totality took place 20 seconds earlier than had been computed,
and took me by surprise while I changed the plates.

Also the first flash came 20 seconds before the time computed ;
fortunately through a window in the tent I observed the rapid
approach of totality, and could start the series of 5 exposures long
before the warning sign agreed upon was given. It later appeared
that the second negative has caught about the second contact.

The second negative shows a great variety of details, which in
the ultra violet have suffered so much from absorption that for those
parts of the spectrum the first negative, taken 3 seconds before
totality, forms avery welcome suppiement. These two spectra together
show between 4 470 and 2 367 350 crescents of very different
length and brightness; in the discussion of the meaning of the
observed particulars the three other negatives may also be used to
advantage. This discussion is reserved, however, for a more detailed
report; | only mention that the enigmatic doubling of the flash
crescents of 18 May 1901 can, in the case considered here, occur
only for wave-lengths above y 484. Though from this it follows
that the doubling may be partly due to the slanting position of the
plate the possibility of the existence of double lines in the flash
spectrum is noways excluded by this.

A closer consideration of this question is also reserved for a more
detailed report.

SuppLement ILI. Report on the operations with the two slit-spectro-
graphs for the solar eclipse of August 30,1905 by
J. H. WittTerbink.

Operations at Leiden. The instruments, constructed for the solar
eclipse of 1901 arrived here such a short time before they had to
be sent off to Sumatra that a thorough investigation of them was
then quite out of the question. This has been made now.

As had been decided upon the siderostat provided with a coelostat
apparatus would serve to feed the three spectral apparatus, and in

(507 )

order to render the mounting more simple, the coelostat mirror would
be used for both slit-spectrographs instead of the southern siderostat
mirror.

The Kelipse Committee had consented to an alteration of the
clock-work of the coelosiderostat, so that the number of the wheels
in outside gearing, of which some were very difficult to get at for
cleaning, was reduced from 5 to 2; I had this constructed by
Mr. Gavtmr. The clock-work was received here in the middle of
July 1905. It has worked excellently.

The two spectrographs were carefully examined and cleaned.

I determined the zeros of the micrometer screws, indicating the
slit-width, by means of diffraction observations, a method which
allows of an accuracy of some microns. The adjustment of the slit
in the principal focus of the collimator object glass, which could not
be easily done with the desired accuracy in a direct way, was
indirectly performed in the following manner. By photographs made
according to the method of Hartmann I determined the position of
the photographie plate in the principal focus of the camera object
glass. Then the collimator was placed as a source of light in front
of the camera object glass, and the same photographs were made
again. From the difference between the foeus found now and the
principal focus found before we could derive how much the slit
had to be removed in order to bring it in the principal focus of
the collimator object glass.

In this experiment it appeared however that both the collimator
and the camera object glass of the large spectrograph had a very
great spherical aberration, and with full aperture they were unfit to
form sharp images, while a diaphragm would cause a loss of light
which, with a view to our purpose, was inadmissible.

Therefore I ordered of Srrinuen, new object glasses; a single
object glass with a field of sharp definition of 2° for the collimator
and a compound one with a field of sharp definition of 15° for
the camera.

Neither of them were in store and in the available time this firm
could only supply object glasses of the first kind of which two pieces
were sent tO me.

Although their fields of sharp definition were too small for the
camera I determined to try them also tor this purpose, as the
middle of the spectrum was of chief interest for the photograph
intended. The spherical aberration was exceedingly small.

Meanwhile the Steamboat Company had sent word that every-
thing had to be shipped 10 days earlier than had been agreed upon,

B15)

Proceedings Royal Acad. Amsterdam. Vol. VIII.

(508 )

so that there remained no time for further experiments in Holland.
The object glasses of Zniss of the small spectrograph were found to
be in excellent order, this instrument had produced very fine spectral
photographs. In order to slide the photographic plate for this instru-
ment during the. flash phenomenon I put the clock-work in order
which in Sumatra had served for the motion of the axis carrying
the four photographic cameras. | devised an arrangement which
let slip the cord, fastened to the plate-holder, with greater velocity
than had been required in Sumatra.

As photographic plates I chose, after experiments for comparison
with four different kinds of Scuinussngr’s plates, his “Sternwarte”
and ‘“orthochromatic” plate. At the last moment I fortunately obtained
two kinds of plates of Capnrr, known to be very good.

Operations at the camp. Besides the mounting of the different
instruments, the operations at the camp included therefore also
several experiments, as: tests for comparison of the old and new
object glasses and tests of the German and Euglish plates.

Owing to a delay in the construction of the pier, and because
I had to take charge of two instruments, whereas according to the
original plan of the expedition each spectral apparatus would be
worked by a separate observer, and also owing to the continual
disturbances from the side of the public, these operations did not
get on at the desired speed, so that at the last moment a great
many things remained to be done and the necessary calmness, which
in America in 1900 and in Sumatra in 1901 so much contributed to
regular proceedings, was entirely wanting.

After tests for comparison we chose for object glasses those of
Sremneit, and for plates the English ones, especially as the ortho-
chromatic plates gave a much more regular spectrum than the
German plates of the same kind.

Assistance, so easily obtained at the previous expeditions was
difficult to get here. Though several weeks before we had asked
for it on all sides, a promise of assistance reached us only a few
days before the critical moment. Not much could be expected from
it, but I myself intended to work the small spectrograph with the
sliding plate-holder and I hoped that the very simple operations with
the large spectrograph would offer no difficulties. Jomt rehearsals as
were held for days together in America and Sumatra were quite
out of the question owing to the above mentioned circumstances.
Yet, though all things were so different from what we might wish
them to be, I still hoped to obtain useful results.

‘3

( 509 )

This, however, has not been the case.

The day of the eclipse. Through the unfortunate concurrence of
three entirely different disturbances, where two of them would not
have been able to prevent success, the results of the two slit-spectro-
eraphs have come to nothing.

The first of these disturbances happened as follows. Some hours
before the beginning of the eclipse, the steel band which transfers
the motion of the siderostat axis to the coelostat axis was broken
entirely without my fault through a movement which was altogether
inadmissible for my part of the siderostat, which we used in common.
At one end of this band, which was fixed between two copper plates
by means of two steel screws, the two holes through which these
screws passed were torn up. This effect could not possibly have
been reached by a stress of 100 kilograms. The fact that this happened
instead of the steel band simply sliding over the steel axis was the
best refutation of the often quoted opinion that this way to transfer
motion should not be reliable. The dust, inevitable in an eclipse
cap, naturally heightened the friction of the band on the axis.
Hence it is evident that the disturbance to be mentioned next would
have had no effect on the instrument if it had been in the condition
as it was before the fracture. Fortunately I had spare bands taken
with me and a new one could be put on, which operation, however,
cost three quarters of an hour of our time which began to grow
more and more precious, nor did this incident contribute to the
quietness so indispensable at an eclipse.

During the first part of the first partial phase, as often as the
sun was visible through the clouds, we could control whether the
image of the sun fell on the slit and it could be easily kept there.

Now, however, something happened which in America and in the
Kast Indies would have been utterly impossible, but proved to be
inevitable in Spain as experience had taught me during a fortnight.
During the second half of the first partial phase, a little more than
a quarter of an hour before the critical moment my assistant admitted
several persons near my apparatus. Against this I was altogether
powerless. It seems that one of these unwished for visitors has pushed
against the coelostat mirror and thus disturbed it, from which may
be inferred that the newly fastened band has more or less given way
at its fastening points and the friction on the axis was not sufficient
to prevent disturbance.

Had not over and above — the third disturbance — the sky been
clouded and the sun for the rest of the time been invisible, I should

(510 )

have detected the absence of the image by means of the controlling
telescope and could have removed the mirror into its proper position,
now, however, the absence of any image was quite accounted for
by the clouds. As a few seconds before the beginning of totality
the sun broke through the clouds the absence of the image was
stated and the displacement of the mirror became manifest; yet
without proper assistance it was impossible to put it in order.

(December 21, 1905).

Koninklijke Akademie van Wetenschappen
te Amsterdam.

PROCEEDINGS

OF THE

Pec TION OF SCIENCES

~<2>-—7

WS) a ORAM Cay Waar es

(Qnd PART)

AMSTERDAM,
JOHANNES MULLER.
June 1906.

+

(Translated from: Verslagen van de Gewone Vergaderingen der
Afdeeling van 30 December 1905 tot 27 April

~~

COON TEN TS:

SSS

Proceedings of the Meeting of December 30 1905

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> »

» > » »
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January 27 1906

February 24

March 31

April 27

KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM,

PROCEEDINGS OF THE MEETING

of Saturday December 30, 1905.

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

' Afdeeling van Zaterdag 25 November 1905, Dl. XIV).

CO ENE eeN yy Se Ss

A. F. Hotieman and F. H. vay per Laan: “The bromination of toluene”, p. 512.

A. Wicumany: “On fragments of rocks from the Ardennes found in the diluvium of the
Netherlands North of the Rhine”, p. 518. (With one plate).

H. W. Bakiuis Roozenoom: “The boiling points of saturated solutions in binary systems in
which a compound ocenrs”, p. 536.

P. van Rompvurcu and W. van Dorssen: ‘The reduction of acraldehyde and some derivatives
of 3. divinylglycol (3. 4 dihydroxy 1. 5 hexadiene)”, p. 541.

P. van Romeurcn and N. Il. Conky: “The occurrence of -amyrine acetate in some varieties
of gutta percha’, p. 544.

W. Karreyn: “The quotient of two successive Bessel functions”, p. 547.

J. P. van per Stok: “On frequency curves of barometric heights”, p. 549.

L. J. J. Muskens: “Anatomical research about cerebellar connections”. (Communicated by
Prof. C. Wiykier), p. 563.

P. van Rompuren and W. van Dorssen: “On the simplest hydrocarbon with two conjugated
systems of double bonds, 1. 3. 5 hexatriene”, p. 565

A. Sits: “On the hidden equilibria in the p,x-sections below the eutectic point”. (Commu-
nicated by Prof. H. W. Baxkiuis Roozesoom), p. 568. (With one plate).

A. Smits: “On-the phenomena which occur when the plaitpoint curve meets the three phase
line of a dissociating binary compound”. (Communicated by Prof. H.W. Baxnuis RoozeBoom),
p. 571. (With one plate).

J. J. van Laar: “On the course of the spinodal and the plaitpoint lines for binary mixtures
of normal substances”. 3rd Communication. (Communicated by Prof. H. A. Lorenrz), p: 578.
(With one plate).

H. A. Lorestz: “The absorption and emission lines of gaseous bodies”, p. 591.

Proceedings Royal Acad. Amsterdam. Vol. VIII.

(7512 )

Chemistry. — “The bromination of toluene’. By Prof. A. F.

Hotueman and Dr. F. H. van prr LAAN.

(Communicated in the meeting of October 28, 1905).

In the reaction between toluene and bromine we have a_ striking
example of the influence exerted on the nature of the product of
reaction by experimental conditions. About this the following is
known:

1. Influence of temperature. In the dark and at a low tempera-
ture there is formed a mixture of bromotoluenes; on the other hand
benzyl bromide is formed at the boiling point of toluene.

2. Influence of light. At a low temperature benzyl bromide is
exclusively formed; the same takes place at the boiling temperature.

3. Influence of catalyzers. Through their action the bromination
takes place exclusively in the core, even in full daylight and at
an elevated temperature.

If we make a closer study of the papers which have appeared
as to this reaction it strikes us, as In so many other cases, that the
virtually known suffers from much uncertainty owing to an insuffi-
cient observance ot the quantitative proportions. When, for instance,
SCHRAMM states that on bromination in sunlight benzyl bromide is
exclusively formed, a doubt arises as to the correctness of this view,
as the only proof he adduces is that the boiling point of his product
lies at 195°—205°; his boiling point limits are therefore so wide
apart that they suggest rather the presence than the total absence
of isomers. As regards the bromotoluenes formed in the reaction,
it was known that these are ortho- and para-bromotoluene. But the
question, in what proportion those are formed under the influence
of the three above factors, has only been made the subject of greatly
varying conjectures and rough estimates. Nothing was known also
as to the nature of the products of reaction which are formed in
the dark at temperatures between the ordinary and the boiling point
of toluene (110°).

There was, therefore, every reason to again study this interesting
reaction and to try to solve the following questions :

In how far is the composition of the reaction-mixture dependent
1. on the temperature; 2. on the action of light; 3. on the presence
of catalyzers.

In my laboratory, first at Groningen, afterwards at Amsterdam,
Dr. vaN ber Laan has made a contribution to the resolution of these
questions by means of a careful experimental research ; he commenced

( 513 )

by making sure of the absolute purity of his toluene and bromine
by means of special methods of purifying; for details his dissertation
and his paper in the ‘Recueil’ (next appearing) should be consulted.

As the composition of the reaction mixture consisting of ortho-
parabromotoluene and benzyl bromide had to be determined, but as
no method for this was available, it was necessary to work out a
suitable process; in order to do this it was necessary to first possess
the three said substances in a chemically pure state so as to be able
to make artificial mixtures for testing the analytical methods.

The preparation of parabromotoluene and of benzyl! chloride presented
no difficulties. The first substance was obtained from paratoluidine
by diazotation, and as this is a solid it could be readily freed from
any adhering traces of its isomers by reerystallisation from ligroin
and thus yield a parabromotoluene also free from its isomers. Benzyl
bromide was made from benzyl alcohol and hydrobromic acid. On
the other hand the preparation of pure orthobromotoluene was not
so easy. This was also prepared from the corresponding toluidine,
but the difficulty was to obtain the latter in a pure condition. This
was overcome in the manner previously communicated (These Proc.
VII p. 395).

In the actual investigation a large excess of toluene was always
taken (8 mols. toluene to 1 mol. of bromine) so as to avoid for certain
the formation of higher substitution products. Besides the three above
mentioned substances the reaction mixture contains, therefore, a large
quantity of toluene: hydrogen bromide is also present and often
also a small quantity of free bromine, especially in the reactions
which were executed in the dark. This reaction product was now
analysed quantitatively as follows: A slow current of air removed
almost quantitatively the hydrogen bromide, which was absorbed in
water and titrated: the quantity thus found is equivalent to and
serves as a measure for the brominated products. In order to free
the liquid from any free bromine, and to determine the amount of
the same, it is poured into a solution of potassium iodide and the
liberated iodine titrated with thiosulphate. The liquid is now washed
with water, dried, and the toluene is distilled off in an airbath
heated by boiling amyl alcohol. On taking the sp. gr. of the distilled
toluene it appeared that this had not carried over any brominated
products to speak of.

After these operations the liquid now only consisted of the bromo-
toluenes and benzyl bromide besides also a small quantity of toluene.
In this mixture the benzyl bromide can be estimated by means of
alcoholic silver nitrate which yields silver bromide quantitatively.

36*

( 514 )

In order to determine in what proportion ortho- and parabromo-
toluene are present, it was necessary to remove the benzyl bromide
from the mixture. This was done by bringing it into contact with
dimethylaniline. There is then formed quantitatively an ammonium
bromide, the bulk of which is deposited as a crystalline mass. By
washing the residual liquid with very dilute nitrie acid the excess
of dimethyl aniline and the still dissolved ammonium bromide
are removed so that we obtain finally a liquid consisting merely of
the bromotoluenes. When dried and distilled in vacuo it is ready
for the determination of the isomers. This was done by determining
the solidifying point of this purified liquid. By means of the solidi-
fying point curve previously constructed by Dr. vax pur Laan, the
composition of the mixture could be at once ascertained from the
said point. By the analysis of a series of made up mixtures he was
satisfied that this method of analysis gives results accurate within
about 1 percent and is therefore sufficiently accurate for the purpose
intended.

With the aid of the method described Dr. vax pur Laan obtained
the following results.

1. Influence of temperature. The flask containing the mixture of
bromine and toluene was kept carefully in the dark. Observations
were made at 25°, 50°, 75° and 100°. At 25° the reaction took
place very slowly and even after a week the bromine had not alto-
gether disappeared. At 50° this was already the case in 3 days.
The subjoined table contains the analyses of the reaction mixtures.
The figures given are each the mean of 3 or 4 concordant determinations.

From this it appears that in the dark a regular increase of the
benzyl bromide content takes place with a rising temperature. From
a graphical extrapolation it appears that benzyl bromide is no longer
formed below 17°, but, on the other hand, above 88° it is the sole
reaction product. These conclusions, however, must still be confirmed
experimentally. The proportion in: which ortho- and parabromotoluene
are formed also alters somewhat in favour of the first-named isomer.
A determination of the sp. gr. of the mixture showed that this does
not contain any of the higher brominated substances. The mixtnre
obtained at 25° had a sp.gr. of 1.3598 at 64°.6 whilst a mixture of
the two isomers in the same proportion shows a sp. gr. of 1.3598
at 64°.

2. Influence of light. As already observed, Scuramm claims to
have obtained exclusively benzylbromide when brominating at low
temperature in full sunlight, although his experimental data create

( 515 )
TABLE I

fang | i Ree
|Composition of the mixture

Composition of the brominatingproduct |
|
Su) ortho para benzyl bromide ortho ++ para
bromotoluene | bromotoluene
25 | B56) | 53.9 | 10.6 Oe | 60.3
SOR ie 23.5 | 82.8 43.17 44.8 58.2
75 B05 =| fe5s | 86.3 45.3 | 54.7
100 — — 100 = 283

some doubi about this. In diffuse daylight ortho- and parabromo-
toluene are also formed according to him; ErpmMann, on the other
hand, stated that benzylbromide is the sole product. The observations
of Dr. Vay per Laan confirm those of Erpmann. In diffuse daylight
the bromination proeeeds very rapidly at 25°; in about 10 minutes
all the bromine has disappeared. The analysis of the product gave
99°/, of benzylbromide. From this it follows that pure benzylbromide
can be readily prepared in this manner. Brinstrr, who attempted
this previously, arrived at the opposite result. This, however, was
caused by the fact that he exposed to the light a mixture of bromine
and toluene in equivalent proportions at the temperature of boiling
toluene. Operating in this manner we obtain indeed a product with-
ont a constant boiling point which on fractionation appears to con-
tain products boiling at higher temperatures. If, however, working
in the light and at LOO°, only one mol. of bromine is used for 10
mols. of toluene, the formation of these higher boiling substances. is
prevented. The excess of toluene is readily removed by distillation,
After a distillation in vacuo Dr. Vax per Laan obtained a produet
solidifving at — 4°.3 of a sp. er. 1.8887 at 65°.5° whilst these con-
stants, aceording to his observations, are 3°.9 and 1.3858 at 65°.5
for pure benzylbromide from benzylaleohol. The benzylbromide thus
prepared contains, therefore, less than 0.5"/, of impurities.

3. Injluence of Catalyzers. As the influence of light is, as we
have seen, very great, all the experiments with catalyzers were made
in complete darkness. Of these were tested: antimonytribromide,
aluminiumbromide, ferricbromide and phosphorustribromide. Of the
first three it is stated that they favour the bromination in the core,
of the latter that it accelerates the formation of benzylbromide. The
observations of Dr, Van per Laan are in harmony with this. The

temperature at which the reaction was tried was 50°, and theaction
of the catalyzers was determined in such a manner that increasing
quantities of them were added and the composition of the reaction
product “determined each time.

A feeble catalyzer was found in antimony tribromide as shown in
the subjoined_ table.

ACS ali tl eae
Temp. 50°; 50 ¢M.* toluene + 3 eM.* bromine. Dark.

Mol SbBr, | aietnee oe es Composition of the brominatingproduct
on 4 mol Bry ortho para ortho | para | benzyl bromide
| bromotoluene bromotoluene |
0:00: | 4.82 {i iiesre 93.5 30/8) =| 43.7
0.0017 | Zo.) aleneiserg 29.4 33.4. | a)
0.0084 | 38.9 | 64.4 4.0 37.8 38.2
O.016,_.| | S83. le metey 26.0 12.0 32.0
0.034 38.9 | 61.4 28 .0 44 A 27.9
| |
0/080) eu = = at 18.7

The quantity of benzylbromide diminishes with increasing quan-
tities of the catalyzer but is not inversely proportional; the decrease
is much less. The proportion of ortho- and parabromotoluene under-
goes but a slight modification.

Aluminiumbromide, however, acts very energetically, as very small
quantities prevent the formation of benzylbromide. The experiments
were conducted by adding a little aluminiumpowder to the mixture
of toluene and bromine, thereby converting it rapidly into” the
bromide. The following figures were obtained :

TB ei ee

Temp. 50°; 50 cM.* toluene + 2.5 ¢.M.* bromine. Dark.

Composition of the

Mol AlBr, Benzyl- mixture.
oni mol Br, | bromide ortho para
| | bromotoluene
as
0 43.7 | 4A.8 58.2
OV002) =| 23a Su Sh Or iees6n
|
0,004 0.5 (?) 44.6 55.4
0.006 0 44.3 55.7
0.017 | 0 | 2 #0 <| 2 som

( 547 )

Whereas SbBr, modifies the proportion of ortho-para slightly in
favour of the para there is present here a much stronger influence
of AlBr, in favour of the ortho.

Particularly interesting here is the influence on the amount of
benzyl bromide. Although with only 0.002 mol. no modification of
those proportions is perceptible, this becomes so pronounced with
double the quantity that practically no more benzyl bromide is
formed. This result is very striking and deserves a closer study.

With ferrie bromide this phenomenon was repeated; this appeared
to be a still more powerful catalyzer than aluminium bromide, as
the limit of its activity is situated still considerably lower as may
be seen from the subjoined table:

I ZX I by a Ae

Temp. 50°; 50 cM.* toluene + 2.5 eM.* bromine. Dark.

Composition of the
Mol Fe Br, Benzyl- mixture:
oni mol Br,| bromide ortho para

bromotoluene

—

() 43.7 418 58.2
0.0007 AQ 8 36.9 63.4
0.004 7.8 — _
0.002 0 36.0 64.0
0.006 . 0 37.9 62.4
0.01 | 0 37.0 63.0

Here, the quantity of ortho is again depressed by the catalyzer.

With phosphorus trichloride as catalyzer Dr. vax per Laan has
only made one experiment which, in accordance with Erpmayn’s
investigatfon, gave an increase in the amount of benzyl bromide.

MAS Beli, EV.

Temp. 51°; 50 cM.* toluene + 3 cM.’ brqmine. Dark.

Mol P Br, Benzyl- Bromotoluenes
on4dmolBr, bromide Sethe para
——$—$—$—$ $ ____—

0 AD 4 41.8 58.2
0.02 D4.7 AL 4 58.6

( 518 )

The quantity of benzyl bromide has therefore, much increased
but the proportion ortho-para has kept fairly well unaltered.

For further particulars as to these researches VAN DER LAAN’s
original dissertation should be consulted. An article by him on
this subject will also appear, shortly, in the “Recueil”.

Amsterdam, Sept. ’05. Chemical Laboratory of the University.
Geology. — “On fragments of rocks from the Ardennes found in

the Diluvium of the Netherlands North of the Rhine.” By
Prof. A. WICHMANN.

(Communicated in the meeting of November 25, 1905)

Ever since the 18 Century, the attention of geologists has been
drawn to the boulders scattered about our heathgrounds and in opposition
to the various and oftentimes curious theories started to account for
their presence there, A. Vosmarr then already expressed the opinion
that they had been transported from elsewhere by “A Mighty Flood”. *)
A little later, A. Brugmans?) and after him $. J. Brvg@Mans *) pointed
to Scandinavia as the original home of these erratics ; but this view,
though shared by a few other scientists, was not generally adopted
until after the publication of J. F. L. Hausmann’s treatise *). It seemed
then as if the only question still remaining to be solved, was in what
way and by what road this transport had been affected. Little or
no thought was given to the possibility that other countries might
be also accountable for their origin.

It was not until 1844 that W. C. H. Srarine, whilst investigating
the nature of these boulders, discovered that at least those composed
of sandstone and quartzite, were found as well in the Ardennes, in
the districts of Berg and Mark, at the foot of the Harz Mountains

1) Jonwannes van Lier. Oudheidkundige brieven, bevattende eene verhandelin z
over de manier van Begraven, en over de Lijkbussen, Wapenen, Veld- en- Eere-
teekens der oude Germaner. Uitgegeyen.... door A. Vosmaer. ’s-Gravenhage 1760,
p. XV, 10, 11, 103.

2) Sermo publicus, de monumentis variarum mutatationum, quas Belgii foederati
solum aliquando passum fuit. Verhandelingen ter nasporinge van de Welten en
Gesteldheid onzes Vaderlands. I. Groningen 1773, p. 504, 508.

8) Lithologia Groningana. Groningae 1781. Preface p. 2, 3.

') Verkandelingen over den oorsprong der Graniet en andere primitieve Rots-
blokken, die over de vlakten der Nederlanden en yan het Noordelyk Duitschland
verspreid liggen. Natuurk. Verhandelingen der Hollandsche Maatsch. van Wetensch.
XIX, Haarlem 1831, p. 341—349,

as in Scandinavia’). It is to be noted that on his first geological map
these diluvial beds are not marked out in separate divisions *).

Two years later, however, his attention was arrested by the pecu-
liarity that, while in Twente and in the Eastern part of Salland and
probably over the whole extent of the Veluwe, the principal con-
stituents of these erratics were quartzite, red or blackish jasper, near
the Havelter hill, before Steenwijk when one comes from the side
of Meppel, one suddenly finds the detritus to consist entirely of
flints. He noticed the same phenomenon near Steenwijk, the Steen-
wijkerwold and even near Vollenhove *). These facts led him to con-
clude that two distinet diluvial deposits had taken place, i.e. one
of. “siliceous material” transported from the Baltic and another
“composed of quartz” derived from the Ardennes.

In 1854 Srarinc had modified his theories. To the siliceous for-
mation he gave the name of “Scandinavian Diluvium’’, and the
quartz, which he no longer regarded as derived from the Ardennes,
received the appellation of ‘“Diluvium of the Rhine’, which also
included the deposits. between the Meuse and the Rhine; and the
beds situated South of the river Lek received the name of Diluvium
of the Meuse. He was careful to add however that: “it would be
wrong to deduce from these appellations that Scandinavia alone was
responsible for the diluvial formation in the North of Holland, and
the Ardennes, or- the mountains of what at present is known as the
basin of the Meuse, for that of one of its Southern parts and the
Rhine for that of the other.” *)

Six years later STARING again proposed another division which he
then considered decisive. Leaving the boundaries of the Scandinavian
Diluvium and those of the Meuse unaltered, the limits of the diluvium
of the Rhine were confined to those parts of the Netherlands lying
between the Rhine and the Meuse. The formation North of the Rhine
and South of the Vecht was indicated by the name of ‘mixed diln-
vium” *), which therefore included the provinces of Overijsel, Guelders,
Utrecht, and the district of the Gooi in North. Holland. The charac-
teristic feature of this diluvium is the presence of erratics from

1) De Aardkunde en de Landbouw in Nederland. Zwolle 1844, p. 14.

*) Proef eener geologische kaart van de Nederlanden. Groningen 1544.

3) De Aardkunde van Salland en het Land van Vollenhove. Zwolle 1846,
Ped; 9) Do:

*) Het eiland Urk volgens den Hoogleeraar Harriye en het Nederlandsche dilu-
vium. Verhandel. uitgegeven door de Commissie belast met het vervaardigen eener
geologische kaart van Nederland. Il. Haarlem 1854, p. 167 m. kaart.

5) De Bodem van Nederland. Il. Haarlem 1860, p, 54—96, PI. L.

(590 )

Seandinavia, from Hanover, from the mountains along the banks of
the Rhine and from the Ardennes; but Srarinc was unable accurately
to define which erraties had been transported by the Rhine and
which by the Meuse.

“By far the largest portion of the quartzites, sandstones, pudding-
“stones and slates, found in those parts of the diluvium, which are
“situated to the South of the Scandinavian drift, are derived from the
“Devonian strata of the Rhine and the Ardennes.” ') Neither did
STARING succeed in proving that the erratics in the diluvium of the
Meuse had originally come from the Ardennes. “The gravel and the
“flints of the Meuse are similar to those of the Veluwe, with the
“important difference, however, that no fragments of plutonic rocks
“are found among them.’’’)

Although for the last ten years the erratics transported from the
North of Europe have been the subject of much careful investigation,
little interest has been bestowed on those derived from Southern parts.
This neglect is due in a great measure to the very nature of those
rocks. The first actual proof that detritus from the Ardennes has
been carried North of the Rhine, was supplied by J. Lorm when
he discovered a Rhynchonella Thurmanni near Wageningen*); but
until now searcely any further progress has been made in the study
of this question.

The difficulty of tracing to their original home the boulders trans-
ported from the Ardennes, lies in the first place in the necessity of
leaving out of consideration, fragments of those rocks which are
represented both in the diluvium of the Rhine and in that of the
Meuse, for it is impossible to determine the exact districts to whieh
they originally belonged. In the second place, it is a well known
fact that the greater part of the Ardennes is very poor in fossils,
so that the chance of finding fossiliferous specimens among the diluvial
erratics is almost nil; — and thirdly, some of the very characteristic
rocks, e.g. the phyllites, are much too soft to offer adequate resis-
tance to the accidents of transportation. However, as I hope to show
in the following pages sufficient material from various formations

1) Wicsapeois

2) 1. cp. 96.

8) Contributions & la géologie des Pays-Bas. Archives Teyler (2) IIIf. Haarlem
1887, p. 80.

Postscript: Frrp. Roemer has already mentioned silicified specimens of Stepha-
noceras coronatum, found in the boulders near Winterswyk, Guelders. (N. Jahrb.
f. Min. 1854, p. 322, 323). These looked exactly like those occurring in the jurassic
layers of Northern France. See also Cl. Schliiter in Zeitschr. d. D. geolog. Ges.
XLIX. 1897, f. 486.

(aovt )

remains to prove that the erratics traceable to the Ardennes may
claim a considerable share in the formation of the mixed diluvium ').

Cambrian system. The principal part of the Ardennes is built
up of layers belonging to the Cambrian system, which A.” Dumont
originally sub-divided into three groups, namely Devillian,'Revinian
and Salmian*). The Devillian and Revinian systems were’afterwards
united by J. GossELeT,*) into one series, called the devillo-revinian,
which consists of phyllites, alternating with bands of greyish black
and dark bluish grey quartzites. These layers may be seen exposed
principally near Revin and Deville, on the banks of the Meuse, near
Roeroi and Stavelot, and also near Givonne, to the north of Sedan. *)
These quartzites are often crossed in various directions by fine veins

of quartz and — a distinetive feature by which they are easily
recognized — they often contain small eubes of pyrite, which in

some cases has been in a greater or lesser degree changed into
hydroxyde of iron. Now and then specimens are found in which
the orginal mineral has entirely disappeared, only the impression of the
cubes being left. J. pe Winpt*) has given microscopical descriptions
of these crystalline quartzites, but has omitted to mention one
special characteristic in which they show great conformity with the
phyllites. In reference to the latter, E. Gurirz was the first to point
out that the enclosed crystals of magnetite and pyrite are sur-
rounded by a zone of quartz, thus forming elongated lenses. *)
From the manner in which these minerals have grown together,
as well as the chlorite, he was led to the conelusion that thev
were coeval. This theory has been refuted by A. Renarp. Although,
with Grinitz, he believes the magnetites and pyrites to have
been formed at the same time as the mass of the rocks, he

}) In all probability this share will be found to be much larger than is thought
at present, because a great many rocky fragments, among others qnartzites and
sandstones, are now ascribed to the diluvium of the Rhine although they are also
present in that of the Meuse.

2) Mémoire sur les terrains Ardennais et Rhénan-Mémoires de l’Acad.-roy. de
Belgique XX. Bruxelles 1847, p. 8.

8) Esquisse géologique du nord de la France. Lille, 1880, p. 19.

4) It cannot be made out which of these localities have provided the boulders.
They are represented in the accompanying map as if they were coming from
Revin, the chief locality.

5) Sur les relations lithologiques entre les roches considérées comme cambrien-
nes des massifs de Rocroi, du Brabant et de Stavelot. Mém. cour. de l|’Acad. roy.
de Belgique LVI. Bruxelles 1898, p. 21, 68.

8) Der Phyllit von Rimognes in den Ardennen. ‘TscHermak’s Mineralog. und
Petrogr. Mitthlg. III]. Wien. 1880, p. 533.

considers the zone of quartz surrounding these minerals to be of
secondary origin, and that pressure on both sides had caused cavities

*) Some time

which afterwards have been filled up with quartz.
before, A. Davprén had already furnished a description of trans-
formed erystals of pyrites found near Rimognes.*) The studies of
other kinds of rocks led to the same conelusion.*) An analysis of
the pyritiferous quartzites of the Cambrian system affords still better
proof of the secondary origin of this quartz, because in this ease
the rock itself is composed of this mineral. When examining specimens,
it is easy to observe the sharp contrast between the two formations.
The quartz which has formed itself around the pyrite, is clear and
transparent, seldom contains enclosures, and is built up of
fibres which stand perpendicular on the erystals of pyrite. The
same structure is seen in the parts which form the veins. L. DE
Dorponor, who has written on the same subject, is inclined to regard
this quartz as chalcedony. *)

By the aid of this data it has not been difficult to prove that
erratics of this kind have been widely dispersed, and it is very
probable that in the course of time their presence will be signalized
from many other places besides those we here indicate.

1. Province Utrecht: Railway cutting near Rhenen, on the river
Lek, Darthuizer Berg, sandpit to the North of Rijsenburg, railway
cutting at Maarn, the heath near the pyramid of Austerlitz, near
Zeist, Heidebosch near the House ter Heide, between the stations
de Bilt and Zeist, to the rear of Houderinge near de Bilt, Soester Berg.

2. Province of North-Holland: Hilversum and the sandy tract to
the North of Larenberg.

3. Provinee of Guelders: Heath near Epe, Bennekom near Wage-
ningen, Eerbeek near Dieren, at several places around Eibergen
Boreulo, Groenlo and Hettenheuvel near Doetichem.

4. Province of Overijsel: Heriker Berg near Markelo.

1) Recherches sur la composition et la structure des phyllades ardennais. Bull.
du Musée roy. d’hist. nat. de Belgique. Il. Bruxelles 1883, p. 154—135.

2) Etudes synthétiques de géologie expérimentale. 1. Paris 1879, p. 443.

8) H. Lorerz. Ueber Transversalschieferung und verwandte Erscheinungen im
thiiringischen Schiefergebirge. Jahrbuch der k. preuss. geolog. Landesanstalt ftir
1881. Berlin 1882, p. 283 —289.

Hans Reuscn. Bimmeléen og Karméen met omgivelser. Kristiania 1888, p. 69, 70.

Aurr. Harker. On “Eyes” of Pyrites and other Minerals in Slate. Geolog.
Magazine (3) VI. London 1889, p. 396, 397.

4) Quelques observations sur les cubes de pyrite des quartzites reviniens. Anu.
Soc. géolog. de Belgique. XXXI. Liége 1903—04. Mém. p. 5085.

( 528 )

It stands to reason that erraties of this type must be more plentiful
still in the district South of the Rhine; in fact, similar quartzites
have been found in the diluvium of the Meuse for a long time past.
In the Province of Limburg they are looked upon as tle most com-
mon kind of erratics. Anpn. Ernxs came across one 3 M. high, 2.6 M.
long and 0.6 M. broad '). According to this author, they are also
found in quantities in the Province of North Brabant, although they
are not so large as those of Limburg. J. Lorimé found rocks of this
composition on the heaths at Mook and at Schaik, also in South Holland
on the beach of Springer in Goedereede and near Rockanje in the
island of Voorne.

“Porphyroids.” But the most conclusive proofs that immense quan-
tities of rocky fragments must have been transported from the Ardennes,
are furnished by the so-called Porphyroids. This rocky formation is
confined to the districts of Revin and Deville, where, more particu-
larly in the neighbourhood of Laifour and Mairus, they form
dikes from 0.1 to 20 M. wide, corresponding to the layers of the
devillo-revinian group. At present only 17 places are known where
this exceedingly characteristic formation *) may be encountered.
Dispersed in a bluish gray or greyish groundmass, may be seen
porphyritic crystals of bluish quartz and of feldspar. Owing to
their peculiar position and their schistose structure, many geologists
have classified these rocks among the series of crystalline
schists, — whilst others have ascribed to them an eruptive origin.
Cu. DE LA VaLike Poussin and A Renarp, who have given the
most detailed description of these rocks, favoured the former view ‘);
Barros, Dauprin, Gosspiet, von Lasaunx and others, on the con-
trary, justly considered them to be quartzporphyry, an opinion which
A. Renarp also finally accepted.

Although these porphyroids can have but a minimum share in the
formation of the Ardennes, they are frequently met with in diluvial
deposits. In Belgium, G. Duwargun only noticed them near Liege *),
which proves that but little attention has been paid to them in that

1) Recherches sur les formations diluviennes du sud des Pays-Bas. Archives
Teyler (2) Ill. Geme partie. Haarlem 1891, p. 23.

2) J. Gosseter. L’Ardenne. Paris 1888, p. 86.

*) Mémoire sur les caractéres mineralogiques et stratigraphiques des roches dites
plutoniennes de la Belgique. Mémoires cour. etc. de Acad. roy. de Belgique XL.
Bruxelles 1876, p. 237—247 (also Zeitschr. d. D. geol. Ges. 1876, p. 750—769).

4) Prodrome d’une description géologique de la Belgique. Bruxelles et Liége
1868, p. 237.

country '), for ApH. Erens mentions not less than 15 gravel-pits in
the neighbourhood of Maastricht in which he found fragments of
these rocks, one being 0.6 M. long and 0.5 M. thick. The most
sasterly place of deposit known at present is Simpelveld’). Not long
ago, Mr. L. Rurren brought me several specimens dug up in the
neighbourhood of Sittard. From observations of Erens, it would appear
that these erratics are scarce in the Province of North Brabant. He
himself found a nice piece at Mook *), and J. Lori a fragment
between Bladel and Postel.

North of the Rhine they have been discovered in the railway cuttings
near Rhenen and also near Maarn (in the latter locality the fragment
'/, M. in diameter), and on the Soester Berg, in the Province
of Utrecht. Another piece was found near Hibergen, in Guelders and
finally Krens mentions having seen in the Geological Museum, at
Leiden, a fragment found in Overijsel: unfortunately he does not
state the exact spot at which it was found ‘).

was over

2. Carboniferous system. Furp. Roemer has given a description
of a few fragments of black carboniferous limestones containing
Productus striatus Fisch. found in the Gooi, near Hilversum and
sent to him for analysis by Srarinc. He came to the conclusion that
they were derived from the carboniferous limestone of the district
between Aix-la-Chapelle and Stolberg °).

SrarInG on the contrary believed them to have been transported
from Visé on the Meuse, in Belgium, and based his opinion on the
similarity of their composition with the limestone found in that part
and also on the almost total absence of this rock from Westphalia.)
Although fragments of carboniferous limestone from Ratingen, N.W.
of Dusseldorf, might have found their way to the Netherlands, the
fact that no traces of the said fossil have ever been observed in
those rocks"), evidently keeps them outside the discussion. It is
true that in the district between Aix-la-Chapelle and Stolberg, the

1) J. Lorié e.g. found several fragments near Lancklaer on the Zuid-Willemscanal.

2) Note sur Jes roches eristallines recueillies dans les dépots de transport dans
la partie méridionale du Limbourg hollandais. Ann. de la Soc. géolog. de Belgigue.
XVI. 1888—89. Liége. Mém. p. 417—420.

3) Recherches sur les formations diluviennes du sud des Pays-Bas. Archives
Teyler (2) Ill. 6!me partie. Haarlem 1891, p. 23, 33.

4) Recherches sur les formations diluviennes. 1]. c. p. 67.

5) Ueber Holliindische Diluvial-Geschiebe. Neues Jahrb. f. Mineralogie. 1857, p. 389.

5) De Bodem van Nederland. Il. Haarlem 1860, p. 96.

7) H. von Decnen. Erliuterungen zur geologischen Karte des Rhemlandes und
der Proving Westfalen. Il. Bonn 1884, p. 216.

( 525 )

Produetus striatus is occasionally met with '), but, like many other
fossils, it is extremely rare.*) The probability of one of these
specimens having been transported to the Gooi becomes therefore
nil. On the other hand, as Srarinc had already pointed out, they are
very common at Visé in Belgium, consequently we are justified in
concluding that the above mentioned fragments must be referred to
that locality.

Other fossil mentioned by Rormer is the Goniatites sphaericus
Mart. (Glyphioceras sphaericum), a specimen of which had been
found near Holten, in Overijsel, and whose original birth-place he
claims to have been the valley of the Roer. This fossil, however,
is found both at Ratingen and Visé: nothing definite can therefore
be said with regard to the place of its origin. I may here mention
that in 1899, Dr. E. Cottixs brought me a fine specimen, well
preserved: and but little polished, which had been picked up in
the gravel of a garden at Utrecht and was very probably brought
from the Lek.

In the railway eutting near Maarn, to the East of Driebergen, I”
found in 1893 a block of crinoidal limestone weighing as much
as 97 K.G. In that same cutting repeatedly were observed pieces
-of compact black limestone. In 1895, fragments of a very beautiful
erinoidal limestone were found in the grounds of the villa Houde-
ringe, near De Bilt, at a depth of abont 1 M. Other pieces of
black and next to these of grayish compact limestone were found
in a railway cutting half way between the stations of De Bilt and
Soest. On the whole, therefore, it cannot be said that rocks of this
type are largely represented in the diluvial deposits under considera-
tion. This is probably owing in a large measure to the sandy nature
of the diluvium of those parts which allows the moisture of the
atmosphere to penetrate to the limestone and gradually dissolve it.
The same physical conditions are probably also responsible for the
paucity of erratics of this description in the Provinces of North-
Brabant and Limburg, and in the Campine. A. Erxns found fragments
of crinoidal limestone near Oudenbosch, *) K. Drnvacx of earboni-

1) H. von Decuen. |. c. p. 211.

2) (. Danrz did not even come across a single specimen in the district of Aix-
la-Chapelle. (Der Kohlenkalk in der Umgebung von Aachen. Zeitschr. d. D. geolog.
Ges. XLV. Berlin 1893. p. 611).

3) L. G. pe Konincx. Recherches sur les animaux fossiles. Le partie. Mono-
graphie des genres Productus et Chonetes. Li¢ge 1847. p. 30.

4) Recherches sur les formations diluviennes |. ¢. p. 67.

( 526 )

ferous limestone in a gravel pit at Gelieren near Genck?) and J.
Lori at Smeermaes and Lancklaer, on the Zuid-Willems canal.

The original home of these various limestones cannot be determined
with any certainty. However, as numerous layers of crinoidal lime-
stone are present in the districts of Aix-la-Chapelle and Stolberg *)
as well as in the valley of the Meuse, more especially near Dinant,
it seems rational that we should in the first place look to these parts
for their origin’). In any ease they must have been transported
along the Meuse, for the district Aix-la-Chapelle—Stolberg is drained
by the Geul, the Inde and the Worm, which all three flow into
the Meuse. :

Finally Rormer gives in his treatise a description of fragments of
phthanite, found near Ootmarsum, in Overijsel, which he thinks
derived from the layers of culm on the lower Rhine. This conjeeture
is not inadmissible, but at the same time the fact must not be
overlooked that this kind of rock is also plentiful in the valley of
the Meuse.

Jurassic System (Oxfordian). In the foregoing pages mention’ has
already been made of a piece of brownish yellow sandy clay, found
by J. Lorn on the Wageninger hill (Guelders) in which was inbedded
a perfect specimen of Rhynchonella Thurmanni Voltz, in every respect
similar to the fossils of this species found at Vieil-Saint-Rémy, to the
South-West of Mézieres in the department of the Ardennes *). This
is the only fossil of this type discovered in our country, although
in’ the diluvium of South Limbourg and Northern Belgium, jurassic

1) Les anciens dépots de transport de la Meuse, appartenant a l’assise moséenne
observés dans les ballastiéres de Gelieren pres Genck cn CGampine. Ann. Soe. géol.
de Belgique XIV. 1886—87. Liége 1887, Mém. p. 103.

Here again, as at Maarn, he ascribed their presence to an “accident”.

2) J. Betsser. Ueber Struktur und Zusammensetzung des Kohlenkalks in der
Umgebung von Aachen. Verhandl. naturh. Vereins Rheinl. u. Westf. XX XIX. Bonn
1882. Corresp. Bl. p. 92.

8) Ep. Duponr. Notice sur les gites de fossiles du caleaire des bandes carboniféres
de Flourens et de Dinant. Bull. Acad. roy. de Belgique (2) XII Bruxelles 1861 p. 293.

Ep. Dupont. Essai dune carte géologique des environs de Dinant |. c. (2) XX.
1865. p. 621, 622, 629.

Ep. Dupont. Carte géologique des environs de Dinant. Bull. Soc. geol. de Fr. (2)
XXIV. Paris 1866—67 p. 672, 673.

Ep. Duronr et Micuet Mourton, Explication de Ja feuille de Dinant. Musée d’hist.
nat. de Belgique. Service de la carte géolog. du Royaume. Bruxelles 1883, p. 9,
26, 33, 34, 53 et passim.

4) Contributions & la géologie des Pays-Bas. Archives Teyler (2) III]. Haarlem
1887, p. 10.

( 527 )

fossils have been frequently met with. We find them already men-
tioned by J. T. BinkHorst van pen Brinkuorst ').

Fr. SeGuers discovered a Rhynchonella and part of an Ammonites
at Genck *). Close to this place, at Gelieren, E. Drenvaux frequently
came across remains of “caleaire a Chailles’*). C. Manaise gave a
deseription of petrified Nerinea found at Rothem and an Isastraea at
Jambes, near Namur‘). A. Erens mentions a few other fossils *) and
finally we have an account of a yellow oolite, discovered by E. van
DEN brogck among the erratics of the Meuse, and here we call
attention to the peculiar siliceous oolites scattered about the plateau of
the Meuse and which probably belong to the jurassic system‘). As
yet no trace of similar oolites has been discovered North of the Rhine,
but J. Lori noticed some in the borings of a well at Mariendaal,
near Grave’). A few weeks ago Mr. L. Rurren found a small pebble
in the diluvium at Kollenberg, near Sittard.

Tertiary system. (Eocene). Very interesting are the accounts of
the discovery of erratics comprising specimens of Nummulina laevigata
Lam. Frrp. Rormer has given a description of a fragment of this
kind derived from Holten, in Overijsel, but believed it to have only
accidentally found its way among tlie erratics*). Sraring made
mention of a couple of rounded-off pieces of hornstone, one of which
had been found on the rising ground of Hellendoorn and the other
on the Steenshul, near Oldebroek, and which he referred to the Alps?
“If we did not know the place where these specimens were obtained,
“we should be rather inclined to think they came from a collection
“in which the objects had been confused and believe these rocks to

1) Esquisse géologique et paléontologique des couches crétacées du Limbourg.
Maastricht 1859. p. 7.

2) Ann. de la Soc. malacolog. de Belgique X. Bruxelles 1875. Bull. p. XXXIV.

*) Les anciens dépdts de transport de la Meuse, appartenant a |’assise moséenne
observés dans les ballastiéres de Gelieren pres Genck en Campine. Bull. Soc.
géolog. de Belgique XIV. 1886/87. Liége. 1887. Mém. p. 102.

+) Sur quelques fossiles du diluvium. Ann. Soc. malacolog. de Belgique X.
Bruxelles 1875. Bull. p. IV.

5) Note sur les roches cristallines |. ¢. p. 413.

6) Ef. van pen Brozcx. Les cailloux oolithiques des graviers tertiaires des hauts
plateaux de la Meuse. Bull. Soc. belge de Géologie III. Bruxelles 1890 p. 404—412.

X. Srarmier. Origine des cailloux oolithiques des couches a caijloux blanes du
bassin de la Meuse. Ann. Soc. géolog. de Belgique XVIII. !890—92, p. CV, 92.

KE. van ven Broeck. Coup d’oeil synthétique sur lOligocéne belge. Bull. Soc. belge
de Géologie VIL. Bruxelles 1893 p, 25, 266,

7) Beschrijving van eenige nieuwe grondboringen, Verhandel. Kk. Akademie vy. W.
Qde sectie. VI, N. 6. Amsterdam 1899, p. 33.

8) Ueber Holliindische Diluvial-Geschiebe. Neues Jahrb. f. Min. 1857, p. 392,

37

Proceedings Royal Acad. Amsterdam, Vol. VIII.

(528 )

“have been picked up near Brussels *)’. K. Martin?) and J, Lorié*)
in fact assign them also to that locality; they forget, however, that
no strata of nummulitie limestone are known to exist there ‘). Their
origin lies much farther South. In 1863 J. GossrLer had already
indieated the original source of these ‘“silex a Nummulites”, of which
a few years later he published a description °). They are dispersed
in large quantities in the arrondissement of Avesnes, in the department
du Nord, more especially in the environs of Trélon*) where, on
account of their hardness, they are frequently used for the paving
of roads.

Since then numerous fragments of this rock have also been found
in Belgium, specially on the plateau situated between the Meuse and
the Sambre, e.g. around Silenrieux, Sivry, Clermont, ete., as well
as in parts lying further West ‘).

The second question which we have to examine, is the period at
which these rocky fragments from the Ardennes have been trans-
ported to districts at present situated North of the Rhine. The view
expressed by StarinG that this transport has taken place before the
deposition of Scandinavian erratics, seems at present also satisfactorily
established, for those carried by the Meuse. In the railway cuttings
at Maarn and Rhenen, rocks of diverse origin lie together in friendly

1) De Bodem van Nederland. Il. Haarlem 1860, p. 89.

2) Niederliindische und Nordwestdeutsche Sedimentirgeschiebe. Leiden 1878, p. 37.

$) Les métamorphoses de I’Escaut et de la Meuse. Bull. Soc. belge de Géologie,
IX. 1895 Bruxelles 1895—96, Mém. p. 60.

4) I. van pen Brogck. A propos de lorigine des Nummulites laevigata du gravier
de base du Laekénien. Bull. Soc. belge de Géologie. XVI. 1902. p. 580.

5) De l’extension des couches & Nummulites laevigata dans le nord de la France.
Bull. Soc, géolog. de la France (3) Il. 1873—74. Paris 1874, p. 5i—58. See also
Ann. Soc. géol. du Nord. 1. 1870—74. Lille, p. 36.

6) Compte-rendu de l’excursion du 7 Septembre [1874] a Trélon I. ¢. p. 681.
Leriche. L’Eocéne des environs de Trélon. Ann. Soc. géol. du Nord. XXXII. Lille
1903. p. 179.

7) Micue, Mourton. Sur les amas de sable et les blocs de grés dissiminés a la
surface des collines famenniennes dans |’Entre-Sambre-et-Meuse. Bull. Acad. roy.
de Belgique (3) VIL. Bruxelles 1884, p. 301—803.

A. Ruror. Sur l’age de grés de Fayat. Bull. Soc. belge de Géologie I, 1887,
p. 47.

L. Bayer. Premiére note sur quelques dépdéts tertiaires de l’Entre-Sambre-et-Meuse.
Bull. Soc. belge de Géologie X, 1896. Bruxelles 1897—99 p. 189—140.

G. Vetce. De Vextension des sables Gocénes laekéniens 4 travers la Hesbaye et la
Haute Belgique. Ann. Soc. géolog. de Belgique, XXV, 1897—98. Liége, p. GLXV.

A. Briarr. Notice descriptive des terrains tertiaires el crétacés de Entre-Sambre-
et-Meuse. Ann. Soc. geolog. de Belgique XV, 1887—88, p. 17,

(529 )

juxtaposition and intermixture, which proves that they must have
been carried together and at the same time to the place where they
ave found at present. From the shape of the front moraine, we con-
clude that the direction of the transport was from the North-East.
The erratics nowadays found at the surface have been gradually
denuded by the action of water and wind. It is therefore evident
that originally these erratics were transported much farther to the
North and East, than their present place of deposit, because they
were seized by the advancing Baltic icestream and carried along
together with the material of its moraine. We are therefore justified
in fixing the period of the transport of the boulders from the Rhine
and Meuse at the commencement of the epoch of maximum glaciation
(Saxonian).

A far greater difficulty presents itself when we attempt to deter-
mine in what way this transport has taken place, for it can only
have been effected by the agency of a river or a glacier. The
hypothesis: that all these boulders should have been carried along
by the Meuse in its downward course, is scarcely admissible. Even
leaving out of account the finding of rocky fragments from the

Ardennes on the strands of Goedereede and Voorne — not to
speak of Suffolk, in England —- there remains a large tract of land

105 K.M. long stretching from Utrecht to Kibergen, over which these
erratics are dispersed in the shape of a crescent. If carried by the
Meuse, its mouths must have extended over a very large area. But
a greater objection to this theory is that, in that case, they must have
been transported across the Rhine (at present the [Jsel) because
rocks of this kind are found at places to the East of this river
(Doetichem, Eibergen, Markelo). Finally, some of these blocks are
so large that they could not possibly have been transported by a
river. Besides, some of them present no marks of polish, which is
another argument against their transport by running water.

For the better understanding of these objections we quote a few
examples from the Province of Limburg and the Campine. A. Erens
found in the environs of Maastricht numerous large blocks of Cam-
brian quartzites, one of which was 38 M. high, 2,6 M. long and
0,6 M. in width, computed to weigh about 12400 K.G."). More
important still are the blocks of sandstone found in the diluvium
of the Campine at Holsteen-Molenheide, near Zonhoven, in the neigh-

bourhood of Hasselt, E. Drtvaux noticed blocks measuring from 4

1) Note sur les roches cristallines |. c. p. 412, 417. Mr. L. Rurren informed
me that in the neighbourhood of Sittard similar boulders reach a diameter of
as, a) ie

37%

(530 )

to 86 M. cub, weighing from 10600 to 95400 K.G.*).. He believed
them to belong to the landenian stage of the eocene system. His
opinion, that they covered the plateau of the Ardennes (where
Cu. Barrois was the first to discover similar masses *), to a height
of 672 M., has been much contested. E. van DEN Broxck classed these
sandstones first among the triassic system"), afterward referred them
to the oligocene system ‘), and finally suggested they might either be
oligocene, miocene or pliocene, but certainly not eocene*). G. DewaLqun
pronounced them to be miocene*), whilst O. van Errsorn sought
their origin in the pliocene system’), more especially in the diestian
group"), but was of opinion that they must be regarded as the
remains of a ‘delta caillouteux”’ *). M. Mourton, on the contrary,
held that they had been formed in the vicinity of their present place
of deposit, by the fusion of the “sable de Moll” *"), an opinion
which cannot be maintained, because similar blocks are present in
the diluvium of Maastricht where no trace of this sand exists 1).

J. GOSSELET compares these rocks with the freshwater-quartzites of the
diluvium of the Rhine and, with reason, thinks that they belong
to the oligocene system ‘*). At all events it is universally admitted
that the Ardennes have been covered by extensive layers of tertiary

1) Description sommaire des blocs colossaux de grés blane cristallins provenant
de l’étage landénien supérieur..... en différents points de la Campine limbour-
geoise. Ann. Soc. géolog. de Belgique XIV. 1886—87. Liége 1887. Mém. p. 117—130.

2) Sur l’étendue du systeéme tertiaire inférieur dans les Ardennes. Ann. Soc.
géol. du Nord. VI. Lille 1879, p. 371.

3) Ann Soc. roy. malacolog. de Belgique XVI. Bruxelles 1880. Bull. p. LXXIV.

4) Note préliminaire sur le niveau slratigraphique’de la Belgique et de la région
dorigine de certains des blocs de grés quartzeux de Ja Moyenne et de la Basse-
Belgique. Bull. Soc. belge de Géologie IX. 1895. Bruxelles Proc. verb, p. 94—99.

5) Les grés erratiques du sud du Démer et dans la région de Heurck. Bull.
Soc. belge de Géologie XV. 1901. Bruxelles 1902. Proc. verb. p. 628.

6) Ann. Soe. géolog. de Belgique. XIV. 1886—87. Liége 1887. Bull. p. 18.

7) Le Quaternaire dans le sud de la Belgique, Bull. Soc. belge de Géolog. XV.
1901. Proe. verb. p. 662.

8) Quelques mots au sujet des divers niveaux gréseux du tertiaire supérieur
dans le nord de la Belgique. 1. ce. p. 632.

°) Contribution & l’Etude des Etages rupélien, boldérien, diestien et poederlien,
l. c. XVI. 1902. Mém. p. 65.

10) Compte rendu de l'excursion géologique en Campine les 23, 24 et 25 sep-
tembre. ]. c. XIE 1899. Mém. p. 205, 213, 214.

11) Aupu. Erens. Nolte sur les roches eristallines 1. c. Pl. XIII.

1) L’Ardenne. Paris 1888, p. 833,

( 534)

system, as has been pointed out by M. Lonesr'), X. Srainupr ?),
J. Corner") and others.

Before stating our reasons for supposing the presence of a gla-
cier in the Ardennes during the second glacial period, we are
willing to admit that J. Gossunur, who of all geologistst knew most
of this mountain range, remarked in reference to this hypothesis :
“on n’en trouve aucun indice sérieux’ *). Indeed we have but few
indications in support of it. The first to draw attention to this ques-
tion was Fr. van Horn, who at the time of the making of the rail-
way line between Tirlemont and Jodoigne, found near Bost blocks of
quarizites from the Ardennes which presented marks quite similar
to the striae caused by glaciers. Van Horny, however, did not feel
justified in drawing from this discovery the conclusion of the former
existence of a glacier °). A year later C. Manaise observed similar
marks on blocks of quartzites on the banks of the Grande Geete,
close to the spot formerly occupied by the Abbey of Ramez-les-
Jochelette, about 10 K.M. from Bost"). G. Dewateur believed to have
seen unmistakable striae on blocks of quartzites in the valley of the
Ambleve, near Stavelot, on the “Hohe Venn’ *). E. Drnvaux also
noticed these horizontally parallel seratches, but believes them to
have been produced by a ‘torrent entrainant et roulant péle-méle
des sables et des cailloux.” *).

Finally, South of Stavelot, on the road to Somagne, G. DrwaLeue
discovered giants’ kettles formed by the agency of glaciers’). It is
regrettable to find that the more detailed study of this subject has
been much impeded by the practice in Belgium of giving the name

1) Les depots tertiaires de la haute Belgique. Ann. Soc. géolog. de Belgique XV.
Liége 1887—88. Mém. p. 59.

*) Le grés blanc de Maizeroul. Ann. Soc. géolog. de Belgique XVIIL Liége
1890—91. Mém. p. 61.

3) Etude sur I’Evolution des Riviéres belges. Ann. Soc. géol. de Belgique XXXI.
1903 —04. Mém. p. 317, 355.

4) L’Ardenne, p. 843.

®) Note sur quelques points relatifs & la géologie des environs de Tirlemont.
Bull. Acad. roy. de Belgique (2) XXV. Bruxelles 1868, p. 645, 664; 1 Pl.

8) Roches usées avec cannelures de la vallée de la grande Geethe. 1. ec. (2)
XXVII, 1879, p. 682—685.

7) Sur la présence de stries glaciaires dans la vallée de l’Ambléve. Ann. Soc.
géolog. de Belgique. XII. 1884—85. Liege. 1885. Bull. p. 157—158.

8) Note succincte sur l’excursion de la Societé géologique & Spa, Sravetor et
LawMersporF en aoul-septembre 1885. Ann. Soc. roy. malacol. de Belgique XX.
Bruxelles 1885, Mém. p. 19.

9) Marmites de géants prés de Stavelot. Ann. Soc. géol. de Belgique. XXY.
1897—98, p. CXXXVIII.

( 532 )

of psendoglacial to all kinds of bosses and scratches which elsewhere
would scarcely be so called, because they do not in the leest resemble
the striae of glaciers *).

This absence of positive characteristics is however easily explained.
Leaving alone the fact that as yet no thorough investigation of the
subject has been made, the condition of the Ardennes themselves
are very unfavorable to research. Its dense forests, fens and heaths
make it diffienlt to reach the surface of the rocks, whose harder
layers are only capable of preserving marks. The reason why so
few traces are found on the sides of the valleys and on the plateau
of the Meuse becomes plain, when we remember that during the
period following the receding of the Northern glacier, the waters
of the Meuse rose 200 M. above the level of the sea, and not only
filled the whole valley but inundated the plateau of the Meuse and
thus destroyed the traces left by the glacier.

Of this we find the clearest proofs in the terraces which have
retained their boulders. *) Besides, exactly the same thing happened
with the Rhine and its tributaries. The sand and small pebbles
carried along by their waters must necessarily have almost entirely
obliterated the marks of the glaciers left on the rocks *).

Striae, however, are not the only evidences of the action of a

1) X. Srarnrmr. Stries pseudo-glaciaires en Belgique. Bull. Soc. belge de Géologie
X. Bruxelles 1896. Pr. verb. p. 212—216.

KE. van pen Broecx. Contributions & l'étude des phénoménes d’altérations
dont l’interprétation erronée pourrait faite croire & Vexistanee de stries glaciaires.
1, c. XIII. 1899. Mém. p. 323—334. Pl. XX.

G. Smorns. Sur une roche présentant des stries pseudo-glaciaires en Condroz.
», Pr. verb. p. 222—293.

) E. Duponr et M. Mourion. Explication de la feuille de Dinant. Bruxelles
1883, p. 100.

A. Rutor. Résultats de quelques explorations dans le Quaternaire de la Meuse.
Bull. Soc. belge de Géologie. XIV. Bruxelles 1900. Pr. vorb. p. 259, 260.

X. Sramier. Le cours de la Meuse depuis Vere tertiaire |. c. VIII. 1894 Mém,
p. 84. Pl. VIL.

E. van pen Broecx. Coup d’oeil synthétique sur l’Oligocéne belge et les obser-
vations sur le Tongrien supérieur du Brabant l.c. VII. 1898, p. 255, 256, 266.

E. van pen Brorck. Exposé sommaire des observations et découvertes stratigra-
phiques et paléontologiques faites dans les dépots marins et fluvio-marins du Lim-
bourg pendant les années 1880—81. Ann. Soc. roy. malacolog. de Belgique XVI,
Bruxelles 1881. Bull. p. CXXV—CXLII. .

3) It might be suggested that the transport of these boulders had taken place by
means of ice-floes, but Mr. Lonesr has demonstrated in the most positive manner
that these ice-masses are incapable of effecting a notable removal. He comes to
the conclusion that among the present climatic conditions no explanation can be

ik

(
2

( 533 )

glacier’ and one might reasonably expect to find in the valleys
some remains of the wall of moraines. That this is not the case
may be accounted for by the supposition that the great Baltie ice-
stream has travelled farther south and in its course also destroyed
these evidences. As there exists a great diversity of opinion with
respect to this forward movement of the ice-stream, it seems necessary
here to state what is known of the dispersion of Seandinavian
erratics in the Provinces of Limburg and North-Brabant and the
Campine.

As long ago as 1778, J. A. pe Luc mentions the discovery of
blocks of granite between Postel and Alfen, and also near Lommel
and. Helchteren'). Subsequently, J. J. p’Omanius p’Hannoy drew
attention to the numerous blocks of granite and other fragments of
“primordial” rocks found on the heath of the Campine. “La quan-
“tité de ces blocs doit étre été immense; car quoiqu’on fasse
“un grand usage pour paver les rues, ainsi que pour faire des
“jetées le long de la mer et des rivieres, on en voit beaucoups
“dans les bruyeres”.*) And Enernspach—Larivibre adds the infor-
mation that some of these blocks of granite measured several
M. cub.*) Somewhat later again, J. G. S. van Brepa mentioned
the finding of two pebbles of granite in the subsoil of Maastricht,
very justly remarking that these rocks must be regarded of later
date than those transported from the Ardennes‘). At that time he
already spoke of blocks of granite found at Oudenbosch, in North-
Brabant °). Stakinc expressed the opinion that these erratics had been
brought there by “some accidental means or other” ‘), although a
short time before Norperr pr War. had recorded the finding, at
Weelde, 10 K.M. to the NNE. of Turnhout and also at Poppel,

found for the transport of the blocks of quartzites from the Ardennes. (Sur_ le
transport et le déplacement des cailloux volumineux de lAmbléve. Ann. Soc.
géol. de Belgique. XVIII. Liége 1890—9i. Bull. p. GVIT—CIX).

1) Lettres physiques et morales sur l'histoire de la terre et de 'homme. IV.
Paris et La Haye 1779, p. 54, 57.

2) Mémoires pour servir 4 la description geologique des Pays-Bas, de la Flandre
et de quelques contrées voisines. Namur. 1828, p. 204, 205.

3) Considérations sur les bloes erratiques et roches primordiales Bruxelles. 1829
(fide P. Cogers. Ann. Soc. roy. malacolog. de Belgique. XVI. 1881, Bull. p. LIV).

4) Natuurk. Verhandel. van de Holl. Maatsch. v. Wetensch. XIX. Haarlem
1831, p. 390.

®) The biggest one originally weighed +5300 K.G. (V. Becker). Het zwerfblok
yan QOudenbosch en zijne omgeving. Studién op Godsdienstig, Wetensch. en
Letterk. Gebied, XXX. Utrecht. 1888, p. 25).

6) De bodem van Nederland If. Haarlem 1860, p. 78.

( 534 )

half-way between the last-named place and Tilburg, of erratics one
of which weighed 200 K.G.’). G. Drnwaqur then again mentioned
two pebbles of granite found in the neighbourhood of Maastricht *).
It is only during the last ten years that a deeper interest has been
taken in the study of this subject, with the result that the presence
of erratics of Northern origin has been ascertained in several places,
as we gather from the writings of C. Bamps, V. Brckrr, E. van DEN
3ropcK, P. Coenrns, BE. Denvaux, G. Drwarqur, A. Erens, O. van
Ertrporn, J. Lorik, A. Renarp and Cr. pr LA VALLin—Poussin.

Another fact worthy of notice is the presence, at these very places,
of boulders derived from the district of the Rhine. The first indications .
of such finds, by G. Dewangun, are rather questionable. They were
fragments of rocks from the lava of Niedermendig, near Andernach,
frequently met with in the valley of the Ambleve, but were believed
to have been fragments of mill-stones, formerly used at Stavelot and
Malmedy. Subsequently KE. Denvavx found a few pieces of lava and
pumice stone in the diluvium of the Campine *); but it was A. Erens
who discovered and described a great number of rocks derived from
the Rhine district, composed of lava from Niedermendig, pumice
stone and Taunus-quartzite *). These were followed at a later
period by trachyte from the Drachenfels, basalt and hornblende-
andesite from the Siebengebirge, and melaphyre and agate from the
basin of the Nahe *). The discovery of these fragments in the North
of Limburg admits of no other interpretation than that these recks
must have been carried South, simultaneously with the detritus from
Scandinavia.

It cannot be denied that fewer erratics from Scandinavian rocks
are found South of the Rhine than North of it. We give the following
reasons in explanation of this fact: 1s*. During the progress of the
Baltic icestream in a South-Western direction, the Seandinavian drift
must already have Jost a certain portion of its material by the mix-
ture of the debris of its own moraine with that of other sources;
2ed. Jt must have suffered further loss by mixing with the moraine

1) Bull. Soc. paléontolog. Bruxelles p. 36. (Séance du 5 Septembre 1858).
2) Prodrome d’une description géologique de la Belgique. Bruxelles et Liége.
1868, p. 237.

8) Les anciens dépdts de transport de la Meuse. Ann. Soc. géol. de Belgique
XIV. 1886—87. Mém. p. 102.

4) Note sur les roches cristallines... Ann. Soc. géolog. de Belgique XVI. 1888—
89. Mém. p. 414, 489—441, 444.

6) Recherches sur les formations diluviennes du sud des Pays-Bas. Archives
Teyler (2) III. 6°™e partie. Haarlem 1891. Tableaux synoptiques I—Y.

A. WICHMANN. “On fragments of rocks from the Ardennes found in the Diluvium
of the Netherlands North of the Rhine.”

Groningen.
Leowwarden “ng

Holter
*Markelo )——

\._ Reriswoude
sMaarry ,

Dusseldorf

Brussel @

9 he eth

Nummul lasre CU end ne
a -

Ne hunch Raman «xxx

>) 4 =
Sodus Mr iad ~.—»—.—x

Yn ES i
ory mb lines Vial Saint dewyy

i eg DVuartule —
Terphyroid ------
(ce) ye G hii

Proceedings Royal Acad. Amsterdam. Vol. VIII.

débris of the glacier from the Ardennes; 3. The melting process
commenced soon after reaching its Southern limit. It was only during
its receding course that the Baltic ice-stream remained tor some time
stationary, and in this period of inaction was formed the front
moraine extending from the South coast of the Zuiderzee to Grebbe
and further as shown by J. Lorm+), over Nimeguen to Crefeld.
The glacierformations, at present situated South of the Rhine, were
afterwards, i.e., during the inter-glacial period, exposed to the turbulent
waters of the Meuse, which, as has been stated above, rose 200 M.
above the level of the sea, at least between Namur and Dinant,
proof of which is afforded by the high terrace. Although this terrace
slopes down towards the North, near Nimeguen, it still reaches a
height of between 50 and 100 M. + A.P.*). Owing to this action
of the Meuse, the erratics found in North-Brabant and Limburg are
generally smaller and more polished than those of the diluvial depo-
sits North of the Rhine. And lastly, a great portion of the glacier
formation has got hidden from view by the large alluvial tract of
the Rhine delta, which has been formed after the breach of this
river at Nimeguen and subsequent alterations of the level by dis-
locations.

Anyhow, it is entirely out of the question to admit that in the
beginning of the quarternary period the Meuse had its outlet into the
sea, a little North of Maastricht and formed there an estuary, —
a theory put forwards by M. Mourton*) and A. Rutor *). As J. Lorik
justly observes, not a single indication exists of the sea having
extended so far inland.

1) J. Lorté. Le Rhin et le glacier scandinave quaternaire. Bull. Soc. belge de
Géologie XVI. 1902. Mém. p. 129—153. N. VIIL

2) lc. p. 131. The high terrace of the valley of the Meuse is generally
considered of pliocene formation, but the presence of Scandinavian ervatics in
places situated farther North, e.g. Mook, Nimeguen, etc., proves that it must have
been formed after the receding of the Baltic ice-stream.

3) Les mers quaternaires en Belgique. Bull. Acad. roy. de Belgique (3) XXXIL.
Bruxelles 1896 p. 671—711. La faune marine du quaternaire moséen revelée par
les sondages de Srrypeek (Meerle) et de Worret, pres de Hoogsrrarren en Gam-
pine. I. c. (38) XXXII. 1897, p. 776—782.

4) Les origines du quaternaire de la Belgique. Bull. Soc. belge de Géologie. XI.
Bruxelles 1897, p. 117.

5) De hoogvenen en de gedaantewisseling der Maas in Noord-Brabant en Limburg.
Verhandel. K. Akad. van W. Tweede Sectie Ill. No. 7. Amsterdam 1894, p. 10,

( 536 )

. "le . , . '
Chemistry. “The boiling points of saturated solutions in brary
systems in which a compound occurs”. By Prof. H. W.
Bakuvis RoozeBoom.

(Communicated in the meeting of November 25, 1905).

In a previous communication ') it has been ascertained what
branches in the three-phase lines for solid, liquid and vapour may
occur in binary systems in which a solid compound appears, namely
for the three cases that:

a. the vapour pressure of the liquid mixtures diminishes gradual
from the component A to the component B;

6. liquid mixtures occur with a minimum pressure;

ce. liquid mixtures occur with a maximum pressure.

For the right understanding of the behaviour of such systems it
is particularly desirable to ascertain what is the order of the pheno-
mena which appear with different mixing proportions of the components
when these, at a constant pressure, are brought from low to high
temperatures.

If those pressures are very low the mixtures, at a sufficiently
low temperature, are completely solid, and on elevation of the
temperature, they pass gradually and, at last, completely into vapour,
therefore simply a sublimation occurs.

If the pressures are sufficiently high (in the case of components
which are not too volatile, 1 atm. is quite sufficient), the solid sub-
stances pass’ gradually and, at last, completely into liquid and these
liquids evaporate at still higher temperatures. In this case, fusion
takes place first and evaporation afterwards.

With moderate pressures, however, the melting and evaporation
phenomena partly coincide, namely when pressures are chosen which
occur on the three-phase lines of the components or the compound.

What cases may be distinguished when no solid compound appears
has been fully investigated previously, by me. ”)

Particular attention has been called to the fact that the three-
phase line of the component 4 may be sometimes intersected twice
at the same pressure, which is possible when this line exhibits the
branches Ia and Id, described in the previous communication. (See
line BD in fig. 1 and 6). In such a case two separate boiling

) These Proc. VII, p. 455. I learned that Dr. Smirs had also come to the
conclusion that the minimum on the three-phase line did not coincide with
point FH.

*) Heterogene Gleichgewichte, Heft 2. p. 338, et seq.

(587 )

points of solutions saturated with solid B occur, one on branch 1b
and another on branch da. At the last point, boiling does not take
place on heating but on cooling. The ¢, «-figures at a constant
pressure have been deduced by me, and the phenomena, in solutions
of salts in water and of sulphur in carbon disulphide, have been
demonstrated by Smits and pe Kock.

The figures 1, 3, 5, 6 show at once that this same case may
also occur in solutions saturated with a compound of the two com-
ponents as soon as their three-phase line shows branch 14 as well
as Ja. Examples of two boiling points of the saturated solution have
not thus far been noticed in binary compounds although they should
be far from rare.

In compounds where, among the saturated solutions, there is
present one with a minimum pressure (Fig. 3), a second boiling
point of the saturated solution might occur with solutions either
richer in A or in #B; in fact a third boiling point at the side of
the solution richer in 4 would be possible if the point D in fig. 3
were situated so low that, at the same pressure, the branches D7),
T,T, and 7,H could be intersected in succession. The saturated
solution would then in succession first disappear, then reappear to
finally disappear once more. Examples belonging to this case have
thus far not been sufficiently studied.

If branch 3 of the three-phase line exists for the solutions richer
in B (GD in Fig. 1 and 6, GH in Fig. 3 and 5), then if this
line is crossed, there occurs at a constant pressure a boiling point
of the saturated solution of a different nature from that on branch 1.
The ¢, z-figure of such a case is quite analogous to that derived by
me‘) for saturated solutions of the component A whose three-phase
line in Fig. 1, 8, 5 always indicates branch 3. On boiling the solution
saturated with A the following transformation takes place :

solid + liquid — vapour.

As solid and liquid now pass together into vapour in a definite
proportion, it now depends on the quantity of those two phases
which of the two disappears at the boiling point. This case occurs
for instance on the three-phase line for ice in systems of water and
little volatile substances as salts, also on the three-phase line for
solid CO, in mixtures of CO, with less volatile substances such as
alcohol.

The same must now also serve for compounds in so far branch 3
occurs therein. Among the binary systems whose liquid-vapour pres-

1) Heter. Gleichg. II. 341 et seq.

(538 )

sure always diminishes from Ato 2, the branch 3 has thus far only
been found with ICI, and ICl, as observed in the previous communi-
cation. From SToRTENBEKER’S experiments, it may be deduced that for
ICL, the branch 8 extends from 34° at 100 mm. to 22°7 at 42 mm.,
for IC] from 22° at 24 mM. to 8° at 11 mM. The peculiar boiling
phenomenon is, therefore, only possible between these temperatures
and pressures, but has not been expressly stated in the solutions
saturated with IC], or ICI.

In binary systems in which a liquid with a minimum pressure
occurs on the three-phase line of the compound, branch 3 must
always appear as shown in fig. 8 or 5. Among the examples cited
in the previous communication, there are sure to be found some
where the simultaneous boiling of the solid phase and the solution
may take place at 1 atm. pressure.

Another kind of boiling-phenomenon may, finally, take place on
branch 2 of the three-phase line of a compound. This branch cannot
occur with the components, for the peculiarity of the branch consists
in this that the saturated solution contains an excess of the compo-
nent 4, whilst the saturated vapour contains an excess of A; the
compound is, therefore, the phase whose composition is situated
between those of the two others. This is, of course, only possible
with a compound and not with one of the components.

According to Fig. 1, 38, 5, 6 of the previous communication branch
2 must oceur with all compounds where coexisting liquids with an
excess of 6 are possible, for it commences immediately at the
melting point.

Now, this is possible with a number of hydrated salts which,
below their melting point, yield saturated solutions with excess of
salt; but the appertaining pressures are then generally so small that
the boiling phenomenon cannot be readily observed. In the ease of
salt-hydrates which occur at a higher temperature so that the equi-
librium-pressure on their three-phase line might amount to | atm.,
the solutions richer in salt seem to be very rare and no example is
known to me.

An example is, however, known if H,O is replaced by NH,. With
the compound NH, Br.3NH,, branch 2 appears and the pressures
are even greater than 1 atm. In this case the boiling phenomenon
has been observed by me.

Branch 2 has, however, been met repeatedly in my previous
researches on gas-hydrates where water is then the component 5. If
we now take those hydrates near solutions with more water the

(539 )

vapour generally contains but little water, and we are dealing with
branch 2.

The conversion now taking place with heat supply at a constant
pressure is:

solid — liquid + vapour.

In all those cases it is, therefore, not the liquid which boils but
the compound. The gas is very plainly seen to emanate from the
crystals lying in the liquid, whilst the latter does not diminish but
increases. The phenomenon has been very plainly observed with the
two hydrates of HCl and of H Br and with those of SO, and Cl,.
With the last two and with HCI.H,O it could be observed at 1 atm.
pressure.

It must also exist with I Cl but limited between 27° at 39 mm.,
and 22° at 24 mm., much more plainly with ICl, where it may
appear between the melting point 101° at 16 atm. and 34° at 100 mm.
Between this a three-phase pressure of 760 mm. occurs at 64°, and
at the said temperature it may, therefore, be observed in an open
apparatus. Solid ICI, breaks up into a liquid with 63 and into a
vapour with 89 atom-percent of chlorine.

That similar phenomena may also appear in compounds which are
very stable at a lower temperature, has recently been demonstrated
by Aten in the case of Bi,S,. This sulphide breaks up at 760° into
a liquid containing 55 atom-percent of S and a vapour consisting
almost exclusively of S. Therefore, the actual melting point of the
sulphide cannot be determined at 1 atm. pressure. A similar behaviour
may be expected of many compounds having a melting point situated
much higher than the boiling point of one of its components, such
as in the case of oxides, sulphides, phosphides ete.

We must point out another peculiarity which distinguishes the
boiling phenomena on branch 2 from those on branches 1 and 3.
The liquids and vapours belonging to the latter are both either richer
in A or richer in & than the compound : consequently the boiling
phenomena concerned are observed in systems consisting of the com-
pound with a smaller or larger excess of one of the components.
On branch 2 however the vapour is richer in A and the liquid
richer in 4, therefore the boiling phenomenon can occur in mixtures
of the compound with A as well as with 4. In the first case such
a system, below the boiling point at the existing pressure, consists
of compound + vapour and the liquid appears only at the boiling
point, in’ the second case, the system below the boiling point eon-
sists of compound ++ liquid and the vapour appears at the boiling

(540 )

point. In the particular case that the compound was perfectly pure,
liquid and vapour should appear both together at the boiling point.

This may be: made plain by the example of I Cl,. The whole
i, a-figure at 1 atm. is schematically represented by fig. 7.

ICl;
Fig. 7.

in which ¢, represents the temperature (64°) in question. In the different
regions G represents vapour and Z/ liquid. The further parts of the
figure are entirely dominated in their relative situation by that of
the three-phase lines. On this entirely depends which branches of a
particular three-phase line will be imtersected at the same pressure.
In fig. 1 (previous communication) a simultaneous intersection of the
branches Ia and Id is only possible on the three-phase line of the
compound. If, however, as with ICl,, the melting point / lies at a
high pressure, a simultaneous intersection of Id with 2 or 3 is
possible. This is why in Fig. 7, besides the boiling point ¢, on branch
2, ¢, also occurs as boiling point on branch 16.

The pressure of 1 atm. is also higher for ICl or I than their
three-phase line, consequently for these compositions, melting and
boiling phenomena occur quite separately and the melting point lines
of IC] and I run quite below the boiling point line.

If we take a pressure somewhat lower than 100 mm. we obtain
a ¢,w-figure 8. For ICl,° we now have again ¢, as boiling point
on branch If and ¢, as boiling point on branch 38. For 1 Cl, melting
and boiling ave still quite distinct but at a pressure below 100 mm,

( 541 )

the three-phase line for solid iodine is intersected both on branch
16 and 1a and therefore the complication in the figure occurs at
the side of the iodine.

Still greater complications may appear when according to Fig. 3
(previous communication) there exist liquids with a minimum pressure
and when consequently the branches 1), 1@ and 16 can also appear
at the side of the liquids richer in B, whose intersection at an equal
pressure may coincide eventually with those of branch 2 or branch
3. When such systems have been more closely investigated it will
not prove difficult to give detailed ¢, v-figures for the same.

Chemistry. — “The reduction of acraldehyde and some derivatives
of s. divinyl glycol (3.4 dihydroay 1.5 hexradieney’. By Prof.
P. van Rompurcu and W. van Dorssrn.

(Communicated in the Meeting of November 25, 1905)

The reduction of acraldehyde (acroleine) with sodium amalgam ')
as well as with zine and hydrochloric acid *) has been studied by
LinneMANN, who states that he has obtained in the first case propyl
and zsopropyl alcohol, in the second case ‘sopropyl and ally] alcohol,
also a substance called acropinacone of the composition C,H,,O,, or
rather a product of non-constant boiling point, of which the fractions
boiling between 160°—170° and 170°—180° gave, on analysis, values
which led to this formula.

Craus*) could not confirm the results of Linnemann as regards the
formation of ¢sopropyl alcohol in the reduction with zine and hydro-
chloric acid.

Griner *) has also repeated LiyNeMANN’s experiments with the object
of preparing acropinacone (divinylglycol) but only obtained very small
quantities of a liquid without constant boiling point which bore no
resemblance to the glycol which, however, was obtained by him
in fairly large quantity by reduction of acraldehyde in acetic acid
solution with a copper-zinc couple. The other products of the reaction
have not been further described by the author.

If we consider the formula of acraldehyde in connection eit the

1) Ann. d. Chem. u. Pharm. 125 (1863) S. 315.
2) Ibid Suppl. I Geers a eeD) Shai

3) B. B. Ill. (1870) S. 404.

4) Ann. d. Phys, et Chim. [6] 26 (1892). p. 369.

views of Tete on the addition of hydrogen to conjugated systems
of unsaturated compounds, then on reducing

CH, CH,
CH we might expect CH ,
| 79 || pOH
C C
Nz Na
an unsaturated aleohol which, however, by intramolecular atomie
pO
shifting would be converted into CH,—CH,—C , propylaldehyde.
NE

On further reduction this would form propyl alcohol, a substance
which actually occurs among the products of the reduction.

Up to the present, propylaldehyde has not been’ found among the
substances formed in the reduction of acraldehyde.

We have, however, succeeded in showing that, although no free
propylaldehyde may be present, a derivative of this substance is
formed under certain conditions so that the intermediate formation of
the said aldehyde is not at all improbable.

First of all the reduction with zine and hydrochloric acid in ethereal
solution according to LinneMANN has been studied, but we succeeded
no more than Griner in isolating a well defined product — besides
allyl aleohol and perhaps smaller quantities of propyl! alcohol ;
generally, the substance obtained, which boiled between 158°—164°,
contained much chlorie.

If, however, we allow zine dust to act on a mixture of acraldehyde
and glacial acetic acid’) then, in addition to allyl and propyl alcohol,
a neutral liquid is formed (b.p. 170°) from which, after fractionating
in vacuo, a product may be obtained boiling between 59°5—60’ at
15 mm. The analysis and the vapour density lead to the formula
CHO;

The compound is not decomposed by potassium hydroxide ; neither
sodium nor phosphorus pentachloride have any action; it cannot be
benzoylated with benzoyl chloride and pyridine. This sufficiently
proves the absence of OH groups.

The said properties, however, render it very probable that the
substance is an ether. By dilute acids it is hydrolysed although but
slowly. An aldehyde-like odour appears but, as the reaction proceeds,
the mass becomes so dark with formation of brownish-black resinous

1) The action of various reducing agents on acraldehyde has been studied. The
results will be published in due course.

( 543 )

products that we have not, as yet, succeeded in isolating well-defined
compounds.

Bromine is readily absorbed by it and that in a quantity which
points to the presence of two double bonds. If we work with a
solution of carbon tetrachloride at a low temperature, but little hydrogen
bromide is formed.

From a substance of the formula C,H,,O, a great many isomers
are, of course, possible. We cannot enter here into a description of
the different experiments made in order to elucidate the structure of
the product obtained, but we may state that we have finally sueceeded
by means of a synthesis, which leaves no doubt whatever.

If, on s.-divinyl glycol which, thanks to the beautiful researches
of GrineR, may be readily prepared, propylaldehyde is allowed to
act for 6 days at 90°, a substance is obtained identical with the one
described above.

(Sp. gr. at 12° of the synthetic product 0.9392
Sse Coho on eoricinal % 0.9416
Refraction at 12° of the synthetic ,, 1.4434
oe eae Oriommale |e. 1.4430.)

As to the synthetic product, propylidene s. divinylethylene ether,
must be given the formula:

CH—O,

1 Pia CH—CH,—CH,

|
CH
|
CH,

the original must also be considered as a derivative of propylaldehyde.
It is, of course, possible that there might be formed at first an
analogous acraldehyde derivative, which afterwards got converted
into a propylaldehyde derivative, but considering the comparative
difficulty with which the vinyl group combines with hydrogen, this
looks less probable.

As one of us (v. R.) explained many years ago, s. divinylglycol
or 3.4 dihydroxy 1.5 bexadiene would form an excellent material for the
preparation of the hydrocarbon CH, = CH — CH = CH — CH = CH,,
otherwise hexatriene 1.3.5.

Different methods which we have tried have not led to the desired

38

Proceedings Royal Acad. Amsterdam. Vol. VIIL

( 544 )

end. At last we think we have succeeded by making use of the
diformate of s.-divinyl glycol, a compound which may be prepared
by heating this glycol for a short time with formic acid.

By fractionating in vacuo, the diformate is obtained as a colourless
liquid which at a pressure of 20mm. boils at 109° and has a sp. gr.
of 1.0747 at 11°. A determination of the formic acid (by saponifica-
tion) gave the amount required for diformate.

In a communication about to follow, the hydrocarbon prepared
from the diformate and the method of its preparation will be fully
described.

University Org. Chem. Lab. Utrecht.

Chemistry. — “The occurrence of B-amyrine acetate in some varieties
of gutta percha’. By Prof. P. van Rompuren and N. H. Conen.

(Communicated in the meeting of November 25, 1905).

Last year, a compound melting at 234° was found by one of
us (v. R.) in the gutta percha of Payena Leerii') of which it could
be stated that it is nof identical with lupeol cinnamate, which occurs
in many varieties of gutta percha; the quantity was then too small
for further research. Since then a little more of that product was
prepared so that it could be proved that on treatment with alcoholic
potash it yields acetic acid and an alcohol melting at 195°.

In these Proc. of June 25, 1905 p. 137 it was stated that the
same product has been found by one of us (C.) in the ‘‘djelutang”
derived from the juice of varieties of Dyera. The identity was shown
by a comparison of the melting points and by melting point deter-
minations of mixtures of the two substances.

A sufficient quantity was now at disposal to determine the nature
of the compound.

In the first place, the substance was recrystallised a few times and
finally obtained in beautiful, long, hard needles which melted at
235° (corr. m. p. 240°—-241°),

On analysis (combustion with lead chromate) the following results
were obtained :

Caleulated for C,,H,,0,

© 81.96, 82.08. C 82.06
1d ae SithoT H 41.14

The compound was found to be dextrorotatory. For the specific

rotatory power in a chloroform solution [@]p = 81°.1 was found.
As stated above, the substance melting at 235° when boiled with
1) B. B. 37 (1904) S. 3443,

(545 )

alcoholic potash yields acetic acid, which was converted into the silver
salt. A silver determination gave 64.2 °/,, theory 64.67 °/,.

The alcohol formed on saponification was a colorless substance
erystallising in long, thin needles and melting at 195° (corr. m. p.
197°—197.°5).

The elementary analysis (with lead chromate) gave:

Calculated for C,,H,,O.
C 84.27, 84.12, 84.32 84.50
El Oi ld Oda 99 11.76

This. aleohol has also a dextrorotatory power. In a chloroform solution
it has [e|p= 88°," and in a benzene solution [@]p = 98°.

On treatment with benzoyl chloride and pyridine, the alcohol readily
yields a benzoate which erystallises in beautiful rectangular little
plates and melts at 230° (corr. m.p. 284°—235°).

After perusing the literature, it now appeared that: the aleohol
melting at 195° is identical with @-amyrine which occurs in elemi
resin and has been investigated and described with great care by
VesTERBERG '). Not only do the melting points of the alcohol obtained
from Payena Leerii-gutta percha and ‘djelutung’, of the acetate
and the benzoate agree perfectly with the melting points determined
by VesterBerRG for $-amyrine and its acetate and benzoate, but in
addition the values found for the specific rotatory power of the alcohol
from ‘‘djelutung” and its acetate differ so little from those which he
states for B-amyrine and its acetate *) that the difference may be safely
ascribed to experimental errors caused by working with dilute solutions.

8-Amyrine has also been found afterwards by Tscuircn *) in the
resin of Protium Carana. It is stated, however, to differ from the
common p-amyrine by being optically inactive, which seems some-
what strange. It should be remarked, however, that the cinnamic
ester of lupeol described by Tscuircn *) about the same period under
the name of erystal-albane was also declared to be inactive, although
we have found this substance having a decided dextrorotatory power.
A further investigation is therefore a desideratum.

Marek °) has obtained from the milky juice of Asclepias syriaca a
substance melting at 232°—233°, the melting point of which could
be raised by repeated erystallisation to 239°—240°. Its analysis led

0?

1) B. B. 20 (1887) S. 1242; 23 (1890) S. 3196.

2) VesTeRBERG states for g-amyrine (in benzene) [z]D = 99°.8]
for the acetate (in benzene) [z]D = 78°.6

3) Arch. d. Pharm. 241 5. 149.
4) Ibid 241 S. 483.
5) Journ. prakt. Chem. Bd, 68 (1903) S. 385 and 449,
38*

( 546)

to the formula C,,H,,0, and on saponification it yielded acetic acid
and an aleohol melting at 192°—193° having the formula C,,H,,0.
The benzoate from the alcohol melted at 229°—2380°.

It can hardly be doubted that Marek has been working with the
acetic ester of B-amyrine. Fortunately, he has not given a name to
the product isolated by him, and hence, has not unnecessarily
increased the already existing confusion.

Undoubtedly, the enormous number of substances said to be obtained
from different resins and milky juices will, on closer investigation,
be reduced to a more modest number and it will often be shown
that pure substances described by different names are one and the
same, but could not be identified owing to incomplete description.
In other cases, names may have been given wrongly to mixtures or
impure substances.

Although it may seem superfluous, it is as well to again point
out how necessary it is, when investigating a natural product, to
purify the components as completely as possible, to filly describe
the properties and particularly to introduce no new names unless
one feels certain of really dealing with a new product.

A short time ago, TscuircH') communicated the results of an
investigation of the components of Balata. From this was isolated a
crystallised substance called «-balalbane melting at 231°, the analysis
of which led to the formula C,,H,,O,

(found C 81.19 H 10.38. calculated C 81.382 H 10.64).

No acids were found by TscurrcH on saponification with aleoholie
potash as he only looked for crystallised acids *). This made one of
us (C.) think that Balata might perhaps also contain acetic esters
and that the @-balalbane might be identical with B-amyrine acetate.

It was not difficult to isolate by Tscuircn’s method the product
melting at 231°.

By repeated recrystallisation from acetone, the melting point rose
to 285°. On saponification, acetic acid was obtained, also an alcohol
melting at 195°. Ester and alcohol mixed, respectively, with 3-amyrine
acetate and B-amyrine gave no lowering of the melting point, so
that «@-balalbane is nothing else but s-amyrine acetate; the name
a-balalbane may, therefore, be struck out.

University Org. Chem. Lab., Utrecht.

1) Amn. d. Pharm. 243 (1905) S. 358.

*) Tscuircu comes to the conclusion that there exist gutta perchas which yield
no cinnamic acid on treatment with alcoholic potash, but I have demonstrated
this fact previously (B. B. 87 8. 5454), (v. R.).

Mathematics. — “Zhe quotient of two successive Bessel Functions.”
By Prof. W. Kaprnyy.

If Jz) and Jz) represent two successive Bessel Functions of
the first kind, the quotient may be expanded as follows:
P+1(2)
Iz)
Of course this equation holds for all values of z within a cirele
whose radius is equal to the modulus of the first root of the
equation /*(2) = 0, zero excepted. Euner and Jacopr have determined
the first coefficients of this expansion; we wish to determine the
general coefficient.
Starting from the known development

=fiet fe + fet

T(z tz 4
(2) 2 a1) — if
ae)
2(v-+-3) — ete.
and putting
= —— & 2(p + p) =a,

the question reduces to the determination of the general coefficient
in the following equation:
v
a +a
a, + ete.

=f,« — fe? + f,2* — ete.

P,,
Let — stand for the approximating fractions of the continued

n

fraction in the first member, and let

Qonti =v, +r,e¢+ria? +... +r, 27

Qn =a +e t pet... tune
A, 4,e fave? -t... tA etl
Qo = eee nee eee to es Te yy ed

Qon—1 =

Qn+2 = &, + q v sic te + oa
Qtr =s +8,¢+...+ 8 25

where
nr n
r= 5 + 1 == z
when » even, and
n+ 1 n+1
r= peorr s=- 9

when nw is odd, then we find

(548 )

att! a," a2)... dn? Anti BaF 1)jh lass os, geen
An—2 Hy 0 bol Ono ONO)

An—1 Xn—1 0 0 sane 00

, ue ; ‘ n—1
In this equation stands for >-— 1 if m is even and for a

when n is odd. If now we replace a, by 2(»-+ p)= 2b, we
obtain the following results. Firstly

O2n-- 1 Bont hn sere cet SO pe batt fati=

2 2 ae
a %,' u! » 6 oly
a, 2%, ts OOO es
an’ tn bn Srao. ny
PENS 2 2 9 2
= x m . .
me » Aan Xn n : 0 :

Aesth tdtent) sel) < oca.g )

an—1 Xn —1 0 6 6 on)
if n is an even number, and secondly

Q2n+4 1 b+! bt OOeo Ona bn+s eae le res es i —

2 2
ay %,' | oe
ae an on sees
! ! ! !
Zeit Cone. | Cee 0 tron
Boot 2 ee 2 2
= (=r oo 3 eae
an+1 ntl r+1 . fr +1 |
2 2 2 2
Ane Oe Wn = 08 eyo ommes 0
An ee ene wad

( 549 )

if 2 is an odd number, where
_. (2n — p— 1) (2n — p — 2)... (2n — 2p)

7 pl = * Oye p42. On —pi— i
: (2n — p — 2) (Qn — pp — 3)... (2n — 2p — 1)
xp = pl bn —p +2... b2n—p—2
. 2n — p — 3) (2n — p — 4)... (2n — 2p — 2
ty = ‘ = pl bn —p+e2... bon—p—3
“4 ao — pi 1) Gap). = ( — 2p 2)
Le p!
en
(2n — p — 1) (2n — p — 2)... (2n — 2p)
An => pl == = by +1 . bon p -1
(2n — p — 2) (2n — p — 8)... (2n — 2p — 1)
Xp = p! by+-1... b22—p—2
(2n — p — 3) (2n — p — 4)... (2n — 2p — 2)
by = p! by +-1.... Oan—p—s

(n—p+1)(n—p)...(v— 2p 4 2)
SS == pl =
It is of importance to remark that
An Nbn, ne Its peo =) (NiO ah os 9 = ete,
and that the determinants in the second members of the equations
(I) and (II) after the substitution 6,=»- p, are respectively poly-

; n(n — 2) (n— 1),
nomia of degrees i and i in »,

Ep

Meteorology. -— “On frequency curves of barometric heights.” By
Dr. J. P. VAN DER STOK.

1. The records of barometric heights, corrected for temperature,
observed at Helder three times a day during the years August 1843
to July 1904, have been chosen as an appropriate material for this
inquiry into the nature of barometric frequency curves. The number
of observations for each month amounts to :

January 5673 July 5673
February 5169 August 5766
March 5646 September 5560
April 5490 October 5766
May 5673 November 5580
June 5490 December 5766

Total 67 252

TABLE I. Frequencies in 10.000 of deviations of barometric heights ; positive and negative being taken together.

| Nov. May
e J sal Febr. | Apr. May June July } Aug. | Sept. | Oct. Nov. | Dee. — renee =e
« | | Webres| face cme Aug.
— 0.5 mm. |] 381 420 4A7 493 612 705 767 704 559 4A9 357 B77 || 1384 472 697
b— 41.5 680 837 798 1028 1176 1486 1493 1405 11410 865 726 755 | 749 950 1390
5— 2.5 7TA2 7179 821 1040 1170 1367 1456 1369 1048 861 750 706 | 737 943 1340
5— 3.5 735 752 841 4010 | 4164 | 1191 1285 | 1292 | 4000 837 737 680 | 726 922 4233
5— 4.5 703 799 826 893 1020 1110 4124 1156 950 771i 684 662) | 742 860 1102
.b— 5.5 679 775 702 907 952 933 998 1016 876 775 686 724 | T16 815 975
5— 6.5 660 683 751 825 809 804 807 806 783 801 7415 660 | 679 790, 806
5— 7.5 647 615 637 4) 733 767 716 602 651 Yblp} 728 655 601 | 630 703 O84
5— 8.5 636 605 606 633 607 572 428 509 684 638 668 | 637 637 640 529
5— 9.5 564 564 579 533 486 402 305 393 550 582 610 553 575 561 396
5—10.5 528 493 532 459 391 262 933 230 468 | 574 557 514 523 508 | 979
FA 498 425 459 362 261 171 179 166 351) lee 452 546 73 ASD 406 194
5—12.5 424 353 406 981 499 105 128 441 983, 347° | 460 4A 420 329 | 436
.5b—13.5 338 342 341 995 426 66 78 73 203 354 | 395 4AD 372 280 8G
5—14.5 267 302 249 | 166 73 4A 42 48 123 27 yale 322 301 199 D1
5—15.5 243, 270 935 102 59 32 28 98 88s 226 | 249" | 967 163 37
.5—16.5 238 213, 238 91 60 22 22 AI) BOM Asse) e495 3 929 128 29
.5b—17.5 995 154 452 BY) i BY) 9 13 9 Sonn |pedd Sree) esd 183 91 16
.5b—18.5 188 127 99 42, 12 3 7 & Sie I AB ly ality 144 6 8
319.5 129 | 116 84 37 10 3 5 4 99 | 43 | 449 | 490 48 5
5—20.5 94 80 Bree 08} 7 3 5 17 a 86 | 88 38 4
5—21.5 76 64 29 Avi 4 4 8 30: |) 62) || 68 a 2
5—99.5 67 56 | 42 12 3 4 40‘) 46 1° 44 56 20
.5—23.5 69 35 16 8 ry ally = -OF8) a) 12
.5—24.5 59 419 18 5 4 10 16 24 9
.5—25.5 42 93 13 7 3) 414 16 | 29 9
5—26.5 29 18 11 ) 4 Odum 19 5
5—27.9 29 29 6 9} 3 9 14 eA 5
.5—28.5 A 14 44 4 Dien| Wy) 17 3 15 4
5—29).5 8 14 (oj; 4 2 2 10 10 10 3
Pan) apa A 9) | 2 | 6 A 9 ey
10 41 3 || 2 a} 6 1
6 a 4 | 2 5 i) 1
4 | 3 De) 3 3
1 1 | | | 2 4
A Or} | 2 4
»—36.5 4 On| | 3 2
y—37 .5 3 0 0 4
27 4 0 | 0 0
38. 4 1 | 5 2) |

(3 1 2 S18 Se She = eae | ee Se ea)
y Op =—|Se= 1h =e = 0 | ¢ are 1s —
inclanvaciroeanlAOVieete eTeem—sl Gis atoh| Wy ot Waele eke o/88 ahcteelecimect
HA @) St] GG Se ee 0 le —|y —1]%3 —
WeeSnecira Mel SCheicte nGwcrel) Pho esty ts ae SNe I)
iceman EC 16h Sh ectt| <6) 0 Sia | ha— Ble
SCR emn OCS asthe Ope sar 0) Be Fam etch —— |e ote Wee
Viet: Clee eres | VP NOW ath 8 fa) bee Lest 0 eG a
SP ap SS a hs a ec aa b= a |
Cece oh Ge OS OG =a Gi Gee) 8 Se: NO tS St ae | ie s=
0 Ce en Cee OTe Omer st-81 OC. ai. at IE Gi 1n0 Ca ces Or — | ‘i
a 99 — [ES = bik tg NS ea KS Sele ae a | --
0 Ge = y OR Se ib SNe hao ih Se SS a a ae | aE
gb + 6 = €¢ PAP NA Oa = es | Se ae aL “ear |i = eS oe |
Cp — Cm VS Cpa be, meas qe gaa re == 1.0) if he) io el ae 1a |
V7 = (Sh eae uS Seite se CS — 16 SG 1) OR S08 Se) Ss ep) oy —
ote oor + LE Pissing aste uRGO) crate Olesen ine al (= | C= | (Gp 6 SP Ss ae i
Gi == 89 + Toba IVS | SS su Lr |06 €b — | 9F 6h ¢ rf | Gy 4 S al
ie = Gob + WG ae | Le +. | €th sae | 87 Soar | Gp || ts Tae Cleat Ochs BS + |
1 — 976 -F Loy OS ey? ale vO 9G a Tg = 1) a5 fie ile | 88 SS | 78
8g HE gor + | 9cr €G aint | OL 0S VE Go ae ie = || GS a Le v6 + | OF il |
6L + LOD + | ‘VEG OD sips Gh igen |eSemea | V6 al | 9F + | 1S — | 08 Vae Ae | elie |
gel. Xa Sr Gp On aries a LY CPataleSGi—NGLete | 88 aL Cis OGi= | | 0g
WY a Ghat GS al BS Wa oe Peo) = Ih Gp SoG. ae || 2 OP SN The ae |
a) 9 — 6 CES || UG Ws | i4e LOS G16. S| G8 kG Se a8 Seis 4 ,—
ho Hay | Wb — || CR A oll dh Oh a ie Sp Wh a ib |) i al Bb i
as Se ee OGY seed Glee ERO aA CO per ROCE SOURS Weel Glen Cita Chet lnGpe——«| =
~~ Need | ORG hae | Oil OO eee FOO OS a NOE al SYactet| Ole 0G ilnSle—N Sh — a
6 (hi = | SSIS Sl OP SP A 2 Oe a 6 Se NS) | Oe =
Qh) == ep S| ip a A Ne SH SS ie SN tg 4) OS S| BS iat
BV | -so9—-ydag | “142d
— jcady—-yor] . :
APIN “AON, ‘00, “AON ‘—pO ‘ydag ‘Say Ane oun Av ady Yate yy, “AQOuT “ue 2
“SUING

"SUOSVOS OY} OJ YOO'OF Ul ‘yWUOUT ATOAa OJ QOQ'OL Ul
‘Mx [eIUOUOdXs 0} SuIploOdV poye[Nowo “nbeay puv setouenbaay podAdosqo usamyoq ‘q—-AA ‘SooUDdIOKT “TT WIV

oe

‘TABLE III. Skew-differences, /—N, of positive and negative deviations.

| | sip | | =i Sums.
£ Jan. | Febr. |March| Apr. | May | June | July | Aug. | Sept. | Oct. Nov. | Dec. Nov. | Mrch.—Apr. May
| Febr. Sept.—Oct. Aug.
5 6 a B00 10 16 go | 43 4|— 46 23 12 15 5A 7 140
3H) 10;— 3 55 | 106 18 139 114 77 48 71 22 62 91 280 348
a5 95 | 56 81 | 122 54 | 445 79 164 | 68 37 | 69 60 20 308
5 93 75 16 135 4h | 444 75 92 90| 61| 50 20), 288 302
5 95 89 42 | 147 | 80 | 79 136 50 86 57 | 86 | 60 | 330 332
5 418 105 7A GPA 61 | 78 133 126 85 83 | 94 86 400 368
5 433 | 197 M4) 454 37 48 74 97 95 | 438 D1 5 356 395
oh) 102 83 16 113 43;— 2 44 | 4A 138 94 60 89 | 334 4AA
5 198 | 408 13 23 26 | — 20 5| 49) 444 62 | 104 97 437 22
By. | 74 | 77 80 | alas 49} — 40) | — 31-|) — 98-7! 38 | 4O 109 | 108 | 368 173 =
Brille 79) Hee 83 75 | — 56 19 45/—31/—42| 49 60 | 4154 75 334 98 =
pele ae | a5 BA | = 45) —47 | — 41 | — 48 | — 45 | 3 3| 4142 ATs 5 964"| S45 =
ata) 50 3p 17 25 6 | — 42 | — 46; — 33! — 9 Th 93 57 232 4O —
(3) {| - SB 12 2 | — 14 | — 19 | — 35) — 30| — 22, — 48 35 20 38 103 3 hy
5 | 4 34 7 | —46-) — 43) |’ —7 82: — 96) — 48+] —'62 |) — 44 | — 33 39 | Sie 2S 57, 2 tw
5 4h | oA 24 | — 33 | — 38 | — 20 | — 22 | — 12 | — 34 | — 57 | — Bt 36 | 90: -="100 —
5 | 5 9) On| ORY = BUN eae Wf ee giB yal an 0 ee Beaty eee By 235) adder GG =
Bir | 96 | = 950 49) 296-49 3 7 einer Ten rea ee ON a 7a | eed (7a | ee =
5 | — 9 D0 a Oe | ON te Sa oem OOM ee OSn l= OO ist One Gill ena gS —
Ais) — 40 | — 14 | — 39 | —19 |} — 7 — 3;— 5| —17| — 33 | — 26 | — 27 | — 107 | — 108 ==
5 | — 32 | — 26) — 23) —17) — 4 — 4)}— 8&8} —16 |) — 38 | — 30! — 196 — 64 —
Ay i Zl, || eS) Ne Oey |), eae S88} — 1/—10/—16 | — 31 | — 38 | — 142 ae it —
5 |— 47|—29}—16|— 8 es 22663) fc gf NB Oey | ee pl es VN
5 »— 389) —19) —18 — 5 [Paw ean OME —— Nd Gat OO Nea OGel, Mee mar
5 OS Ot altte |e — 3!—14) — 16! — 30 | — 107 — 37
5 SSO) |) S35 Gl |) — 4/— 2] —10} — 20} — 75 49
5 — 99) DB 16 | "2 ee eet Oe | — ae ra Ol ee 8G = 9()
5 == 19" | es I leh ee SI — 2/— 2; —17| —13 | — A — 16
S50 Sil da 6s) = 48 | | — 2|}— 2}—10/—10}— 39|) — 44
“yo fea | ee), | 9) — 6|—12}— 36) —-44
al) Sai es 8) | ==) 2 | — 35/2 96, Zo s3
SG Se RS | 2 5 | 90) |) 2 4
a5 — 4/— 3 | | — 2}/— 3/— 12
eStart ee = VE |
SS a) 0 = Oe |
jes 2 0 | See |
5 |— 38 0 | Onl =S.53
5 |— 1 0 Oe
— ij;— 1 ea

( 5538 )

In registering the observations the decimals have been omitted, so
that the number of occurrences corresponding with a height of P
mm. includes all values between P + 0.5 and P— 0.5 mm.

Owing to this simplification the amount of labour is less than
would appear from the great number of data. The next work to do
was to multiply the frequency numbers with a factor such that the
total number for each month amounted to 10.000. The frequencies
thus obtained correspond with expressions for the probability of
occurrence expressed in 10.000"s parts of unity. Then the average
height was calculated and, by means of simple, linear interpolation
the whole curve shifted in such a manner that the new frequencies
correspond with deviations from the average value expressed in
multiples of whole numbers. This has been done not only with a view
of abridging the computations of the moments of the second and
third order but principally in order to obtain an evaluation of the
skewness of the curves, which may be defined as the inequality of
frequency for equal positive and negative deviations from the arith-
metical mean. If of such a series of data the frequencies corresponding
with equal deviations are taken together, no account being taken
of their sign, the skéwness is eliminated, and the numbers obtained
in this way may be considered as belonging to a symmetrical curve
(Table I).

For this curve we calculate the factor of precision (stability) and
investigate in how far the actual curve agrees or disagrees with the
curve of the normal exponential law (Table II).

As has been mentioned above, the inequalities of frequencies for
equal deviations of opposite sign have been taken as a measure of
the skewness.

Tables I—III show, separately for each month, the sums and differ-
ences thus formed. The numbers of Table I added to those of Table III
will give twice the number of frequencies corresponding to positive
deviations, their differences being twice that corresponding to negative
deviations. The values given for Winter, Summer and Spring-
Autumn are obtained by taking together the corresponding numbers
in the same Tables; consequently they are not quite identical with
the numbers which would have been obtained if the frequencies for
these seasons had been calculated from the absolute heights, instead
of, as has been done here, from the deviations; in the latter the
annual variation has been left out of consideration. The annual varia-
tion, however, being very small, this will not influence the results
to an appreciable degree.

2. Table IV shows the results of the treatment of the frequencies
given in Table I, as indicated. If the deviation from the arith-
metical mean is denoted by «, then:

jee ea 1 1 2 M?
Me —— , d= ——_, h= , t= , x= —_..
n—l1 n My2 OVYna oe

TABLE IV.

uM 3 h eee
Jan. 10.261 mM. | 8.272 mM, | 0.0689 | 0.0682 | 3.081
Febr. 9.522 7.597 | 0.0743 | 0.0743 | 3.444
Mrch. || 8.969 7.194 0.0788 | 0.078% | 3.409
Apr. 7.280 5.864 | 0.0971 | 0.0962 | 3.083
May 6.218 5.022 0.1136 | 0.419% | 3.067
June 5.391 4,322 OM312> |) (OM305" 1) 3x412
July 5.276 4.469 0.4340 | 0.4354 | 3.204
Aug. 5.374 4.300 | 0.4816 | 0.4312 | 3.495
Sept. 6.972 | 5.602 0.1014 ., 0.4007 | 3 098
Oct. 8.372 | 6.832 | 0.0845 | 0.0826 | 3.003
Nov 9,490 9.006 | 0.0745 | 0.0725 | 2.974
Dec. 10,085 8.173 | 0.0701 | 0.0690 | 3.045

From this summary it appears that the frequency curve of baro-
metric heights, as derived from observations made at Helder, shows
systematic departures from the normal curve corresponding to the
exponential law. For all months (except February and July) h is
greater than /’; in February these factors are equal and the curve
is nearly a normal one, in July 1’ >h.

In agreement with this result the calculated value of a is always
(except in the two months mentioned) less than its true value; the
departures from the normal law are greatest in winter, smallest in
summer time.

It may be noticed that the departures from the normal curve,
given in table II, are generally of an opposite sign to those which
are found in the great majority of series of errors: whereas for
the latter the rule holds that small deviations occur oftener than is
required by the normal law (in which case /' > and a cale. > 2),

id

(555°)

here the reverse obtains, the frequency of barometric heights showing
a deficit for small and a surplus for moderate deviations.

In an earlier paper (this volume p. 314) I have shown that, in
taking together series with different factors of steadiness, each series
occurring with equal subfrequency, we must expect to find too great
a number of small deviations.

From this follows the apparently somewhat paradoxical conclusion,
that a sum of frequency numbers as those of barometric deviations,
all showing negative differences for small deviations, may, when
taken together, lead to a resulting curve in which these differences
have vanished or even turned positive.

This conclusion is of some importance because an investigation
into the frequency of barometric heights, in which the different
months are not treated separately, may lead to normal curves (the
skewness being left out of account) whereas in fact no normal curve
exists and appears only as an artificial consequence of the combi-
nation of incomparable frequency numbers.

The exceptional behaviour of the months of February and July might
then be explained by assuming that the different series of barometric
curves corresponding with different winds (barometric windrose) are
more differentiated in these two months than in the other ones.

A second remark is that frequency numbers as given in Table I
cannot be accepted as a measure for the variability of the atmo-
spheric pressure in the course of a month, at least not if we adhere
to the conception of this variability as generally admitted.

On the one hand we have here to do with the superposition of
two kinds of variability, 1st the secular variability as shown by the
variability from year to year of monthly means and 2°¢ the vari-
ability from day to day, which might be called the interior variability
for the month in question ; it is the latter definition which corresponds
with the usual conception.

On the other hand, daily means or observations taken at fixed
hours are by no means to be regarded as being independent of
each other.

The questions, therefore, arise: how can we separate the two kinds
of variability, and to what degree are daily mean values of baro-
metric observations to be taken as dependent upon each other in
the different months.

For a knowledge of the climate of a place the latter question is
of importance ; it might also be formulated thus : what is the average
duration of a barometric disturbance, a question which can hardly
be answered by means of direct investigation.

( 556 )

TABLE V.

| spe | ee L. | tate | : ae as Tin pas

| | u —
008s sis | 496° | +19 | 14.5—45.5 118 472 | —%
05—4.5 || 4025 996 | +29 | 415.5-16.5 125 134. | <9
4.5— 2.5 || 4004 962 | +39 | 46.5—17.5 100 404 |) ae
2.5— 3:5 256 | 985 + | 17.5—18.5 75 79 =
3.5— 4.5 888 875 | +413 | 18.5—19.5 58 50° |. <a
4.5— 5.5 846 826 + 20 19.5—20.5 43 | AS — 2
5.5— 6.5 me | 757. | — 4 || 0524.5 33°. |" 99.1) = ataaae
6.5— 7.5 oa | oe | — 2 | o1.5-99.5 25 93° -| Se
i585 591 | 608 | —47 | 22.5—93.5 21 46. |. oa
8.5— 9.5 519 526 | = al) 28o—aeb5 13 49. |) es
95-105 || 497 | 459 | —92 | 94.5—95.5 13 7°) eae
105-115 || 363 | 98e |; 121 3) 95 866 10 5 | “ee
44.5—12.5 304 | 322 | —48 | 26.5—97.5 7 3. | ace
12. 5—13.5 Oye 968 | —24 | 27.5—298.5 6) 2|+4+4
13. 5—14.5 49° | 216 | —25 | 98.5-ete. 13 4 |aaeg

The first problem is identical with the calculation of the prob-
ability of an event a+ 6, when a and 6 follow the normal law and
are independent of each other.

This problem of the superposition of two laws of errors has been
already treated by Brssrn*) and subsequently D’OcaGNE *) gave a
general solution for the superposition of several groups of errors.

It appears that, if 7 be the factor of stability of the secular and
h, that of the interior variability, the resulting deviations also follow
the normal law, the new factor 4 being determined by:

Th,
V 7? ayes

From the values of 2 given in Table IV and those of H caleu-
lated from monthly means we can, therefore, deduce that of A, :

h —_

1) Untersuchungen tiber die Wahrscheinlichkeit der Beobachtungsfehler. Astr.
Nachr. XV, 1838.

2) Sur la composition des lois d’erreurs de situation d’un point. GR, Acad.
sc. CXVII, 1894.

( 557 )

ut! Hh
SA VAEEe Se

By the following reasoning the second problem may also be easily
solved.

The total mean value for a given month, as calculated from
monthly means, must be the same as that deduced from J cor-
responding daily means.

The mean error (incertitude) of the total mean is, as monthly
means may be considered to be independent of each other:

a e?| M,
n(n—1) Vn
For the mean incertitude deduced in the same manner from
observations made three times each day :
M,
VN
too small a value would be found as these observations are certainly
not independent of each other; therefore, if the number of obser-

vations which, on the average, constitute an independent group be
called p, we must have:

ee Se el)

: M, M,Vp

If we wish to express the average duration of a disturbance D
in numbers of days, we have, in our case:

IE N NM? N

ee re ee <a San ove a2
eee 3in eH 3in (?)

Table VI shows the values of the interior variability A, thus
caleulated by means of form. (1) and the duration D of a barometric
disturbance.

It appears from these results that, on the average, the duration
of a barometric disturbance at Helder is in:

Winter 6.90 days
Summer 4.89) -,,
Spring-Autumn 6.04 _,,

or in round numbers resp. 7, 6 and 5 days in winter, spring-
autumn and summer.

3. It would perhaps not be impossible, and it certainly must be

( 558 )

TABLE VI.

He. “Ni bee hy, D
| |
| = Ee ee
Jan. 0.4411 0.0689 0.0787 71.52
| b
Vebr. 0.1458 0.0743 | 0.0864 7.46

=)

March|| 0.1682 | 0.0788 | 0.082 | 6.89
Apr. || 0.2105 | 0.0974 | 0.1094 | 6.49
May || 0.3019 | 0.1136 | 0.1296 | 4.46
June 0.3181 0.1312 | 0.4440 | 5.19
July || 0.3392 | 0.1340 | 0.4459 | 4.92

Aug. || 0.3330 | 0.1316 | 0.4432 | 5.00

Sept. || 0.2350 | 0.1014 | 0.1493 | 5.74
Oct. 0.13 | 0.c845 | 0.0996 | 5.12
Nov. || 0.1892 | 0.074 | 0.0810 | 4.81
Dec. || 0.4449 | 0.0701 | 0.0806 | 7.82

the final aim in. inquiries of this kind to come to a rational expres-
sion for the frequency of barometric deviations as a function of the
distance of centres of depressions and of their average depth and
extent but, even if we assume the most simple relations between
pressure and distance of the centrum, we must expect to find rather
complicated, exponential expressions, which can be treated only by
expansion into series. It is, therefore, desirable to summarize the
characteristics of the frequency curve in an empirical formula of
the form :
eH! (A + Be + Cz? + Dei + Ext), . . . . ()

The constants of this formula can be easily determined and, if we
sueceed in establishing a rational expression, there will probably be
no difficulty in indicating their meteorological meaning.

The frequency curve, positive and negative deviations being taken
together (Table I), is then represented by the expression :

ZF = ¢—-B* (A Cz? 4 Ext) 2 ee ee
which represents a symmetrical curve, and the formula for the
differences of Table HI becomes:

Y = 2e—-H" (Be + Da). 3
If, as in our ease, the deviations « are departures from the arith-
metical mean,

: (559 )

es) ie.) ao
F [Zee s=1l 4 [2 CPi) Ta [ee di —— lie
0

0

0
[ee dai== a, ete. ; [re dit =—» ih eee)
0 6
For the determination of the constants of form. (4) we then have
the four relations:
C 13H HT

A = —
a 2F? att 4H* Vn

Vm ES,
samp et

s ree al (7)
Aa Se aa got = pees
QUE ATT Vx

2H? 4/T*

Multiplying resp. by 1, —3,-+3and—1 and adding, we find:

Mal +bH—c=0

—— Eee

bu, SU, 1
a= > SS SS SSS
l,Va uy UV x
or, because :
1 1 1
SSS = =——, Hs = =
Ee ae ot > Ope i? Vx

Ee SH 377.
Tommie

From this equation possible values for H can be derived, but not
in an advantageous manner as the quantities /, 4’ and h" generally
are only slightly different.

In practice, i.e. if we coine to expression (4) by expansion of a
theoretical formula, the problem will probably be less difficult, as
the constants H and A or H and / will not be independent of each
other, and it will be possible to reduce the four equations (7) to three
or two.

In this preliminary investigation we confine ourselves to the most
simple case that H = which, as it will appear, leads to satisfactory
results.

Putting

1 = Oslin hy. eG)

we find:
39
Proceedings Royal Acad. Amsterdam. Vol. VIII.

( 560 )

AYyYar=h(1—3 K)
CVa=12K
EY a= 4 EK Ss. ee
The position of the points of intersection of the observed frequeney
curve with that caleulated by assuming the simple exponential law
to hold good (the points where in Table Il the numbers change their
sign) is determined by the equation :

(A + Ca’ + Ha‘) Ya —h =),

or:
3 3}
ati —_—@a@ Ee Woe een tee 10)
/? are (
0.525 1.651
— a ;
; h - h

In fact Table I] shows that there are no more than two well
defined points of intersection, which justifies the omission of higher
powers than the fourth in form. (4).

Tabel VII shows the values of the constants of (4) and the values
of a calculated with the help of form. (9) and (10).

It is evident that, if form. (4) and the values of its constants
determined in the way indicated give a good representation of the
observed facts, the values of the coefficient must be nearly equal

TABLE VII.

| |
A |
C iD} zy ay

| Calculated, Observed,

| 7 eT
Jan. | 377107 | 381><10—* 227><10-" | — 36><10-" | 7.62 | 23.96
Febr. | 419 | 420 | 0 0 | 7.07 | 99.99
March) 438 | 417 142 — 3) | 6.66 | 20.95
Apr. | 532 | 493 310 — 482 | 5.44 | 47 00
May | 620 | G12 | 662 | — 456 4.62 | 14:53

| | | | |
June | 728 705 | 532 = Arr | 4.00 | 19.58
|

July | 779 767 | — 1581 | + 41008 3.92 | 19).32
Aug. | 734 | 704 588 |= 9320 3.99 | 19.55

| |
Sept. | 560 | 559 | 4QI OS. 5.48 | 16.298

| | |

Oct. | 444 | 419 | 940 — 994 6.21 | 19.54
Nov. | 386 | 357 772 | — 4148 | 7.05 | 22.46

Dec. 377

i

( 561 )

to the frequencies corresponding with the deviations O—O.5 mm.,
as given in Table I, so that the greater or less degree of agreement
between these values may be taken as a criterion for the proposed
assumption 7 = h.

In order to show that this agreement is fairly s&tisfactory, the
observed frequencies between the limits O and 0.5 are given once
more besides the calculated values of A.

If we compare the situation of the intersection points as shown in
Table II and as calculated according to form. (10), we see that the
situation of the first point of intersection agrees well with the
observed facts, but that the second points «@,, as calculated, cor-
respond with greater deviations than occur in reality.

As this second point of intersection naturally coincides with small
frequencies the degree of precision of which is questionable, it seems
difficult to decide whether these differences may be ascribed to
insufficiency of material, to the omission of a possible fourth term in
form. (4), or to an error introduced by the supposition 7 = / ; as the
calculated values of @, are jointly too great, the latter cause has to
be regarded as the most probable one.

4. The fact that in Table IL], in which a measure is given for
the skewness of the curves, except for ¢—0O, only one zero-value
occurs, proves that in form. (5) the addition of a third term is cer-
tainly not required. The calculation of the constants B and Das well
as the determination of the point of intersection 3 can, therefore,
easily be made.

As:

os)

| Yedx = 0

0
we find immediately :
a 3D A "
gg on bo, tee ee)
whereas :
a BieD
Vida = ob Se Sy Se cat eee LD)
pe h'

0
denotes the surplus of positive over negative deviations.
If we take the absolute sum of positive and negative deviations
as a measure for the skewness s:

lid n P

— Oe 2f Y du = [ Fue = | Vidar — vp;
0 0 0

39*

( 562 )

OF:

3
>

str=2 | Vide 0°: "2° 32) anes
e
The situation of the point of intersection 3 is determined by the
equation :
BE Dp? = 0's eo See > ee
By (11) and (12):
B=3h?, D=—2h'r-... » . . one
Boh? = Sin 1. Uae = eee
With the help of (13) we find from these values :
s=p (1 + 4e—h),

t Py 99, Laos a
vd P —= 2a 6 Vt
By means of the values » or s, to be taken from Table III, the
constants of form (5) as well as the position of the point of inter-
section ean, therefore, be determined ; we choose yr, so that a com-
parison of the calculated and observed values of ‘/, or?/, may serve
as a criterion for the method followed in calculating the constants

of the empirical formula.
TABLE VIII.

Calculated.

= Ghserved:

Pee Ee he Se ee
Jan. 707 <40-* | 4505 5<405*| 4015<40- > | = 32$¢4057 7) Saree
Febr. 606 | 4184 | 400 ety | ae
March | 467 923 ears eee OG | 45.5
Apr. | 639 1277 | 484 | — 444 | 12.6
May | 493 | 576 163 — 444 | 40.5
June | 483 | 668 | 249 — 286 | 9r3
July | 486 | 998 | 262 aig | <a
Aug. 426 | 908 | 995 | — 256 ee 8)
Sept dlaaes | 1073 | 4143 et 124
Oct. | 429 | 748 Wan e>) Se 14.5
Nov. 599 1467 100 = 37 | 16.4

Dee. 605 1309 89 — WwW iss

Mean 528 10538

563
( )

The average values of y» and s show a satisfactory agreement
with the form. (17):
s 10538

See 99
Y 528

we

From the aggregate values given in Table III for three seasons
we find:

Sums.

yp t pans s p/n
Winter 3849 1340 5189 1297 oe 2.87
Spring-Autumn 2959 937 3896 9.74 3.16
Summer 2380 747 S27 7.82 3.19

For the values of 8 in these three seasons :
Observ. Tab. II] Cale. Tab. VII

Winter eG 17.05
Spring-Autumn 14 13.68
Summer 95 9.55
Anatomy. — “Anatomical research about cerebellar connections.”

By L. J. J. Muskens. (second communication). (Communicated
by Prof. C. WINKLER).

A comparative examination into different species of mammals I
have thought desirable in order to get information about the course
of the axis-eylinders arising from the cortex cerebelli. The develop-
ment of our knowledge in this matter in the last 15 years has
resulted in that at the present time the following question has been
placed in the center of discussion: do the strands of fibres, which
form the superior Crus cerebelli, arise from the cortex cerebelli stric-
tiore sensu or have we to regard the basal cerebellar nuclei as an
undispensable intermediary for all these cortico-fugal nervefibres ¥ On
the one hand we find in some rodentia in the lobus petrosus cere-
belli exclusively-cortex and white matter (squirrel), on the other hand
we find in others (rabbit) equally a part of the nucleus dentatus
situated in the pedunele of that lobe. In both animals the lobus
petrosus is situated in a separate bony hole. We find in this lobe
therefore a very fortunate opportunity for operative procedure therein,
leaving the other neighbouring central structures and also the semi-
circular canals intact. We can here in a comparative physiological
way find an answer on the above question and at the same time
avoid a large cranial aperture,

( 564 )

Since Maron stated, that after large lesions as hemi-exstirpation
of the cerebellum a number of nerve-strands degenerate up to the
mesencephalon and down to the spinal cord, it is notable, that subse-
quently Manam, Ferrier and Turner, R. Russeut, THomas and
especially Props and vAN GrHucHTEN have more and more directed
their attention to smaller and smaller lesions, so that it became
more and more clear, that most of the degenerations, found by Marcut,
were caused by affection of neighbouring parts. Finally have CLARKE
and Horsiry recently succeeded in stating definitely, that all fibres of
the superior crus cerebelli do not arise from the cortex, but from
the basal nuclei. Their material was larger than that of any of the
precedent investigators and only very limited exstirpations, mostly
without any lesion of the nuclei, were used. If the lesion was
limited and the cerebellar cortex exclusively hurt, never the dege-
neration was found further than the nuclei. They stated moreover,
which parts of the cortex are directly connected with special parts
of the basal nuclei.

Independently of this result the examination of my own material
(experiments on the lobus petrosus in different rodentia) tends clearly
to reinforce their conclusion. Whereas in the case of the squirrel (where
only cortical and white matter in the lobus petrosus cerebelli —- inex-

actly called floceulus — can be hurt) the degeneration stops short in
the lateral part of the dentate nucleus, we find in the rabbit always
a part — especially and exclusively the middle third part of the
superior crus cerebelli on cross section — degenerated. These dege-

nerated fibres could be followed in the series of sections up to the
lesion. Here, in the case of the rabbit, we had removed a number of
ganglioncells, situated in the peduncle of the lobus petrosus and
being contiguous to the nucleus dentatus.

We see therefore that as well the Marcui-work in the same spe-
cies as experiments in kin animal groups lead to the same
answer to our question viz. that only the ganglioncells of the basal
nuclei and not the cells of PurkinsK, have to be regarded as the
origin of the degenerations after the cerebellar lesion. The last reserve
left in this matter by Epicrr can therefore, so it appears to me,
be abandoned.

In accordance with the above investigators and also with my
former communication in’ These Proc. VIL p. 202 about experi-
ments in rabbits I could not find in the spinal cord of .the
squirrels, examined, any degeneration. Regarding the middle cere-
bellar peduncle, the relations are more complicated and need further
research.

Chemistry. “On the simplest hydrocarbon with tivo conjugated

systems of double bonds, 1.3.5. hevatriene.” By Prof. P. VAN
RompurGn and W. van Dorssen.
In 1878 Tinpen*) advanced the hypothesis that the terpenes might
be derivatives of a hydrocarbon of the formula :
(OlRL, = (Ciel —= Cleh = Ola CH= CHE

At the meeting of the Assoc. franc. pour lavane. des Sciences in

2

Paris 28 Aug. 1878, Franxcuimont pronounced the same opinion and
suggested that this compound might, perhaps, be obtained by elimi-
nating of the two chlorine atoms from acrolein chloride. The efforts
made by Gne of us (v. R.) many yearsago to prepare that hydrocarbon
in this manner did not prove successful. The researches on terpenes
Which afterwards definitely led to the result that, in the ease of
these substances, we are dealing with cyclic compounds made the
above cited hydrocarbon recede into the background.

The views of Trie_e on conjugated systems of double bonds, and
the researches originated therefrom, in addition to the studies on the
aliphatic terpenes myrcene and ocimene, hydrocarbons in which the
existence of three double-bonds has been proved by different inves-
tigators, have again drawn our attention to the 1.3.5 hexatriene,
because it would represent the simplest hydrocarbon in which oceur
three double linkings that also form two conjugated systems.

One of us (v. R.) has pointed out previously that one of the
methods which might lead to the desired product consists in the
action of metals on 3.4 dichloro-1.5 hexadiene.

The investigations of Greiner *) have acquainted us with the ana-
logous bromine compound which is formed by the action of phos-
phorus tribromide ons. divinyl glycol. We have treated this substance,
prepared according to-GRriNER’s directions, with metals but have not
yet succeeded in preparing the hydrocarbon in that way. There was
however, another way still at our disposal to gain our objeet, namely,
by starting from s. divinyl glycol and converting this into a formic ester.

It is known that the formates of polyhydric alcohols, in which oceur
a QOH-group and a formic acid-residue connected with two C-atoms
linked together, yield, on heating, unsaturated compounds with eli-
mination of carbon dioxide and water. It was now obvious to prepare
the monoformate of divinyl glycol. We endeavoured to do this by
heating this glycol with oxalic acid but obtained, mainly, brownish

4) Journ. chem. Soc. 1878. p. 80.
2) Ann, d. Chim. et d. Phys. [6] 26 (1892) p. 305,

( 566 )

compounds not looking fit for further investigation. By cautious
treatment with formic acid the diformate was, however, readily
obtained (see p. 544).

In order to convert this into the hydrocarbon, a reaction was
applied which one of us had previously used for preparing allyl
alcohol from the diformate of glycerol, and which consists im heating
that compound with glycerol.

And, indeed, a mixture of the diformate of divinyl glycol with
the glycol when heated slowly, first at 165° and then gradually to
200°, evolves carbon dioxide and a little carbon monoxide and yields
a distillate consisting of two layers, the upper one of which consists
of-a hydrocarbon.

The triformate of glycerol, like the diformate of diviny] glycol,
may be distilled without notable decomposition by heating it some-
what rapidly at the ordinary pressure. Recently one of us (v. R.) found
however that it is decomposed by prolonged heating at a temperature
a little below the boiling point and it then yields the same decom-
position products as the diformate of glycerol.

If now the diformate of s. divinyl glycol is heated at 165° and
the temperature allowed to rise very slowly, an evolution of gas is
observed and in the receiver is collected a liquid consisting of two
layers. The upper layer again consists of a hydrocarbon identical
with the one cited above.

Probably, the simplest way to explain this reaction is to assume
that the diformate contains a litthe monoformate which is decomposed
in the desired sense, with formation of water which in turn regene-
rates. monoformate from the diformate. Finally, a residue consisting
of glyeol (respectively, polyglycols) is obtained and in the distillate
a little formic acid is found, besides water, whilst the gases evolved
consist of carbon dioxide and carbon monoxide. The last method
appears to give a better yield than the first one.

The hydrocarbon formed is separated and distilled, the portion
distilling up to 95° being collected. It is then dried over a piece
of caustic potash, which also removes traces of formic acid and then
rectified a few times over metallic sodium.

It then forms a colourless, strongly refractive liquid with a slight
pungent odour; in contact with the air it appears to slowly oxidise.
The boiling point lies between 77°—82°, the main fraction boils
between 78°,5—80° (corr. ; pressure 766 m.m.)

The analysis and the vapour density gave values leading to the
composition ©, H,..

For the physical constants of the main fraction was found ;

{ 567 )

Spec. gr.,) 0,7565

Np, 1.49856.

If we calculate the molecular refraction from these data, with the
aid of the formula of Lorentz—Lorenz,; we find JZR = 31,08,
whilst for C,H, is found J/R= 28.53 assuming that the hydrocarbon
possesses three double bonds, and making use of the atomic refrac-
tions of Conrapy') and the increment for the double bond.

The difference of 2,5 between the calculated and found molecular
refraction is a striking one. According to Brinn *) excesses always
occur with substanees with a conjugated system of double bonds.
In the aliphatic terpene ocimene, an excess (to the extent of 1.76)
is also found, and this assumes an extraordinarily large proportion
in the case of allo-ocimene. *)

As regards the structural formula of the hydrocarbon obtained,
its formation from

CH,=CH—CH—CH—CHB=CH,
|
OH OH
by the elimination of the two OH-groups by means of formic acid
points to the formula:
CH,=CH—CH—CH--CH—CH,
which indeed represents 1.3.5-hexatriene.

A glance at this formula shows that it may appear in two geo-

metrical isomeric forms, namely in the c/s and trans form‘):

CH,=CH—CH CH,—CH—CH
|| and Tl
C,H—CH—CH HC—CH=CH,.

If, with Txrein’), we accept partial valencies the formula of
1.3.5-hexatriene should be written:
CH,=CH—CH=CH—CH=CH,

Unsaturated hydrocarbons with a conjugated system readily take

1) Zeitschr. physik, Chem. 3, 226.

2) B.B. 38, 768.

3) CG. J. Enxtaar, Dissertation 1905, Compare literature on the subject p. 87.

) Probably, the hydrocarkon is a mixture of both. In the fractionation, besides
the main fraction, a distillate could be obtained boiting between 77.5? and 78°.5
(sp. gray 0.7558, nny 1.494 MR 30.8), also a final fraction boiling between
80°—82° (sp. gryg 0.7584, no 1.503, MR 31.2). We hope to repeat the expe-
riment on a larger scale.

6) Ann. 306. 94,

( 568 )

up hydrogen on treatment with absolute alcohol and metallic sodium,
In the reduction of our own hydrocarbon, 2.4 hexadiene might be
expected in the first place, although, a priori the formation of other
hexadienes is not to be excluded. In the 2.4 hexadiene

CH, — CH = CH — CH = CH — CH,,
we have again, however a compound with a conjugated system
which might be further hydrogenated to hexene 3.

In fact, our hydrocarbon when treated with boiling absolute
wecohol and metallic sodium takes up hydrogen. The study of the
product (or products) of the reaction is not facilitated by the contra-
dictory statements found in the literature about the hexadienes. A
future Communication will treat more extensively of this reaction and
also of the original hydrocarbon whose structure we will try to deter-
mine also by other methods. We may state further that a dibromine
addition compound has been prepared melting at 89—90° and a
tetra-compound melting at 115°.

University. Org. Chem. Lab. Utrecht.

Chemistry. — “On the hidden equilibria in the p,w-sections below
the eutectic point’. By Dr. A. Smits. (Communicated by Prof.

Hl. W. Bakuvuis Rooznpoom).

The p,v-sections of binary systems in the neighbourhood of the
eutectic point have been fully discussed by Bakuuis Rooznpoom *); in
this the course of the solubility isotherms in the unstable and metastable
region were, however, not examined. This problem could only be
taken in hand after van per Waans’ paper *) on: “The equilibrian
helween a solid body and a fluid phase, especially in the neigh-
hourhood of the critical state” had been published.

Availing myself of this paper I shall discuss the just-mentioned
problem, and show briefly in what way the stable region is connected
with the metastable and unstable region.

If for the two substances A and £ the volume in solid: state is
larger than in liquid state, these substances will have negative melting-

: é Gy) . : :
point curves, 1. e. 2 will be negative, and the melting-point curve
will therefore pass to lower temperatures with increase of pressure. If

') Die Heterogene Gleichgewichte 2, 159 (1904),
2) These Proceedings Oct. 31, 1903, 439,

( 569 )

this case occurs, the eutectic melting-point curve, furnished by the system
A+ B will generally present the same course. This case is rare.

As, however, Bakunuis Roozmpoom already observed '), a negative
eutectic melting-point curve is also possible, when only the melting-
point curve of one substance is negative, provided the negative course
of one melting-point curve be stronger than the positive course of the
other. To this belong all cryo-hydrate lines.

In the P,7 projection fig. 1 it has been assumed (which, however,
is of minor importance here) that the negative course of the eutectic
melting-point curve results from negative melting-point curves of the
substances A and A.

The particularity attending the negative course of the eutectic
melting-point curve, is this, that a p,v-section corresponding with a
temperature below the eutectic point, will contain a region for S4 + L
and a region for Sp+ L, separated by a liquid region L. The
limits of this liquid region are given by solubility isotherms, which
according to VAN pER WaAats’ theory, are portions of two continuous
curves indicating the fluid phases which can coexist with the solid
substance A respectively 4, and which have been called de solubility
isotherms.

The regions for S4 + G and Sy + Lresp. Sg + Gand Sp + 1
below: the eutectic point being separated by a region for S4 + Sp,
the question which I wished to solve came to this: ‘what is the
course of the two solubility isotherms in the region for Sy + Sp”.

In order to answer this question we first examine what is the
p.e-section which corresponds with a temperature above the eutectic
point, but below the melting points of the two components. The
temperature which I have chosen for this purpose, is denoted by
é, in the P,7-projection. The p-v-seetion corresponding with this is
represented in fig. 2. As van per Waats has proved that the solu-
bility isotherm has two vertical tangents for the case x, < ry, but
only one vertical tangent for the case v, > vy two continuous solu-
bility isotherms with one vertical tangent have been drawn in this
p-#-section; for the one solubility isotherm this vertical tangent lies
at the liquid point “4, and for the other at the vapour point G.
We see further that the branches which separate the liquid region
L trom the regions for S4 + L and S,-+ LF diverge towards higher
pressure. The portion of the liquid-vapour-region 4 —-- G, which may
be realized in stable condition, lies between the two three phase
pressure lines SyQ@L and Spl G. If we now examinea pre-section,

1) Loc. cit. p. 418,

(570 )

corresponding with the eutectic temperature, denoted by ¢, in the
P,7-projection, we get what is represented in fig. 2. The two three
phase pressure lines S4—-- (+ Land Sg + L+G have both
descended, the former, however, stronger than the latter, and they
have finally coincided.

The two solubility isotherms intersect besides in the unstable region,
also in the points G and “4. While the point of intersection ( indi-
cates the possibility of a coexistence of Sy + S, + G, the second
point of intersection 4 indicates the possibility of a coexistence of
S;, + Sp +L, and when at a definite temperature, as is the case
here the two points lie on the same pressure line, this means that
at that temperature the four phases Sy -+ Sz + L + @ can coexist,
provided the pressure be equal to that indicated by the horizontal
line which joins the four coexisting states. At a higher pressure the
regions for S4-+ and S;4-+ L are separated by the triangular
region for L.

In order to get a clear idea of the form which the px-section
assumes at a temperature 7,, lying somewhat below the eutectic
temperature, it is necessary to draw the metastable branches of the
lines for S4+ Lap + Gap, for Sp t+ Lapn+Gipg and for L4 + G4;
as has been done in fig. 1. We see then, that the situation of the
first two three phase lines is just the reverse of that of the stable
branches. For the stable branches that for S;+ Lan + Gz lies,
namely, above that for Spt Lap + Gp, tor the metastable branches
the reverse is the case. If, taking this into consideration, we now
draw the p.-section corresponding with the temperature ¢,, we get
fig. 4, from which we see that the first point of intersection of
the two solubility isotherms has moved upwards, and the second
downwards. The first point of intersection denotes, as has been
said, the coexistence of Sy + Sp-+ G, and the second the coexistence
of S4a+ Sp-+ LZ; at constant temperature these three phase equilibria
are only possible at one pressure, because we have here a system
of two components, hence for pressures between the two points of
intersection mentioned there must be change of the three phase
equilibria into a two phase system, where the two three phase
pressure lines form the limits of a new fio phase region, viz. for
Sa+ SB- :

The second point of intersection of the solubility isotherms which
causes the occurrence of the three phases S4-+ S,-+ Z lies here
in agreement with the dotted line traced in the P,7-projection for
the temperature f, at a pressure below that of the supercooled liquid
of pure A.

(571 )

It is further te be seen in this p,v-section, that the two metastable
three phase pressure lines for S4 + G-+ L and for Sp+L+e@a
lie above the stable three phase pressure line for S4-+ Sg -+ G,
and that the first lies between the two others. At the same time we
see that the character of the solubility isotherms does not change,
the only modification which is brought about for each of the isotherms
compared with the usual case is this that the metastable part is
enlarged.

If we now take a temperature which lies still somewhat lower,
viz. ¢,, we get a p,w-section as represented in fig. 5. All the three
phase pressure lines have diverged, and descended, except that for
Sy+tSp,+ 4, which has strongly ascended. The second point of
intersection lies now, in agreement with what the dotted line for the
temperature ¢, traced in the P,7-projection shows, far above the
point indicating the vapour tension of the supercooled liquid of A.
The metastable part of the two solubility isotherms has greatly in-
creased, and with it the region for Sy + Sz. With further decrease
of temperature the character of the modifications in the p,v-section
remains the same, so that it is unnecessary to examine another.

If we had applied the same considerations to the case that the
eutectic melting-point curve has a positive course, we should, with the
exception of the unstable region, have found but one (lower) point
of intersection for the solubility isotherms, for the branches which
gave a second (higher) point of intersection in the case under dis-
eussion, recede continually from each other.

I have not represented this latter case, as it yields nothing speciai.

The case treated shows once more, how the examination of the
equilibria which are hidden from our eves, may contribute to widen
our insight into those accessible to experiment.

Amsterdam, December 1905.

Anorganic Chemical laboratory of the University.

Chemistry. — “On the phenomena which occur when the plaitpoint-
curve meets the three phase line of a_ dissociating binary
compound’. By Dr. A. Sirs. (Communicated by Prof. H. W.
Bakuuts RoosgeBoom).

1. In a previous paper’) I have already pointed out, that the
interesting systems metal-oxygen, metal-hydrogen and metal-nitrogen,
to which we may still add many of the systems metal-halogen, and
metaloxyde-acidanhydride, belong to the type ether-anthraquinone,

) Zeitschr, f physik. chem. 51, 193 (1905.)

but they are more complicated, because here the components may
combine.

Now from a chemical point of view it is of the highest impor-
fiance to examine also these more complicated phenomena, in order
io obtain in this way a general insight into the phenomena of equi-
librium for the case that compounds are raised to high temperatures,
and placed under such a pressure that critical phenomena are found
with saturated solutions. As yet any insight into. this was wanting.

By bringing the results of my investigation on ether-anthraquinone
in connection with the cases lately discussed by me in a paper:
“Contribution to the knowledge of the PX and the PT-lines for the
case that two substances enter into a combination which is disso-
ciated in the liquid and the gas phase” '), I have succeeded in
arriving at a clear conception of the above mentioned phenomena.

In all the cases which | shall shortly discuss here, | start from
the supposition that the compound under consideration is miscible
with both components in fluid state in all proportions. On the whole
our knowledge as to this is exceedingly slight, nor is there the least
certainty on this head for the substances which I shall adduce here
as examples. 7

2. First of all I shall consider the case, that two substances
A and B yield a dissociating compound A,, B,, the melting point of
which les above the critical temperature of the substance A. This
case is met with in the system CaQ—CO,. If now the solubility of
the compound A,B, in A is still shght at the critical temperature
of A, the continuous plaitpoint curve, which starts at the critical point
of A (CO,) and terminates in the evitical point of B (CaO) will meet
the solubility curve of A,B, (CaCO,) in fluid A (CO%) in two points.
That the point p exists has already been demonstrated by Dr. BicHNER *);
in temperature this point lies only slightly above 81", the solubility
of CaCO, in fluid CO, being still very slight at this temperature

This case has been represented in Fig. 1. The upper half of this
diagram contains the projection of the spacial figure on the PT-plane;
the lower half represents the projection of the tivo phase regions *)
coevisting with solid substance, and the plaitpoint curve. The com-
bination of these two projections seems to me the simplest way of

1) These Proc., June 1905, p. 200.

2) Thesis for the doctorate, 106. (1905).

3) AL first 1 gave the name of three phase regions to these regions because,
though they indicate only f¢wo phases, a third coexists with them. It seems,
however, better to me to speak of teva phase regions coexisting with solid substance,
which term I shall use henceforth.

(573 )

representation for a first investigation of these problems. For the sake
of clearness [ must draw attention to the fact, that in the T-X-pro-
jection the lines aE, Ep, qFE’ and E’c are the solubility curves,
Whereas @E,, E,p, qgF’E,’ and E,’c represent the vapour lines. In
the P-T-projection, however, we get one three phase line tor each
pair of two corresponding lines for the liquid and gas phases coexist-
ing with solid substance. These three phase lines are indicated
by A+L-+G, A,B, +L-+G and B-+1.+4G in the P-T-projection.

The first meeting of a solubility curve with the plaitpoint curve
takes place in p and the second in g. According to VAN DER WAALS’
theory a continuous transition from the solubility curve into the
coexisting vapour curve takes place in these two points. If we take once
more the system CO,—CaQ as an example, p indicates the critical
point of the saturated solution of CaCO, in fluid carbonic acid, and
q the critical point of another solution saturated with CaCO, witha
much larger concentration of CaCO,.

Between these poimts p and q a fluid phase may oecur alone or
by the side of solid’ A,, B, (CaCO,), and in the neighbourhood of
these points the phenomenon of retrograde solidification must present
itself. 1 will further emphatically point out here, that it is assumed,
as is easily seen in the T-X-projection, that near the melting point
the difference of the volatility of the components is not so
large as to prevent the occurrence of a vapour of the composition
of the compound. The point F’, where the composition of the vapour
is the same as that of the compound, is the macimum-sublimation
pomt and the point F, where the concentration of the liquid is the
same as that of the compound, is the mdéndmem melting pout, or the
melting point under the three phase pressure '). What I did not yet
show im my previous paper is this that two lines start from the
points Foand FF’, which pass contimuously into each other at K.
These lines form the continuous bounding curve of the two sheets
of the PTX-surface for the composition of the compound. The con-
tinuous bounding curve touches the plaitpoint curve in K, so that
K denotes the critical point of the dissociating compound. That this
point K does not constitute a special point of the continuous plait-
point curve is due to the fact that when the compound is assumed
to dissociate, the critical poimt of the liquid compound does not
essentially differ from that of the liquids with other compositions.

In fig. 1a the projection of the two phase regions coexisting with
solid substance is represented, and also that of the plaitpoint curve

') These Proc., June 1905, p. 200.

( 574 )

on the p-z-plane: further the solubility isotherms corresponding with
the temperatures of the points p and q are indicated, from which
the phenomenon of retrograde solidification appears clearly.

3. In the case discussed the situation of the points p and q
depends on different properties of the compound and its components.
In special cases it will, therefore, depend on this, on what part of
the three phase line of the compound the point q lies. Undoubtedly
there will be many cases where this point falls below the melting
point. Probably this case will occur the sooner the more the volatility
of the two components differs. In this paper, however, I continue to
assume, that a vapour of the composition of the compound may exist.

In this different cases may present themselves, which each call for
a separate discussion. So highly remarkable phenomena make e. g.
their appearance, when the plaitpoint curve cuts the three phase line
of the compound between the melting point and the maximum subii-
mation point. I shall, however, discuss this case and some others in
another paper, and restrict myself now to the phenomena, which
occur, when the point of intersection g, as has been drawn in Fig. 2,
lies not only below the melting point of the compound, but also
below the maximum sublimation point. A/so im this case the possi-
bility is excluded that the compound melts, and the only way in which
the solid compound can vanish, is by evaporation.

The line for solid A, B,-+ G, which would touch the three phase
line A, B,-+L-+G in the maximum sublimation point, if this
point existed, runs on uninterruptedly to infinity, at least when no
further compleations appear.

The T-X-projection occurring in fig. 2 may contribute te elucidate
some points. As is to be seen there, the two phase region L’, gH’
coexisting with the solid compound, does not possess any liquid
or vapour of the composition of the compound, which is in harmony
with the supposition, that the points F and F’ are wanting.

In fig. 2a I have traced the projection of the two phase regions
coexisting with solid substance, and of the plaitpoint curve on the
p-x-plane. Further there are some solubility isotherms in this dia-
eram, which require a few words of explanation,

The curve /Gec/’ denotes the solubility isotherm for a temperature
somewhat below that of the point g. If we now consider the tem-
perature of the point g, we get a solubility isotherm which touches
in g, and which has two more points of inflection, as is indicated
by the curve f, G,q,g/'. At a higher temperature we get a solu-
bility isotherm, which does not touch any more, and from which
the two points of inflection may disappear.

( 575 )

4. In the third place I will point out what I have already demon-
strated in a previous publication '), that when the tension of acom-
pound is smaller at its melting point than that of the components,
a three phase curve may occur with a very peculiar shape, viz
with one minimum and two maxima.

Let us now consider the case that the melting point of this com-
pound lies above the critical temperatures of the components, then
the very peculiar phenomenon may present itself, that what occurred
once in the system ether and anthraquinone, is here to be realized
twice, and that the solubility curve which runs from one eutectic
point to the other, meets the plaitpoint curve feur times, which
appears in the PT-projection fig. 3 as a four times repeated inter-
section of the three phase curve A,,b,-+-L-+-G and the continuous
plaitpoint curve DALd in the points p,q, q' and p’.

It appears from the PT and TX-projections that for all possible
concentrations a range of temperature may be pointed out, within
which the solid compound can only coexist with a fluid phase.
When, however, which is conceivable, the portions cut out of the
three phase line have no range of temperature in common, the
temperature regions for solid + fluid, lie above each other, and so
we have no synmnetrical phenomena for any temperature on both
sides of the line for A,B, in the PT-projection.

The systems hydrogen-water and oxygen-water belong to the type
ether-anthraquinone when the components are miscible in all propor-
tions. Each of these systems will then yield a point p and a point q.
Supposing, which is, however, highly improbable, that by the appli-
cation of a catalyser we could bring about equilibrium between
oxygen, hydrogen and water vapour at any temperature, we should
get a continuous three phase line for ice + L-—+ G as is indicated
in fig. 38, and also one continuous plaitpoint curve. The equilibrium
with water, however, lying theoretically almost quite on the side of
water at lower temperatures, we should commit a practically un-
appreciable error, when we tried to realize at these lower temperatures
the diagram drawn here by starting in one case from ice, resp.
water -+ hydrogen, and in another case from ice, resp. water +--+
oxygen.

This example, however, is not suitable for illustration of the
assumed case, because for this purpose we require a compound
which appreciably dissociates at its melting point. I have only men-
tioned the system H,—QO, to show how remarkable this system is.

It is very probable that systems are to be found, with which the

1) loc cee

40

Proceedings Royal Acad. Amsterdam. Vol. VIII.

( 576 )

supposed case may be realized without excessive experimental
difficulties. This may sueceed with NMH,—HC/. A system for which
fig. 3 holds, presents also this particularity, that we have here a
P, T, X-surface of two sheets with a minimum curve bounded on the
upper side by a continuous plaitpoint curve, which, in consequence
of the great difference between the critical temperatures of the
compound and the components might possibly have the shape
described here.

Prof. Van per Waats was so kind as to draw my attention to
the particularities of the P, T, X-surface of two sheets, which may
be derived directly from those of a surface with a maximum curve,
by simply reversing everything. The minimum curve, i.e. the locus
of all points for which the concentration of liquid and vapour are
the same, forms here the lower boundary of the projection of the
P, T, X-surface of two sheets on the P, T-plane. This curve is repre-
sented in fig. 3 by the dotted line LZ’, which touches the plaitpoint
curve at LZ, and the continuous three phase line at NV. This point
NV, lying between the minimum Jf in the three phase line and the
maximum sublimation point /”, as I have shown in a paper forwarded
to the Zeitschr. f. phys. Chem. towards the end of September, is
a point where the concentration of the vapour is equal to that of
the liquid, and is therefore at the same line a point of the minimum
curve, which becomes metastable on the left of NV. The peculiar
feature in the P,T, X-surface of two sheets drawn here manifests
itself, when the bounding curves are traced for different concentrations.

It appears then, that if we come from the side of 4, the con-
centration of the point Z is the first, at which the bounding curve
presents some particularity. At this concentration we get, viz., two
bounding curves, which starting from Q and 5S, terminate at L in
a so-called cusp, as is here once more separately represented.

L

«
Ss

With a concentration somewhat richer. in A we get now two
bounding curves which pass continuously into each other. The con-
{inuous transition takes place where the bounding curve touches the
plaitpoint curve. Further this continuous bounding curve shows this
particularity that the two branches touch each other near the eritical

point, and form in this way a loop, as is separately represented below.

ur

72
(he

The point of tangency m lies on the minimum curve.

With concentrations still richer in A, the character of the bound-
ing curves remains the same, only the point m shifts along the
minimum curve towards AN, so that, when we choose the concen-
tration corresponding with the point .V, the bounding curve gets
this shape, where the vapour branch as well as the liquid branch
touches the three phase line at J.

NH

If we now pass on to greater concentration of A, we get again
bounding curves of the usual form, for the point of tangency m lies
now in the metastable region. If the critical point of the bounding
curve, coincides with the maximum temperature of the plaitpoint
curve then m lies at the absolute zero point. Leaving further parti-
cularities undiscussed, I will only just point out that the minimum
curve, beyond the point N towards lower temperatures, lies below
the three phase line, which is necessary, because the supersaturate
solution has a smaller vapour tension than a saturate one and it is
wanted for the realisation of the metastable branch of the minimum
curve that the solid substance does not make its appearance.

Now as to the T-X-projection on fig. 3 we may still remark,
that in accordance with the foregoing remark the liquid line gf” Nq’
euts the vapour line gf Nq’ in N at a temperate and pressure
lying somewhat below that of the maximum sublimation point £”,
but slightly above that of the minimum point M of the three phase
line. In NV vapour and liquid are therefore of the same concen-
tration, but this is not the case at the minimum J/.

In fig. 3a the projection is represented of the two phase regions
coexisting with solid substance on the p,v-plane, which diagram does
not call for further elucidation.

Amsterdam, December 1905.

Anorganic-Chemical-laboratory of the University.
40%

(578 )

Chemistry. — “On the course of the spinodal and the plaitpoint
lines for binary mixtures of normal substances.” By J. J.
vAN Laar. (Third communication). (Communicated by Prof.
H. A. Lorentz).

1. In my last paper’) on the above mentioned subject I discussed
the general equations of the spinodal and the plaitpoint lines, viz.
RT =/(v, 2) and F(v,«)=0 (derived in a previous communication *))
for the special case 6,—=6,, i.e. 2 =6, when a denotes the ratio

ry
2

of the critical pressures a and @ that of the critical temperatures —
1 1
of the components. (The Aigher critical temperature is always 7’).

I started from van per Waats’ equation of state, where > was
assumed to be independent of v and 7, while further in the quadratic
equations :

b = (l—«)? b, + 2a (1—«) b,, + 2? d,
le = (l—«)? a, + 2a (l—«a) a,, + #7 a,
it was assumed that =
eee > Ol) Ser (Ain woo 6 oo (((t))
which reduces the above expressions to
b=(1l—a)b, + 2),
a= (1—«) Va, + «# Ya,)’.

Henceforward we shall indicate by the name normal (binary)
mixtures such mixtures, the components of which are not only simple,
but where oth the relations (1) may be considered as satisfied.

The discussion in question led to the occurrence of tvo separate
branches of the plaitpoint line (see plate loc. cit.), which present
a double pomt at a definite value of @ (fig. 4). If 6< 2,89
(when 6, =%,), we have the normal shape, represented in fig. 2; if
6 > 2,89, we find the abnormal shape, represented in fig. 1, which
as yet has been only considered possible for mixtures, of which at
least one of the components is associating (abnormal). (C,H, -+- CH,OH,
C,H, + H,0, SO, + H,O, Ether + H,0).

The possibility of a third case was also briefly mentioned (see
fig. 3), examples of which have been described inter alia by Kurnen
(C,H, + C,H, OH, ete.) ; but this case was not further discussed, nor
the connodal relations and three phase equilibria, which, for the

1) These Proc., June 1905, p. 144. A

*) These Proc., April 1905, p. 646.

(ono)

rest, were already known. (The chief points had already been
previously described by Korrmwra and van per Waats).

In a later paper’) the place of the double point, the knowledge
of which is important, because it indicates the separation of two
very different types, was determined for the perfectly general case

b, Phy, and the discussion of the shape of the plaitpoint line was

<
extended to the case += 1, i.e. to the case which is of frequent
occurrence, that the critical pressures of the two components are
equal. In this latter case if was inter alia found, that not before
6>9,9 the case of fig. 1 loc. cit. is found.
I further derived from the perfectly general expression :
ele a(una)) ashen C4 (Una) 10

: Med aE? on aS a0 sa LEN ae
of the plaitpoint line also the initial course, viz. Pla , chiefly
1 Av 0

in connection with opinions expressed previously on this point.

As I remarked before (loc. cit. p. 84), van per Waats had already
drawn up the differential equation of the plaitpoint line, and drawn
a series of general conclusions from it. Also in a few papers of very
recent date") he has demonstrated in his own masterly way how
far we may get with general thermodynamical considerations and
general relations, derived from the equation of state. But seeing that
VAN DER WAALS himself in his Ternary Systems IV (These Proc. V,
p. 1—2) with perfect justice emphatically points out the absurdity
of the often prevailing opinion as if an equation of state should not
be required for the knowledge of the binary systems, I have consi-
dered it not unprofitable to transform the differential equation of the

1 Ce a NOE Or (ev
plaitpoint line, viz. -~+ ~-({ .-] , where / represents the second
Oz dv\0a pr :
member of RT =/(v,«) — the equation of the spinodal lines —

by means of the equation of state into a finite relation F(v,2), which
in combination with kT = /f(v, x) expresses the plaitpoint line in the
usual data 7',v, x, This enabled me to get acquainted with new par-
ticulars concerning its course (inter alia its splitting up into two
separate branches), and to examine this course in its details more closely

1) Arch. Teyner (2) X, Premiére partie, p. 1—26 (1905),

2) These Proc. VIII, p. 144.

5) These Proc. VII, p.271—298. The first mentioned paper was cited by me
(loc. cit. p. 34), so it has by no means “been overlooked”, that already ten years
ago vAN DER Waats determined the principal properties of the critical line. (ef,
y. D. Waats loc. cit. p. 271).

(580 )

than has been done up to now. I also pointed out (loc. cit. p. 15)
that already before me Korrmwera has tried to find a finite expression
for the plaitpoint line, but has not fully succeeded in this. His dis-
cussion extends after all only over the special case’) 6, =b,—6,,, a,=a,
(but a,,—=a,), whereas in my ,paper cited it was assumed in the

discussion that 6, = 0,, but that a, S a, (anda, = V/ay0,). Konmewnc’s

paper is of the highest importance, specially with regard to the
connodal relations, which are often so intricate, and to which we
shall presently come back.

The equation of the plaitpoint line once being derived in the above
mentioned finite form, it was hardly .any difficulty to derive also

l fadT,,
for the expression T\ ae ) om the side of lower critical temperature
ax
1 0

ry) p

an accurate expression, in which on/y the quantities @ = 7a and x =—
1 Pr

oceur. In Van per WaaLs’ paper mentioned by me in the paper
cited, again only the general differential equation for the expression

mentioned is given. (cf. (9) p. 89).

2. Some important points are left for discussion.

1st The discussion of the transition case at the double point, with
regard to the shape of the spinodal lines ete; and the discussion of
the possibility of the 3*¢ case (loc. cit. fig. 3).

2nd The treatment of the special case 6 = 1.

3¢ The different connodal relations in the three chief cases and
in the transition case.

4th The particularity of the cusp at R,, R, and R,' in the p,7-
representations of the three cases (loc. cit. la, 2a and 3a).

5th The question concerning the occurrence of a minimum critical
temperature, and in connection with this of a maximum vapour
pressure. ,

Let us in accordance with our last paper (loc. cit. p. 144) begin
with the fifth point.

a. Minimum-critical temperature.

In this paper I derived the formula:

late oe iene Ley Gy nae 3
Tae) P= aL ak he)

1) Arch. Neérl. 24 (1891), p. 297, 324, 337 and 341.

Putting A < 0, we get:

i.e.
Ed
¢<7--
(‘ih a=
or
4nVa
O< : eee on)
(8 Va—1)
This gives the following synopsis :
Pe onan eee ee ee GAN = 10 16 25
6 << ; 's 1 22) 2 1 1"/,; LORE Dl aa a 49°

6 always being assumed >1 (7', is the lower of the two critical
temperatures), a minimum critical temperature can only oceur, when
a, i. e. the ratio of the two critical pressures > '/,,-

If ~="/,, this takes place for ali values of 6; if = '/,, only
for values of @ between 1 and 2; ete. ete. (For a =1,a minimum
occurs in the above series of extreme values for 6, viz. 6 = 1).
Now in by far the most cases 2 will probably he between 1 and 4,
so that 6 will always have to be quite near 1, if a minimum critical
temperature is to be found.

Let us take as an_ illustration the normal substances C,H, and
N,O, investigated by Kugnen. There

74 273 + 36
OD nn ae ee
45 273 + 35

== 150.0"

According to the above rule, 6 has to be smaller than 1,04, if 7’,
is to be minimum. This is the case here. Kurnen found really a
minimum value for 7%.

We also call attention to the fact that when 6,=0,, so 7= 6,
no value of 6 exists >1 satisfying the inequality (3). For @=a—1(a,=a,,
5, =b,) the two members are equal, and the line of the critical
temperatures is a straight line. The foregoing is in perfect concordance
with what we have derived in a previous paper with regard to this
point (loc. cit. p. 43).

Also in the special case =1 evidently not a single value of

@ exists greater than 1, which satisfies (3). But in the case d= 1

there is always a value of x conceivable, yielding a minimum for

(582 )
T,. Evidently in this case Ya must be greater than */,, as
A(//2)*—9(V a)? 4-6 ax —1= (Va—1)'? 4-1),
and hence 2 >7/,,, in agreement with what has already been

found above.

Maximum of vapour pressure. As is known, this will occur at
higher temperatures, when at /ower temperatures in the ease of a three
phase equilibrium the three phase pressure does not lie between the
vapour pressures of the two components, but is greater than either. The
concentration «, of the vapour lies then between the concentrations
x, and x, of the two liquid phases. On the side of the lower critical
temperature x, >, will always have to be satisfied.

Let us now try to determine the condition for this.
For equilibrium between the phase 1 and 38 we have evidently
when pu, and gy, represent the molecular potentials of the two components:

(Ua), = (Ua)s 3 (Ud), = (42)5
or

02
C.—(2—« x) +RTlog(1—e#,)= C -(2-°E), +RTlog(1—a,)

02 : s 02
Cy— (2+a -a)5) +RTlogtz, =Cy— (2+« —3)55 | +RTloga,
“/, z/, ;

where 2 = | pdv — pv, and C, and C) are functions of the tem-
perature.
Subtraction of the two equations yields :
02 17, 1-4, 92
— -+ RT log — + RT ie
dw, c, Om, vs

or

1l—e, 2, 1 [foe 02
log : = ee :
eo a lS IE Oa’, a

as has been repeatedly derived before, inter alia by van Der Waats,

02
Now we found before fol Fa (lc. p 649 formula (3) and p. 650):
ae

02 2Va a
—= at (Va, — Ya) — ¢ fe ? (o, — b,).

Henee we have for «= 0, ES OV (Hie

02 d2 1
5.) = 2Va(Va,—Va,) | — —— |} —a,(6,—4,) be :
Ow, Ow, s—0 vy 052 B.°

( 583 )

1
so that we get at /ow temperatures (when— and — may be neglected
; Vs Us

and v, =, may be put):

ae 1 [a,(b,—),) 2Ya(Va,—V4,)
log — =— ———. —_ —— bee (4)
ot) ae aD Oe b

1 1

From this we see already, that when b,=6, (7=8@), so

Zane
Va, > Va, (because 6 must be larger than 1), then (1 **) is always
&
1/0

negative, i.e. <2. Hence just as little a three phase pressure >

than the two vapour branches, as a minimum eritical temperature.
Let us now proceed te derive the condition for «, >, from (4).

= Reet oe Va

Then (sividing by ——) we must get:

= dy

Va
b (6,—),) = 2(Va,—V4,),
|
1. 6;
bs WSS ae A ih\c
b, Va,
Ose Va, 0
or as by = 5 anc Va, = Was
6
—+1>2——,
from which follows :
a
G Sed a ees ee co ak ee Be a
~ 2VYa-1 ©)

Hence this condition is another than the condition (8) for the
minimum critical temperature, and we shall at once examine in
how far the two conditions include or exclude each other.

No more than for 2—4@ does a value of @ satisfy the above
Inequality for r—=1.If 6 = 1; then, provided x >'/,, m—-2Yx+1
must be > 0; and as this will always be satisfied, x, will be >,
for 6=1 on the side of the first component, when a>'/,. (We
found only then a minimum critical temperature for G=1, when
Sse)

We can now easily prove, that always :

An [x A
Gli/z=1)? > 2/21
when 2 >'/, For the above leads to:
(8 Va—1)? >4 Va (2 Va—1),

i.e. to m—2Ya+1> 0, which is again always satistied.

( 584 )

Hence we have for «> 7*/,:

If there is a minimum critical temperature, then also av, >a, but
not necessarily vice versa); if not a, > a,, then there is v0 minimum
of 7'.. (Again the reverse need not be true).

If x should be < '/,, then never x, >2,, while 7; is only minimum,
when (3) is satisfied, viz. if 7 >7'/,,. But this exceptional case, viz.
that for 6 >1 the value of 2 remains below */,, will be very rare.

It appears therefore convincingly from the above, that the two
conditions include each other often, but by no means always.

Just one example: Ether + H,0.

273 + 364 195
== = Taya iS = —
Here 6 = 373 198 3 Ono — aga SAD 7or == 12-30 ene
; “ 51,0
second member of (3) becomes therefore = ak = 1,39. As therefore

6< 1,39, there will be a minimum critical temperature, and hence
also xv, >, according to the above rule. In fact the second member
of (5) = 1,46, and @ being < 1,39, so a fortiori 6 << 1,46.

What is found, is in harmony with: experiment, as the three phase
pressure was found larger than the vapour pressure of ether.

Let us now take C,H, + H,0. :

Here the three phase pressure” was found smadler than that of
C,H,. Let us now examine if the inequality of (5) predicts the same. As

273 + 364 ; 195
6= ——__ = 2,07, r7=—_=4,31, Va = 2,08, so we find
273 + 35 45,2
4 :
for >—-—~ the value-1,36. And so 2,07 is not << 1,36 now. Here
2Va—l1

too the rule holds again.
According to the above rule there is now not a minimum critical
35,9

7 OF
30

temperature either. The second member of (3) becomes now =1,31,

and 2,07 is still less < 1,31 than < 1,36.

The two examples are illustrations of the jist principal type,
where a plaitpoint curve runs from C, to A, and one from C, to C).

The reader will observe, that water serves here as 2"' component,
so a very abnormal substance. But we must bear in mind, that in
the neighbourhood of «=O, where both the rules hold, the liquid
phase consists almost entirely of ether (vesp. C,H,), so that the water
present may be considered as almost perfectly normal on account
of the extremely high degree of dilution.

For the sake of completeness we mention that two other known
examples, which with those mentioned are about the only ones

known, or rather investigated, which belong to Type 1, both follow
the rule derived.

With C,H, -+CH,OH @ is viz. 1,69,7=1,63, so on account
of x=0, «x, cannot be >«w,. And with SO,-+ H,O, 6=—1,49,
m= 2,47, Va =1,57, hence the second member of (5) = 1,15. And
t,49 is not < 1,15, so a, is also not > .w,. This implies again that
no minimum critical temperature is found.

So the fact that of the four mixtures C,H, -+- CH,OH,, C,H, + H,O,
SO,-+H,O and ether + H,O only the last has a three phase
pressure greater than the vapour pressures of the two components,
is in perfect harmony with the theoretical derivations given above.

3. Let us now briefly discuss the third point, viz. the connodal
relations. As we are guided by the different figures of the adjoined
plate, a few words will suffice. The essential part was already given
by me in a few suggestions in one of my last papers (loc. cit.
p. 37 at the foot and p. 38 at the top; p. 44 at the foot and p. 45
at the top: p. 48 in the middle), where I referred to Korrewse’s
well-known papers, with regard to the neighbourhood of the points
R, and R,, and to some papers by vAN DER Waaus, with regard to
the points R, and &#’, with the third principal type.

Now we may add to this, that recently van per Waats [in the
Proceedings of the same Meeting as in which my first paper on the
spinodal and the plaitpoint lines was published (Meeting of Mareh 25
1905)| has given an addition to his former considerations concerning
the just mentioned third type, in agreement with what Korrrwrea
derived for this case already 14 years ago (loc. cit. p. 316—318,
figs. 30—35). We have reproduced this course of transformation in
our figs. 9, 10 and 11, but now in connection with our former
considerations on the course of the plaitpoint line. So also in
other cases.

a. Principal type I (igs. 1—6).

In fig. 1 we see the gradual transformation of the principal

(transverse) plait, when the temperature falls from t= —= 2,37
0

at CU, to 0,80. (These numerical values relate to special ease b,=6

2)

but when 6, $3, the relations are modified only numerically, as 1

have demonstrated in the above cited paper in the Arch. Teyler).

Te the : : E
7, is the temperature of the point C,, and is put == 1. a=
1

( 586 )

is here = 4. (ef. for these and other data the already repeatedly
mentioned paper in these proceedings).

The plaitpoint ? has strongly shifted to the side of the small
volumes ; there is always equilibrium between a gas phase 3 anda
liquid phase 2, which is comparatively rich in the 2™¢ component.
With smaller volumes the gas phase 3 is practically equal to a
liquid) phase, but the transition is gradual. (The full-traced border
curves of tbe plaits in their , v-projection, on which the straight
node lines rest, represent everywhere the connodal lines; the dotted
lines always represent the spinodal curves; the plaitpoint line is
indicated by crosses).

At t=1,6 and t=1 we see the connodal lines in the figure.
If r is somewhat below 1, e.g. 0,98, a connodal line arises running
at a short distance round C,, while the large connodal line shifts
its plaitpoint further to C,. At t= 0,97 the two plaits meet in a
homogeneous double point*). At still lower temperatures we have an
open plait, of which the two branches of the connodal line recede
towards the right and the left, and which is traced for r= 0,8. Up
io the highest pressures, 7, and , continue to differ, and it is no
longer possible to mix the two phases to one homogeneous liquid
phase by pressure, however great. With values of 7’ between 7,
and 0,977, the homogeneity reached at a certain high pressure was
again broken at still higher pressure, after which the two phases
diverge more and more up to a certain limit.

In fig. 2 an important moment has been represented. At t= 0.63
the spinodal curve douches namely the plaitpoint line C\A in R,,
and from this moment a new closed connodal line begins to appear
of the shape as is represented in fig. 3 (x = 0,62) within the connodal
line proper. The spinodal line touches that isolated curve twice, i.e.
in the plaitpoints p and p’' [all this has been fully explained by
Kortewne (loc. cit.)|, whieh for r= 0,63 coincide to a so-called
“point double hétérogene” in R,*). The connodal line in question
does not yet present, however, realizable equilibria, beeause that line
lies on the y-surface above the tangent plane to the connodal line
proper, which determines the phases 8 and 2.

1) In fig. 1 the spinodal lines seem to touch each other in this double point ;
of course this has to be an intersection.

2) It need hardly be mentioned, that every time only one, after the contact at Ry
two points of the plaitpoint line correspond with the temperature of the spinodal
and connodal line under consideration. All the other points of the plaitpoint line
which is every time projected as a whole, belong to other, lower and higher tem-

peratures.

( 587 )

Fig. 3a@ gives an enlarged, schematical representation of that iso-
lated connodal line, where some straight lines represent the “hidden”,
non-realizable equilibria. The points a and a’, and in the same way
6 and b' are corresponding points. The ‘tail’ at 0’ is always directed
towards the side of the plaitpoint (which has already disappeared in
our diagram) of the principal plait, the “point” at @ lies on the
opposite side.

We point out that the shape of the spinodal line, as is drawn
in figs.3 and 38a, implies, that it touches the plaitpoint line in the
peculiar way, indicated in fig. 2. In the immediate neighbourhood of
Rk, the uppermost portion lies left of the common tangent, the lower-
most portion on its rzglt.

At somewhat lower temperatures, in our example at r= 0,61
fig. 4), the isolated connodal line begins to towch (in M7) the eonnodal
line proper, and from this moment one of the two new plaitpoints,
viz. p, will become the plaitpoint of a new branch plait, which has
thus arisen from the principal plait in the way described above.
Cf. e.g. fig.5, where r= 0,60. The point p’ is always unrealizable,
and this continues so down to the absolute zero, where the plaitpoint
line terminates in A. On the other hand all the plaitpoints P from
M to C, will form realizable plaitpoints of the new plait.

In fig. 4 phase 3 begins to split up into two new phases, the gas
phase proper 3, anda new liquid phase 1, rich in the 1st component
of the mixture. There is a three phase /ine, the beginning of a three
phase triangle (see fig.5), which continues to exist from this point
down to the lowest temperatures.

In fig.5 it is also seen how the connodal line which passed on
uninterruptedly before, but which is now broken off in the angles
1 and 3 of the three phase triangle, proceeds on the w-surface.
With this corresponds the well-known “ridge” on the connodal line
at 2.

At t=0,59 the new plaitpoint P reaches the lower critical tem-
perature C,, and from this moment the branch plait is always open
on the side «=O, and this continues so down to the lowest
femperatures.

The p,v-representations are omitted for want of space.

Fig. 6 gives the p,7-diagram of the plaitpoint lines. Noteworthy is,
that we meet with a cusp in the line C,A at R,, where the spinodal
line touches the plaitpoint line (ef. fig. 2). We shall prove this further
on. As we have already shown in our former paper, the pressure p
approaches —27p, at A, where 7’=0O. (This derivation holds

( 588 )

S

evidently also for the general case that ZU Comparison with

fig.4 teaches us, that the point 1/, where the three phase pressure
begins, lies at a temperature Jower than that of R,. If the three
phase pressure lies between the vapour pressures of the two com-
ponents (the full-traced curves starting from C, and C, represent the
vapour tension lines), in other words if 2, > «,,, then fig. 6 holds;
if on the other hand «, >, and the three phase pressure always
higher than the vapour pressure of the two components, then fig. 6a
holds. The line C,& shows then a minimum. (In the figure the three
phase pressure line is always denoted by AAA).

b. Principal type TT.

After what has been discussed above, the relations for this type
may be made sufficiently clear even without diagrams. At a tem-
perature somewhat lower than that of #,, where the spinodal line
again touches the plaitpoint line (now C,A) a three phase equilibrium
again prevails. Now the gas phase 3 does not split up into 3 and
1, as with type I, but the liquid phase 2 into two liquid phases 2
and 1. Just as with type I the plaitpoints from J/ (between R, and
C) to A were unrealizable (cf. also fig. 6), those from / (now between
R, and (C,) to A are now also unrealizable. The three phase equi-
librium formed continues to exist down to the lowest temperatures.
Here the same phenomenon of the minimum critical temperature in
the neighbourhood of C, is met with as with type I. At tempera-
tures lower than 7’=0,96 7’, the two liquid phases 1 and 2 are
no longer to be mixed to one homogeneous phase by pressure,
however great.

The successive p, v-lines are again omitted.

Finally we find in fig. 7 the p,7-representation. The three phase
pressure line lies here between the two vapour pressure lines, so
that z,< #, on the border near 70.

ce. Principal type ILI.

The possibility of this type for mixtures of normal substances will
be examined separately afterwards. When it occurs (inter alia for
mixtures of C,H, with C,H,OH,, ete,, for triethylamine and water),
then the plaitpoint line C,C, has the shape drawn in fig. 8.

If we pass downward from the higher critical temperature at C,,
a double plaitpoint will again oceur at A, at the temperature indi-
cated by ¢,, henee formation of an isolated connodal line as in fig. 3,
at somewhat lower temperature. This goes on till at ¢, the closed

(589 )

curve in J/" begins to get outside, i.e. outside the connodal line
proper of the principal plait, at which the phase 3 begins to split
up into 3 and 1 just as in fig.4. This splitting up itself is repre-
sented at ¢, in fig.9. A three phase equilibrium has formed then
just as in fig.5. The shape of the different connodal lines is still
quite the same as in the analogous case in fig. 5, only the plaitpoint
P of the principal plait had already disappeared there. This course
has already been given by Kortrwrc, as was mentioned above, and
VAN DER WAALS, too, has accepted it in one of his last papers (loc.
cit.) on the transformation of a principal plait into a branch plait
and the reverse.

The three phase equilibrium established is however not of long
duration as we shall see. At still somewhat lower temperature ¢, a
very interesting transformation takes place (see fig. 10), also men-
tioned by Kortrewee (loe. cit. p. 318, fig. 34), and later by vAN DER
Waais (Le.). The small letters @, 6, c,d and a’, b', c, d' placed in fig. 9
give a clear idea of the transformation.

Still somewhat lower, at ¢, (fig. 11), the plaits have reversed their
functions; the branch plait of fig. 9 has become a principal plait, and
reversely the principal plait has been transformed into a branch
plait. We may notice that the “tail” at 4 is always turned to the
side of the principal plait, both in fig. 9 and in fig. 11. Also the
“ridge” has changed its place after the transition of fig. 10.

And then the further transformation resumes its normal course.
There comes a moment, at ¢, (represented in fig. 8), that the isolated
connodal line of fig. 11 begins to retreat within the connodal line
proper of the principal plait. This takes place in J/’, and the three
phase equilibrium, which accordingly has been of very short dura-
tion, finishes. The two phases 1 and 2 have again coincided, and
after this we have only coexistence of 3 and 2, as before, and as
with type II before J/ in the neighbourhood of /,. The plaitpoint
P of the principal plait continues to exist for some time more, but
will soon also disappear (at (,) ') Also the closed connodal line
remains past J/' still for a short time within the connodal line
proper, gets smaller and smaller, and disappears at last at /,', where
the spinodal line touches the plaitpoint line once more (fig. 8 at ¢,).
The temperature 7,, is the lower critical temperature of the two
components, that of C,, and at still lower temperatures we begin
gradually to approach the second plaitpoint line CA.

') The temperature of R’y (and M') may also be lower than that of C3. This
really occurs for the above mentioned mixtures. The point P of the principal
plait has then already disappeared before 1 and 2 coincide at M’.

( 590 )

At ¢,, contact of a spinodal line and the plaitpoint line takes
place for the third time, viz. at the branch C,A mentioned. Again at
somewhat lower temperature a three phase equilibrium will be found
at J/ by the repeated splitting up of 2 into 1 and 2, and now for
good and all, down to the lowest temperatures. All this is quite
identical with the case treated with type IL.

Theoretically of importance for this remarkable third (very ab-
normal) principal type is therefore this, that after the two liquid
phases 1 and 2 have become identical at J/’ (¢,), there must again
take place splitting up of the homogeneous liquid phases into two
separate phases with sufficient lowering of the temperature, viz. at
M, somewhat below R, (ef. also fig. 12).

We point out that the point J/ in fig. 4 and 6, and in fig. 7 isa
so-called wpper mixing-point, i.e. that at temperatures higher than the
temperatures corresponding with that point the two phases 3,1 or
2,1 will form one homogeneous phase. The same thing is also the
case for the points J/ and J/” of figs. 8 and 12. Above the tempe-
rature of Jf 1 coincides with 2, above that of M7" again 1 with 3.
But the point J/' is there a so-called dower mixing-point, for at
temperatures /ower than that of J/' the phases 1 and 2, distinct at
higher temperatures, coincide to one homogeneous phase.

For the plaitpoint line CC, of the third type (fig. 8) all the points,
lying between J/" somewhat before 7, and J/' somewhat beyond
?’,, are not to be realized. They form again the series of hidden
plaitpoints p’, indicated in the figs. 9—11.

The p, x-representations are again omitted.

In the figs. 12 and 12a the p,7-representations of the plaitpoint
line are drawn of the type mentioned. We again notice the three
cusps R,, R, and h’,. In fig. 12 the three phase pressure lies between
the vapour pressures of the components; in fig. 12a above them.
CR, has then again, as in fig. 6a, a retrogressive course.

We shall put off the discussion of the remaining points to a

following paper. Those points are: a. The transition case between
type I and II with the double point; %. the discussion of the possi-
bility of the occurrence of type IIL; ¢c. some remarks on the special
case 61; d., the proof, that in the p, 7-representations the different
points #,, R, and fh’, are cusps.

( 591 )

Physics. — “The absorption and emission lines of gaseous bodies.”
By Prof. H. A. Lorentz.

(Communicated in the Meetings of November and December 1905).

§ 1. The dispersion and absorption of light, as well as the
influence of certain cireumstances on the bands or lines of absorp-
tion, can be explained by means of the hypothesis that the molecules
of ponderable bodies contain small particles that are set in vibration
by the periodic forces existing in a beam of light or radiant heat.
The connexion between the two first mentioned phenomena forms the
subject of the theory of anomalous dispersion that has been developed
by SeLiMeyer, Boussinesg and HeLmuourz, a theory that may readily
be reproduced in the language of electromagnetic theory, if the
small vibrating particles are supposed to have electric charges, so
that they may be called electrons. Among the changes in the lines
of absorption, those that are produced by an exterior magnetic field
are of paramount interest. Vorer') has proposed a theory which
not only accounts for these modifications, the inverse ZEEMAN effect
as it may properly be called, but from which he has been able to
deduce the existence of several other phenomena, which are closely
allied to the magnetic splitting of spectral lines, and which have
been investigated by Hato *) and Gexst*) in the Amsterdam labo-
ratory. In this theory of Vorer there is hardly any question of the
mechanism by which the phenomena are produced. I have shown
however that equations corresponding to his and from which the
same conclusions may be drawn, may be established on the basis
of the theory of electrons, if we confine ourselves to the simpler
cases. In what follows I shall give some further development to
my former considerations on the subject, somewhat simplifying them
at the same time by the introduction of the notation I have used
in my articles in the Mathematical Encyclopedia.

1) W. Vorer, Theorie der magneto-optischen Erscheinungen. Ann. Phys. Chem.
67 (1899), p. 345; Weiteres zur Theorie des Zeeman-effectes, ibidem 68 (1899),
p- 352; Weiteres zur Theorie der magneto-optischen Wirkungen. Ann. Phys., 1
(1900), p. 389.

*) J.J, Hato, La rotation magnétique du plan de polarisation dans le voisinage
d'une bande d’absorption, Arch. Néerl., (2), 10 (1905), p-. 148.

8) J. Geest, La double réfraction magnétique de Ia vapeur de sodium, Arch.
Néerl., (2), 10 (1905), p. 291.

4) Lorentz, Sur la théorie des phénomeéenes magnéto-optiques récemment décou-
verts Rapports prés. au Congrés de physique, 1900, T. 8, p. 1.

41

Proceedings Royal Acad. Amsterdam, Vol. VIII,

(592 )

§ 2. We shall always consider a gaseous body. Let, in any point
of it, © be the electric force, the magnetic force, } the electric
polarization and

Dy Ss Wie Ake ane eee
the dielectric displacement. Then we have the general relations
0. 0.5, 1 0Dz O62 ‘0H2. 1 0D,

oy ti, 50g is crs Onan Oe Oz eimene oer

ieee -_ = = een (2
Ow Oy deat pha (2)

ag, UG, ine een, cen eine
Oye «202 ee eae Of 202. sn fen me soe (One:
0g, 0€,, Ie Gay
ft St See stia set Sealer aaa
Oa Oy c) xt

in which ¢ is the velocity of light in the aether.

To these we must add the formulae expressing the connexion
between & and }, which we can find by starting from the equations
of motion for the vibrating electrons. For the sake of simplicity we
shall suppose each molecule to contain only one movable electron.
We shall write e for its charge, m for its mass and (xX, y, 2) for
its displacement from the position of equilibrium. Then, if 4 is the
number of molecules per unit volume,

p= Nex, Y= Ney, V2 Nez. - ee

§ 3. The movable electron is acted on by several forces. First,
in virtue of the state of all other molecules, except the one to which
it belongs, there is a force whose components per unit charge are
given by *)

€, + af,, €, + aP,, E+ eP-,
a being a constant that may be shown to have the value '/, in
certain simple cases and which in general will not be widely different
from this. The components of the first force acting on the electron
are therefore
e (&, +a Y,), e (€, +a Py), e (€- =e a p.) sty te ake (5)

In the second place we shall assume the existence of an elastic
foree directed towards the position of equilibrium and proportional
to the displacement. We may write for its components

=f XPS Vfiy, NaS NZi, see eye we eee)

/ being a constant whose value depends on the nature of the molecule.

') Lorentz, Math. Eneycl. Bd. 5, Art. 14, §§ 35 and 36.

( 593 )

If this were the only force, the electron could vibrate with a
frequency 7,, determined by

Vega eee he ey Pore onalie’ a(t)

In order to account for the absorption, one has often introduced
a resistance proportional to the velocity of the electron whose com-
ponents may be represented by

dx dy dz
Sa) A a) a Ere ee a (2)
if by g we denote a new constant.

We have finally to consider the forces due to the external mag-
netic field. We shall suppose this field to be constant and to have
the direction of the axis of z. If the strength of the field is H, the
components of the last mentioned force will be

eHdy _ eHdx
Chait c dt’

It must be observed that, in the formulae (2) and (3), we may
understand by 9 the magnetic force that is due to the vibrations
in the beam of light and that may be conceived to be superimposed
on the constant magnetic force H.

(eee shee, arse Felon)

§ 4. The equations of motion of the electron are

ax dx eHdy
eet F, + ar) — fx — g— pases oo
% dt? e(Es 7, 2s) f g dt a olde
ay dy eHdx
eae ,\ ey aL ai pe
nae Cyd OW) IVa 9 dt c dt
hay 2 a s dz
Dearie eel Parad Bite Sree

These formulae may however be put in a form somewhat more
convenient for our purpose.

To this effect we shall divide by e, expressing at the same time
x,y,z in $., Py, P-. This may be done by means of the relations (4).

Putting

m ; ice gk,
ae Were Wel ee (10)
we find in this way
OP: 0s H 04,
m' : =€,+aP,—/f'S—g Px P, :
ot Ot cNe Ot
0? P, oo 5 (OD, H 0 ..
af x }, = gE, JL a Py ae f' Py =1g }, ~~ p. ;
oe d¢ cNe Ot
0? N- : ON.
Se Seen

41*

( 594 )

The equations may be further simplified, if, following a well
known method, we work with complex expressions, all containing
the time in the factor e’"* If we introduce the three quantities

— fo Np GeO) a oo 8. (bil)

try

(CeO aime - Ch cm so) oo. o - ((IZ))
and
nH
>= Att Rt oem Fis Pichi Bees
i cNe

the result becomes

f= (E+ in) We — iS Py,

©, = (§ + i 4) Py +15 Pr, iy Sha eae
€, = (E + im) P:.

§ 5. Before proceeding further, we shall try to form an idea of
the mechanism by which the absorption is produced. It seems difficult
to admit the real existence of a resistance proportional to the velo-
city such as is represented by the expressions (8). It is true that in the
theory of electrons a charged particle moving through the aether
is acted on by a certain force to which the name of resistance may
be applied, but this force is proportional to the differential coefficients
of the third order of x, y,2 with respect to the time. Besides, as
we shall see later on, it is much too small to account for the absoerp-
tion existing in many cases; we shall therefore begin by neglecting
it altogether, i.e. by supposing that a vibrating electron is not subject
to any foree, exerted by the aether and tending to damp its vibrations.

However, if, in our case of gaseous bodies, we think of the mutual
encounters between the molecules, a way in which the regular
vibrations of light might be transformed into an inorderly motion
that may be called heat, can easily be conceived. As long as a mole-
cule is not struck by another, the movable electron contained within
it may be considered as free to follow the periodic electric forces
existing in the beam of light; it will therefore take a motion whose
amplitude would continually increase if the frequency of the incident
light corresponded exactly to that ofthe free vibrations of the electron.

In a short time however, the molecule will strike against another
particle, and it seems natural to suppose that by this encounter the
regular vibration set up in the molecule will be changed into a
motion of a wholly different kind. Between this transformation and
the next encounter, there will again be an interval of time during
which a new regular vibration is given to the electron. It is clear
that in this way, as well as by a resistance proportional to the velo-

(595 )

city, the amplitude of the vibrations will be prevented from surpas-
sing a certain limit.

We should be led into serious mathematical difficulties, if, im
following up this idea, we were to consider the motions actually
taking place in a system of molecules. In order to simplify the
problem, without materially changing the circumstances of the case,
we shall suppose each molecule to remain in its place, the state of
vibration being disturbed over and over again by a large number of
blows, distributed in the system according to the laws of chance.
Let A be the number of blows that are given to N molecules per
unit of time. Then

N

Gat
may be said to be the mean length of time during which the vibra-
tion in a molecule is left undisturbed. It may further be shown
that, at a definite instant, there are

molecules for which the time that has elapsed since the last blow
lies between #® and # + dd.

§ 6. We have now to compare the influence of the just men-
tioned blows with that of a resistance whose intensity is determined
by the coefficient g. In order to do this, we shall consider a mole-
cule acted on by an external electric force

aetnt
in the direction of the axis of z.

If there is a resistance g, the displacement xX is given by the

equation

m —=—f

dt? 4

so that, if we confine ourselves to the particular solution in which
x contains the factor e'™', and if we use the relation (7),

d& :
X—g— + aee"!,
dt

ae Z
x (PIE BA oe woe oo (Q'h)
m(n,?-— n?) + ing

In the other case, if, between two successive blows, there is no

resistance, we must start from the equation of motion
ax : :
m—=—fx+aeent
dt? .

whose general solution is

aeeint sier Z
x = ——_ + Cye' mot + (CAG SEUDUN Og * on ic (16)

+s m(n,*—n*)

By means of this formula we can calculate, for a definite instant
t, the mean value x for a large number of molecules, all acted on
by the same electric force a ent, Now, for each molecule, the con-

: . xa :

stants C, and C, are determined by the values of x and immediately
; dx sist

after the last blow, i. e. by the values x, and ae existing at

0
the time ¢—®, if ® is the interval that has elapsed since that blow.

We shall suppose that immediately after a blow all directions of
the displacement and the velocity of the electron are equally pro-

1 Y dx
bable. Then the mean values of x, and (=) are 0, and we shall
at
0
find the exact value of x, if in the determination of C, and C,, we
dx :
suppose Xx and 7 to vanish at the time ¢— #.
Cc

In this way, (16) becomes

int
eas aee qi 1 1 +. 2 et(my—n)> — Z a ee= 3 e—i(no$n) Ss},
m(n,7—n"*) 2 ny 2 ny

- PS ears :
From this X is found, if, after multiplying by —e * dd, we inte-
tT

erate from ®=O0 to F=o. If w isan imaginary constant, we have

1 - uo—— 1
f° Sao = a
T 1—wur

Hence, after some transformations,

— e .
x z nt. 3s eer

: 1 2 imn
TON A oe pate
Tt T

If this is compared with (15), it appears that, on account of the
blows, the phenomena will be the same as if there were a resistance

determined by
2m
C= = ae oe) nee

and an elastic force having for its coefficient

CAE ees SS)
T

(597)

Indeed, if the elastic force had the intensity corresponding to this
formula, the square of the frequency of the free vibrations would

1 :
have, by (7), the value n,? + ma The equation (15) would then

take the form (17).

In the next paragraphs the last term in (19) will however be
omitted.

As to the time rt, it will be found to be considerably shorter than
the time between two suecessive encounters of a molecule. Hence,
if we wish to maintain the conception here set forth, we must sup-
pose the regular succession of vibrations to be disturbed by some un-
known action much more rapidly than it would be by the encounters.

We may add that, even if there were a resistance proportional to
the velocity, the vibrations might be said to go on undisturbed only
for a limited length of time. On account of the damping their amplitude
would be considerably diminished in a time of the order of magnitude
m

—. This is comparable to the value of t+ which, by (18), corresponds
q

io a given magnitude of yg.

§ 7. The laws of propagation of electric vibrations are easily
deduced from our fundamental equations. We shall begin by sup-
posing that there is no external magnetic field, so that the terms
with ¢ disappear from the equations (14).

Let the propagation take place in the direction of the axis of z
and Jet the components of the electromagnetic vectors all contain
the factor

Coie (alee ees age pe te (20)
in which it is the value of the constant g that will chiefly interest
us. There can exist a state of things, in which the electric vibrations
are parallel to O X and the magnetic ones parallel to O Y, so that
€,, P., D, and H, are the only components differing from 0. Since

differentiations with respect to ¢ and to z are equivalent to a
multiplication by cm and by —ing respectively, we have by
(2) and (3)

1 : i ee

q Dy = — Day q &2 = — Dy

C «
Hence

De &,

and, in virtue of (1),
P= (c9* — 1)...
The first of the equations (14) leads therefore to the following

(598 )

formula, whieh may serve for the determination of q,

1 :
eg? “od Seana eer Pach tanto (rl
q ae, (21)
Of course, q has a complex value. If, ii xand w real, we put
1 —ix
(22)

I ’

Ww

the expression (20) becomes

in ( 1")
wnt t — —— }z
a
é ’

so that the real parts of the quantities representing the vibrations

contain the factor
ake
; ere

multiplied by the cosine or sine of
w
It appears from this that w may be ealled the velocity of propa-

gation and that the absorption is determined by x. If
nx

eae

Ww
(index of absorption), we may infer from (23) that, while the vibra-

: 1
tions travel over a distance i? their amplitude is diminished in the

ratio of 1 to —.
é
In order to determine w and x, we have only to substitute (22)

in (24). We then get

CAMs gy 3
whe it) LSet i

or, separating the real and the imaginary parts,
= 2 cr-2 y

ae dane li 2
o

from which we derive the formulae

eae
i i 24
o? & + 93 §2 4 9? a (24)

§ . (85)

| /ELN ER re
& + 7? Le ze 7"? Es

in. which the radical must be taken with the positive sign

(599 )

If the different constants are known, we can ealeulate by these
formulae the velocity and the index of absorption for every value
of the frequency mn; in doing so, we shall also get an idea about
the breadth and the intensity of the absorption band.

§ 8. In these questions much depends on the value of 4. In the
special case § = 0, i.e. if the frequency is equal to, or at least only
a little different from that of the free vibrations, we have on account

of (25)
2 US Ea ai es v2F
q

From what has been said above, it may further be inferred that
2c

along a distance equal to fhe wave-length in air, i.e. , the

n
amplitude decreases in the ratio of 1 to

2acx

e

Ww

Now, in the large majority of cases, the absorption along such a

2arcx
distance is undoubtedly very feeble, so that ——- must be a small
W@W
c7x?
number. The value of —~ must be still smaller and this ean only
@ .

be the case, if 4 is much larger than 1.
This being so, the radical in (25) may be replaced by an approxi-
mate value. Putting it in the form

: We ees

we may in the first place observe, ea since 7 is large, the numerator
2§+1 will be very small in comparison with the denominator,
whatever be the value of §. Up to terms with the square of
2&+11

52 gt a?’
S

we may therefore write for the radical

1
lao ies 2 Q (ze 2)\2
2B fy 8 (+7)
and after some transformations
Gaoee 47y7?—4§—1
ot 9 (eee
As long as § is small in comparison with 7?, the numerator of
this fraction may be replaced by 477. On the other hand, as

2§+1 1028+)
(

2

( 600 )

soon as € is of the same order of magnitude as 4? or surpasses
this quantity, the fraction becomes so small that it may be neglected,
and it will remain so, if we omit the term — 46 in the numerator.
We may therefore write in all cases

cx 7

o> BRE)

so that the index of absorption becomes

, ee aoe

This formula shows that for §=O the index has its maximum
value

be ee eo een ae

and that for §—= + v7, it is »?-++1 times smaller.

Fhe frequency corresponding to this value of § can easily be cal-
culated. If @ may be neglected, a question to which we shall return
in § 18, (11) may be put in the form

ea ON (ng eae) BE ROTTICE SPIOTTGe rd Gigs. (ae)

Hence, for § = = 7
m (n® — n,?) = == on = = vag),
or, on account of (10) and (18),
2mvn

min? —— 1) = an A
0 “ T

2yn
n®? —n,? = +—_.
= :
If n—n, is much smaller than 2,, we may also write

FS es = o 6 oo ero oe (28)
T

The preceding considerations lead to the well known conclusion,
somewhat paradoxal at first sight, that the intensity of the maximum
absorption increases by a diminution of the resistance, or by a lengthen-
ing of the time during which the vibrations go on undisturbed. In-
deed, if gy is diminished or t increased, it appears by (10) and (412)
that o) becomes smaller and by (27) 4, will become larger. This result
may be understood, if we keep in mind that, in the case m= Mo,
the one most favourable to ‘‘optical resonance’, in molecules that
are left to themselves for a long time a large amount of vibratory
energy will have accumulated before a blow takes place. Though
the blows are rare, the amount of vibratory energy which is converted
into heat may therefore very well be large.

( 601 )

In another sense, however, the absorption may be said to be
diminished by an increase of t (or a diminution of g), the range
of wave-lengths to which it is confined, becoming narrower. This
follows immediately from the equation (26). Let a fixed value be
given to §, so that we fix our attention on a point of the spectrum,
situated at a definite distance from the place of maximum absorption,
and let 4 be gradually diminished. As soon as it has come below
§, further diminution will lead to smaller values of /, i. e. to a
smaller breadth of the band.

If g is very small, or t very large, we shall observe a very nar-
row line of great intensity.

§ 9. The observation of the bands or lines of absorption, combined
with the knowledge that has been obtained by other means of some
of the quantities occurring in our formulae, enables us to determine
the time t and the number NV of molecules per unit volume.

I shall perform these calculations for two rather different cases,
viz. for the absorption of dark rays of heat by carbonic dioxyd and
for the absorption in a sodium flame.

As soon as we know the breadth of the absorption band, or,
more exactly, at what distance from the middle of the band the
absorption has diminished in a certain ratio, the value of + may be
deduced from (29); we have only to remember that in this formula,
nm is the frequency for which the index of absorption is »? + 1 times
smaller than the maximum 7,.

AnestrOM') has found that in the absorption band of carbonic
dioxyd, whose middle corresponds to the wave-length 4—= 2,60 u,
the index of absorption has approximately diminished to 4%, for
4 = 2,304. This diminution corresponding to v = 1, we have by (29)

1

—-=n—N,
Tt

if m, and n are the frequencies for the wave-lengths 2,60 and 2,30 «.

In this way I find

t= 10-4 see.

In the case of the absorption lines produced in the spectrum by
a sodium flame, we cannot say at what distance from the middle
the absorption has sunk to 4 /,. We must therefore deduce the value of
t from the estimated breadth of the line. Though the value of »
corresponding to the border cannot be exactly indicated, we shall

°
1) K. Anestrém, Beitriige zur Kenntniss der Absorption der Wirmestrahlen
durch die verschiedenen Bestandteile der Atmosphiire, Ann. Phys. Chem. 89 (1890),
p. 267 (see p. 280).

( 602 )

probably be not far wrong, if we suppose it to lie between 3 and 6;
this would imply that at the border the index of absorption lies be-

l
tween 0 i, and i. If therefore n relates to the border, the for-

QO7

1 1 1
mula (29) shows that the limits for —are-— (n—np) and i (1—10).
a 0

In Hat.o’s experiments the breadth of the D-lines was about
1 A. #. The relation between » and the wave-length 2 being

2% ¢
i ,
2
we find for that between small variations of the two quantities
2a
dn = — d2.
22

Hence, if we put di=0,5 A. #.=0,5 X 10-8 em., we find
n — Mm = 0,26 X 1012,
from which I infer that the value of t lies between 12 * 10—!? and
24 « 10-2 sec.

§ 10. In the case of carbonic dioxyd the number .V may be
deduced from the measured intensity of absorption. In ANastrém’s
experiments this amounted to 10,6 pCt. in a layer, 12 em. thick,
and for 2= 2,60. The amplitude being diminished in the proportion
of 1 to eo in a layer whose thickness is z, and the intensity of
the rays being proportional to the square of the amplitude, we have

e-24o = 0,894,

and
ko = 0,0046.
Now, by the formulae (27), (12), (40) and (18)

Ne?

ieee eee
4om

Ae 4 em Ips
eT

Here r and i, are known by what precedes, As to the charge 6,
it is, in all probability, equal to that of an electrolytic ion of hydrogen.
lt is therefore expressed in the usual electromagnetic units by
the number 1,3 * 10-29, and in the usual electrostatic units by
3,9 <10- The unit of electricity used in our formulae being
V42—3,5 times smaller than the common electrostatic one, we
must put

os 14X0040:-. .- gi eee)

( 603 )

In the case of the infra-red rays whose absorption has been
measured by AnGstrém we are probably concerned with the vibrations
of charged atoms of oxygen or carbon. The mass of an atom of
hydrogen being about 1,3 >< 10-4 gramme, I shall take

Meo el Oman.

The result then becomes

N=6 > 1017.

§ 11. The above method is not available for a sodium flame.
Hato has however observed that the value of N for this body may
be deduced from his measurements of the magnetic rotation of the
plane of polarization and Grrst has shown that the magnetic double
refraction in the flame may serve for the same purpose. In what
follows I shall only use one of Ha.1o’s results.

In the first place it must be noticed that in the case to be con-

-

. a: s :
sidered, § is much larger and 5 aga much smaller than unity. The
SS) |
radical in (24) may therefore be replaced by
§
i
Sista te
and the formula becomes
y 5
= ities ea
w 2(s* + 1’)

Now, if there is an external magnetic field, the velocities of pro-
pagation , and w, of right and left circularly polarized light can
be caleulated by a similar formula. We have only to replace § by
§$—S and by §+5.') From the results

enya Z
<=] + —— eal ES + - pists Shee es
o, CA Cie | O, 25 + $)? + 777]
we find for the angle of rotation per unit length
1 ieee n cas s+5
a ar Ge ) ia pea ae as ELiea aes) eY)
2"\o, o) 4clG—S' +e ELS +7

In order to determine NN hy means of a measured value of g,
we begin by observing that, in virtue of the equation (28), for
which we may write

each value of § determines a certain point in the spectrum whose
distance from the middle of the band is proportional to &. At the

') See Lorentz, Sur la theorie des phenoménes magncéto-optiques, etc., § 16.

( 604 )

border of the band (if there is no magnetic field) § has the value
yy, the coefficient » being some moderate number, say between 3
and 6 (§9), and for one of the components of Zerman’s doublet we
have §=¢. In the magnetic field used by Hatio the distance of
the components from the middle of the original line amounted to
0,15 A. #., half the breadth of the line being 0,5 A. E., as has
already been said.
We have therefore the following relation between 4 and ¢:
Gi ig 0 lonkOyo
3,90

YP

je 5 silehi ameter rat belied (Se)

On the other hand, a point in the spectrum, at which the angle

of rotation per unit length was approximately equal to unity, was

situated at a distance of 1,6 A. EL. (= of the mutual distance of the

two D-lines) from the middle of the original line. Fhis being 10 times

ihe distance from this line to one of the components, we have
approximately

SONG:

On substituting this value and (32) in the formula (31), it appears

that the terms 4? may be omitted. Hence, if (13) is taken into account,

0,005 . = 0,005 33

= = eres Deo
Zp ’ ae ’ H (33)

or since g = 1 is,
Ne= 200H.
The strength of the magnetic field in these experiments was 9000
in ordinary units, or

9000
— —___. = 2600

in those used in our equations. Taking for e the value (30), I finally find
N=4 X10",

§ 12. The value of 4 may likewise be calculated, both for the
carbonic dioxyde and for the sodium flame. In the first case we can
avail ourselves of the formula (27), in which &, is now known ;
the result is

n a

— SS =

= DPC IDES
2¢ k, Ak,

7N

For the sodium flame we first draw from (33)

( 605 )

nr fu

— 0,01 — — 500
2

$= 0,005

c
and we then find by (32) the following limits for 7
550 and 270.

These results fully verify our assumption that 9 would be a
large number.

Finally we can compare the values we have found for + with
the period of the vibrations. In this way we see that in the flame
some six or twelve thousand vibrations follow each other in uninter-
rupted suecession. In the carbonic dioxyd oa the contrary no more
than a few vibrations can take place between two successive blows.

§ 13. After having found the number NV of molecules in the
sodium flame we can deduce from it the density d of the vapour of
sodium. In doing so, | shall suppose the molecules to be single atoms,
so that each has a mass equal to 23 times that of a mass of hydrogen.
Taking for this latter 1,3 >< 10-*4 gramme, I find

ahe= NB S< MVE

This is not very different from the number 7>< 10-9 found by Hato.

Haro has already pointed out that this value is very much smaller
than the density of the vapour really present in the flame; at least,
this must be concluded if we may apply a statement made by
FE. Wispemann, according to which a certain flame with which he
has worked contained per em’. about 5X10—7 gramme of sodium.
Perhaps the difference must be explained by supposing that only
those particles that are in some peculiar state, a small portion of the
whole number, play a part in the phenomenon of absorption. This
would agree with the views to which Lrenarp has been led by his
investigation of the emission by vapour of sodium.

It must be noticed that the value of N we have calculated for
carbonic dioxyd warrants a similar conclusion. In the experiments of
Angstrom the pressure was 739 mm. At this pressure and at 15°C.
the number of molecules per cm*. may be estimated at 3,2 < 10!9.
This is 50 times the number we have found in § 10.

§ 14. An interesting result is obtained if the time + we have
calculated for carbonic dioxyd is compared with the mean lapse of
time between two successive encounters of a molecule. Under the
circumstances mentioned at the end of § 13, the mean length of the
free path is about 7 > 10~° cm. The molecular velocity being
4 >< 10+ em. per sec., this distance is travelled over in

( 606 )

IE Sine Oe! 0 Tsees

i.e. in a time equal to 18000 times the value we have found for r.
We see in this way that it cannot be the encounters between mole-
cules, by which the regular succession of vibrations comes to an end.
It seems to be disturbed much more rapidly by some other cause
which is at work within each molecule.

In the case of the sodium flame there is a similar difference
between the length of time + and the mean interval between two
encounters.

§ 15. We shall now return for a moment to the resistance that
has been spoken of in § 5, the only one that is really exerted by
the aether. This resistance is intimately connected with the radiation
issuing from a vibrating electron, and if a beam of light were
weakened by its influence, this would be due to part of the incident
energy being withdrawn from the beam and emitted again into the
aether. Of course, this could hardly be called an absorption. But,
apart from this objection, we can easily show that the resistance in
question is much too small to account for the diminution of intensity
that is really observed. Its component in the direction of a is

e dx
6ac dt’

or, for harmonie vibrations of frequency 7,
Dee (thoi

6zeé dt ’
Comparing this with (8), we find

n? e?

ar 620

This amounts to 2,0 10~-2! for ecarbonie dioxyd (for the wave-
length 2=2,60u (§ 9)) and to 4,0 10-°° in the case of the
sodium tlame. These numbers are far below those which result from
(18), if we substitute the value that has been calculated for +r. We
then get, for carbonic dioxyd 4,0 x 10-9, and for the sodium flame
a number between 1,2 X 10—!6 and 0,6 & 10—1!¢.

§ 16. It has already been shown in § 8 that an increase of ¥
broadens the absorption band, diminishing at the same time the ab-
sorption in its middle. Indeed, in many eases we may say that
the broader the band, the feebler is the absorption for a definite
kind of rays.

The question now arises what is the total amount of energy

( 607 )

absorbed by a layer of given thickness z, if the incident beam con-

tains all wave-lengths occurring in the part of the spectrum occupied

by the absorption band. In treating this problem, I shall suppose

the energy to be uniformly distributed over this range of frequencies,

so that, if we write /dn tor the incident energy, in so far as it

belongs to wave-lengths between 7 and n+ dn, J is a constant.
The total amount of energy absorbed is then given by

ce)

Asif a — 2-3) du eam sates marian (OA)
0
Now, if the coefficient g and the time r were independent of the

density of the gas, both § and 4 would be inversely proportional to
NV; this results from (10), (12) and (28). The equation (26) shows
that under these circumstances and for a given value of n, & is
proportional to NV. The value of A will therefore be determined by
the product .Vz. This means that the total absorption would solely
depend on the quantity of gas contained in a layer of the given
thickness, whose boundary surfaces have unit of area; if the same
quantity were compressed within a layer of a thickness }z, the
absorption would not be altered.

The result is different, if g and + depend on the density. In order
to examine this point, I shall take z to be so small that 1
may be replaced by 242 — 2h?z?, so that (34) becomes

A =225 Jef — 2 {ean
0 0

Let us further confine ourselves to an absorption band, so narrow,
that we may put

e — 2ke

cf mameeG (So)

—— EM (a) ye es oe ie (SO)
n y

—nq, k= — ———

1 = Magi b= 5 Ret ina Boer (37)

Introducing §, instead of 7, and extending the integrations from

=—x to $=-+%, as may indeed be done, I find from (35)

caw A 1
A= 2 ——— 27},
2em' 4eq'

or, on account of (10),

Ss

aK

xl
A

2em

1
Ne* z — —— (Ne? z)? :
4eq

Two conclusions follow from this result. First, the absorption in
an infinitely thin layer of given thickness does not depend on the
42
Proceedings Royal Acad. Amsterdam. Vol. VIII.

( 608 )

value of g. In the second place, if the layer is so thick that the
second term in the formula has a certain influence, for a given
value of Vz, the amount of absorption will inerease with g. It will
therefore increase by a compression of the gas, if by this means the
coefficient g takes a larger value. An effect of this kind has really
been observed by Anestrom ') in his experiments on the absorption
produced by carbonic dioxyde.,

This result could have been predicted by theory if the idea that
the succession of regular vibrations would be disturbed by the colli-
sions between the molecules had been confirmed ; then, by an increase
of the density, the time + would become shorter and the formula (18)
would give a larger value for the coefficient g. As it is, the vibra-
tions must be supposed to be disturbed by some other cause (§ 14)
and we can only infer from Anestrow’s measurements that the intlu-
ence of this cause must depend in some unknown way on the density

of the gas.

§ 17. Thus far, we have constantly assumed in our calculations
that the coefficient 4 is very much larger than unity ; this hypothesis
has been confirmed by the values given in § 12 and, to judge from
these numbers, it would even seem hardly probable that aj can in
any case have a value equal to, or smaller than 1. Yet, there is a
phenomenon which can only be explained by aseribing to a a small
value. This is the dissymmetry of the Zerman effect, which has been
predicted by Vorer’s theory *) and has shown itself in some experi-
ments of Zwnman*). In so far as we are here concerned with it, it
consists in a small inequality, observable only in weak magnetic
fields, of the distances at which the two outer components of the
triplet are situated from the place of the original spectral line.
Whereas in strong fields the position of these components is deter-

mined by the equations $= +S§ and §=-—§, it corresponds to
s$=0 and §=—1, if the magnetic intensity is very small.

Vorer has immediately pointed out that the dissymmetry can only
exist, if 4 is not very large. Yet, from the fact that the effect could
scarcely be detected by Zmmman, he concludes that the coefficient must

1) Axasrrim, Uber die Abhingigkeit der Absorption der Gase, besonders der
Kohlensiiure, von der Dichte, Ann. Phys., 6 (1901), p. 168.

2) Vor, Uber eine Dissymmetrie der Zreway’schen normalen Triplets, Ann.
Phys., 1 (1900), p. 376.

5) Zeeman, Some observations concerning an asymmetrical change of the spectral
lines of iron, radiating in a magnetic field. These Proceedings, H (1900), p. 298,

( 609 )

have been rather larger than unity. In my opinion, we must go
farther than that and aseribe to a a value, not sensibly above 1,
my argument being that the dissymmetry can only make itself felt,
if the difference between the distances from the original line to the
two components in question is not very much smaller than the breadth
of the line.

We know already (§ 9) that §—O at the middle of the line and
$=ry at the border. Now, if 4 were sensibly larger than 1, the
places corresponding to §=0O and §=1, i.e. the places occupied
by the two components in a weak field, would lie within the
breadth of the original line; it would therefore be impossible to
discern the want of symmetry.

§ 18. Whatever be the exact value of 7, ZeeMan’s experiments
on this point show at all events that under favourable circumstances
a displacement of a line, corresponding to a change from § = 0 to
$=, or to a change

1
ee eee ee es... (38)

9 '
om Nn,

of the frequency, is large enough to be seen. But, if sueh is the
case, we shall no longer be right, if we discuss the value of §, in
omitting quantities that are but a few times smaller than unity.

A quantity of this kind is the term « in the equation (11), which

1/ ‘
/,;, and

as has already been mentioned, is but little different from
which we have omitted in all our calculations. If we wish to take
it into accougt, we shall find that all that precedes will still hold,
provided only we replace n, by the quantity 7',, determined by

if — @ =m Hie Oth iote ok ae oe hd at LSE)
Indeed, (28) may then be written in the form

&=m (n',? — n’),
and the place of maximum absorption, the middle of the line, will
correspond to the frequeney 7,, exactly as it formerly corresponded
to the frequency 7',.
Now, by (7) and (10)

and by (39)

1 3 a ' a
a a Me yal Uy ae aD oe (40)
m mr ym
or, on account of (10),
a N e
Ny =N%— 5 Ge NOP a ecru in a0)
2n, m

We learn from this equation that an increase of the density must

( 610 )

vive rise to a small displacement of the absorption line towards
the side of the larger wave-lengths. A shift of this kind has been
observed by Humpnrnys and Monier in their investigation of the in-
fluence of pressure on the position of spectral lines. However, as
the formula (41) does not lead to the laws the two physicists have
established for the new phenomenon, I do not pretend to have
given an explanation of it.

Nevertheless we may be sure that in those cases in which the
dissymmetry of the Zeeman effect can be detected, the last term in
(41), which in fact is of the same order of magnitude as the expres-
sion (88), can have an influence on the position of a spectral line
that is not wholly to be neglected.

On the other hand, it now becomes clear that, in the case of a
large value of 4, the term @ in (11) may certainly be neglected, its
influence on the position of the middle of the lme being much smaller
than the breadth. *)

§ 19. We shall conclude by examining the influence of the last
term in (19), which we have likewise omitted. If we replace / by

m : : : . * m' ;
f+— and, in virtue of (10), /’ by f’ +—, which I shall denote by
: z = 4

(f’), and if this time we neglect the term @, the formula (11) may
again be written in the form (28). Indeed, if we put

" (7 1 :
es ee So. oto oo | (4)
m T
we shall have
§ = m' (n'",? — n’)

1
n'ont Sede SS oho, 0 (283)

Ante

an equation which shows that the absorption band lies somewhat more
towards the side of the smaller wave-lengths than would correspond
to the frequency m, and that its position would be shifted a little,

0
if the time + were altered in one way or another (§ 16). These displa-

1) Prof. Junius has called my attention to the fact that in many cases the absorp-
tion lines are considerably broadened by the change in the course of the rays that
can be produced in a non-homogeneous medium by anomalous dispersion. In the
experiments of Hato, I have discussed, this phenomenon seems to have had no
influence. This may be inferred from the circumstance that the emission lines of
his flame had about the same breadth as the absorption lines,

( 611 )

cements would however be mueh smaller than half the breadth of
the band. This is easily seen, if we divide the value of m", —n
caleulated from (43) by the value of m— n
The result

that is given by (29).

is (ef. §12) a small fraction, because 7,7 is equal to the number of
vibrations during the time +, multiplied by 2 a.

(January 25, 1906).

=

KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM.

PROCEEDINGS OF THE MEETING
of Saturday January 27, 1906.

—aS (i Co—

(Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige
Afdeeling van Zaterdag 27 Januari 1906, Dl. XIV).

€ OWN aN eS:

F. M. Jarcer: “Contribution to the knowledge of the isomorphous substitution of the
elements Fluorine, Chlorine, Bromine and Iodine, in organic molecules”. (Communicated by
Prof. A. P. N. Francurmonr), p. 614. (With one plate).

L. van Iravur: “On catalases of the blood”. (Communicated by Prof. C. A. PEKELHARING), p. 623.

L. van Iratire: “On the differentiation of fluids of the body, containing proteid”. (Commu-
nicated by Prof. C. A. PEKELHARING), p. 628.

O. Postma: “Some remarks on the quantity H in Borrzmann’s “Vorlesungen iiber Gastheorie”.
(Communicated by Prof. H. A. Lorenz), p. 630.

F. A. F. GC. Went: “Some remarks on the work of Mr. A. A. Puts, entitled: “An enumeration
of the vascular plants known from Surinam, together with their distribution and synonymy”,
“ p. 689.

W. Karreyn: “The quotient of two successive Bessel functions”, (2nd paper), p. 640.

H. J. Zwiers: “Researches on the orbit of the periodic comet Holmes and on the perturbations
of its elliptic motion”. (Communicated by Prof. H. G. van pe Sanpe Baxknuyzen), p. 642.

L. S. Orysrem: “On the motion of a metal wire through a lump of ice”. (Communicated by
Prof. H. A. Lorentz), p. 653.

Pp. P. C. Hoek: “On the Polyandry of Scalpellum Stearnsi”, p. 659.

Jan pe Vries: “A group of complexes of rays whose singular surface consists of a scroll and
a number of planes”, p. 662.

Crystallography. — “Contribution to the knowledge of the isomor-
phous substitution of the elements Fluorine, Chlorine, Bromine
and Iodine, in organic molecules’. By Dr. F. M. Janerr.
(Communicated by Prof. A. P. N. FRancuimonr).

(Communicated in the meeting of November 25, 1905).

Some time ago a paper was published by Gossner ') on the erystal-
forms of Chlorobromonitrophenol, Dibromonitrophenol and Lodobromo-
nitrophenol being an experimental contribution to the knowledge of

1) B. Gossner, Krystallographische Untersuchung organischer Halogenverbin-
dungen. Ein Beitrag zur Kenntniss der [somorphie von Cl, Br und J. Zeitschr. f.
Krystall. Bd. 40. (1905). 78—85.

43

Proceedings Royal Acad. Amsterdam. Vol. VIII.

(46149)

the isomorphous substitution of the halogens C7, Br and J in organic
molecules. The author first gives a short résumé of the chief series
of inorganic compounds where Cl-, br- and /-compounds have been
compared in regard to their crystal-form. Even in cases where a
direct analogy in form does not occur an isodimorphism may be
always proved to exist.

The /-compounds differ in most cases from the others as regards
their behaviour.

Only a few complete series of analogous halogen derivatives of
organic compounds have been investigated and in no case as to
their mutual behaviour in the liquid state.

A complete crystallographical investigation was made of : p-Chloro-,
p-Bromo- and p-lodoacetanilide*), the melting points of which are
respectively, 179°, 1674° and 181°. The Bromo- and the Jodo-com-
pounds are both monoclinic, the Chloro-compound differs and is
rhombic. The Sr- and the /-compound present in symmetry and
parameters a distinct analogy with the rhombic C/-compound; the
plane of cleavage is, however, a totally different one °*).

-Cl-compound: Rhombo-pyramidal.

a:b:¢=1,3347 : 1: 0,6857 ; 8 = 90°0'. Cleavable towards {100}.

Br-compound: monoclino-prismatic. *)

a:b:c=1,3895:1:0,7221; 8=90°19'. Cleavable towards {301}.
J-compound: monoclino-prismatie. *)
a:b:c=1,4185:1:0,7415; 8=90°29'. Cleavable towards {301}.

GossNER °) proved that the C/-compound is dimorphous and also
that it possesses a more labile monoclinic form. On the other hand,
the Br- and J/-compounds are certainly also diémerphous but here
the rhombic modification is the more labile. The more labile and
the more stable modifications possess very analogous parameters,
although their molecular structures are different. He thinks however
that the irregular positions of the melting pomts may be satisfac-
torily explained from all this.

On the other hand, in the series Chlorobromo-, Dibromo- and
Lodobromonitrophenol, all three derivatives are directly-isomorphous
with each other. (Structure: (OH):(NO,): Br=1:2:4; Cl, Br
and J on 6).

1) B. Gossner, Z. f. Kryst. 38. 156—158. (1904).

?) Fets, Z. f. Kryst. 32. 386 (1900); Idem 32. 406.

8) Miace, Z. f. Kryst. 4. 335; Fers, Z. f. Kryst. 87. (1903). 469; Witson, Z. f.
Kryst. 36. 86. Abstract; Panepranco, Z. f. Kryst. 4. 393.

4) Sanson, Z. f. Kryst. 18. 102.

®) Gossyer, Z. f. Kryst. 88. 156—158.

( 615 )

This is the first properly investigated series of halogen-substitution
products in organic chemistry where C/, Br, and J replace each
other in a directly isomorphous manner.

Notwithstanding this complete isomorphism there occurs here a
remarkable abnormality in the position of the melting points, just
as in the case of the isod¢morphous p-Halogen acetanilides. This
abnormality cannot, therefore, be explained in the manner described
above; in fact it is quite incomprehensible:

Cl-compound: m.p. 112°C. Spee. gr. 2,141 Mol. Vol. 118,7
Br- i Mie PLPlaAds (Cbs amasaode oy Elia ND fed
I- = m.p. 104° C. - ee G4 natn eel 29103

In this case it is the /-compound which exhibits an abnormal
melting point.

From all this it is evident that there is still something strange,
as regards the mutual morphotropous relations of the halogens, at
least, in the case of organic compounds. Some facts relating thereto
will therefore be communicated in what follows.

I have, frequently, published papers on the Methyl esters of p-
Chloro-, and p-Bromohenzoic acid *). The Chloro- and Bromo-deriva-
tive each appeared to possess a different form, whereas the melting
point line of binary mixtures should lead to the conclusion that an
isodiémorphism was present here, with a melting point line of the
rising type, although it seemed impossible then to define by physico-
chemical methods the mits of mixing for the two kinds of mixed
crystals.

In order to treat the existing problem as fully as possible, I
prepared first of all the corresponding /lwore- and Lodo-compound.

p-Fluorotoluene kindly presented to me by Prof. HoLueman was
oxidised with KMn0O, in alkaline solution, the p-Fluorobenzoic acid
was separated with HCl and then esterified by means of methy]
alcohol and hydrogen chloride. The ester, which has a strong odour
of aniseseed oil, is a liquid rendering measurements impossible, but
on the other hand the acid could be measured erystallographically.

p-Toluidine was diazotised and converted by means of KT into
p-lodotoluene, this was distilled with steam, recrystallised and oxidised
as directed to p-Jodobenzoic acid. In the same manner, p-Aminobenzoic
acid was converted by diazotation ete. into its acid and this was

1) Jazcer, Neues Jahrb. f. Miner. Geol. und Palaeont. (1903). Beil. Bd. 1—28;
Zeits. f. Kryst. 38. (1903). 279—301.
43*

( 616 )

purified by sublimation. Both Jodobenzoic acids thus obtained were
then esterified by means of methyl alcohol and HCl.
The product so obtained was purified by repeated reerystallisation
from boiling alcohol until the melting point became constant at 114°.
The methyl ester of p-Lodobenzoie acid m.p. 114° crystallises from
ether + alcohol in colourless needles, having a faint odour of aniseseed
oil, which are very neatly formed, and exhibit the form of fig. 8.

Rhombo-bipyramidal.
a:6:c0=1,4144:1 : 0,8187.

Forms observed ; a = {100}, predominant, very strongly lustrous,
sometimes with delicate, vertical stripes; p = {210}, very sharply
reflecting; 6 — {110}, narrow, often absent, but yields very sharp
reflexes; v = {122} and += {011}, well-developed; o = {112}, very
small and often absent altogether.

Habit: flattened towards {100}, with tendency parallel to the c-axis.

Angular measurements:
Measured: Calculated:
a: p = (100) : (210) =*35°15?/,’ —
b:v = (010) : 422) —*51 49 —
b : p = (010) : (210) = 54 44"/,

v :v = (122) : (122) = 76°23’ 76°22’
b: r= (010) : (011) = 50 24"/, 50 41°/,
a:v = (100) : (422) = 77 29 77 23
v:v = (122) : (122) = 25 42 25 41
r:r == (011) : OT1) = 79 12 Us) alal
v: 7 = (122): 014) = 12 50'/, 12 37
p:r = (210) : (011) = 68 23 68 33
CEO 22) ie OU 16 43°/,
0:0 = (4112) : (112) = 43 3 42 557/,

Cleavable towards {010}.

The optical axial plane is {001} with the }-axis as first bissectrix.
The apparent axial angle in @-monobromonaphthalene is about 80°;
the dispersion is @<v. On a, p and 6 orientated extinction.

The sp. gr. of the crystals is: 2,020 at 10°; the equivalent volume
= 12957.

Topic axes %: p: o = 6,8179 : 4,8208 : 3,9464.

From the above it follows that the /-compound is perfectly iso-
morphous with the analogous Br-compound. By way of comparison

(617 )

some of the chief data observed in both compounds are placed here
in juxtaposition :

p-lodobenzoic Ester : p-Bromobenzoie Ester :
Rhombo-bipyramidal Rhombo-bipyramidal
a:b6:c=1,4144: 1: 0,8187 a:b:c=1,3967 : 1 : 0,8402
Forms : Forms :

{100}, {010}, {O11}, {210}, {112}, {122}. {100}, {010}, {011}, {210}, {112}, {122}
On {100} delicate stripes On {100} delicate stripes
parallel c-axis. parallel c-axis.

Cleavable along 0. Imperfectly cleavable along 0.
Axial plane is {O01}; 1st Diag. is . Axial plane is {010}; 1st Diag. is 6.
Angles: Angles:
ap ='39 16" a: p = 34°56’
b:v = 51°49’ b= >ietO/
0:0 = 42°55 ete. 0:0 = 43°50’ ete.

The dispersion in the /-derivative is of an opposite character to
that in the Br-compound; the apparent axial angles are almost equal
if that of the /-derivative is measured in e-Bromonaphthalene and
that of the Lr-derivative in oil of Cassia.

It seems remarkable, that in our case the Bromo- and /odo-com-
pounds behave in an analogous manner and that it is the Chloro-
compound which exhibits here a deviating character.

In order to show the further relation of the three compounds the
binary melting point lines were determined and represented in fig. 9.

The melting point line Lr-/-compound does not deviate markedly
from the straight line, the difference is really negligeable. The lowering
of the melting point of the /-derivate is, therefore, practically directly
proportional to the number of added molecules of the Br-compound.

The melting point lines C/-/- and C/-Br-compound take an analogous
course, that is to say, all the melting points lie between the lowest
and the highest melting point. Both melting point lines belong to
the rising type of Roozesoom, which may occur in isodimorphous
substances. The lower branch and the mixing limits could not be
found by thermometrical methods. The existence of these two branches
may indeed be proved, and they are even situated at some considerable
distance from the top branches — at least at the side of the com-
pounds having the highest melting points — as was found by Dr.
Bb. R. pe Bruyn. It is, however, not possible to determine this line
with sufficient accuracy. The progressive change of the cooling-curve
is of such a nature that a discontinuity is observed from which we

( 618 )

may draw the above mentioned conclusion that the lower branch of
the melting point line — at least at the side of the rhombic mixed
crystals — is situated at a fairly considerable distance from the upper

Fig. 9. Binary melting point lines of the three halogenised benzoic methyl esters.

branch. This change in direction of the cooling line is, however, so
slight, that the true situation of the point on the lower branch cannot
be indicated with certainty.

The determination of the mixing limits by an investigation of the
solid phases, which are in equilibrium with solutions of known con-
tent, met with difficulties of an analytical character. An effort was,
therefore, made to determine those mixing limits by the erystallo-
graphic process. For that purpose solutions were prepared of mixtures
of the two esters, for instance of the ch/oro- and the bromo-ester,

( 619 )

in ether + alcohol, and the homogeneous mixed crystals obtained
on slow evaporation were individually investigated crystallographic-
ally and then their melting point was taken, namely the temperature
at which the last particle of solid matter disappears in the surrounding
fused mass. If one may assume that this last solid particle, in each
of the cases investigated, is really in stable equilibrium in regard
to the fused mass, the temperature thus found, by comparison with
the already found upper branch, indicates the molecule-percentage
composition of the mixed crystal under investigation.

If we take little of the Br-ester and much of the Cl-ester, we
obtain from the alcoholic solution monoclinic mixed crystals which
possess quite the form and angular values of monoclinic Cl-ester
itself. Of such crystals the melting point never exceeded 46'/,°. If
the proportion of the components is reversed mixed crystals of a
thombic form are deposited quite analogous to the Br-ester. These
erystals gave melting points from 79*/,° down to 47° ; but not lower.

Assuming that the melting point of the end terms of the mono-
clinic series does not differ practically from 47°, it then looks as
if rhombic mixed phases may exist which, at 47° as transition tem-
perature, attach themselves zmmediately to the monoclinic terms. I
have found, however, that rhombic mixed crystals with various melt-
ing points kept together in a closed tube for four months become
turbid and partially opaque with a rough surface as soon as their
melting point falls be/ow 65°. It is also remarkable that the rhombic
mixed phases of this kind are more and more badly formed and curve-
planed, and that they become more distorted, as if existing in a kind of
enforced condition, when their composition begins to differ from that at
65° towards the monoclinic side. It seems to me that when accepting
the above hypothesis, all rhombic mixed phases below 65° represent
metastable conditions, which, in the solid state, are very slowly
broken up, to be partly converted into monoclinic terms.

That is to say the melting figure takes schematically the form of
fig. 9; the said metastable conditions are then points situated onthe
extended part of the lower branch to the right, which indicates the
composition of the rhombic mixed crystals coexisting with the fused
mass. The stable hiatus in the mixing series then extends from 18°/,
to 60°/, of the 5r-compound.

From all this it follows that in consequence of the very slow
conversion of the mixed crystals, no sharp determination of the
mixing limits can be made in this manner when less than 60°/,
Sr-ester of monoclinic character is present.

In the system C/-ester + /-ester the matter is still more troublesome.

( 620 )

There, the end term of the monoclinie mixing series is situated
still much closer to the axis than in the case mentioned. In conse-
quenee of the very great difference in solubility of the C/-, and the
[-compound we never obtain here from alcoholic solutions anything
else but rhombic very delicate needles, while the monoclinic phases
erystallise so indistinctly that they are quite unsuitable for a serious
investigation.

The Br-, and the /-ester readily erystallise together in all propor-
tions with angular values which differ but little from those of the
components. No optical anomalies could be found in such mixed
phases. From this it follows that to those two halogen substitution
products belongs au analogous molecular structure. Their molecular
volumes in the solid condition agree indeed very well; the difference
is smaller than between that of the C+, and Br-compounds.

As regards the lowering of the melting point of the compound
melting at the higher temperature by addition of the one melting
at the lower temperature, this is not proportionate to the number of
added molecules, as in the system Sr- + /-compound. In the
mixtures of C/- and J-ester, the observed values are always situated
on a curve which occurs above the line of the proportionate lowering
of the melting point; in the system C/- and Sr-ester, on a two-
periodic curve which occurs below the said straight line.

It must also be observed that the mixed phases deposited from
alcoholic solutions possess a larger content in the compound melting
at the higher temperature than the solution from which they have
formed. For instance, from a solution containing 20°/, of Bbr-ester
and 80°/, of Cl-ester, mixed (rhombic) crystals were at first depo-
sited which melted at 57° corresponding with a considerably higher
percentage of the br-compound.

The Chloro-compound which is monoclinic with:
a:6:c=1,8626 :1:3,4260, and B= 64°18’

and the forms:
a={1003, c={0013, r={1023, p—f210}, <= f0113, o=fT11}, a=f111}, o= {113}.
presents a habit which is not at all like that of the two other deri-
vatives, although that habit, as shown in fig. 1—5, is in a high
degree variable, according to the choice of the solvent and tempe-
rature of crystallisation.

The habit of the Br- and /-compound is on the other hand per-
fectly analogous ; in the /-ester it is, moreover, very constant under
different conditions of crystallisation (fig. 8) whilst it is still some-

( 624 )

what changeable in the Br-ester (fig. 6 and 7), although no longer
so strong as in the C/-derivative. With an increasing atomic weight
of the halogens, the changeability of the crystal-habit, owing to a
change in conditions of crystallisation, decreases considerably and
gradually.

The sp. gr. of the three compounds, their equivalent volume and
their topic parameters are :
Cil-ester: d,,o = 1,382; V=123,37 .y:p:w=5,1731 : 2,7774 -9,5153.
Br-ester: d,,o= 1,689; V = 127,29 .y: w:w—6,6611 : 4,7691 :4,0070.
J-ester: d,,o = 2,020; V=129,70.x: w:w = 6,8179 : 4,8203 : 3,9464.

It must be remarked here that the melting points of the three
esters increase regularly by 35'/,° notwithstanding the difference in
erystalform : 44°—79*/,°— 114°.

The above admits of no other explanation than the assumption
that all three halogenised esters are dimorphous. The C/-ester must
still exist in a more labile rhombic form and the Br- and J-esters
in a more labile monoclinic form. In one of Bruni’s communications *)
a “monoclinic”? p-Bromobenzoic Methyl Ester is described by an
Italian investigator with the object of proving an “isomorphism”
with the analogous p-Nitrobenzoic ester. The given measurements
have, however, absolutely no connection with those applied to the
p-Chlorobenzoate, so that this monoclinic form can in no ease be
the one intended. Moreover, none of the measured angular values
of the p-bromo-derivative agree with those obtained by myself. It
appears to me doubtful whether the measurements mentioned in
Bruni’s paper are really correct or it may be that the operator has
really not been working with p-bromobenzoic Methyl ester at all.
All efforts made by me to obtain from this substance a crystal form
different to the rhombic one proved fruitless, whilst in the Italian
treatise, the supposed “monoclinic” form is represented as a perfectly
stable one which, therefore, occurs continuously.

In order to prove an eventually existing dimorphism of these sub-
stances, I have made use of LEHMANN’s microscopical method with the
aid of the crystallisation microscope constructed by him. It appeared,
however, that in none of these cases a positive result could be
obtained. I think that in the case of each of these substances, I can
notice two different ways of crystallisation under the microscope,
namely long, rather delicate needles and also parallelogram-limited

1) Brunt and Papoa, Gazz. Chimie. Ital. (1904). 84a. 133—143; Rendic. Lincei
(1903). 5a. 12. 348.

( 622 )

flat needles which exhibit higher interference colours and like the
first named extinguish normally on the longitudinal direction. Howe-
ver, the difference — if present at all — is so indistinct that taking
into consideration the inclination of these substances to alter their
erystal-habiousit under var circumstances in so high a degree, I dare
not conclude to an already proved dimorphism, Experiments with
mixtures richer or poorer in Chloro-ester also exhibited the same
properties. It is, therefore, quite possible that we have two forms
for each of the three substances, but this has not yet been proved
and also it could not be ascertained whether in the given circum-
stances both eventually present forms stood to each other in the
position of monotropy or enantiotropy.

A few short data as to the halogenised benzoic acids deserve mention.

1. p-Chlorobenzoie acid m.p. 243° has been measured by FELs
(Zeitschr. f. Kryst. 32, (1900) 389). It is monoclino-prismatic, with
a:6:c¢=1,2738 : 1: 3,3397, and @ = 78°24}'. The forms observed
have intricate symbols; besides a= {100} and c = {001}, he finds
d = {207}, o = {111}, e = {233}, wu = {822}, v = {411}. Sp.gr. = 1,541.

2. p-Fluorobenzoie acid m.p. 182° synthesised by me is also
monoclino-prismatic. If to the forms occurring here we assign
the symbols: a = {100}, c = {001}, r = {203}, s = {403}, and ¢= {043},
the indices being therefore analogous to those given above, the
parameters become :

GU Ci TACT A S20
== (S467

Although there exists here an undeniable difference in habit, I
think I may still conclude that there is a direct isomorphism of
the Chloro- and Fluore-compound. A distinct plane of cleavage was
not found. The melting point of p-Pluworobenzoic acid is also elevated
by addition of the Cl-compound.

Angular values:

Cl 160.00,
71S) == i'5236)
Sel Oedely
6-2) —WoelG,
C21g, — 16720)

Gg 204

The habit is thin-tabled towards c. with a rectangular circum-
ference. The crystals were obtained from alcohol + ether and were
generally badly formed. The extinction on ¢ is orientated.

( 623 )

- 3. p-Bromobenzoic acid, m.p. 252° was obtained by me in_ tiny
crystals from ethyl acetate + benzene but they were very badly
formed. They are monoclinic and probably quite isomorphous with
the two other acids. The angle of inclination amounts to about 78'/,°.

4. p-lodobenzoie acid has not as yet been obtained in measurable
crystals owing to its little solubility in most of the organic solvents.
Its melting point is situated at 267°, therefore higher than that of
the Br-derivative. A direct isomorphism with the three other halogen

benzoic acids is not improbable.

Physiology. — “On catalases of the blood”. By L. van Ivan.
(Communicated by Prof. C. A. PrxnnHarine.)

(Communicated in the meeting of December 30, 1905).

The discovery made by Tutnarp that bloodfibrine possesses the
property of decomposing hydrogenperoxide has also been extended
to defibrinated blood, by Scuéxpuin (Journ. f. prakt. Chemie 89,
22). It has found a practical application in the judicial investi-
gation on bloodtraces and has been the object of manifold scientific
investigations. A resuming report precedes the investigations by
Senter (Das Wasserstof/superoxyd zersetzende Enzym des Blutes.
Zeitschr. f. physik. Chemie 44 |1903)| 257—318) to whose work we
refer the reader. Senter calls the enzyme which he has isolated
from blood Haemase whereas I myself prefer to use the name
of catalase, which has been given by Lorw (Catalase, A new enzym
of general occurrence, Report N°. 68 U. S. Depart. of Agriculture.
Washington).

Although the catalases, those enzymes which are able to split
H,O, in water and oxygen, are universally scattered in the vegetable
and animal kingdom, it has as yet not been possible to isolate one
of these bodies in state of purity.

Although different phenomena indicate that there exist more than
one catalase (apart from Lorw’s a- and (-varieties) it jhas been
impossible as yet to discern them.

The following communication gives a new contribution to the
properties of the catalases of the blood, which may perhaps lead to
a differentation of the catalases, and which at least gives an opportunity
of dividing the catalases of some animals into two groups.

To Mr. C. J. Konine at Bussum I owe the communication that
human-blood diluted from 1—1000 heated at 638° for half an hour,

( 624 )

still contains a quantity of catalase whereas ox-blood under the
same conditions no longer contains any catalase after half an hour.
Want of time prevented the above mentioned gentleman from pene-
trating any further in this matter, so that he left the treatment of
this subject to me.

The loss of activity of the catalase of the blood by heating has
been investigated for ox-blood by Senrgr (/. c. p. 293). Those
investigations show that a diluted bloodsolution loses its activity in
a quarter of an hour at 65°, that the rapidity of decomposition is
considerably smaller at 55° and that the solution after having been
heated for three hours at 45° still contains 60°/, of the catalystic
power, which it possessed originally.

Moreover it appeared that the loss of activity is not proportional
to the present quantity of the enzyme, but that this phenomenon
takes place with constant rapidity. I thought it useful to investigate
the observed phenomenon with regard to some species of blood *)
more closely and to see at the same time if it was possible to
render the catalasereaction serviceable to the distinguishing of different
species of blood.

Method of investigation. 5cM* of the different species of blood in
a dilution of 1—1000 are heated for half an hour at 63°, then
cooled down to 15° and mixed with 3 cM®*. of a hydrogenperoxide
solution of 1°/,;. The mixture is put into a fermentation tube, such
as are used at the investigation of urine on glucose. If the mixture
still contains catalase the developing of oxygen begins within a few
minutes so that by the development of this gas it is indicated whether
catalase of the blood is present or not.

Human- and monkey-blood (Macacus cynomolgus) investigated in
this way appeared to contain still catalase after having been heated
at 63° for half an hour, whereas the blood of horses, oxen, pigs,
goats, sheep, rabbits, cavies, rats, hares, chickens, pigeons, fish (flounder)
and frogs did not show any reaction after the described treatment
with H,O, and did not split off oxygen within 3 hours.

Now the blood of some of these animals contains only a small
quantity of catalase, but in the liver this substance is present in
greater quantity. According to Barrett and Stern (Compt. rendus
138 |1904], 923-924) 10 mG. blood of a frog produces after being
mixed with H,O, of 1°/,, 7.5 cM.* oxygen in 5 minutes, whereas

1) For the providing of species of blood I am indebted to Messrs. W. C. Scutmmen
and M, G. pe Brury, of the veterinary school at Utrecht and to Dr. J. Birtikorenr,
director of the zoological gardens at Rotterdam.

( 625 )

an equal quantity of the liver liberates 295 cM.* oxygen in the
same time.

The liver of a frog was mixed with purified sand, and the mixture
thus obtained was shaken with water. A drop of the decanted
liquid called about in a H,O,-solution a turbulent development of O.
If 5 eM.° of the liquid was heated for half an hour it lost the power
of decomposing H,O, quite, so that also with a considerable original
catalytic power the above mentioned time is sufficient to make that
power disappear.

In order to get an insight into the rapidity with which the catalase
of the blood loses its activity I put into practice the following method
of investigation for some species of blood.

5 eM.* of the bloodsolution (1—1000) were put in some test-tubes;
the tubes and their contents were heated for some time varying
from O—110 minutes in the thermostate at 63°, then cooled down
to 15° and mixed with 10 or 20 cM.* of a H,O,-solution of 1°/,.
The action having taken place, for 1’/, hour at 15°, the catalase-
action was interrupted by adding 10 eM.°* diluted sulphuric acid
and the quantity of hydrogenperoxide, which had not been decom-
posed was immediately titrated back with */,, N. Kaliumpermanganatie
solution. While the not heated bloodsolution indicates the quantity
of H,O, which is decomposed by 5 milligrams of the used blood-
species, it could be investigated at an arbitrary point of time in how
far the catalytic action had been weakened by the heating.

At the used degree of concentration an oxidation of the catalytic
may originate by the H,O, (Senrer /.c. 279) but I would not reject
the advantages which are offered by larger concentration as it was
not wanted to get in the first place absolute figures.

In the table mentioned below the results of my investigations are
written down while the graphic representation gives a more ample
survey.

It is peculiar that here as well as at the blood investigations of
Untennutn and those of Nutsser and Sacus (Berl. klin. Wochenschr.
1905 N°. 44) the bloodspecies of related animals (man and monkey)
show a- relation with regard to catalytic power concerning the
absolute strength as well as the greater resistance against the increase
of temperature.

I think I may deduce from these investigations that the catalases
occurring in the blood of different species of animals are not identical.
My own observations, it is true extend to some individuals of the
different species of animals only, but from reports of Baretn and

( 626 )

Action of diluted bloodsolutions (14 — 1000) on hydrogenperoxide solutions (containing 1 pet. H,O,) after 5 cM’.
of the bloodsolution having been kept at 63° for the mentioned number of minutes and cooled down
to 15°. Time of the influence of the H,O,, one and a half hour at 15°.

Time of Number of milligrams H,O, decomposed by Developed quantity O in cM*. (0° — 760 mM.
the heating} H M H = =e]
at 63° in | Human Lies cats Horse Ox Goat | Pigeon || Human aa Ese aN Hones | Ox Goat | Pigeon
; Blood key blood blood Higodalle blood Bide" blood key blood blood blood blood blood
minutes. blood | (arterial) | (venous) blood | (arterial) |(venous)
107.8 407.3 66.6 43.9 20.7 8.8 0.7 3.55 3.53 2.49 41.44 0.68 0.29 0.02
93.5 Cael 4.9 0 3.07 0.32 0.16 0
76.5 84.5 0 0 4A 3.4 0 2.52 2.77 0 0 0.43 0.10 0
68.9 2.2 423 2.26 0.07 0.04
48.3 | 57.6 0 0 4.2 0.5 4.58 4789 0 0 0.04 0.016
-— 0.2 0 — 0.006 0
98.9 | 44.4 0 0 0.95 1.45 0 0
| 46.8 | 262 0.55 | 0.86
eed OOS Gem 0.33 0.55
aeeBeT 1080 0.29 | 0.33
ly = 5542 26.3 0.48 | 0,20
3.9 0.13 a
3.0 3.6 0.10 0.12
| 2.2 2.9 0.07 0.095
A 4 0.046 |

( 627 )

Haire (Soc. biol. 57 [1904] 264) it appeared already that the same
organs of animals of the same species contain mostly about equal
quantities of catalase and that this quantity is not dependent on the

mG. H,0,
an”
Number of milligrams H, 0, decomposed
10) V)° : .
by five milligrams of different bloodspecies
. after their having been heated at 63° for
S. some minutes in a dilution of 1 : 1000.

6S 90 95 100 ws 9
Human blood.
se nneee eee eee eee Monkey blood.
ee eee Horse blood (arterial).
eee oie Se ._~Horse blood (venous).
seawsine seas eras OXs bloods
eee Goatablood:

44-4:4-4-+-4-4-4-4-¢ Pigeon blood.

temperature of the body nor on the metabolism. To illustrate the
catalytic power of the bloodspecies which I investigated quantitati-
vely I add a short survey,

( 628 )

Quantity of oxygen in cM. (0°—760 mA.) obtained from the
action of 1 cM.* blood on a solution of H, O, of 1°/,.
Man 710
Monkey 706
Horse (venous) 288
Horse (arterial) 438

Ox 136
Goat 58
Pigeon 4

Utrecht, December 1905.

Physiology. — “On the differentiation of fluids of the body,
containing proteid.’ By I. van Irani. (Communicated by
Prof. C. A. PEKELHARING).

(Communicated in the meeting of December 30, 1905).

For the research of blood, sperm and other fluids of the body,
containing proteid, the phenomenon, that even traces of these sub-
stances are able to decompose hydrogenperoxide, has been used for
a long time already. If a drop of a hydrogenperoxide-solution is
put on an object on which blood or sperm was dried up, develop-
ment of gas is observed, even by the presence of traces of blood or
sperm which give rise to the forming of froth, when blood is
present. As the experiment can already be taken with some fibres,
and the splitting off of oxygen can be traced under the microscope,
it has often rendered me good services in examining on behalf of
the court of justice, at the preliminary-examinations. If then no oxygen
is liberated and the object on which dubious stains occur, has not
been exposed to a temperature higher than 65°, it may be concluded
from the fact that no reaction occurs, that blood and sperm were
absent.

It was obvious to try to render the result of the experiments
communicated in the preceding paper, serviceable to the distinguishing
of the blood of men, resp. monkeys, from the blood of other species
of animals. Not every one is in the possession of the serum of
Ustennut or of the sera recommended by Nrisser and Sacus (Berl.
klin. Wochensehr. 1905, number 44) for the anti-complementary
influence, while it may be moreover of great value to the court of

( 629 )

justice to have within some hours the certainty whether bloodstains
appearing on an object, originate from human blood (in our country
of course monkey blood may as a rule be left unconsidered).

It is now indeed possible to show the presence of human blood
(resp. of the monkey) in older stains with the aid of the catalase of
the blood. I wish to draw the attention to this that the presence of
blood must have been proved by a preceding microscopical, chemical
or spectroscopical investigation because other fluids of the body too
(sperm and milk) cause a reaction of catalase.

The method of investigation is simple. If the dubious blood trace
is dried on some tissue, a piece of it is extracted with water at the
ordinary temperature and the extract is divided into two parts. One
part is mixed with a solution of hydrogenperoxide of 1°/, and the
mixture is put into a fermentation tube. The other part is heated
for half an hour in a waterbath at 63°, then cooled down to 15°
and after having been mixed with a solution of H,O, it is also put
- into a fermentation tube. If within some hours oxygen develops in
both tubes, it may be concluded that human -— (resp. monkey-)
blood is present; the quantity of oxygen in the second tube is of
course smaller than that in the first. When active catalase of the
blood is present the splitting off of oxygen begins soon after the
mixing, and is finished in some hours.

If however only in the first tube oxygen is split off and the
second tube does not show any development of gas, it follows that
the catalase of the blood has become inactive by the heating to 63°
for half an hour, and that the dubious blood does not originate
from man or monkey.

[I was in the opportunity of applying these experiments to fresh
bloodstains of man, dog, ox and horse and to bloodstains on linen
from the year 1903 originating from man, oxen, horses, goats and
pigs. The old bloodstains gave the same results as the fresh blood.

If we dispose of more bloodstains we can follow the process of
the reaction somewhat quantitatively by preparing for instance a
larger quantity of extract with water, dividing this in parts of 5 eM.*,
heating this to 63° during different periods in test-tubes and mixing
it with H,O, and titrating it, as has been communicated in the
preceding paper. The peculiar process of the reaction, graphically
expressed, does not give a representation of the absolute quantity
of the catalase of the blood which is present, but is so characteristic
that human- (and monkey-) blood can be easily distinguished from
that of another species of animal, even in the dried state.

dt
Proceedings Royal Acad. Amsterdam. Vol. VIII.

.

( 630 )

It is hardly necessary to mention separately that the reaction of
the catalase can be made serviceable to the distinguishing of mother’s
and cow’s milk.

Cow’s milk which has been heated to 63° for half an hour no
longer possesses the property of decomposing H,O,, a property
which mother’s milk still possesses in a rather considerable measure
under the same circumstance.

In a mother’s milk which I thank to the kind interference of
Prof. Kouwrr I found the following results, but it should be
observed that the action of the milk on the solution of hydrogen-
peroxide found place in the nitrometer of Lunem and that I read off
the volume of the developed oxygen after twelve hours :

5 eM.* mother’s milk not heated gave 24.8 cM.’ oxygen.
Bie Bs ms » heated to 63° for <«

quarter of an hour gave 18.5 __,, 5
Dana ss i eo ore nc ae ome os
Dy 3 55 , for an hour ee Oe 3

Utrecht, December 1905.

Physics. — “Some remarks on the quantity H in Bourzmany’s
“Vorlesungen iiber Gastheorie’.’ By O. Postma. (Communi-
cated by Prof. H. A. Lorentz).

(Communicated in the meeting of December 30, 1905).

§ 1. It seems to me that some of the views advanced in one of
the first paragraphs of the above mentioned work, are inaccurate;
and it may be desirable to draw attention to this fact, because
several considerations of BorrzMann and others are based on them.
l1 mean § 6 on the “Mathematische Bedeutung der Grosse 1”.

In the ease of a gas the molecules of which are all of the same

type, this quantity is represented by [P1f do. Now, in § 5, assum-
a

ing that the gas is of the simplest nature and that the motion of
the molecules is ‘“molecular-ungeordnet”, BottzMann has shown that,
in general, the quantity /7 decreases by the collisions and is minimum
in the stationary state.

Such a gas would therefore move of its own accord to the
stationary state i.e. with Maxwe’s distribution of velocities, as
BoL?zMANNn shows further on.

Now it is demonstrated in § 6 that the quantity AH has also

( 631 )

another meaning, and that the circumstance that /7 is minimum
implies that the probability of the corresponding distribution of velo-
cities, indicated by the function /, is maximum. Afterwards the
connection between H/ and the entropy is indicated in § 8 on the
“Physikalische Bedeutung der Grosse H”

The meaning of // in question being very incompletely derived
in §6, we shall have to consult Vol. 76 of the Sitzungsberichte der
Wiener Akad., to which Botrzmann refers, and Vol. 72, to which
he refers in Vol. 76.

In § 6 p. 40, Botrzmann begins with the following reasoning:
“Fiir alle Zusammenstisse, fii welche der Geschwindigkeitspunkt
des einen der stossenden Molekiile vor dem Zusammenstosse in einem
unendlich kleinen Volumelemente lag, befindet sich derselbe, wie
wir sahen, bei Constanz aller anderen, den Zusammenstoss charakte-
risirenden Variabeln nach dem Stosse wieder in einem Volumelement
von genau gleicher Grésse. Theilen wir daher den ganzen Raum
in sehr viele (€) gleichgrosse Volumenelemente @ (Zellen), so ist die
Anwesenheit des Geschwindigkeitspunktes eines Molekuls in jedem
solehen Volumenelemente mit der Anwesenheit in jedem anderen
Volumenelemente als ein gleichméglicher Fall zu betrachten, gerade
so wie friiher der Zug einer weissen oder einer schwarzen oder
einer blauer Kugel.”

So it is as if the velocities were assigned to the molecules by
taking for every molecule a slip of paper from a box, which box
would be filled with slips of paper each indicating a unit of volume
of the “whole space”. The probabilities a priori ave therefore equal
that the components of the velocity §, 1,5 lie between two values
which differ d&, dy, ds.

Here at least something has been adduced to account for the fact
that these probabilities are equal, which has not been attempted in
Vol. 76 of the W. S. We have to derive it from the fact, that at
a collision the “points of velocity” skip from a certain volume into
one of the same size (ef. the “daher” of the quotation). For me
this has, however, by no means convincing force; for that one point
always skips from a volume to one of the same size does not prove
that it can just as well be found in any volume of the same size.
We shall presently show that this can hardly be assumed.

3ut let us first proceed. Let us assume n molecules are to have
a velocity, then the probability a priori that of them n, have their
“point of velocity” in the first volume w, n, @ in the second volume

n!

(n, ) !(n, w)! Sie

ete. is proportional to 7 = , Where (n,-++-n,+...) =n.

44*

( 632 )

p\P
If we now assume that for p! may be taken v2p x( 2) , we get
é
IZ= — w(n, ln, + nzlny +... +C, and so Z is maximum when
w (n,ln,+tn,ln,+..) is minimum or when [ress dE dy dS

is minimum, or when // is minimum, if we have a simple gas.

The distribution of the velocities /(§ 46) for which His minimum,
is therefore also that with the greatest probability, concludes BoLTzMANN.
So the stationary state with Maxwe.w’s distribution of velocities is at
the same time the most probable (p. 42). f

This, however, follows by no means from the above, for the
stationary state is that, for which the change of // with the time in
consequence of the collisions = 0, whereas the most probable state
here is that for which every conceivable variation of the numerator
of 70:

Not before the condition is taken into consideration that the kinetic
energy of 2 molecules must have a definite value, as BourzMann does
in Vol. 76 and 72, it can appear whether the result is the same.

Now

fo +0 +o

H= af i) fre nB)Uf(E 18) dE dy ds

—o —m—o
must be minimum, while the conditions
+o +a +0

n= f freroaeae

—o —wm —@
and
=f2@ Site
pea i f (+ + SF (ENS) dé dy a
pt Sea ae)
exist, when m is the mass of every molecule, and Z the kinetic
energy of m molecules.
In Vol. 72 p. 450 Bonrzmann gives the solution of this, and it
appears that when no external forces exist,
If(Equs) +atu.4m(S? + 77+ 67) =—0
where 4 and uw are constants which are still to be determined.
From this follows Maxwnr.u’s distribution of velocities.
But what probability problem has now been solved? I cannot see
that any has been solved but the following: From a box with slips
of paper, each indicating a volume element, one has been taken at

( 633 )

random for each of the n molecules of a certain quantity of gas;
the “point of velocity” of the molecule was every time placed in
the volume element extracted. The mn velocities which have been
extracted chanced to be such that the sum of the energies of the
molecules has a definite value 2. What distribution of the velocities
among those n molecules is now a posteriori the most probable?

The most probable is therefore the distribution of Maxwrxi. But
this is a problem without importance for the gas theory. For it is
easy to see that the mean velocity indicated by the slips of paper
in the box is infinitely large, so that if from this 7 slips of paper
are taken at random, in general the mean velocity, which is indicated
by them, is also infinitely large, and there is only an infinitely small
chance, that the energy of the molecules becomes finite. If we
now see that of every finite gas-mass the energy is finite, we cannot
assume that the velocities would have been assigned to the molecules
in the way mentioned above. The chances a priori for every velocity
must, therefore, not be considered as equal.

The mean velocity in the box may be calculated as follows.

If in the unity of volume there are c points of velocity, then in
a spherical shell with radius 7 and thickness d7 there are: 4 27? ¢ dr.
The sum of the velocities now is 427° cdr, and so for a sphere
with radius r—=ar‘c. The number of points of velocity in the

. : 3
sphere is 3 arc, so the mean velocity = a”

For the whole space, therefore, the mean velocity is infinitely
large. In this way it is proved that the hypothesis of the equality
of the chances a priori is inaccurate, and so also the result that
Maxwet.’s distribution of velocities is the most probable state.

§ 2. Of course nothing is said here in derogation of the proof,
that Maxwett’s law holds in the stationary state, which BoLtzmMann
gives in the §§ preceding § 6 and in § 7. But it is ineorrect to
speak of transition of probable to improbable states when the meaning
is from stationary to non-stationary states. This incorrect view gives
rise to wrong considerations when BoiTzMANn discusses the fiction
of the reversal of the’ molecular velocities in the last part of § 6.

It is assumed there, that a gas has originally a “molecular-unge-
ordnet” but “improbable” distribution e.g. all molecules have the same
velocity. The gas moves now to the stationary state with MaxweL1’s
distribution of velocities. But before it is reached, all velocities are
reversed, which causes the same conditions to be passed through
but now in reversed succession. This will cause H to increase. Is

( 634 )

dil
this not incompatible with § 5, where it is proved that = ee only
a

be negative or zero? No, says Bourzman, for the reversed motion is
not one for which this theorem holds, because the motion is ‘molecular
eeordnet”. For the molecules, which a molecule with a certain velocity
meets, are not taken at haphazard from the whole number but
their velocities are connected with that of the molecule under con-
sideration. This is specially clear when at the moment of reversal
the motion had not yet lasted long.

Now, however, BourzMANN meets with another dfficulty, which is
to be removed. Does the increase of H not also clash with the laws
of probability, as the smallest /7 gives the most probable state ?

No, for the increase of 7 is only improbable, not impossible.

This difficulty seems to me to have only been raised by the in-
correct view discussed above. The smallest // is not the most pro-
bable. Moreover we do not do justice to the subjectivity of statements
concerning probability, when we speak of a transition from pro-
bable to improbable states, as if objective properties of substances
are expressed in this way. BonrzMann loses repeatedly sight of this;
particularly at the end of the second part of his “Gastheorie”.

In my opinion the views on this matter of Dr. A. PANNEKORK, occurring
in these Proceedings, Vol. VI, p. 427) are not perfectly correct either.

The latter assumes also that in the above mentioned case of
reversal the reversed motion is “molecular-geordnet’, and tries now to
make clear what this means. With perfect justice he says, that it
does not mean, as seems to be sometimes assumed, that the state
may be calculated beforehand; this might also be done in the original
ease if the initial state was known. Now, however, we get the im-

1) Another remark on this subject. Under 2 we read: “one more remark, however
is to be added”, on which something follows, that does not supplement what has
been said, but is in direct opposition to it. Moreover, the author seems to con-
found the collisions in the fictitious system (after reversal of the motions) and what
BotrzmMann calls the collisions of the opposite kind.

For it is not correct that the points Q,Q,', RR,’ return to P, P,'in the reversed
system; by reversal of the volocity we get a point of velocity lying diametrically
opposite to the first.

Also 3 gives rise to different questions. As e.g. is it allogether correct that in
the statistical way of treatment the direction of the normal of collision is consi-
dered as independent of the velocities? It can certainly not be independent of the
relative velocity? And further: does the fact, that in the calculations itis assumed
that the molecules do not hinder each other when colliding against a third, give
sufficient justification for calling the radius of a molecule small of the first order
with respect to the distances of the molecules ?

( 635 )

pression that Dr. Pannekork considers as the distinctive feature which
renders the original motion “ungeordnet”’ its dissipating influence, and
that which makes us call the reversed motion “geordnet” its bringing
the velocity points nearer together. When in consequence of the col-
lisions the ‘points of velocity” get dissipated the state would be ‘“unge-
ordnet’, when they draw nearer to each other, it is ‘“‘geordnet’’. But
this holds only in this special case, and it might just as well be just
the reverse.

For what does “molecular-ungeordnet” mean. This appears when we
examine the place where BorirzMann introduces this idea. We find it
p- 20 in the formula (17): Zp) = ® F, dw,. Here ® represents the
sum of the contents of all the oblique cylindres, into which a mole-
cule of ihe 2°¢ kind must get in order to collide with one of the
ist kind. The formula now expresses that the molecules of the 24
are, in proportion to the volume, as numerous in all these cylindres
together as in the whole gas mass, or that these cylindres constitute
a quantity taken at random from the gas mass with regard to the
molecules of the 2°" kind.

Now in my opinion an “ungeordnete” distribution might very well
be imagined, in which the points of velocity are more dissipated
than in the stationary state. And of a gas in such a state the points
of velocity would be brought nearer together by the collisions till
Maxwetu’s distribution of velocity is reached.

§ 3. With reference to the foregoing Prof. Loranrz was so kind
as to direct my attention to the work of Jrans on the kinetic
theory *). In this work a derivation of Maxwett’s distribution of
velocities occurs, which is called a new one by the author, but which
essentially agrees with the reasoning of BorrzmMann in the above
mentioned § 6 on the ‘“Mathematische Bedeutung der Grésse H”’,
though the outward form is quite different. It is true that an impor-
tant improvement has been made, which for the first time renders
it in reality a derivation of the law; it is viz. not only demonstrated
there, that the most probable state of a gas is that, for which the
distribution of velocities in question occurs, but also that the chance
is very great that a state will make its appearance, differing but
very little from the most probable: for when it is only known that
a state is the most probable, its probability may yet be so very small,
that it does not say anything as to whether that state will oceur or not.

Accordingly Jans calls this most probable state the “normal state’,
in which he is now perfectly justified.

1) “The dynamical Theory of Gases” by J. H. Jeans; Cambridge, 1904,

( 636 )
The ‘normal’ state is now the same as the “stationary” state.
However, the same objection applies to this derivation as to that
of BoLrzMAaNn.

Jeans calls his method “The method of General Dynamics” in
opposition to the usual one, with the aid of collisions, which he
calls “The statistical Method”. This name, however, does not seem
very appropriate to me; the considerations here are just as much
statistical as in the usual method. Dynamics do not play any part
in it but this, that the state of a gas with N molecules, so deter-
mined by 6 N-coordinates and components of velocity, is represented
by a point in a 6 N-dimensional space, and that now the change
of state of the gas runs parallel with the motion of this point in the
generalized space.

A great number of possible states gives therefore a great number
of points, and their changes a great number of orbits, the general
course of which is to be studied. Instead of with these mathematical
points we may also imagine the generalized space to be filled with
an homogeneous liquid, the motion of which we must examine,
which then according to the author is a “steady-motion” in hydro-
dynamic sense, the stream-lines of which are determined by the
property that their energy is constant.

This, however, brings us about to the end of the dynamic conside-
rations. They form an illustration, but nothing is proved by them.

The author now examines, what part of the generalized space is
taken up by points representing systems of a certain state. But this
is the same as what BoitzmMann calls the probability of a system ofa
certain state. Both represent the proportion of the number of systems
of equal possibility possessing a certain property, to the total number
of systems. The objections to be made to the expression for the
probability hold also for that of the part of the space.

JHANS treats successively two problems :

1. What part of the generalized space is oceupied by the systems
with a certain distribution of the coordinates of the molecules (or
what chance is there of a certain distribution of density of the gas)
and in connection with this: how are the systems distributed in
that space with regard to the distribution of the coordinates.

2. What part of the generalized space is occupied by the systems
with a certain distribution of velocities of the molecules (or what
chance is there of such a distribution) and how are the systems
distributed in that space with regard to the distribution of velocities ?

Only the first problem is fully treated by Jrans; for the second,
the most important, we are referred to the first.

( 687 )

It is then assumed that the gas is inclosed in a vessel with
capacity 2, divided into 7 elements @, so that nw=2. We imagine
now a certain distribution of density, at which a, molecules are
placed in the first element of volume, @, in the second ete. The
number of ways, in which NV molecules may be distributed over the
n elements, so that every time this distribution of density exists is

N! -
- For each of these ways every molecule must be
©, Oe oat
placed in a certain element of volume from the m and so the
| th
representative poimt’ is restricted to the — part of the whole
n

D>*

sentative points will occupy the fraction n—‘ of the whole of the
generalised space’. This is in somewhat different words nothing but
“the chance of each of the combinations is n—%”’, and the reasoning
resis evidently on the assumption that each molecule has every time
an equal chance to any place in the vessel.

The representative points of the systems with this distribution of
velocities occupy therefore together a part of the generalized space

N!

= = ey n—N (which therefore represents the total chanee ; an
$A, 2 222 Ay:

generalized space, in the same way with the following, so “the repre-

expression agreeing perfectly with the chance of a certain distribution of
denoities in § 1). After a similar reduction as in Bo.tzmann follows

1/
5 . : 12" :
from this: the part of the generalized space (chance) = -oNE,,
r—
(22) 2
sn
F ieee 1 Nis
where K,=—S as-+ — } log — in the above mentioned distri-
NN sone 2 N
s—T

2
Ka as a special value of the general funetion:

: IL RCD Y
Ke afi log da dy dz,
Q Lead) ;
0 0

integrated over the vessel, where p represents the molecular density
as function of the coordinates of an arbitrary point, and », the
mean density throughout the vessel. A’ is a function corresponding
closely with Bonrzmann’s H, specially -when we leave out the
constants and write:

K ={{f> log v dex dy dz just as /GL was ff J log f da dy dz;

vis the density function, just as / is the function of velocity.

bution (A). Now, neglecting 4 by the side of a, we may consider

( 638 )

K is now minimum when a, =a, = ete. or when » = constant.
It is obvious that this also means “the part of the generalized space’,
is maximum or the chance is maximum. So on the above assump-
tion the most probable distribution is that of uniform density.

Now Jrans proves further, that also by far the greater part of
the generalized space contains systems which differ infinitely little
from these with minimum A, so that this state may be called the
normal one. Expressed in the other way this is, that the chance is
infinitely great of a state deviating infinitely little from the most
probable state. Though Jrans’ proof does not seem faultless to me
(no sufficient attention is paid, in my opinion, to the order of mag-
nitude of infinitesimals) yet the result seems to me to follow from
Bernoutt’s theorem, provided “systems differing infinitely little’ is
taken in the proper sense.

So Jeans concludes: it is clear that the gas-masses with uniform
density will represent the ordinary case.

The second problein might be treated in the same way. Instead
of the molecules which are to be distributed over the elements of
volume of the vessel, we have now the velocity points of the
molecules which are to be distributed over the elements of volume
of the whole space. We get now in the same way for the part of
the generalized space occupied by systems with a certain distribution

v)
of velocities, the expression niles n—N, but now WN is infinitely
G15! . ++ On
large. According to the other mode of expression this is again the
chance to that distribution of velocities.

The treatment of the problem is further the same as that of the
first, but now we have to do with the quantity H/. And finally it
may be proved, that by far the greater part of the generalized space
is occupied by systems which differ very little from that with mini-
mum HZ or the normal state is that for which H is about minimum,
from which, taking into account the condition that the energy = L,
Maxwenv’s distribution of velocities follows.

Now it is, however, clear that the same objection may be raised
to this reasoning as to that of BourzMann.

The above expression for the part of the generalized space (or
ihe chance) rests on the assumption that the representative points
are distributed uniformly throughout the generalized space also here,
or that for every molecule the chance that the point of velocity
vets into a certain element of volume, is independent of the place
of that element. What now does the condition, that the energy
= /, mean? Either that attention has been paid to it in the distri-

( 639 )

bution of velocities or not. If no attention has been paid to it, it
is not to be accepted that the energy always becomes finite (see § 1);
if attention has been paid to it, the chance a priori can no longer
be taken equal for each element of volume, and the above expression
is faulty, and so also the further reasoning.

So it seems to me that also this derivation of Jeans must be
considered as incorrect’).

Botany. — Some remarks on the work of Mr. A. A. PULLE,
entitled: “An enumeration of the vascular plants known from
Surinam, together with their distribution and synonymy.” By
Prof. F. A. F. C. Went.

Mr. Putin has worked out the botanical material collected by the
expeditions of the last years, of one of which he was a member
himself. He has also tried to render our knowledge of the flora of
Surinam more complete by incorporating into his work the older
collections which are preserved at Leyden, Utrecht, Géttingen, Berlin,
Kew Gardens and in the British Museum.

In this way a total number of 2100 vascular plants appeared to
be known for Surinam and although it may be said with certainty
that this number is far from representing the real number of species,
occurring in our colony, yet we must appreciate that here for the
first time a comprehensive idea is given of the flora of Surinam.

Without entering into further details it must be mentioned that
the author is led to the important result that phytogeographically
Surinam belongs to the Hylaea, the region of the Amazon river,
with the exception perhaps of the still unknown territory west of
the Wilhelmina range. The Hylaea would then extend from the
mouth of the Amazon river over French Guyana and Surinam and
gradually form a narrow littoral strip in British Guyana, finally
passing into the Orinoco district. As a consequence of this the
conception must be given up that across Surinam there is found a
continuous savanah district, such as occurs in Demerara and more to
the west; where savanahs are found in our colony their presence
must be entirely attributed to local influence of the soil.

1) Jeans’ derivation occurs for the first time in the Philos. Magazine VI, 5, 1903,
under the title of “The Kinetic Theory of Gases developed from a New Standpoint”
p. 597. That also the “molecular ungeordnet’’ hypothesis is implied, which Jeans
denies, is proved by Burpury in the same magazine VI, 6, 1903 in an article
on “Mr. J. H. Jeans’ Theory of Gases” p. 529.

( 640 )

Mathematics. — “The quotient of two successive Bessel Functions’.
(2nd paper). By Prof. W. Kapreryn.

In our preceding paper we gave the value of the general coefficient
of the expansion

I’¥1(z)

D2)

cl A a a a

Now we wish to draw the attention to a couple of relations which
exist between these coefficients. The first is obtained from a particular
integral of the following differential equation of RiccaT1

du
cE

pee ei ik a i teeta URS conc. {(IU))

Putting
a

20 4+1)4+u,
this differential equation reduces to
du,
2 ale he aie Zz —10;
Repeating this process, it is evident that the equation (1) is satisfied
by the continued fraction

C—

a2
=e 33
2(y--1) — —_ 2?
ee
2(v-+3) — ete.
: q DP+1(z)
which represents the value of — z ‘OVD
Introducing therefore

u=—f,2—f,2'—f,2' — ete.

in the equation (1) we have
2@+ D7, =1
9: (ick 1s lL Nee ue eee eee
where irl wows
The second relation may be deduced from our former equation

antl as” oss ay? anti fri = (= 1)!

An 2 Xn—2 tn 2A, 9...00

Ane Xn—1 0 0 Foro 0) (0)

( 641 )

where

a =2(» + p),
_ Grp = 1) ...(2n — 2p)
a p!

(2n — p — 2)... (2n — 2p — 1)
p!

ap

ap +1...@2n—p—1

Ap +1... 42n—p—2

¥ __(2—p +1)... (~— 2p + 2)

p ——~ Ap +1... —p+1

p!

n ; . a—l ;
and / stands for coz 1 when 7 is even or for —j— when n is odd.
a

Putting a, = 2b, this equation may be written
S21 BH 6.9... ba? Ont fart = Dn - - . --- (2)
and it is found that the determinant DD, satisfies the condition

r—l—n—2
10% _— nb, One by Drs a SAS Lou ew b, ihe bn 4 b, a bn Dy—2 a
—2 .n—3 .n—4 =
ai - _ = b, sald e b, ~ + bn—1. by > + bn—2 Dus es

the last term being

Ba]
NEE Sear Yee et ee eerie ey meee Ee oy)
9 1 1 af l

when 7 is an even number, and
n—l
(a SU) EPIT aa a TL ne st ei
2 2
when 7 is odd.
Substituting in this equation D, by their values from (2) we get
this second relation between the coefficients /,

a n—l

: ae (n—p)...(n—2p) fn+-p

i —— —1)p : JOE
det = ) ESR a wey arene
P=

Finally we will show that from the recurrent relation between
the determinants D, the value of
Lim Jn ==
n=o Sn+1
may be deduced. For the series

father thet.

is converging when

or when

|z2|< Lim WA ip :
n= Jn+1

Now
Lim Va= = Lam 2 be ae bn} wrt —@
Jn+1 Dn
therefore
iB b, see bn 41 Dr CaN
Lin - =
um D, 5
(MenieO sts lvq Os oo Oy IO —o) :
Li See 7B ss =($) ete.

and finally

a 6
(:)
~) ore, “ SWEee

Wet bet 3

— ete.

b
Hence it is evident that @ is a root of the equation /’(z2)=0 as
might be expected.

1

Astronomy. — “Researches on the orbit of the periwdie comet
Holmes and on the perturbations of its elliptic motion.” By
Dr. H. J. Zwiers. (Communicated by Prof. H. G. van DE
SANDE BAKHUIJZEN.)

In 1902, after the reappearance of the comet Holmes in 1899—
1900 I published in full the results which I had derived from the
investigation of the observations after its return.*) With the most
accurate elements which I had been able to deduce from its appearance
in 1892 —93 I had calculated in advance the perturbations arising
from the action of Jupiter and of Saturnus and at first also of
the earth and thence I have derived a system of elements for 1899
September 9.0 mean time Greenwich, which served as a basis for
an ephemeris published in No. 3553 of the Astron. Nachrichten.
By means of this ephemeris the comet has been rediscovered at
the Lick Observatory and the relatively small difference between the
observed and the computed place proved that the elements of the

1) Recherches sur l’orbite de la cométe périodique de Holmes et sur les perturbations
de son mouvement elliptique, par Dr. H. J. Zwiers. Deuxieme mémoire. Leyde,
E. J. Brill, 1902.

( 643 )

orbit found for 1892 and the computation of the perturbations which
had been based on them were very nearly correct.

The observations in 1899 and 1900 furnished me with sufficient
material to apply to the elements such small corrections as brought
the remaining differences between the predicted and the observed
positions within the limits of ordinary errors of observation. The
system of elements obtained thus, which satisfied both the appearance
of 1892—93 and that of 1899—1900 and which in my “Deuxiéme
Mémoire” p. 78 has been recorded as ‘Systeme VII’, must naturally
furnish the basis for further investigations. Therefore I shall give
it here in its general features.

System VII.
Epoch 1899 June 11.0 mean time of Greenw.
Osculation 1899 September 9.0 _,, Pee 3
M, = 22661" 3264
w= 516" 188791
log a= 0.558 1820.0
p= 24°17! 23"54
e= 0.41135382
i= 20° 48' 9"84
m= 345 48 38.06 } 1899.0
Sb = 331 43 18.24
i= 20°48) 10°29
mw = 345 49 28.27 } 1900.0
S = 331 44 8.95 |

Although the corrections which had to be applied to the elements
in consequence of the new observations were small, I immediately
after the publication of those researches resolved to repeat the compu-
tation of the perturbations between 1892 and 1900 with the new
elements and to extend it to all the planets of which the disturbing
effect could not a priori be neglected as being insensible. This
elaborate investigation, which necessarily required a new discussion
of the two appearances of the comet, was however only partly
finished when in 1905 the preparation for the third appearance had
to be taken in hand.

I have then started from system VII, which though not perfect, yet
satisfied all practical demands. I did not venture, however, to use
those elements without more for the computation of the places at
the return of the comet in 1906. It is true that the disturbing planets,
especially Jupiter, whose influence is by far the greatest, remained
at a considerable distance during the entire revolution of the comet,
yet the feeble light of the comet in 1899—1900 and the difficulty

( 644 )

experienced by most observers to properly identify the comet in the
midst of numerous faint nebulae near the apparent orbit, made me
fear that such a rough ephemeris of the apparent places for 1906
might prove insufficient for rediscovering it and observing it.

In the autumn of 1905, I therefore resolved to derive the pertur-
bations which the comet would suffer on its path between the perihelion
passages of 1899 and 1906. The original plan of also computing the
perturbations arising from the action of Saturnus had to be given
up through lack of time. And so Jupiter remained the only disturbing
planet. The method I chose was that of the variation of the elliptic
constants; I also chose an interval of 80 days, because former
investigations had shown that the accuracy, attainable by it was
more than sufficient for my purpose. In former researches we have
always adopted the rule that for each new epoch the small varia-
tions which the elements had undergone during the course of the
last interval were to be applied to them. The computations required
for this implied, however, an amount of labour not to be underrated,
and as in this case the computations could have only a preliminary
character I could leave aside these small corrections by which in this
case only small quantities of the second order were neglected. Thus
the above mentioned system VII was used as a basis for the com-
putation of perturbations for the entire revolution. The places of
the disturbing planet are taken from the Nautical Almanac; the
longitudes only were reduced to the equinox of 1900.0 by applying
the precession. The neglection of the small corrections for nutation
and for the variation in the obliquity of the ecliptic cannot have
any perceptible influence on the perturbations caused by the planet.

Instead of the elaborate tables of perturbations I shall for shortness
communicate only the summed series, namely the quantities “7 for
the mean daily motion and the quantities // for the other elements.
By working out each table the reader will be able to form a judgment
on the accuracy reached. The initial constants printed in big figures,
which in the construction of the tables were derived from the first

dE : x : ;
values of a (Z representing one of the 6 elements) and from their
at

differences up to f/V are chosen so that the integrals disappear for
1899 September 9 as lower limit. Up to 1900 February 16 the
derivatives could be borrowed from the tables which I have commu-
nicated in my Deuxieme Mémoire ps. 26—32; with regard, however

du
to the interval chosen now I had to multiply = by 4, and the

other derivatives of the elements by 2.

TABLES OF THE JUPITER PERTURBATIONS.

1899

1900

1901

1902

1903

1904

4905

1906

Dates t roy “ M - ,
di }
raat? : n | = 94.5780 n n ,
pHa + 4 Uae 7.382 See 0.476 413.677/— 29.200
seu rie ed eg MS kaa
eepee + 0.7604 3.297 1S se 2505+ 2.601/— 4.611
un th do l eaeaaere 2.530;— 3.150; 4.197
eee 1.018;— 13.652; ey pene 5.973,— 13.460/+ 411.287
an 0.203 — 25.174) BEN cs 6.752\— 28.616|4 48.978
July 26 + 1.867— 37.406 ot edie 5.822;— 47.169\4- 26.379
hey: + 5 118\— 49.494 oe euegeee 4.954\— 67.123/4 36.268
ee + 9.397 — 60.732 Ruheh ese 6.051,— 86.5164 48.205
TENGE + 14.489, — 70.570 OS Cer 10.874|— 103.621/4+ 62.168
eee ie 20.130) 16. Ca Crs 20.923)— 117.008|4 77.946
Aug. 30 + 26.049— 84.681 ie — 37.388|— 125.552/4 95.199
ae + 31.931.— 88.702 ete — 61.130|— 128. 426/44. 113.505
win & + 37 aloe 90.817 Rea elie 92.674|— 125.094|4- 132,387
April 27 dee cad th 4+ 31.9635) Se ea Heo
July 16 + 46.560 — 90.624 i ee 179.598|— 99.047/-4 169.854
Oct. 4 aaa 5 + coals pee eae ee
ee + oe 87.924 i es: 295.802|— 48.453/4+- 203.590
Ronis + 52.337,— 87.192 i ey iy 362.844|— 15.304\4 217.997
ane + 51.916 — 87.735 i eas 434,953|-- 21.975|4- 230.081
Ang. 20 mo es 90.131 i. a 508.617|+ 62.385|4 239.783
ee + #8 ae 94.838 a Seem 584.410-+ 104.856/4- 246.855
ee es + 45.193, — 102 143) i Oe er, 660.082) 148 .314\4- 254.221
; + 41.996 — 112 115) |— 734.136 /-+ 191.759/+ 252.903
ee + 38 875|— 124,566) a em 805.216) 234 308) 952.011
a ; =f 36.183 — 139.032) 4 ee 872.176|-+ 275. 384|4 248,718
ue + 34.243 — 154.764 e peo 934.1934 314.438|+ 243.294
a : ++ 33.309— 170.745 E wee 990.396 + 351.479|4+ 235.703
coe + 33.518|— 185.739 Hoey —1040.430|+- 386.444|4- 226 250
ee “+ 34.834 198.384 naaaaey —1083.499/+ 41912914 244.885
ah + 36.981|— 207.384 ere —1117.855|-++ 448.580|4- 201.767
rae + 39.372|— 211.999 fimo 111 214) 472..756|4- 187.961
: + “.116|— 212.537 eee —1152.415|4+ 490.526/4+- 176,824
a E + “.312/— 242.305 sev a —1158.665)+ 508.036|-4- 173.909
e@ oD d

( 646 )

By means of these tables it is not difficult to integrate the pertur-
bations for an arbitrary epoch according to the known expressions
of the mechanical quadrature. As a new osculation epoch I have
chosen

1906 January 16.0 mean time Greenwich

and I have found:

Ai=+ 4034 A $= — 3'32"48
|
Ap=+ 1°258874 {fF = + 883'5368
A,M = — 1147'7070 Age = +8" 2408 |

Ag=+3' 2"01
hence the new elements become:

epoch and osculation 1906 January 16.0 mean time Greenwich
M, = 1266412143
u = 517"447665
log a = 0.557 4267.74
gp = 24° 20' 25"55
e= 0. 412 1574
t= 20°48'50"63
m = 3845 5730.35 } 1900.0
SX) = 331 40 36.47

From these disturbed elements we derive for the time of perihelion
passage
1906 March 14.1804 mean time Greenwich
while the original system VII, without regard to the perturbations
during the period since 1899 June would give
1906 March 13.8083.

If we take into account that the small retardation of not yet 9
hours is compensated by an increased longitude of the perihelion of
8', we find a posteriori confirmed, what could have been foreseen,
that the perturbations during the second revolution have only slightly
affected the places of the comet in space.

By reducing the elements 7, a and §% to the mean equinox of
1906.0 I find

t= 20°48'53"30 |
w= 346 231.63 » 1906.0.
SQ = B31 45 40.75 |

In order to compute from these elements an ephemeris I have
derived the following expressions for the heliocentric coordinates of
the comet referred to the equator:

( 647 )

x = [9.993 7731.9] sin (v + 77° 37! 24"85)
y = [9.876 2012.2] sin (v — 20 58 31.25)
z = [9.832 7001.5] sin (v — 1 47 16.19)

The coefficients in square brackets are logarithms; the quantity
denotes the true anomaly of the comet.

By means of the expressions above given the heliocentric coordi-
nates have been derived from + to 4 days for mean noon at
Greenwich; the coordinates of the sun were taken from the Nautical
Almanac after having been reduced to the mean equinox of the
beginning of the year. In the reduction of the mean places to
apparent ones the aberration terms are omitted, because, as it is
known, the influence of the aberration for the bodies of our solar
system can be more simply accounted for by subtracting from the
times of observation the equation of light. In the two following
tables which contain the apparent places of the comet in @ and
d I have therefore added in column # for each date the equation
of light expressed in mean solar days. The 4 column gives the
logarithms of the geocentric distance. As first date I have chosen
May 1st because I had derived from a preliminary computation that
before that time there would be no chance to discover the comet
owing to its small apparent distance from the sun and its large
distance from the earth. The possibility did not seem excluded,
however, that by means of powerful telescopes or sensitive photo-
graphic plates the comet might be discovered in January 1906.
Therefore I have derived positions for that month and sent a short
ephemeris to Prof. Kreutz, who in a circular has communicated
it to astronomers. To give a clear idea of the apparent orbit of the
comet and also because the published places were not perfectly
correct owing to a small reduction error, I here shall give the
correct results from 4 to 4 days. Up to now (February 14) no tidings
about the discovery have arrived, at which we need not wonder
if we consider the cloudiness and especially the southern and
generally unfavourable position.

The next table gives the apparent positions of the comet for the last
8 months of the year. The direct computations have been made from
4 to 4 days; between them one date has been interpolated taking
into account the fourth differences.

As a measure for the probable brightness we generally calculate

the quantity H= Pe Although on account of the irregular varia-

tion of the comet’s light it is not certain that the brightness will be
45*

( 648 )

PLACES OF THE COMET BEFORE THE CONJUNCTION.

1906 apparent « apparent 0 | log p | 3 H
Jan, 4 | 90°'%5"4'65 | — 9193 4.7 |. 0.47858 | 0.017373. | 0.0930
5 53 18.18 | — 20 26 48.4 48066 456 | 0299
9 | 2 193.9% | — 1999 45.4 48257 533 | 0299
13 9 46.66 | —4183024.6 | 48431 603 | .0228
47 1758.35 | —4173017.8 | .48590 668 | 0298
2 26 8.26 | —16 2858.8 | .48733 726 | .0298
25 3416.26 | —15 2698.4 | 48860 778 | .0297
29 42.22.19 | —4149250.3 | 48974 g24 | .0297
Febr. 2 50 25.91 | —1348 7.8 | .49067 863. | .0297
6 58 27.36 | —121224.5 | .49147 06 | .0297
10 | 922 696.56 | —41 543.5 | 49013 993 | .0997

proportional to H, 1 for completeness have added this quantity to
the table from 4 to 4 days. In 1892—93 this so-called “theoretical
brightness” varied between 0.075 and 0.012.

Because the elements adopted for 1900 might still require small
corrections, and as up to 1906 only the principal perturbation by Jupiter
has been taken into account, it is not improbable that when the
comet happens to be discovered there will be some difference between
the observed and these computed places. In order to facilitate the
search for astronomers who possess the needed instruments for
finding it, I have repeated the calculation of the places first on the
supposition that the comet will pass through its perihelion 4 days
earlier, and secondly that it will pass 4 days /ater than would
follow from the most probable elements. Although the adopted
latitude of + 4 days will probably be much larger than the
real error in the accepted time of passage through the perihelion
I give the results as obtained from direct calculation. The following
table contains the variations in right ascension and declination for
the two suppositions; column A loge gives the corrections which
would have to be applied to the 5 decimal of log @ from the ephe-
meris communicated before.

( 649 )

APPARENT PLACES OF THE COMET FROM MAY 1 TO
DECEMBER 31, 1906,
ror O® MEAN TIME AT GREENWICH.

1906 a 8 logy p s H
TTS Fama
May 1 0 40 15.28 + 12 49 44.3 0.47733 0.017 322 | 0.0240
3 44 0.82 + 13 25 36.3 47632 282
5 AT 46.23 +14 41 21.2 47528 241 0241
7 51 34.54 36 58.4 AT42A 199
9 55 16.77 + 15 12 27.8 ATHA2 156 0242
11 59 1.94 47 48.8 .47200 111
13 4 2 47.03 + 16 23 1.3 47084 066 0243
415 6 32,06 58 4.7 46966 O19
17 10 17.02 + 17 32 58.6 46844 0.016 972 0244
19 14 1.90 + 18 7 42.8 46719 923
24 17 46.67 4216.7 -46591 873 0246
23 21 31.32 + 19 16 39.8 46460 822
25 25 15.84 50 51.9 46326 770 0247
27 29 0.20 + 20 24 52.4 -46189 17
29 32 44.40 58 40.9 46048 663 0248
31 36 28.40 -+- 21 32 17.0 45904 608
June 2, 40 12.22 + 22 5 40.5 45757 552 . 0250
4 43 55.83 38 51.0 45607 495
6 47 39.23 + 23 11 48.3 45453 437 0252
8 54 22.42 44 32.4 45296 378
10 55) bod + 2417 2.4 45137 317 0253
12 58 48.06 49 18.9 AAITA 256
14 2 2 30.46 + 25 24 21.5 44807 194 0255
16 6 12.54 BB) Rs) 44637 134
18 9 54.18 + 26 24 43.6 ALAGB4 067 0257
20 43 35.40 56 2.8 44287 001
22 47 16.13 + 2727 7.41 44AO7 0.015 935 0259
24 20 56.31 57 56.2 43923 868
26 24 35.89 + 28 28 30.0 43736 799 0261

June 28

July

Aug.

bo

=

17 35.80
20 56.25
24 14.64
27 30.86
30 44.79
33 56.32
37 5.28
40 11.54
43 14.91
46 15.20
49 12.25
52 5.84
54 55.77
57 41.84
4 0 23.84
3 1.59
5 34.86

+ 98 58 48.2
-+ 29 28 50.8
58 37.7
+ 3028 8.8
57 24.2
+ 31 26 244
5b 8.4

+ 40 14 47.
38 20.5
+4 211.5
25 50.9
49 19.0
+ 42 12 35.9

42538
42326
A114
41892
41669
41442
1212
40978
40740
40499
40254

065
0.014 987
907

0.013 988

0.0264
0266
-0269
0271
0274
-0277
0284
-0284

ae
0291
-0295
0300
0304
0308

.0313

Oct.

hm is

4 8 3.48
10 27.94
12 45.92
14 59.28
lyf FAO
19 9.03
21° 4.88
22 54.34
24 37.12
26 12.92
27 41.47
29 2.48
30 15.71

33 6.20
33 45.85
34 16.56
34 38.08
34 50.16
34 52.64
34 45.25
34 27.94
34 0.56
33 23.06
32 35.43
31 37.75
30 30.16
29 12.87
27 46.15
26 10.32

S

+ 46 16 53.

+ 47 19 18.

59 34.
+ 48 19 14.
38 33.
57 31.
44916 3.

34 9

Bl 46.
+50 8 51.
% 2.
4A 43,
56 23.
+ 51 10 49.1
4 6.
37 44.
49 0.
59 5A.

.33512
33212
.32913
32615
.32320

32027
31737
31450
31168
30891
30618
30351
30092
29840
29595
29359
29134
28919
28715
28523
28345
28181
| 28034

0.013

0.012

0.014

191

0323

.0329

0334

0339

0345

.0350

0356

0361

.0366

.0370

0375

.0378

0381

0383

0384

1906 | a é log s | H
Oct. 30 | 424 95-75 at 59 9 “A 0.27897 | 0.010 971
Nov. 1 22 32.88 18 25.4 27779 944 | 0.0384
3 20 32.19 2% 0.8 .27678 916
5 18 24.96 32 95.2 27595 895 .0383
7 16 9.72 37 35.5 27530 879
9 13 49.28 4A 29.2 27484. 867 .0380
11 41 23.70 4k 4.0 27457 861
13 8 53.82 45 18.4 Q7451 859 .0376
15 6 20.53 45 14.0 27466 863
17 3 44.79 43, 1A.2, 27502 872 .0374
19 4 7.59 40 48.9 27560 886
oy 3 58 29.94 36 35.4 27640 906 .0365
93 55 52.74 By Peal Q7742 932
25 53.17.00 % 8.9 27865 963 .0357
27 50 43.59 16 0.8 28010 | 0.044 000
29 48 13.36 6 89.9 .28178 (42 .0848
Dec. 4 45 47.10 + 5156 9.2 98368 090
3 43, 25.56 44 32.6 28578 144 .0337
5 4A 9.42 31 53.9 28810 204
y| 38 59.30 18 47.3 29062 269 .0326
9 36 55,80 3 47.5 29334 340
11 34 59.40 + 50 48 29.0 29627 ‘AT 0314
13 33 10.58 32 26.7 29939 499
15 31 29.73 15 45.9 30270 587 0302
17 29 57.19 + 49 58 31.6 30619 681
19 28 33.49 40 49.4 30984 779 .0289
14 27 17.93 22, 43.8 .31365 883
93 26 41.50 4 20.5 31763 993 0275
25 25 13.95 + 48 45 43.9 32175 . | 0.012 107
27 24 I 29 26 58.7 22601 226 0262
29 93 45.29 8 88 .33039 350
34 23 14,07 + 47 49 18.0 33489 479 0249

( 653 )

VARIATIONS OF a, J AND log @ FOR THE ALTERED TIME OF PASSAGK

THROUGH THE PERIHELION.

7 =—4 days T=+4 days
1906 -
De Aé A loge Aa Ad J log p
May 5 |43°13°48 | + 38'55°2| + 931 | — 31342 | — 39 97.7 | — 233
» 24 + 3 22.45 | + 36 23.2 | + 294} — 3 22.12 | — 37 38 eer 297
June 6 + 3 33.07 | + 33 10.6 | + 355 | — 3 33.23 | — 33 58.3 | — 359
>» 92 | +3 46.12] + 2919.9] + 413] — 3 46.62 | — 3043.4] — m8
July 8 | 44 1.95 | + 2455.8] + 469] —4 2.30] — 2553.4] — 476
» 24 }+ 418.58) + 20 4.4 | + 521] — 4 20.42} — 21 4.4 | — 529
Aug. 9 | -+ 4 38.61 | + 14 54.2 | + 567 | — 4 4.55 | — 15 55.9 | — 576
» 2% 7+5 249)/+ 9 39.3} + 606] —5 6.87 | — 10 41.7} — 616
Sept. 10 | +5 31.97| + 44.9| + 632] — 5 38.299] — 5 45.9] — 642
» 26 7+ 6 9.74] + 39.2 | + 640 | — 618.02 | — 1 49.4 | — 649
Oct. 42 7+ 655.91 | — 4 27.14} + 621] —7 5.99 | + 4.4 | — 627
>» 8 |+744.03|— 93.3] +566] — 754.54] — 120.9] — 569
Nov. 13 | -+ 8 15.71 | + 415.2 | + 475 | — 8 23.95 | — 619.7 | — 474
» 29 |} + 8 10.71 | + 10.42.4 | + 361 | — 8 14.80 | — 12 50.9 | — 356
Dec. 415 + 7 29.94 (ee 45 44.7 | + 247 | — 7 30.69 | — 17 37.3 | — 241
Leyden, January 1906.
Physics. — “On the motion of a metal wire through a lump of ice’.

By L. S. Ornstem. (Communicated by Prof. H. A. Lorentz).

In a well known experiment on the regelation of ice a metal
wire charged with weights is placed on a lump of ice. It moves
slowly through the ice, while on the upper side new ice is formed;
after a short time the motion takes place with uniform velocity. This
phenomenon is explained by the fact, that if we increase the pressure
the meltingpoint is lowered.

In order to calculate the velocity of the wire I shall consider an
infinite circular cylinder which is moved through an infinite lump

( 654 )

of ice by a force perpendicular to its axis. The phenomenon is
the same in each normal section. I suppose round the wire a
layer of water whose thickness is small in comparison with the
diameter of the wire. At the bounding surface of water and ice
there is a pressure, which decreases from the lower to the upper
side of the boundary. This pressure depends on the force by which
the wire is acted on pro unit of length. As the motion is very
slow the temperature in each point may be supposed to be the
meltingpoint corresponding to the pressure existing in the point.
The flow of heat, determined by the distribution of temperature
is the same as if the wire were at rest. At the upperside of the
bounding surface of ice and water heat flows away and water is
frozen, at the lower side the ice is melted by the heat that is carried
towards the surface. If we can determine the quantity that is melted
we shall be able to determine the velocity acquired by the wire.

Let Mf be the centre of the circular section of the wire and R
the radius, the boundary between ice and water being a circle of
radius R -+ d.
, The pressure at the
circle A’ B'C" in any point
i may be represented by
the formula

P=p.t+ bos,

gy being the angle between
the radius JZ’ and the line
M A which has been taken
for axis of ordinates. The
corresponding temperature
is

dt
i=1,+2(2) cos @ ,
dp),

dt
(=) being: the change of
dp),

the meltingpoint per unit increase of pressure near 0° C.

Let k,, be the coefficient of conductivity within the circle ABC,
that of the layer of water, and &, that of the ice without A’B'C’.
The differential equation for the temperature is in every one of
these fields

C

kh

2

0%t Oi ae
dz? | dy?

The conditions at the limits of the fields are:

teat ABC re Gt. (eS 7 (Coy
On 1 On 3
dt

Beat Al BIC’ t= tot, 120 (=) cos Pp,
dp 0

3. at infinite distance ¢t, = t,.
The normal at ABC coinciding with the radius.
The formulae:
t, =t, + B,rcos y in the wire,
Ss
t, =t, + B, rcosg + = cosg in the layer of water,

Y

fe ll. — 608 —p in the surrounding ice

satisfy the equations r being the distance from the point M. For
the coefficients I find the relations

B= B,+

ky B,=4( 2, — a)

G Crees e) 1
Wea = (Edy yap), Rea
Neglecting powers of d/R I find

dt
Cre (5) to
Ri AR+ dk, +4/alk, kr
dt
(F) ety

OS ———————
; 2(R+ a)ik, +2/r(k, —-,)}
For an element E’F” of A’B’C’ the flow of heat into the ice
towards the surfaces amounts to:

C,
a eset
if we write dg for the angle E’ MF”. Hence the total quantity of
heat conducted through the ice towards the surface A’S’ per unit

of time:

cos pdg ,

n/2

Bee d o(2).
ee ae

In the layer of water the flow of heat per unit of time is for 4’ F"

( 656 )

Cm
(Rd)?

— (R + d) dg cos @ k, (», —-
and for A’ B’ totally

F pala DAK es se 1c Paes
me Nets = (Raeayty ae dp) byte, EE

Of course as much heat is lost at the surface B’C” as is conducted
towards A’ A’; and the melted and frozen quantities of ice and water
will therefore be equal. W being the quantity of heat that is required
for the melting of a gramme of ice, the melted quantity is

k, —4/R(k,—k dt
2 E ke, —*/ ht, ~hy) + | b (=)
a k+4/ fk, —h,) dp 0
W. ‘
If S, is the specifie gravity of ice, the volume of this quantity is:
k, —4/ p(k, —k lt
2| &, ipa lB) +k, ei b
k,+4/pr(k, —&,) dp 0
ea WS,

On the other hand, if the cylinder moves with a uniform velocity
v a volume

2Rv.
is melted. So we find for the value of v

E = ae a (Fy

Sais /R(k, —k,) dp 0

~ RW 8,

To express 6 in the force P acting per unit of length of the
cylinder we have only to notice that an element EY = Rdg is
acted on by a foree per unit of surface p cos gy = (p, + 0 cos g) cos gy.
Hence :

Te

[es 2f( cos p + b cos? g) Rdpy = abR

The velocity C in case P= 1 is found to be

dt ks tana he hk
dp 0 ; ka +4/p(h, = k,) ;
c= i neers (7
ak? WS,
We can find another expression for ¢/p if we pay attention to the
motion of the water. If we conceive the wire to be at rest but

the ice moving along it, we shall see at the limit A'S’ water con-
tinually streaming into the channel AB A'S’ while it streams out of

( 657 )

it and freezes at the part B'C" of the surface. The velocity of the
ice being v we find for the quantity of water entering through /'/”’
(R + d)v cos » dg.

This is also the difference between the quantities flowing across
FF' and EE’ upwards.

This quantity can also be determined by means of the hydro-
dynamical equations. Take for axis of € a circle with radius R + 4d
and for axis of 4 a radius of the circle. The forces acting on an
element KZOP are in equilibrium. Writing uw: for the velocity
parallel to the axis of §, w for the coefficient of viscosity, neglecting
the velocity w, and taking the intersection of the §& circle, with
EE’ for origin of coordinates we have:

Ou dbsing
Donan a ee
At the circle AB, uz = 0, at A'B', ue =v sing, therefore :

bsingpy? vsing sin @ b d?
Se Se)

hea ete wor d 2 4k
and the quantity streaming across LZ' is
+4/a
: (roa? vd) ,
fe Gr iat sty
=)

the difference between the quantities of water flowing across FF”
and HE" will therefore be

Ios? = rad)
Gita! aye eee?
and we have, neglecting powers of 4/p:
bd’
oa (112)

In the experiments the wires become curved. I suppose the wire
to be perfectly flexible and the stress to have the same value S in
all its parts; the force per unit of length perpendicular to the wire
is given in each point by
dw

ds’
dw being the angle between two consecutive tangents to the curve.
The curvature being not large we can use the coefficient given by
(J) to find the normal velocity arising from this force. This velocity is

S

dw

Sa
Cee

( 658 )
In a time dt the element ds of the wire describes a surface
d
C S— de dt.
ds

If the wire at the ends is vertical the whole wire will therefore
describe an area
dw
dt | CS—ds =n CS dt.
ds
0
Now if the velocity of the wire is v, and the distance between
the vertical ends d,, the same area will be vd, so that we have

acs
v= (IIT)
d,
or if the angle between the ends is 2a, and P the weight at each end,
2aCP
oS (1II,)
d, sina

We shall next consider the form taken by the wire if it descends
as a whole with uniform velocity. It is determined by the condition
dw dz

CSS = =
ds ds

or
dw _ mw dz

ds d, ds
As odw=ds, @ being the radius of curvature, this equation becomes
dy

da? 1

are eh
1 eae

Taking the axis of 2 horizontal at the highest point of the line,
the axis of y vertical downwards we have for «= 0,

dy
== VS =
Y dx
therefore
dy 4
SS = ==
da d d

( 659 )

In order to find the formula (//*) for curved wires we can put,
approximately, for 6 its value at the point z=0 y=0.
So that we may put for

By this the formula (II?) gives

= aa (3) Sees ech ky Se SAODEG
12ud,\ R

S being equal to the weight hanging at each end.
If the angle between the tangents at the ends is 2a, we have
other formulae. The equation of the curve becomes

and the velocity, if P is again the weight at each end
2aCP

d,sina

(IIIa)

By the hydrodynamical method the same velocity is found to be

2aP fie
fon 12aud,sina\R eee

Dr. J. H. Meersure has made a series of experiments, of which
he will communicate the results at a later opportunity. The agree-
ment with the theory is not very satisfactory. It must be noticed
however that d is very small. The roughness of the surface of the
wire will therefore greatly increase the resistance to the motion of
the water, so that the result of the hydrodynamical method can no
longer be considered as correct.

Zoology. — “On the Polyandry of Scalpellum Stearnst’ by P. P.
C. Horx.

One of the largest forms of the genus Scalpellum which is so
rich in species is Scalpelluwm Stearnsi, Pitssry from shallow water
near the coast of Japan.

This species is represented by two varieties or sub-species in the
collection of Cirripedes made by the Siboga Expedition in the waters
of the Dutch East Indies and handed over to me for description. Both
forms agree in the main with PinsBry’s species — they differ, however,

( 660 )

in some regards from one another as well as from the Japan species.
I made the acquaintance of the latter by studying a few samples
which were kindly lent me from the Berlin museum by the Director
(Prof. K. Morsius) and by the curator of the Crustacea Department
(Prof. W. WeELTNER).

Apart from Pinssry'), the Japan species has also been named
and deseribed by Fiscunr*); one of the two varieties from the
Malay Archipelago has of late again met with the same fate from
ANNANDALE *), who tried to introduce it into the literature of the
Cirripedia as a new species.

Yet, though we dispose at present of three names and _ three
fairly extensive descriptions for this species, a very curious: pheno-
menon in the life-history of the reproductive period of this Scalpellum
has hitherto escaped the attention of its describers ; for I can hardly
believe that they could have discovered this peculiarity and yet
not mentioned it in their papers.

Piuspry says of this species (and Fiscuerr in this regard quite agrees
with him) that it was found in shallow water in Japan. The speei-
mens of the Berlin Museum were from Nagasaki and apparently also
from coastal waters. Those of the Siboga Expedition are from four
different stations the depths of which range from 204 to 450 m.
ANNANDALE had a single specimen at his disposal, caught in Bali
Straits at a depth of 160 fathoms, about 290 m.

Scalpellum Stearnsi belongs to the unisexual species of the genus:
the large specimens with fully developed capitulum of a length of
about 5 em. and with (for a species of Sca/pellum) very long pedun-
cles (of 5—9 em. length) are the females. The males (which should
not be called “complemental” males in this case) are looked for in
vain at the place they ordinarily occupy, viz., at the inner side of
the scutum, near the occludent margin, a little in front of or above
the adductor muscle, in a duplicature of the sac or mantle which
covers the valves of the capitulum on their inner surface. They are
not to be found there — and I think this explains why they escaped the
attention of the earlier describers. Darwin discovered that the little
males in one of the species (in Sc. rostratum, Darwin) were attached
as three little parasites to the body of the bermaphrodite, close
under the labrum, between it and the adductor muscle almost in
the median line of the body — but even at that place they are not

1) Proceed. Acad. Nat. Sci. Philadelphia. 1890. p. 441—443.
2) Bullet. Soc. Zool. d. France. XVI. 1891. p. 116—118.
5) Memoirs Asiatic Soc. of Bengal. I. N°. 5. 1905. p. 74—77.

( 661 )

to be found in Se. Stearns?. I noted, however, that that part of the
sac or mantle, which unites the two scuta behind or beneath the
adductor muscle and which can be better seen by moving the two
seuta slightly from one another, in the largest and oldest specimen
of the collection, showed a crusty and grainy surface — just as if
a Flustra or other Bryozoon were attached to it. Investigating a part
of that crusty covering I easily found that each grain represented
a male and that over a hundred of these were attached to the same
female. Each male is inclosed in a kind of capsule (a thickening of
the mantle) and that part of the mantle-surface which is opposite
the head-end with the prehensile antennae forms a little elevation
over the surface of the capsule. They are in parts so closely
placed as to flatten one another mutually. Their dimensions are
0.7 < 0.5 mm. they are even small for males of Scalpellum.
Their structure agrees with that of the males of several other species
of this genus: round about the opening of the mantle, at the extremity
of the little elevation over the surface of the chitinous capsule, four
rudimentary valves are observed. What I think, so far as my experience
goes, is characteristic for this species, is that short rudimentary
tentacles are attached to the surface of the mantle between (alternating
with) the small valves, little appendages — which of course have
nothing in common with the articulated antennae or other limbs of
the Cirripedes. Should any doubt remain, as to whether these little
parasites really represented the males of this species, these tentacles
might be used to dissipate it. A few small, quite young females, in
which the capitulum however was already furnished with calcareous
valves and the whole appearance of which corresponded with an
early condition of fullgrown females, were found attached to the
surface of the capitulum of one of the large specimens. Now, these
little females are furnished with the same tentacles. They are
embryological organs, which of course may have importance froma
morphological or phylogenetic point of view, but which have dis-
appeared in the fullgrown females. In the young females they occupy
the same place as in the males, viz. at the free extremity (the tip)
of the capitulum attached to the chitinous surface between the two
ealeareous plates which represent the terga, near the anterior extremity
of the orifice — in the females large, in the males relatively much
smaller — which gives entrance to the cavity in which the animal’s
body is lodged.

I do not believe that examples are known in animals so highly
developed as Cirripedia of such a pronounced polyandry as in this
species of Scalpellum. As a rule, the number of males found attached

46

Proceedings Royal Acad. Amsterdam. Vol VIII.

( 662 )

io the ecapitulum of the female or of the hermaphrodite is one at
each side only, in some species it is two or three and the largest
number I have observed was five. How can we explain that there
is a species with such a large number as the case mentioned? I
have tried in vain to find an explanation. We do not know much of
the habits of these animals. It is hardly admissible that the great
number of males should be connected with the depth at which they
live, for (1) the same species which is found in the Malay Archipe-
lago at a depth of 200—400 m. lives in the Japan sea in shallow
water, and (2) we know species living in coastal waters and others
found in depths of over 1800 m., all of which have two males
only. A connection exists no doubt between the place where the
little males are found attached and their great number —~ but I
am at a loss to understand what the relation may be. The eggs
of these Cirripedes are fecundated at the moment they are excluded
and form two leaves (the so-called ovigerous lamellae) which remain
in the sack or mantle-cavity of the female until the eggs hatch out.
If the males are attached at the margin of the mantle-cavity, the
chance that the eggs will be impregnated is of course larger than
in the case when they are attached at a greater distance, as in
Se. Stearnsi. So it is easily understood that in the latter case a
greater number of males would be required — but why did they
choose for attachment a place which is less favourable for impregnation?
Because they were so numerous and did not find space enough at
the ordinary place?

Mathematics. — ‘A group of complexes of rays whose singular
surfaces consist of a scroll and a number of planes’. By
Prof. JAN DE Varies.

1. The generatrices of a rational scroll can be arranged in the
groups of an involution /,; to this end we have but to arrange
their traces on an arbitrary plane in the groups of an J,. If we
consider each pair of lines /,/' of J, as directrices of a linear con-
eruence, it immediately occurs to us to examine the complex of
rays I’ which is the compound of the o' congruences determined
by it.

Let the scroll 0” be of order n and let it have an (n—1)-fold
directrix d. The generatrices 7 form a fundamental involution J,—1,
each group of which consists of the (w—1) right lines, coinciding
in a point of d. This /,-, has evidently (n—2) (p—1) pairs in

( 663 )

common with the given /,; so on d lie as many points of intersec-
tion H of pairs /./' of the involution /,. Each ray through a point 1
belongs to the complex I, likewise each ray in the connecting
plane / of the right lines /,/'; i. 0. w. the complex has (n—2) (p—1)
principal pots H and (n—2) (p—1) principal planes h.

2. On an arbitrary plane @ a rational curve cv with (n—1)-fold
point D is determined by 9”. The rays of the complex lying in e
envelop a curve (@) of class (n—1)(p—1), the curve of involution
(director curve) of /', in which the points of ce" are arranged by
the given /,. *)

So the complex is of order (n—1) (p—1).

The line of intersection of @ with a principal plane / being a
ray of I, the curve of the complex (@) touches all principal planes.

If @ is made to pass through a right line / of 9”, then (@) splits
up into the pencils having for vertices the traces L' of the (p—1)
rays conjugate to / and into a curve of order (m—2) (p—41), the
curve of the involution of 7", on the curve c’—! which @ has in
common with ge” besides. So a tangent plane of 0” is a singular
plane of TP.

The singular surface consists of a scroll and the principal planes.

When a tangent plane contains one of the principal points it passes
in general through the directrix d, therefore through all principal
points. Then (@) splits up into (w—2) (p—1) pencils (H) and (p—1)
pencils (L’).

Of the n—1 generatrices / through a point H, two, /, and J),
form a pair of J/,. If we bring @ through one of the remaining
right lines /, (k=1 to n—38), then (a) consists of (p—1) pencils with
vertices L',, the pencil (#7) and a curve of class (m—2) (p—1)—1.

In an arbitrary plane through H the curve of the complex (a)
consists of the pencil (H) and a curve of order (n—1) (p—1)—1.

3. The rays of the complex through an arbitrary point A
envelop a cone (A) of order (n—1) (p—1) passing through the
principal points.

If A lies on 9” cone (A) consists of (p—1) concentric pencils and
a cone of order (n—2) (p— 1).

If we assume A in a principal plane / then only one pencil
separates itself from the cone of the complex.

1) I’, has (n—1) (p—1) pairs in common with the involution J, which an
arbitrary pencil determines on c”.

( 664 )

If A is taken on the line of intersection of two planes h, two
pencils are separated from the cone. Three pencils are obtained
when A is point of intersection of three principal planes.

If we take A on the curve c"-? which a plane 4 has in common
with g” then (A) consists of p concentric pencils and a cone of
order (n—2) (p—1)—1.

If A is a point of intersection of @” with two principal planes
the number of pencils evidently becomes (p-+-1).

4. The curve of the complex (a) is of order (p~—1) (2n+-p—6)’).
It possesses 4 (j — 1) (p — 2) (n — 2) threefold tangents *) which are
transversals of as many triplets of right lines belonging to a group
of J,. The cone of the complex (A) possessing evidently as many
threefold edges, the scrolls each having three conjugate right lines /
as directrices form together a congruence y of order and class
4 (p — 1) (p — 2) (n — 2).

Kach principal point # is for this congruence a singular point of
order (p — 2); the singular cone is broken up into (p— 2) pencils.

Kach principal plane / is a singular plane of order (p— 2) and
contains (jp — 2) pencils of rays of congruence.

5. The right lines resting on four lines / belonging to a group
of /, form a scroll enclosed in I, of which the order is going to
be determined.

Each transversal ¢ of three conjugate right lines /,,/,,/, and the
arbitrary right line a@ intersects still (7 — 8) generatrices m of 0”.
To each of these right lines m can be made to correspond the (p — 3)
right lines 7 forming with 7, /,,/2, a group of J,.

To each ray / belong (p —1), triplets /,,/,, 7,, so 2(j~—1), trans-
versals ¢ and therefore 2 (nm — 3)(p—-1), rays m.

The congruence (1,1) of the right lines resting on m and a has
with the congruence y in common (n — 2) (p —1) (p — 2) rays ¢,
so that to m are conjugate (nm — 2)\p—1)(p — 2)(p—8) right
lines JU’.

Now each transversal of four lines / belonging to a group of J,
evidently gives four coincidences of the correspondence (/' , m).

1) The characteristic numbers of the curve of involution of an Jp on arational
c" are found in the dissertation of Jon. A. Vreeswik Jr. (Involuties op rationale
krommen, Utrecht 1905, page 38). :

*) See also my paper “Complexes of rays in relation to a rational skew curve”
(These Proceedings, VI, page 12). —

( 665 )

Consequently the scroll of the transversals of quadruplets of the

1
involution is of order 7 (p — 1) (p — 2) (p— 3) (4n — 9).

Each principal point and each principal plane of I bears

1
ee 32) (p—9) right lines of this seroll.

6. If @” possesses also a single directrix e all principal planes of
I pass through e and the complex is in itself dual.

Al
If o” has a nodal curve Jd of order 3 (” — 2) (n —1) each gene-

ratrix 7 rests in (m— 2) points on gd, and is thus cut by (mn — 2)
right lines /. By this the generatrices are arranged in a symmetric
correspondence of order (n— 2), having with J, given on 9” in
common (n—2)(p—1) points H. So the complex has again
(n— 2) (p —1) principal points and as many principal planes.

In like manner the order of I remains the same. But now the
curve of the complex can break up on account of its plane contain-
ing two or three principal points by which two or three pencils
are separated. Besides @ can contain still’a right line /. So here
the degenerations of (@) are dually opposed to those of the cone (A).

(February 21, 1906).

has E70 Sanh igh ae ee Bild ios,

yes a AKO uit yi

BOR tie &, Sad
Pe) a) :

pean : erie tig: pan ®
, = ‘

He ate exe bs
ea) AIS meen 3

KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM.

PROCEEDINGS OF THE MEETING
of Saturday February 24, 1906.

—DOGe

(Translated from: Verslag van de gewone vergadering der. Wis- en Natuurkundige
Afdeeling van Zaterdag 24 Februari 1906, Dl. XIV).

S(O) AN ah aH aN EBS

W. H. Junius: “A new method for determining the rate of decrease of the radiating power
from the center toward the limb of the solar disk”, p. 668. (With one plate).

A. F. Horreman: “On the nitration of ortho- and metadibromobenzene”, p. 678.

J. J. Buanksma: “The introduction of halogen atoms inio the benzene core in the reduction
of aromatic nitro-compounds”. (Communicated by Prof. A. F. Hotteman), p. 680.

F. A. F. C. Went and A. H. Buaauw: “On a case of apogamy observed with Dasylirion
acrotrichum Zuce.”, p. 684.

J. C. Kapreyn: “On the parallax of the nebulae”, p. 691.

J. J. van Laar: “On the course of melting-point curves for compounds which are partially
dissociated in the liquid phase, the proportion of the products of dissociation being arbitrary”.
(Communicated by Prof. H. W. Baxuvis Roozrsoom), p. 699.

C. J. Exkraar: “On ocimene and myrcene, a contribution to the knowledge of the aliphatic
terpenes”. (Communicated by Prof. P. van Rompurcn), p. 714.

C. J. Exkraar: “On some aliphatic terpene alcohols”. (Communicated by Prof. P. van
RompurcH), p. 723.

H. B. A. BockwinkeL: “On the propagation of light in a biaxial crystal around a centre of
vibration”. (Communicated by Prof. H. A. Lorenrz), p. 728.

K. Martin: “On brackish and fresh water deposits of the river Silat in Western—Borneo”, p. 742.

Physics. — “A new method for determining the rate of decrease
of the radiating power from the center toward the limb of
the solar disk”. By Prof. W. H. Junius.

(Communicated in the Meeting of January 27, 1906)

The brightness of the solar disk is known to diminish considerably
from the center toward the limb. Although this prominent feature
of the solar phenomenon should be among the first accounted for in
every theory of the Sun, it leads to problems presenting so many
difficulties, that a satisfactory explanation is, until now, altogether
wanting. And even the empirical study of the law according to
which the radiating power varies across the disk, is not very advanced.

What we know about the question is founded on researches in

47

Proceedings Royal Acad. Amsterdam. Vol. VIII.

( 668 )

which either a photometer, or a thermopile, a bolometer or a radio-
micrometer was used for exploring an image of the Sun. The results
obtained by different observers are rather discordant’). This may be
partly due to instrumental or accidental errors, but there is also a
systematical error which must have influenced similarly all of the
results thus obtained, and which proceeds from the scattering of the
rays by the terrestrial atmosphere. In any point of an image of the
Sun is not only to be found the radiation coming from the corre-
sponding point of the disk, but, besides, some diffused radiation
proceeding from other parts of the disk. This disturbing effect will,
of course, vary in magnitude with the condition of our atmosphere,
but it will always act in a levelling way, parts of the image lying
near the edge receiving more diffused radiation from the middle
parts of the disk, than receive the central parts of the image from
the marginal parts of the disk.

We may completely avoid this source of error by using a method
in which the radiating power of the different parts of the disk is
calculated from observations made on the occasion of a total eclipse
of the Sun.

Let us suppose the curve, representing the intensity of the solar
radiation from the first until the fourth contact as a function of time,
to be exactly known’). The curve will show us by how much the
total radiation has increased or decreased between any two epochs.
Every (positive or negative) increment is exclusively due to rays
coming from that strip of the solar disk through which the Moon’s
limb has appeared to move between those very epochs.

Suppose the time after third contact to be divided into equal
intervals of, say, 2 minutes, and the position of the Moon’s limb
at the end of each interval delineated on the solar disk, then the
latter will be divided into 39 narrow strips, successively contributing
the Anown quantities a, b, c, d,.. to the total radiation.

Now, let us distinguish 2 concentric zones on the solar disk and
denote by «2, v3 , . . wv, the radiation coming from these zones per

1) Of. J. Scuremer, Strahlung und Temperatur der Sonne, p. 43—49 (1899),

2) It is well known that, at Burgos, the observation of the eclipse of August 30,
1905, has not been favoured with a clear sky (Cf. the Preliminary Report in the
Proceedings of the Meeting of November 25, 1905). Nevertheless, the measurements
of total radiation have yielded some results of sufficient accuracy to justify that,
in our present investigation, we make use of the radiation curve then secured.
Further particulars regarding the observations will soon be published in the
complete report on our expedition.

( 669 )

unit surface. (According to results obtained by Laneiny and by Frosr
we shall suppose the radiating power to vary only with the distance
from the center, not with the position angle). One of the. strips will
contribute to the radiation :

D0 te 0 BBY te Oni Livy

if it cuts out of the first zone an area d,, out of the second zone
an area d, etc. The next strip contributes :

C= 8) ta a Ese =. ee En Ly
and so on. We get 39 equations from which ~,, w,, ....7, may be
resolved.

Determination of the coefjicients of the n unknown quantities.

I have found the coefficients d,, J, .. . &, &, . .. by weighing.
On a piece of excellent homogeneous paper the solar disk was drawn
and divided into a suitable number of concentric zones, which were
intersected by ares representing the Moon’s limb in its successive
positions. The following astronomical data, necessary for making the

drawing, have been kindly procured to me by prof. A. A. NiLAND.

contact I II II IV

position angle 293°,4 104°,5 304°,9 114°,9

local time 23533" 105 02 51™58s 0° 55™ 39s 2h 12™ 44s
Moon’s radius : Sun’s radius = 132,8 : 126,8.

Now the strips were carefully separated from each other and
weighed (for subsequent control). Then each strip was cut along
the zone circles, and the pieces were weighed separately. In order
to make the pieces recognizable, the zones had all been differently
painted, each with a narrow line of water-colour. The weighings,
which were accurate to half a milligram, gave the coefficients of
the unknown quantities 2, @3....a,. So the unit of area, adopted
for measuring the surface of the solar disk, corresponds to a piece
of our drawing paper weighing 1 milligram.

The breadth of each of the outer five concentric zones was '/,,
of the Sun’s radius; then came seven zones with breadth */,, of the
radius each, leaving round the center a circle with radius */,,. The
average distances of the zones from the center, expressed in thou-
sandth parts of the radius, will now be used as indices «,8.... of
our 13 unknown quantities; so these will be written :

Ura Veoor M5009 “4009 Yao0r Ya001 Vio0r Yor

47%

Vorss oes. Usrs1 Ysass Ur7zs)

+o 3Q BO 8 ~ ads. WE a o oes
3 a

TET ae es ca SS Te NTS SSS SDS ESTOS iE

126 ay;

28 a5
18 @,,+
13 73+
10 @,,5+
8B ty
7 Bors"

v

2e55+69,50,,,+ 2

@, +66

Lis pol

Geos

7

Byyg +19

925
59 @y5,+84 Berg 1 eas
29 59, +90,50,,, +77 ®aag—b 1,52,,,
19 w,,,+27,52,,,+46
14 ®oa,+19 Borst 28 ®aa,+40
10 #,,,+12 «,,,+15 #,,,+18
8 Byagt 9 ty, +10,52,,,+12,52,,,4+80
6,52 595+ 7 Bargt 8 55+ 9
6 oat 7 a 8 Beast 8
6 Deaeaie 6,528.75 7 CANA E 7

6 Boast 6,52 475+ 7 Beas 7 Bypg 715,52; 99+ 16,5099 +17, 52559 18,52 49-F 18,5255) 21,525, + 20,52, 5,

6

Boog 6,5%;3+ 7

Opa 7

Byy5-+ 16

@yy,+15

®roo
av

Bagot 08 Leno

Brop 48 Leno Ol eoo

Bopp t28,5% 9740 Bey t45 yoo

Broo t2l By $25 Aggo +33 Loot 36 #555

Begg Ll, D@ ego + 19,545. +22,02,9.+26,57,,,+31 «,,,

Byop $15,527 g99 + 16,520,917 Heo t17,585,+18 2,,,+19 @ 19, +82,

St ova)

On p. 670 the equations are written out. We have confined our-
selves to 13 equations ; increasing this number would not have led
to greater accuracy, as the values of a,b,c... had to be found from
the radiation curve, that is by graphical interpolation, in which pro-
cess it is understood that a// of the observations have already been
taken into consideration.

Determination of the constant terms of the equations.

Table I contains the results of the observations made at Burgos
with our actinometer. The second column gives the galvanometer
deflections, from which the numbers of the third column, representing
the intensity of the radiation, are calculated *).

Owing to the clouds there are large gaps in the series of obser-
vations; but nevertheless, after the results had been plotted down,
we saw that there was only little room left for fancy when drawing
the radiation curve in such a way, that closest agreement with the
observational data’ was obtained. As a matter of course the curve
has not been drawn between the series of points, but so as to join
the highest points, for the observed values could only be too small.
Only one exception is made to this rule, the value found at 0% 17™ 3s
being very probably too high by some error or instrumental dis-
turbance.

The middle part of the radiation curve has been reproduced on
the annexed plate. For determining a, 6, c, . . . we have used the
part included between 0% 55™ and 1" 37", which was very carefully
constructed on a larger scale. It deserves notice that the relative
accuracy of the small ordinates (corresponding to few minutes after
totality) is nearly as great as that of the larger ones, because
ihe galvanometer deflections from which they were calculated are
all lying between 118 and 347 scale divisions. Table II refers to
this part of the radiation curve. In the second column are given
the ordinates of the curve at the epochs 0? 55” 40s and every
two minutes later; the unit corresponds to an intensity = 1000.

1) Particulars concerning the connection between the numbers of these two
columns will be found in the forthcoming report on the Dutch expedition. The
method and the instruments used al Burgos were the same that are described in;
“Total Eclipse of the Sun, May 18, 1901. Reports on the Dutch Expedition to
Karang Sago, Sumatra, N'. 4: Heat Radiation of the Sun during the Eclipse”, by
W. H. Jus. The numbers of the third column are proportional to the total
radiation coming from a circular patch of the sky, 3° in diameter, with the Sun
in its center,

(6

#9)

PARSE Ee ae

Galvano- | Intensity
Time. meter- of Time.
deflections | radiation.
hy mis h ms
92 98 48 | 280 4750000 0 20 48
36 (0 931 1444000 || 2nd contact 51 58
38 33 287 1794000 53 53
54 28
46 58 287 1794000 55 18
51 38 270 1688000 || 3rd contact 55 40
53 49 260.5 1631000 55 58
56 8 278.5 1745000 57 58
58 33
93 4 58 256 1610000 59) 13
te) 283.5 1786000 59 53
9 56 284.5 1792000 ab Alsi}
11 44 275 1736000 2 28
1st contact 33 8 3 33
35 48 296 1430000
38 3 256.5 1625000 7 38
40 38 269.5 1709000
4A 38 270 1712000 21-45
42, 48 270.5 1715000 WP} °33
4h 0 260 1649000 93 3
45 33 259.5 1646000 23 58
46 38 256.5 1627000 94 53
4759} 248.5 | 4566000 95 53
48.53 95025 1589000. 96 53
50 8 249 1580000 97 53
5133 241 1529000 28 58
53. 8 233.5 1483000 30 8
55 3 997 1442000 31 8
56 33 226 1435000 32 11
58 23 216.5 1376000 9} |)
34 20
(0) 9/08} 4192 1222000 35 25
8 53 184 4170000 36 34
10 28 177 1127000.
44 43 Al 7/i lets) 1091000 9 4 58
43 43 165.5 1054000 Gy fs
14 58 159 1013000 || 4th contact 12 24
ATS 150 956000 13 18
19 28 136 867000 14 20

Galvano-
meter-
deflections.

428.5

—
ie)
ler)
ou

Intensity
of
radiation.

819000

9
13
33

600?
23000
419160
42700
55700
74800

108800
97700

207000

635000
665000
676000
722000
745000
776000
805000
832000
865000
897C00
926000
950000
981000
1007000
1937000
1060000

1506000
1581000

1648009
1657000

( 673 )

But this observational curve has to be corrected, owing to the
circumstance that in the lapse of time considered the Sun’s altitude
has diminished. We may proceed as follows. Apart from a possible
influence of sun-spots or faculae there is no reason why the eclipse
eurve would not be symmetrical if the Sun’s altitude (and the con-
dition of our atmosphere) reinained constant. Between 23" and 1"
the variation of altitude is very small. Now taking 0" 53™ 50s as

TABLE II. TABLE III.
Ordinates Or Ginates
: f
Time ees corrected |Increments
radiation eadintion
curve.
curve.
Eis | Radiation per unit surface
0 55 40 0 0 90.4 of the concentric zones of
4U.1—a@
57 40}. 20.4 20 1 the solar disk.
32.4=)
59 40 59/5 52.5
Sonor Ie coe = —i)s4 bi3)5)
AE AAO) 915.0 91.0
45.5=d Long — 0.2166
} 3 40 136.5 136.5
50:5:—"e Lors = 02501
5 40 187 187
Sie fi Las = 103023
7 40 240 244
DOr e==9; Xy75 = 0.3290
9 40 296 297
Sjsie =e Lino = O 3488
11 40 354 355
1) Se Xo, = 0.3662
13 40 4A2 4 4.
0) ep =i Vata tol 3)
15 40 472 474
61 =& Lanne = Obs
17 40 532 535
C257 Ly,9 = 0.4278
19 40 594 597
62 =m " Lan = 0.4240
21 40 655 659 7
62 =n 1p) = 0.4380
23 40 717 721
62a9—"0 x) = 0.4388
25 40 776 783
61. a= p
27 40 834.5 844.5
64. =¢4 =
29 40 891.5 905.5
60.5=r
31 40 947 966
60 = Ss
33 40 | 41001 1026
. 59a
35 40 | 1053.5 1085.5

the epoch of mid-eclipse, we draw a horizontal line through a point
m corresponding to that epoch. The line cuts the descending branch
of the curve in /; we make mn—m/ and thus find a point 2 of
the hypothetical radiation curve for constant altitude of the Sun.
Acting in a similar way for a few more points, we get an idea of
the magnitude of the smoothly increasing correction which is to be
applied to the ordinates of the ascending branch. K. ANesTRém’s
measures of the intensity of the radiation for different altitudes of
ihe Sun‘) have also been considered in determining the correction.

The third column of Table II contains the ordinates of the corrected
curve; in the fourth column are given their successive increments
which, of course, are the values to be assigned to the absolute terms
of our equations.

Results.

The solution of the equations leads ‘to the numbers of Table III;
the results are plotted down in fig. 2 on the plate. Through these
points ,we have drawn a curve satisfying the condition that its
curvature should gradually diminish; it shows us the law of variation
of the radiating power from the edge toward the center of the solar
disk. Putting the ordinate at the center equal to 100 and expressing
the other ordinates in the same unit, we get numbers comparable
with the results obtained by other investigators.

The comparison with the spectro-photometric observations by
H. C. Voern?) and with the measurements of total radiation made
with a radio-micrometer by Wutson*) and with a thermopile by
Frost *), is given in Table IV. We add in Table V the results of a
spectro-bolometric investigation by Very *), as these numbers have
been used by Very and by Scuusrer °) in testing their explanations
of the phenomenon.

According to Frost’s measurements the total radiation appears to
diminish from the center toward the limb in about the same pro-
portion as the radiation of wave-length 650uu, whereas my numbers
show a decrease very similar to that exhibited by rays of wave-

1 K. AvastRom, Intensité de la radiation solaire a différentes altitudes. Recherches
faites & Ténériffe 1895 et 1896.

2) H. C. Voaet, Ber. d. Berl. Akad. 1877, p. 104.

3) W. E. Wuson, Proc. Roy. Irish Acad. [3], Vol. 2, p. 299, (1892).

t) E. B. Frost, Astron. Nachr. 130 (1892), p. 129.

6) F, W. Very, Astroph. Journ. 16 (1902), p. 73.

6) A, Scuuster, Astroph. Journ, 16 (1902), p. 620; 21 (1905), p. 258.

( 675 )

TAB Tok. IV.

Distance} H. C. VoGEL’s spectro-photometric measurements. Total radiation.
ae | Receiver in solar | Eclipse-
of \405—412|440—446| 467 —473)/510—515)573—585)658—666 image, curve.

disk. fa wp py pe py py Witson | Frost | Juius

0.0 100.0 | 100.0 | 160.0 | 100.0 | 100.0 | 100.0 | 100.0 | 100.0 | 400.0
99.8 | 99 9 99).9 99:9 99.8
6 99.4 98.6
98.8 98.4 96.6
3 96.3 94.0

94.5 £67 95.3 93.6 90.3
8 92-5 89 8 85.5

84.5 91.0 88.7 84 6 79

0.7 74.4 77.8 80.8 80.0 5
0.75 69.4 | 73.0 76.7 75.9 | 80.1 88.1 73.3
0.8 63.7 | 67.0 | 74.7 | 709 | 74.6 | 84.3 | 88.9 | 77.9 | 70.4
0.85 | 56.7 | 59.6 | 65.5 | 64.7 | 67.7 | 790 || 63.5
0.9 RIE a lei50)- Deal s50e6y | 56.610) 25920) | 974.0 | 74.9 | 68.0 | 55.0
0.95 347") 3o20 45.6 | 44.0 | 46.0 | 58.0 | | (60.5) | 44 0
| 1.0 AStONk00\ 46.0" |-- 46.0 | “25.0; || 30:0 ease | (24.0)
| | |

AS Bals i *V.

Distance F, W. Very’s spectro-bolometric measurements.
from
center. MG6pp | 46842 | 550pe | 6l5ue | T8l yp 1010 pe 1500 pw

0.5 85.8 90.2 93.3 94 8 94.4 94.3 9559)
0.75 74.4 76.4 83.4 84.5 88.5 89.4 95.0
0.95 47A 46.2 58.7 68.1 74.9 76.5 85.6

length 510uu. At first sight the evidence is in favour of the results
obtained by Frost, because the maximum of the curve representing
the energy in the solar spectrum (or perhaps rather the ‘center of
gravity’ of the enclosed surface) lies closer to 650ue than to 510m. _
But this argument fails; for the measurements of Vocr and those
of Frost are all disturbed alike by atmospheric diffusion. Had the
spectro-photometric observations been free from this influence, then
the rate of decrease of the radiation from the center toward the

( 676 )

limb would doubtless have been found quicker for all wave-lengths,
and, very probably, the distribution for the region 650uu would have
proved to agree better with my results than with the uncorrected
values of Frosr.

Winson’s measurements seem to have been influenced by other
causes of error still, besides atmospheric scattering, as his numbers
are greater than those obtained by Frost, and harmonize not as well
as the latter with the spectro-photometric series.

The observations of Vrry have given considerably greater ratios
in the marginal regions than those of Voc. Mr. Very himself points
out the difference, and remarks that the bolometer has an advantage
over the eye in the red where the heat is great; but I may suggest,
on the other hand, that instrumental errors (reflection or scattering
of light by prisms, lenses, tubes, ete.) are easier discovered and
corrected in spectro-photometvie than in spectro-bolometric work.

It seems to me that observing an eclipse-curve by means of a
very simple but sensitive actinometer, without lenses or mirrors,
must yield results concerning the radiation of different parts of the
solar disk which deserve more confidence than the values hitherto
obtained in other ways. I wish to lay stress upon the advantages of
our method, rather than on the reliability of the numbers secured
at Burgos under not very favourable circumstances. In a clear sky
the shape of the eclipse curve will easily be found with very great
accuracy.

The same method will also be applicable with radiations covering
limited parts of the spectrum, if we only put suitable ray-filters
before the opening of one of the diaphragms in the actinometer. It
may even be possible, in a future eclipse, to use an arrangement
which brings several ray-filters by turns before the opening ; thus,
when disposing of a quick galvanometer, one would be able to
simultaneously determine, with one actinometer, the eclipse curves
for rays belonging to five or more regions of the spectrum, and the
results would be independent of selective atmospheric scattering.

Remarks on the hypotheses used for explaining the distribution of
the radiating power on the solar disk.

The diminution of the intensity of radiation toward the limb is
almost generally ascribed to absorption of the rays by the solar
atmosphere '), and it is supposed that, in absence of that atmosphere,

1) J. Scuetnern goes as far as to say: “Eine andere Deutung des Lichtabfalls ist
nicht zulissig.”” (Strahlung und Temperatur der Sonne. p. 40).

the photosphere would show itself as an equally luminous disk. But
then it appears to be impossible to find such values for the thiek-
ness of that atmosphere and for its coefficient of absorption, as to
give a law for the rate of diminution of brightness, consistent with
observation. Very!) e.g. when attributing the effect to absorption
only, arrives at the absurd result that we should have to assume
that the absorptive power toward the limb is smaller than that nearer
the center. He, therefore, suggests the existence of other influences
which, combining with the absorbent process, would reconcile theory
to observed facts. Diffraction by fine particles, columnar structure
of the solar atmosphere, irregularity of the photospheric surface, are
thus introduced.

ScHUSTER *), on the other hand, is of opinion that the difficulty
which has been felt in explaining the law of variation of intensity
across the solar disk is easily removed by placing the absorbing
layer sufficiently near the photosphere and taking account of the
radiation which this layer, owing to its high temperature, must itself
emit. He then really finds values for the absorption and the emission
of that layer, harmonizing with the results of Vury’s and Wixson’s *)
measurements, and also with the properties of the energy curve of
the spectrum of a black body at different temperatures. But, for all
that, serious doubts as to the correctness of the premise and the
conclusions must subsist.

Indeed, the calculations of Scnuustrr as well as those of Very,
Witson, Lanciny, Pickerinc and others, concerning the same subject,
are based on the assumption that the light travels along straight
lines through the solar gases, whereas everybody who has duly
noticed A. Scumipt’s “Strahlenbrechung auf der Sonne” will at the
least have to give in that rays coming from the outer zones of the
disk must have followed curved paths through the solar atmosphere.
By this circumstance the said calculations lose their convincing power.

And besides, the fundamental idea that a considerable portion of
the photospheric radiation should be absorbed by a thin atmosphere,
encounters a difficulty of greater importance still. This point, I
think, has also first been moved by A. Scumipt. What becomes of
the absorbed energy accumulating in the atmosphere? According to
ScHusTeR e.g. (l.c. p. 322) the atmosphere transmits largely ‘/, of

1) F. W. Very. The absorptive power of the solar atmosphere. Astroph. Journ.
16, p. 73—91, (1902).

2) A. Scuuster. Astroph. Journ. 16, p.d20—327, (1902); 21, p. 258—261, (1905).

3) W. li. Witson and A. A. Rampaur. Proc. Roy. Irish Acad. [3], 2, p. 299—
334, (1892),

( 678 )

the radiation emitted by the photosphere ; so it stops almost */,, and
only a small fraction of this absorbed energy leaves the Sun in the
form of radiation, emitted by the atmosphere itself. After all, more
than half of the radiation coming from the photosphere is retained
by the absorbing layer, and we cannot suppose it to go back to
the interior without violating the second law of thermodynamics.
As long as it has not been shown how the solar atmosphere may
get rid of that immense quantity of energy continually supplied and
never radiated, similar considerations will remain very unsatisfactory.

Our problem appears to be much less intricate when viewed from
the stand-point taken by Scumipr'), though the mathematical treat-
ment will not be easy. A uniformly luminous sphere surrounded
by a concentric, perfectly transparent refracting envelope, will offer
the aspect of a disk the brightness of which diminishes towards the
limb. This has been established approximately by Scumipr for the case
of a homogeneous, sharply limited envelope. It is easily understood
that a similar result must be obtained when assuming a transparent
atmosphere of gradually decreasing density and refractive power ;
but then, of course, the rate at which the luminosity varies on the
disk will depend on the law of density variation. We may proceed
a little farther, and accept Scumipt’s hypothesis that the incandescent
core of the Sun is not a sphere with a sharp boundary, but a gaseous
body the density and radiating power of which are smoothly dimi-
nishing along the radius. In this way, I think, we dispose of pre-
mises from which it seems possible to derive an explanation of the
general aspect of the solar disk without involving into such serious
difficulties as were hitherto encountered.

Chemistry. — “On the nitration of ortho- and metadibromobenzene.”
By Prof. A. F. HoLieman.

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

After the disturbing influence which the halogen atoms exercise
on each other’s directing influence in regard to the nitro-group, had
been noticed in the nitration of the dichlorobenzenes, it was necessary
to extend this research to the nitration of the dibromobenzenes so
as to be able to find the connection between the results with the
dichloro- and dibromocompounds and to compare the same with the
result of the nitration of the corresponding monohalogen benzenes.

1) A. Scummr, Physik. Zeitschr. 4, 282, 341, 453, 476 ; 5, 67, 528. (1908 and 1904),

Lc Se eee

W. H. JULIUS. A new method for determining the rate of decrease of the radiating power

1200000

1100000

1000000

900000

300000

700000

600000

500000

400000

300000

200000

100000

from the center toward the limb of the solar disk.

Middle part of the radiation curve obtained during the solar eclipse
of August 39. 1905.

EEE
pieeesees

Oh "10 20 20 40 50 1 10 20 20 40 50

bo

Radiating power across the solar disk.

( 679 )

The necessary experiments have been considerably delayed, because
it appeared that the ortho- and meta-dibromobenzenes had not as
yet been obtained in a perfectly pure condition, and the search for
a good method absorbed much time. We have at last suceeeded in
preparing m-dibromobenzene from perfectly pure m-bromoaniline by
diazotation in a dilute hydrobromie acid solution, according to a
direction given by Erpmayn for another purpose. J/eta-dibromobenzene
has a sp. gr. of 1.960 at 18.5°, and solidifies at — 7°. It is true
that F. Scuirr incidentally mentions (M. 11, 335) that he has met
with m-dibromobenzene solidifying at + 1°, without saying how he
has obtained the same, but there is good reason for doubting the
correctness of this statement. In this case, the product obtained by
me and my coadjutors (Sirks, Sturrer) with its 8° lower solidifying
point should contain about 16°/, of impurities. In the nitration of
our m-dibromobenzene, however, a product is obtained having a
sp. gr. such as was to be expected from a mixture of the isomers
(Br : Br? : NO,*) and (Br’: Br’? : NO,*) brought together in the propor-
tion indicated by the solidifying point, so that a contamination of
our preparation with such a large quantity of another substance is
altogether out of the question ; moreover, on distillation our preparation
yielded two fractions within one degree which both possessed
practically the same sp. gr. and solidifying point.

QO-dibromobenzene which was obtained im an analogous manner
from o-bromoaniline, had a sp. gr. of 1.996 at 11° and solidified
at + 6°.

The preparation of the six dibromonitrobenzenes was carried out
in a manner analogous to that of the six dichloronitrobenzenes,
described by me in the “Recueil” 28, 357.

The composition of the products of nitration of the dibromobenzenes
was determined from their solidifying point and their sp. gr. and
led to the results united in the subjoined table with the composition
of the products of nitration of the dichlorobenzenes. The temperature
of the nitration was 0°. (See p. 680).

In ortho-dibromobenzene the disturbance of the directing power of
the one halogen atom owing to the presence of the other one is,
therefore, much less than in the case of orthodichlorobenzene because
in the first one 18.3 and in the second only 7.2°/, of by-product
is formed, whilst monobromo- and monochlorobenzene yield, respec-
tively, 29.8 and 37.6°/, of by-product. On the other hand, the
disturbance caused by the entry of the nitro-group between the two
halogen atoms in m-dibromobenzene is very nearly equal to that in
m-dichlorobenzene, therefore much Jarger in regard to the ortho-

( 680 )

Quantity of ne of by-prod. in

by-product in °/, 100 parts of main prod.
o-CatliClemsan| 7.2 | Tes
m-CO,H,Clo 4.0 | 4A
CuBr, 0. wee 22.4
m-C,H,Br, 4.6 | 4.8

; f

C,H,Cl 29.8 | 42.0
C,H,8r 27.6 60.5

compounds. One would feel inclined to attribute this to ‘‘sterie
disturbances” introduced into Organie Chemistry by V. Mrier, were
it not that the specific volume of chlorine and of bromine in the
dichloro- and bibromovenzenes differs but little.

Perhaps it is rather the atomic weight of chlorine and bromine
which has some connection with the above. For further particulars
concerning this research the “Recueil” should be consulted.

Amsterdam, Org. chem. Lab. of the University, January 1906.

Chemistry. — “The introduction of halogen atoms into the benzene
core in the reduction of aromatic nitro-compounds”. By
Dr. J. J. Buayksma. (Communicated by Prof. A. F. Honimman).

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

Some time ago I cited and communicated some experiments *)
which showed that, in some cases, in the reduction of aromatic
nitrocompounds, halogen atoms may be removed from the benzene
core. In 1901 an article by Pinnow’) appeared in which a
fairly large number of cases are mentioned, where halogen atoms
are introduced into the benzene core in the reduction of aromatic
nitrocompounds. Pixnow endeavours to find the conditions under
which this secondary reaction is as much as possible prevented in
order to prevent formation of halogenised amidocompounds as by-
products, alongside the amidocompounds.

1) Proc. 30 March 1904, Recueil 24, 320.
*2) Journ. fiir Prakt. Chem. (2) 68, 352.

( 681 )

So when I obtained 5-chloro-4-6-dibromo-2-amido-m-xylene as by-
product in the reduction of 4-6-dibromo-2-nitro-m-xylene, I tried to
CH Cs
br / NO, nN NH,
oy Hy 1G ay CH,
Br Br

introduce halogen atoms into the core, taking the simplest case,
namely, the reduction of nitrobenzene with tin and hydrochlorid acid.

As is well-known, various intermediate products are formed in
the reduction of nitrobenzene to aniline. The formation of chloro-
aniline from nitrobenzene may be explained in the following manner: ')

C,H,NO, + 4H — C,H,NHOH + H,0
C,H,NHOH + HCl = C,H,NHCI + H,O
C,H,NHCI > CIC,H,NH, (0. + p,).

The fact that, in the reduction of nitrobenzene, phenylhydroxylamine
occurs as an intermediate compound, has been demonstrated by
BaMBERGER, Who has also proved that, on boiling phenylhydroxylamine
with hydrochloric acid, o- and p-chloroaniline are formed *). It has
also been proved by L6s that o- and p-chloroanilines are formed in
the electrolytic reduction of nitrobenzene in alcoholic hydrochloric
acid solution *). The object of the experiments to be described was
to try and conduct the reduction of nitrobenzene with tin and hydro-
chloric acid in such a manner that instéad of aniline, as much as
possible chloroaniline was formed.

The experiment had, therefore, to be carried out in such a way,
that the phenylhydroxylamine formed was not at once further reduced
to aniline, but to give this substance an opportunity to be converted
into chloroaniline, under the influence of hydrochloric acid. The
conditions were also to be such that the phenylechloroamine C,H,NHCI,
which is formed intermediary, could be readily converted into chloro-
aniline.

The intramolecular conversion of phenylchloroamine into 0- and
p-chloroaniline is, however, but little known, as the first substance
is very unstable but the conditions under which acetylchloroanilide
is converted into p-chloroacetanilide have been closely investigated.
It has been shown that this reaction is very much accelerated by
increase of the temperature and also by addition of hydrochlorid acid ‘).

1) Lop, Die Electrochemie der Organischen Verbindungen p. 166, 3e Auflage (1905).
2) Ber. 28, 451. Bampercer and Lacurt, Ber. 31, 1503.

3) Ber. 29, 1896.

4) Benper, Ber. 19, 2273. Buanxsma, Recueil 21, 366, 22, 290.

( 682 )

If, on account of the analogy between phenyl-chloroamine and
acetylehlorophenylamine, we assume that in the case of the first sub-
stance the velocity of the conversion into 0- and p-chloroaniline is
also strongly accelerated by elevation of temperature and addition
of hydrochloric acid, the conditions for obtaining chloroaniline instead
of aniline, in the reduction of nitrobenzene with tin and hydrochloric
acid, are as follows:

1. Slow reduction, or addition of tin in small quantities at the
time, in order not to at once reduce the phenylhydroxylamine to
aniline.

2. Excess of hydrochloric acid so as to rapidly convert the phenyl-
chloroamine formed into chloroaniline.

3. The reaction should take place at the boiling temperature, as
elevation of temperature also promotes this conversion.

The experiment was now conducted as follows:

10 ee. of nitrobenzene were dissolved in 100 ce. of aleohol and
200 ce. of 25 °/, hydrochloric acid were added. This solution was
boiled over the naked flame, whilst 15 grams of tin were added
through the reflux condenser in small portions. Each time, after
adding a small amount of tin, the boiling was continued until every-
thing had dissolved before adding a fresh portion. The experiment
lasted six hours. The unaltered nitrobenzene was now removed by
steam, the residue was rendered alkaline and the aniline and chloro-
aniline recovered by distillation in steam.

In this way, 6.5 gram of oil were obtained. The greater portion
of this oil was distilled between 182° and 225°, the residue solidified
in the distilling flask, and proved to be p-chloroaniline (m. p. 70°).
The oil consisted of aniline and o- and p-chloroaniline.

From a chlorine determination according to Carius, it appeared
that the mixture consisted of 55°/, of chloroaniline and 45°/, of aniline.

If the reduction experiment was made with SnCl, and HCl (0+ :)
chloroaniline (53°/,) were formed together with aniline. In this case,
the stannous chloride was also added in small portions, so as to
vive the intermediary formed phenylhydroxylamine an opportunity
of being converted into o- and p-chloroaniline. Nitroso-benzene gives
the same result ’).

In the same manner, the reduction of nitrobenzene with tin and
hydrobromie gave a mixture of aniline and (0- and p)-bromoaniline.

At present it is still difficult to predict which aromatic nitro-

‘) Cf. Gotpscumprt, Zeitschrift fiir Phys. Chem, 48, 435.

( 683°)

compounds will yield a large quantity of halogenised by-products on
reduction with tin and hydrochloric acid. It would be necessary to
know something more about the reduction velocity of the nitrocom-
pounds *) (and of the intermediary formed hydroxylaminederivatives),
and about the intramolecular conversion velocities of the halogen-
phenylamines.

It is known, for instance, that o0-nitrotoluene gives a large amount
of chlorinated by-product on reduction with tin and hydrochloric
acid *). The o-tolylbydroxylamine formed as intermediate product is,
therefore converted here into 5-chlorotoluidine, and the reduction ex-
periments of GoLpscumipt *) on o-nitrotoluene are in agreement with
this. GoLpscumipt has shown that, with increase of the temperature
the reduction velocity also increases, whilst an elevation of temperature
also increases the conversion velocity of the halogenphenylamines.
It now appears that this last reaction is the most strongly accelerated,
for the amount of halogenised by-products increases with elevation of
the temperature ‘*).

Resumé. It has been shown that the reduction of nitrobenzene with
tin (or Sn Cl,) and hydrochloric acid may be carried out in such a
manner that p-chloroaniline occurs as the main product. The cause
of this must be explained by the fact that, in the reduction of
nitrobenzene, phenylhydroxylamine occurs as an intermediate product.
As on reduction of all aromatic nitrocompounds, hydroxylamine
derivatives are formed as intermediate compounds, we shall generally
notice on reduction of such nitrocompounds with tin and hydrochloric
acid, besides amidocompounds, also halogenised amidocompounds
(with halogen atoms o- or p- in regard to the NH, group), whilst
the quantity of these two last substances will be dependent on the
conditions under which the reduction is carried out. In some cases
no halogen atoms are introduced, but they are even eliminated from
the benzene core °).

I hope to record more fully further experiments in the Recueil
later on.

Amsterdam, January 1906.

1) See the note on the preceeding page.

2) Beitstemn and Kiintperc, Ann, 156, 81. Hotteman and Junaius, Chemisch Week-
blad II. 553.

3) l. c.
4) Pinnow, I. ¢.
5) Recueil 24, 320.

Proceedings Royal Acad. Amsterdam. Vol. VIIL.

( 684 )

Botany. — “On a case of apogamy observed with Dasylirion
acrotrichum Zucc.” By Prof. F. A. F. C. Went and A. H.
BLAAUW.

In the summer of 1904 a specimen of Dasylirion acrotrichum Zuce.
was in bloom in the Utrecht Botanical Garden. The home of this
tree-like Liliacea is in Mexico; on a short stem it bears a bundle
of flat leaves with thorny margins. Although the plant is pretty
often cultivated in European botanical gardens it is very seldom seen
in bloom. Hence constant attention was paid to the here mentioned
specimen. The inflorescence was two metres long; the principal axis
was ramified and had a great number of steeply erected lateral axes
in the axils of bracts; each of these carried some 50 to 150
unstalked female flowers. Dasylirion is dioecious so that male flowers
were entirely absent.

Each flower had a perianth consisting of six green leaflets and a
pistil; this latter consisted of a triangular ovary with a short style
and three stigmas. The ovary was unilocular and had on its bottom
three ovules.

After the flowers had finished blooming it seemed as if some
ovaries began to swell. As there could be no question of fertilisation
in the absence of male sexual organs, it was thought that perhaps
a new case of apogamy or parthenogenesis was present here. The
ovaries were now regularly examined ; they more and more assumed
the appearance of little fruits, looked like small nuts provided with
three wings and strongly reminded one of the fruitlets of Rheum.
It appeared that many ovules swelled, but never more than one in
each ovary. Not nearly in all flowers this phenomenon was observed,
in no more than 10 to 40 percent it was at all visible.

For a detailed investigation these ovules were now fixed in
FLemMina’s fixing solution (the weak solution) and then washed in
the usual manner and gradually placed in strong alcohol. This was
done for the first time on August 15; from 158 ovaries 49 ovules
were obtained, i.e. 31 percent. This was a maximum, however, for
when later material was collected in the same way on August 22,
September 3, 10, 138, 19 and 25, October 8 and 22, November 12,
December 15 and 24 and on January 19, 1905, each time more and
more ovules appeared to be unfit for use, as they began to wrinkle.
Such as looked more or less swollen were fixed; among these some
had grown thicker and finally the impression was that some seeds
had ripened. But ultimately not a single germinable seed appeared
to be on the plant and after January 19 no material fit for investi-

(685 )

gation could be got. Nothwithstanding this the preserved material was
examined, since it was possible that only the unfavourable conditions
under which Dasylirion lived in the Botanical Garden at Utrecht,
were the reason why no ripe seed was formed.

On microscopical examination phenomena were indeed observed
which seemed to point to apogamy or parthenogenesis, but the mate-
rial proved insufficient to obtain a consistent result. Leaving apart
even the already mentioned fact that not a single ripe seed was
produced, the number of ovules in which ultimately anything parti-
cular could be observed, was extremely small. For microscopic
examination revealed that most ovules which outwardly showed
nothing abnormal, were yet already in all stages of disorganisation.
_ Although we are unable to offer a finished investigation, yet it
seemed desirable to us to publish what we have seen. For Dasylirion
blooms so seldom in Europe that for us the chance of finishing our
investigation is practically nihil, while now at least attention has
been drawn to it, so that perhaps in the mother country of the plant
some one may feel inclined to re-examine it.

Moreover the number of known cases of apogamy or partheno-
genesis is so small that there is every reason to publish each new
ease. And finally the material examined by us presents some points
which deserve attention for special reasons.

The fixed material was embedded in paraffin, cut with the micro-
tome and then stained, as a rule with saffranine only, sometimes
with saffranine, gentian violet and orange G.

The ovules of Dasylirion are anatropous and furnished with two
integuments ; the outer one consists, besides of an exterior and inte-
rior epiderm, of cells, situated rather irregularly in 2 to 4 rows;
towards the chalaza it is much more strongly developed. The inner
integument consists of two layers of closely adjacent cells. The
micropyle is formed by the inner integument only, the edges of

which are strongly swollen — the cells are larger and the thickness
is bere about four cells — and are closely adjacent, so that they

only leave a narrow slit between them.

The tissue of the nucellus is small-celled near the chalaza, but for
the rest it consists of large cells with very little protoplasm and
apparently very much cell-sap. The more peripheral cells are smaller,
their cell-walls are perpendicular to the integument, especially near
the micropyle, but the others are greatly lengthened in the direction
of the chalaza so that they have become tube-shaped. These tubes
are often more or less bent, so that longitudinal sections present an
appearance which is rather difficult to disentangle. The swelling of

48*

( 686 )

the ovules was in many cases to be ascribed to the strong turges-
cence of these nucellus-cells only ; in older stages also the cells or
the outer integument began to increase their volume, evidently also
by the increase of the cell-sap only.

These strongly lengthened nucellus cells at first caused us to believe
that more than one embryosac is formed, but an accurate examination
of the preparations finally gave us the conviction that only one
embryosac is found. Certainty on this point will be obtained only by
investigating the development and for this purpose the collected
material was unsuitable, for also in the youngest ovules the embryosac
was already completely formed. It is long-drawn, somewhat in the
shape of a dumb-bell, at the base extending near the chalaza, at the
top near the micropyle surrounded by a single layer of nucellus cells.

Now it appeared that in the great majority of these embryosaes
nothing particular could be observed; sometimes a little protoplasm
or more or less disorganised and swollen masses, but no egg-appa-
ratus, no polar nuclei and no antipodal cells, so that’ presumably in
nearly all the ovules a disorganisation had already taken place before
they were fixed.

Only a few ovules presented more particularities and these we
shall describe here, in the first place those where a young embryo
was found.

In an ovule, collected on August 22, there is found at the top of
the embryosac and filling this part of the latter entirely, a cellular
body with eight normal looking nuclei, making the impression of an
embryo. The rest of the embryosae is empty and only some disor-
ganised masses lie in it; of an endosperm nothing can be seen, no
more than of antipodals or embryosac-nucleus ; concerning this latter,
however, the possibility must be granted that it has fallen from the
preparation during the staining, although we do not think this probable.

In an ovule, collected on September 10, the top of the embryosae
is filled by a cell-mass of some 20 to 30 cells, the walls of which
are strongly swollen; the nuclei are small and are in a state of
disorganisation as well as the rest of the protoplast. The whole makes
the impression of a more or less disorganised embryo. Further there
is in the embrosac a pretty large quantity of protoplasm in which
we could find no nuclei.

Finally we found in an ovule, collected on August 22, a still
larger cellular body, reminding us of an embryo. It consists of about 40
cells, the contents of which are still more disorganised, with swollen
cell-walls which strongly absorb staining substances. Having regard
to the former two preparations we are of opinion that this also

( 687 )

must be looked upon as an embryo, the development of which has
already for some time been stopped and which is now in progress
of disorganisation. Also here nothing peculiar was further found
in the embryosae.

Of course we looked also for the presence of an egg-apparatus,
especially in the younger stages, but there is only one preparation
in which anything of this kind can be detected. It is an ovule,
collected on August 22, where in the top of the embryosac three
cells are found, two shorter ones with distinct nuclei and a third
which is larger with disorganised cell-contents in which the nucleus
can still be discovered, however. We believe this to be the egg,
the others synergids. Here also nothing else is found in the embryosac
except prcetoplasm, which stains strongly.

In 10 other ovules an endosperm was observed in various stages
of development. It must be stated at once that in none of these
anything of the nature of an embryo is seen. Although it may be
objected that for some ovules the series of sections is not complete,
yet this is certainly not the case with the majority. Especially where
the micropyle is seen in the section, the embryo would be sure to
be observed if it were there, but also in this case no trace of it
can be found. So we arrive at the conclusion that here an endosperm
has been formed without the embryo having developed.

An ovule, collected on August 15, shows the smallest quantity
of endosperm. The upper part (’/, to */,) of the embryosac is filled
up with it. The shape of the embryosac has been changed; it is
swollen, has become cylindrical or somewhat broader towards the
bottom, has a thickness of O,4 mm., while the nucellus has a
maximum diameter of 1,0 mm. The lower part of the embryosac in
which no endosperm is found, has entirely collapsed and has evidently
been squeezed by the surrounding cells. This same shape of the
embryosac was met with only once without an endosperm having
been formed in it, namely in an ovule, collected on the same day.
In the lining protoplasmatic layer no nuclei could be seen, but still
we believe that this was a first beginning of the formation of an
endosperm. Now the endosperm of the just-mentioned ovule consists
of thin-walled cells of varying size; normal nuclear divisions occur
but also nuclei of abnormal size with a number of nucleoli, indicating
fragmentation. At one of the sides of the embryosac the formation
of the endosperm has not yet been completed.

Curiously enough the next stage in the development of the endos-
perm was observed with an ovule, fixed on December 15. Here the
greater part of the tissue of the nucellus has been displaced, so that

( 688 )

it forms only a narrow layer round the endosperm, somewhat
thicker near the chalaza (greatest thickness of the embryosac 1,2 mm.,
of the nucellus 1,5 mm.). Here also the lower part of the embryosac
is not filled, but is entirely abortive. The endosperm-cells are of
rather unequal size, most nuclei do not look normal, but still divisional
stages occur; in the more: peripheral cells small grains which strongly
absorb staining substances appear outside the nucleus. As in some
other cases, the impression is got here that the formation of the
endosperm takes place rather irregularly, as if in various spots within
the embryosae pieces of endosperm-tissue would form which grow
towards each other so that seemingly more than one endosperm lies
in the embryosac. At any rate this seems to be so when one limits
his attention to one preparation; by comparing, however, the different
successive sections of one ovule there finally appears to be only one
mass of endosperm. The formation of the endosperm begins in the
lining of the wall of the embryosac and from there proceeds inwardly ;
in this process the cavity is gradually filled up, the endosperm now
meets itself from various sides and it is these divisional lines that
remain visible. .

That the formation of an endosperm starts indeed at the periphery
of the embryosac, appears e.g. from an ovule, collected on Septem-
ber 19. Here the size of the whole endosperm is greater than in
the already mentioned ovules (diameter 1,35 mm.), so that only a
very narrow layer of nucellus-tissue is visible all round, mostly at
the chalaza (greatest diameter of the nucellus 1,4 mm.); but the
whole endosperm is hollow and in this cavity remnants of the proto-
plasm of the embryosae are visible. The endosperm-cells are here
of very different sizes and so also the nuclei vary much. Some of
them look normal, show karyokinesis, others are enlarged, have
assumed all sorts of capricious shapes, the number of nucleoli has
vreatly increased and a number of fragmentation stages can be observed.
- Two ovules, collected on September 10, show a. still further
developed endosperm. The nucellus tissue has been more displaced,
the shape of the endosperm-cells is pretty regular, their cell-wall
is somewhat thickened, the nuclei are almost normal; in any case
there is much less indication of fragmentation than with the just
mentioned ovule.

In an ovule, collected on September 19, the endosperm is so
strongly developed that of the nucellus tissue hardly anything remains
visible. This also applies to the cases which will be described
presently. The endosperm-cells have strongly thickened but still
fairly gelatinous walls; the contents of the cells consist of a number

( 689 )

of small grains which stained very strongly and which somehow
make the impression of nucleoli; of a nucleus nothing is found any
longer, unless we apply the name to some thick, coloured masses.

Three ovules, fixed on December 15, all showed the same picture.
A strongly developed endosperm is present with very thick cell-
walls, absorbing saffranine more or less, and protoplasts which are
entirely foamy and in which nothing of a finer structure is found.
This endosperm must evidently be reckoned among the horny ones;
it was extremely difficult to cut. Sections of the ovules could only
be made after treatment with hydrofluoric acid. It is not impossible,
of course, that the foamy appearance of the protoplasts must be
ascribed to this treatment, although we do not think this probable
on account of other experience with this method. In the endosperm
some fissures are visible, the last remnants of the cavity of the
embryosae.

Finally an ovule with an endosperm was found among the material
collected on January 19. Here also cutting was only possible after
treatment with hydrofluoric acid. The endosperm is entirely dis-
organised, borders of cells can scarcely be recognised. No more than
in the preceding cases we think this must be ascribed to the manner
of treatment.

We have now described all cases of formation of an endosperm,
observed by us. It will have been noticed that the order is not
chronological, the arrangement was such that we gradually proceeded
from the least developed to the complete endosperm. From this it
follows already that the formation of an endosperm takes place very
irregularly with these ovules, sets in now sooner, then later, and
that the endosperm may pass into disorganisation at various stages
of development.

Summarising, it appears that with Dasylirion acrotrichum an endo-
sperm is formed without fertilisation. This endosperm finally disorga-
nises ; it may do so already at a pretty early stage of development,
but it may also first attain its complete development. But an embryo
could never be found together with such an endosperm. From this
it does not follow, however, that it could never be formed together
with an endosperm, especially since in three ovules — in which,
to be sure, no endosperm was formed — in the top of the embryosac
a cell-body was found which we take to be an embryo, which how-
ever very soon passes into a state of disorganisation.

One may now ask to what cause this disorganisation must be
ascribed. It might be suspected that the circumstances of this Dasylirion
were abnormal. Although we grant that these were different from

( 690 )

the conditions in the mother country of the plant, yet we must
remark that the plant was in the open air for along time before and
after it had bloomed during the very hot summer of 1904 and that there
was no question of this specimen being sickly. We venture another
supposition: to us it seems that this plant makes, so to say, an
aitempt to apogamous development, but that these endeavours do not
succeed. For this would plead that the endosperm develops here
independently of an eventual formation of an embryo and that the
embryo is sometimes planned, but never grows to any considerable
size. If this be the case, in the mother country of the plant similar
phenomena should be observed, but at the same time normal ferti-
lisation and seed-formation. We ought to know the development of
the embryosae, in order to know why the apogamy is unsuccessful
here, even though the plant makes an attempt in this direction. If
in the embryosae mother-cell a reduction division has taken place,
this would be very easy to understand and it would also explain
the greater facility with which the endosperm is formed. For, after
fusion of the two polar nuclei the normal number of chromosomes
of the 2z-generation (not, of course, of the endosperm) would be
re-established again; we have tried to determine this number and it
seemed to us to be 20 to 24. But as long as we do not know how
the endosperm is formed this determination is of little value; for
we owe to Trnvus') the knowledge of a case of endosperm formation,
with Balanophora elongata, where the endosperm nuclei are formed
by division of one of the two polar nuclei. It is, to be sure, the
only case on record where an embryosac fills with endosperm,
without a normal embryo being formed. In this respect the ovules
of Dasylirion, described by us, could be compared with Balanophora.
On the other hand there is this great difference, that with Balanophora
an embryo is later formed from part of the endosperm and of this
there is no question with Dasylirion.

We put the word apogamy at the head of this communication
because it leaves unsettled whether here phenomena of parthenogenesis
were indeed observed. It is an open question to what extent the
development of an endosperm without previous fusion of the polar
nuclei with one of the generative nuclei of the pollen tube can be
brought under one of these conceptions. Those who will not use
the word fertilisation in the case of endosperm formation, like
STRASBURGER, will object to it; those who embrace the opposite view,

1) M. Trevus. L’organe femelle et l’Apogamie du Balanophora elongata Bl. Ann.
du Jardin botan. de Buitenzorg XV. 1898 p. 1. See also J. P. Lorsy, Balanophora
globosa Jungh. Ann. du Jardin boten. de Buitenzorg 2me Série I. 1899, p. 174.

( 691 )

like GuieNarp and Bonnier, will think the use of these terms
admissible. Although we incline towards this latter opinion, we shall
not dwell on this point here.

But we think it desirable to point out that a closer study of
unfertilised ovules, especially of dioecious plants will perhaps yield
surprising results. Since we know through Loxzs that chemical stimuli
may cause the development of an egg, the possibility must be granted
that this may also be the case with higher plants. When a normal
fertilisation does not take place, such chemical stimuli would at any
rate render a beginning of development possible. Looked at from
this point of view the case of Dasylirion is perhaps important, but,
as we stated already at the beginning of this communication, only
an investigation in the natural place of occurrence of the plant can
give an answer to this and allied questions.

Astronomy. — “On the parallax of the nebulae’. By Prof. J. C.
KK APTEYN.

Up to the present time we know hardly anything about the distance
of the nebulae. On the whole they do not allow of the most accurate
measurement, and as a consequence direct determination of parallax
is generally to be considered as hopeless. A few endeavours made
for particularly regular nebulae have not led to any positive result.

The proper motions (p.m.) seem more promising, at least for the
purpose of getting general notions about the distances of these objects.

Spectroscopic measurements of radial motion show that the real
velocities of the nebulae are quite of the order of those of the stars.
Therefore, as soon as we find the astronomical proper motion of
any nebula, we conclude, with some degree of probability, that its
distance is of the order of that of the stars with equal p.m.

Meanwhile it may be considered to be a fact, most clearly brought
out just by the observations presently to be discussed, that as yet
p.m. of a nebula has not been proved with certainty in a single case.
It does not follow that these p.m. are necessarily very small. The
time during which the position of these bodies has been determined
with precision, is still short, the errors of the observations are large.
The effect of these errors on the annual p. m. may easily amount
to 02 or 0"3,

We might endeavour to lessen the influence of the errors of
observation by determining not the individual motions but the mean
p.m. of a considerable number of nebulae.

( 692 )

If this succeeded we might then compare this mean p.m. with
the mean p. m. of different classes of stars, the mean distance of
which is known with some approximation or, better perhaps, with
the mean radial velocity of the nebulae determined by the spectro-
scope. The comparison would lead at once to ideas about the real
distances.

Unfortunately the mean of a great number of observed p.m. will
not be materially more correct than the individual values, if the total
proper motion is small. The cause of this lies in the fact that in such
a case the effect of a determined error of observation is not at all
cancelled by an equal but opposite error of observation. Suppose for
instance two nebulae both having in reality a p. m. of O01. For
the first let the error of observation be 0"10 in the direction of the
p- m. For the second assume an equal error in a direction opposed
to the p.m. The observed p. m. of the first nebula will be 0’11,
that of the second O"09. Taking the mean of the two we are not
brought nearer to the real value.

For this reason we shall not be led to any valuable result in
this way, even if our material consists of very numerous objects, as
long as the errors of observation exceed the real p. m.

The difficulty here considered would vanish if, instead of the total
p- m., we could avail ourselves of some component of the p. m.,
which in different direction would have different sign. In this case,
if systematic errors can be avoided or determined, the accuracy would
increase as the square root of the number of objects included.

Such a component of the p. m. is that in the direction towards
the Antapex. From this component we may derive the mean paral-
lactic p. m. which is a measure of the mean parallax.

I will not here stop to consider the hypothesis involved. It must
be sufficient to state that it assumes that the sum of the projections
on some determined direction of the peculiar p. m. vanishes in the
case of very numerous nebulae or, which comes much to the same
that the peculiar p. m. may be treated as errors of observations.

Let ;

h be the linear annual motion of the solar system;

© the distance of a nebula from that system ;

4 the angular distance of this nebula from the Apex of the solar
motion ;

v, t the components of the observed p. m. in the direction towards
the Antapex and at right angles to that direction ;

p the component of the peculiar p. m. in the direction towards
the Antapex.

( 693 )

The parallactic p.m. shall then be :

h
—sniA=v—p.

Q
If this equation is written out for each individual nebula and if,
after that, we take the mean of all the equations, the quantities p will

h
disappear and we obtain the mean value of —, which is the secular

parallax.

Or rather :

As we may treat the quantities p as if they were errors of obser-
vation, which mix up with the real errors of the observed quantities
v, we may write out for each nebula an equation of the form

h
SS SUA Ue hs te oh ey) tah eee ot (LN)

9

If then we assume that the distance @ is the same for all the
nebulae, we may solve the whole of the equations (1) by the method
of least squares.

I have long wished to apply this method in order to get some
more certainty about the position of the nebulae in space, but I have
been restrained by the extent of the work connected with such an
enterprise.

The difficulty has disappeared since the publication, a few years
ago, of a paper by Dr. MOnnicnmeyer assistant at the Observatory of
Bonn (Verég. der Kin. Sternw. zu Bonn. N°. 1). In this paper all
the materials available at the time of its appearance have been
brought together in a way which, for my purpose, leaves little to
be desired.

This paper contains the observations of Dr. MénnicuMryer himself.
They bear on no less than 208 objects, mostly chosen among such
nebulae as can be measured with considerable or at least moderate
precision. Dr. M6nnicumeyer has collected besides, all previous obser-
vations of these objects. I have confined myself to the observations
of those nebulae for which all the observers have used the same
star or stars of comparison. I have further rejected the observations
of those objects for which Monnicnmryrr did not succeed in deter-
mining the personal errors. The observations which thus have served
for the investigation are those of MéynicumryeEr’s paper pages 59—70,
from which have been excluded, in the first place, those objects
which in the list of pages 15—17, second column, have been denoted
by the letter M; further the planetary nebulae, the clusters and the
ring-nebula h 2023.

-

( 694 )

There remain 168 nebulae.

A good judgment about the accuracy of the observations may be
obtained by the probable error derived by Méynicumeryrr for his
own observations on page 9. For the other observers I have availed
myself of the data contained on pages 18—25,

The accuracy was found little different for the several observers
with the exception of RimxKrr.

I therefore simply assumed the weights to be proportional to the
number of observations. For Rimker only the weight was reduced
in the proportion of three to one. For Scumipt the number of obser-
vations is not given. For reasons given by MOonnicHMEYER they are
“immerhin etwas fraglich” (I. ec. page 14). The results of Scumipt
got the weight of only a single observation for that reason.

An. overwhelming majority of the observations has been made
between 1861—1869 and 1883—1893. It was possible therefore in
nearly every case to contract all the observations in two normal
differences from which ‘the proper motion and its weight could be
derived at once without any serious loss of accuracy.

From these p.m. I then derived the components t and v, assuming
for the position of the Apex, the coordinates

Az = 273°, D5 = + 29°5.

The whole of the materials was divided into the three classes of
Monnicumerer. They are described by him on page 9 of his paper
in the following way:

Class I. Nebulae with starlike nucleus not fainter than 11 mag-
nitude ;

Class II. Nebulae with moderately condensed nucleus not fainter
than 11 magnitude ;

Class III. Difficult objects, in the first place irregular nebulae
without any sharply marked point; furthermore all very faint objects
and the very oblong nebulae.

Most of the objects have been classified by Méynicumnyrr himself
on page 9 of his paper. The nebulae wanting in this list have been -
classified by myself, in accordance with the descriptions on p.p. 27—54,
as follows: h 693, 1088, 1225 in Class I; 4 421, 1017, 1212, 1221,
1251, 3683 in Class II; 2 316, 1461 in Class III.

The p.m. as derived are relative p.m.; they are the motions
relative to the comparison stars. MOnnicHMrEYER has investigated the
p-m. of the comparison stars themselves; he has found a sensible
p.m. for only 7 of the objects used for my investigation. The
following table contains his results for these 7 stars.

( 695 )

Star of used for pests :

mag. (a Us in are Dv Sin 4
Comp. nebula er. circle

S$ V 1 VW |

415 6.0 h 132 |+ 0.0140 | — 0.089 | 0.227 + 0.225 | 0.94
90 8.8 h 805 (4+ .0237) — _ .4170 352 —— ee Gaede OO)
129 6.4 hyd i 0170) — a 197 yl + .255 |. 0.97
164 Tea h 1329 i— .013 00 .192 + .167 | 0.99
4168 9.5 M 90 |+ .014 00 204 — .180 | 0.98
208 4.7 II 542 |— .0050;-+ .010 075 + .055 | 0.80
942 6.6 h 2050 |— .01384|]— .4152 199 — .197 | 0.45

These p.m. were applied by Moénnicumeyer before he derived his
definitive differences in @ and d (Neb.-Star). In no other case a
correction for the p.m. of the comparison stars was applied.

The majority of the observers used the ringmicrometer.

The principal error to be feared for observation with this micro-
meter is the personal error in right ascension. MO6nNiIcHMEYER has
devoted the utmost care to their determination. Notwithstanding this
it may be considered a fortunate circumstance that this error has no
influence on the result for the mean parallactic motion, at least in
the ideal case that the nebulae are distributed uniformly over the
right ascensions from O to 24 hours.

For it seems highly probable that the distance of the nebulae is
not systematically different in the different hours of right ascension.
This being so the personal error will vitiate the parallactic p.m. of
the nebulae at the same distance in right ascension on both sides
of the apex, to the same extent but in opposite directions.

It is true that the distribution in right ascension is far from being
uniform ; still we may be sure that whatever residual personal
errors may still exist in the materials of MOnNIcHMEYER, must appear
considerably diminished in the result. Meanwhile I have tried to
obtain some idea about the possible amount of these residual errors
in the following way.

I computed the average proper motion in right ascension for each
hour separately. Taking the simple mean of all these hourly averages
we may expect to get a result in which not only the peculiar proper
motions, but, as explained just now, also the parallactic motions
shall have vanished.

( 696 )

This final result may therefore be assumed to represent the residual
influence of the personal errors on the p.m.

For the value mz of this mean I find

Hx = — 0.5000 4

In deriving this result the hours with many nebulae did not get
any greater weight than the hours with only a few objects. Owing
to this cause the final weight is found to be only 0.4 of what it
would have been had the distribution been uniform.

We shall get a result of appreciably greater weight if in the first
place we combine by twos the hours lying symmetrically in respect
to the apex. In these mean values the parallactic motion is already
eliminated; we may therefore further combine the twelve partial
results having regard to their individual weights.

In this way I find

tu = + 0.50006.

It thus appears that MOnnicomnyer has succeeded remarkably well
in getting rid of the influence of the personal errors.

As mentioned just now these errors appear still further diminished
in the result for the parallactic motion.

There thus seems to be ample reason for neglecting any further
consideration of them. In order to enable the reader to get at once
a pretty good insight in the accuracy really obtained, I have divided
the whole of the material not only into the three classes fof
MOonnicumeyer, but I have subdivided each of them into a certain
number of sections, each of about the same weight.

I thus got the following summary. (See p. 697).

The values of rt have been included in the table merely in order
to show that in them too no traces of any personal error are visible.

In order to get the yearly parallaxes 7, I have divided the secular

h
parallaxes — by 4.20; this number being, according to CampBELL’s
Q

determination, the number of solar distances covered by the solar
system in a year in its motion through space.

The probable errors were derived in the hypothesis that the com-
ponent v is wholly due to errors of observation.

If we compute the probable error of one of our 13 results from
their internal agreement we get 0."023. This number differs very
little from the values directly found. Here again we have an indication
that systematic errors must be small.

The last row of numbers contains the simple averages of the 13
individual results.

cs a pun t — pee. mu pees
hu ham ae i i ui U}
0.0 — 5.33 | 13 | + 0.014 | — 0 039 | + 0.023 | — 0.009 | + 0.0055
5.33—10.57 | 12 | — 0435) + .051 .022 | -+ .012 i)
I 10.57—12.22 | 10 |— .045|-+ .034 023 | + .008 5°
A222 12.45 9 |— .004]}— .027 022 | — _ .006° 5
\} 42.45— 0.0] 10 | — .008})-+ .013 023 | + .003 5°
0.0— 9.50} 12 | + .021 | + 0.014} + 0.019 | 4+ .003 4s
| 9.50—11.140 | 10 | — .004|— .016 O19 | — .004 4s
II 41.40--12.46 | 44 | — .008} — .037 020 | —_ .009 5
| 12.46—12.28 | 12 |-+ .019}— .040 .020 | — .0095 5
12.283— 0.0] 44 | + .0005} — .040 020 | — .0095 5
0.0 —12.44} 20 | + .030 | + 0.016 | + 0.019 | + .004 4s
Ill 12.14—12.32 | 16 | — .046 | + .038 .019 | + .009 4s
42.32—0.0/ 19 |+ .016 | — .036° O18 | — .009 4
Simple Wy i W) i v
mean of | 168 — 0.004 — 0.005 + 0.005 — .0013 + 0.0012
13 results

We thus finally get for the mean yearly parallax
— 00013 + 0"0012 (168 nebulae). . . . . (3)

This is the parallax relative to stars of comparison the mean

magnitude of which is
8.75

Meanwhile, as mentioned before, MOnNicHMEYER applied p. in. to
7 of his 183 stars of comparison.

If he had refrained from doing so, we should have found the
parallax O"0004 smaller. We thus have in conclusion:

Mean parallax of the 168 nebulae relative to stars of comparison
of the mean magnitude 8.75.

— 0'0017 + 0012 (p.e.). . 2. . 2. @

In N°’. 8 of the Publ. of the Astr. Laboratory at Groningen the
mean parallax of the stars of magnitude 8.75 was found to be

COCCS rg we ee yl bse ta)

To this value we might apply two corrections :

( 698 )

1st. Because, since the publication of the paper mentioned, our
knowledge about the sun’s velocity has made considerable progress ;

2.4, Because in its derivation a slight mistake was discovered.

I shall not apply any correction, however, because the two cor-
rections nearly compensate each other for the magnitude 8.75. There
is a fair prospect of the possibility of materially improving the values
given in Publication 8 before long. It seems advisable to wait for
such improvements before we alter these determinations.

If for this reason we provisionally adopt the value (5) we get:

Mean absolute parallax of the 168 nebulae

0"0046 + 0'0012 (p.e.) . . \.. 7) eeu

This result is somewhat less reliable, however, than (5) because
of the additional uncertainty in the absolute parailax of the stars of
comparison.

The value (6) agrees nearly with the mean parallax of the stars
of the tenth magnitude.

I shall not insist on the exact amount brought out for the parallax.
I shall only direct the attention to the fact that from observations
covering only a period of somewhat over thirty years, we get a
probable error of hardly over O".001. If this is the ease with visual
observations we may look for really excellent results by photography.

The best measurable nebulae must be generally the smaller ones.
The number of these which can be photographed is enormous.

With his Bruce-telescope (opening 40 centim., foe. dist. 202 centim.)
Max Wo.rr obtained in 150 minutes a single photograph of the
region near 31 Comae, containing 1528 measurable nebulae (Publ.
Konigstuhl T p. 127).

This richness of material will enable us to confine ourselves provi-
sionally to those nebulae which allow of a very accurate measurement.

Personal errors must disappear because we shall certainly succeed
in nearly every case in making our pointings on the same point for
the several epochs. The peculiar p. m. will be the more thoroughly
eliminated the more extensive our material; especially if this material
is distributed over the whole of the sky. Errors in the precession
have no influence at least on the value of the relative parallax.

I am convinced that by photography we may obtain, even within
ten years, results which will far surpass in accuracy those of the
present paper. Thus we may hope, in the near future, to reach
a fairly satisfactory solution of the vexed question respecting the
position of the nebulae in space.

The same treatment to which we have here subjected the nebulae
may of course also be applied to other objects. We have already

( 699 )

undertaken that of the Helium-stars and might perhaps afterwards
try the same method for the stars of Pickrrine’s 5'* Type.

In concluding it is only just to say that, whatever be the merit
of the present investigation, it belongs mainly to Dr. MOnnicuMEyeER.
As compared with his careful and elaborate labour, that spent on
the derivation of the present result is quite insignificant.

Chemistry. — “On the course of melting-point curves for compounds
which are partially dissociated in the liquid phase, the proportion
of the products of dissociation being arbitrary’, by J. J. van
Laar. (Communicated by Prof. H. W. Bakuuis Roozepoom).

1. It is well known, that a liquid mixture of e.g. two compo-
nents 14 and 5, which can form a compound 4A,, B,,, reaches its
maximum point of solidification, when the ratio of the molecular
quantities of the two components is as r,:»,, in other words when
there is no excess of one of the products of dissociation of the com-
pound A,, B,,.

Expressed differently: when we determine the points of solidification
of a series of liquid mixtures of A, 6 and the compound with
increasing excess x of one of the products of dissociation of the

dT
compound under consideration, then (S)=0 for the curve of soli-
ax 0

dification or melting-point line thus formed.

Hence the melting-point curve of a compound, with increasing
addition x of one of the products of dissociation, will have an
horizontal direction at 2—0O, as soon as there is but the
slightest dissociation of the compound in the liquid phase. If there
is no dissociation at all, the admixture may be considered as an
alien, indifferent substance, and the initial direction of the melting-
point curve will show all at once the normal descending course at
Ba 0}

As will also appear from the following computation, the initial
horizontal course will of course pass the sooner into a descending
course, the slighter the dissociation of the compound is.

7

. . . . a . . .
The peculiarity mentioned of (=) becoming zero with the slight-
av Jo

est trace of dissociation of the compound, was already proved by
Prof. Lorentz in 1892, on the occasion of an investigation of
STORTENBEKER On chlorine-iodides'). Prof. van per Waats too has

1) Z. f. Ph. Ch. 10, bl. 194 et seq.

49
Proceedings Royal Acad. Amsterdam. Vol. VIII.

( 700 )

proved this property, induced by a statement made by Lr Caareirer*).
The proof given does not directly- bear, however, on the case that
in the liquid phase also the compound (van DER Waats’ so-called
complex molecule of salt and water) is found by the side of the
products of dissociation.

2. Here follows another simple and quite general proof of the
property in question, in which specially the condition in the liquid
phase is taken into consideration, in which by the side of the com-
pound the products of dissociation occur in varying quantities.

Let us suppose there ¢hree kinds of molecules:

those of the compound A, 5, ; number n, = 1—a
those of A ; ma n, = ?,4
those of B ; ¥ n, = v,a-+ 2.

So @ is the degree of dissociation of the compound, and « the
excess of B e.g.

Now from the property, that the molecular potentials of these three
substances, viz. , @, and m@,, are homogeneous functions of the Om
degree with respect to the numbers of molecules, follows immediately :

du, du, du, 0: 1
me de 5 i dx a "3 le = ) : $ j : ‘ ( )

Here the differentiations with respect to @ are to be taken total,
so that e.g.:
du, Ou, | Ou, da
dx 0a da dx’
i.e. at constant temperature.
[The above property is proved (loc. cit.) as follows. We have viz.
in consequence of the mentioned peculiarity of the functions p,, 4,
and u,:

Th Ou, Ou,
Se ta aE |
One, Ou, On, San
ea St tegen On am |

Sh also 2! eine equaliteree= area lie Oe ad
So also .~ being equal to .—, ete. (on uy aie and uw, =a)

2 1
1) Verslagen Kon. Akad. van Wetenschappen (4) V, p. 385 (1897).
2) These and the following properties were already proved by me in 1894. See
Z. f. Ph. Ch. 15, p. 459 et seq. (“Ueber die genauen Formeln, etc.”).

( 701 )

Ou On, Ou,
2 es — =0
0 On, eo On, - On,

1

Ou Op, Ou,
ee 1 0
Cooma Oa HOR |

So if we pass from the variables n,, m, and n, (of which there
are only two independently variable) to the variables @ and x, we
have also:

Ou, Ou, Ou, \*)
= — = 0
See eae |

nets ue Ou,

stm scan Cage oC

da
The first equation mluuied by a and added to the second,
Ak

gives immediately (1). |
Now follows from the equilibrium of dissociation :
Sula Pie, aF Yu, 0

1) We can easily test the truth of these simple properties by supposing the
functions zg’) «'; and y«’g constant in:

—_
Le = Be ae Re log = =, + RE log
ete
N

v,a
N’

iy ie

0
Then we have immediately (having divided by RZ’) after differentiation yan
a

taking into consideration that

N=1+0,+7,—lfe4-«=1+4 6a+2,

for the first member:

it 0 ; : o
—y- a teele- x tO +]

are: N\—

(1 — a)

=(—149, +r) —S—a+natrata)=0—LxN=0.

0
After differentiation aE we find for the first member:

(1—a)

=| #8

— {tent

aL
ae aN
=1—5(l—etre+mat )=1-X N=0.

And according to what has been proved, this will continue to be true, also
when p'o, #’; and y's are still functions of z and a.

49*

( 702 )

immediately, after total differentiation with respect to x (7 constant) :
du, du d
pp

1 Uy
— — ee en ee eR
da * da Tus da C)

And from (1) and (2) follows, that when n,:n,=v,:7, (i.e. z=0),
we have necessarily

du,
Oe oe he ean a ee
de (3)

: dy
So the becoming zero of ar, is the primary moment, on account of
Lv

dT
which also (a) will have to be O in the presence of a solid phase:
anv

with change ef « (with which also @ changes) the mol. potential
of the unsplit Be does, namely, not change when «=0. {This
property will evidently also continue to hold for an arbitrary number
of splitting products].

dT ca ; re
That now also ea = 0, follows from the condition of equilibrium:
au /,

— H+ Hp = 0,
when w is the mol. potential of the solid phase. Total differentiation
with respect to 7’ yields viz. :

d da =
ar Sp ge tL T te) aa

; ope , 0 0 da if =(2 0 da
in which 57, 18 again | 5+ 35 app 2M4 ae = | ae fears 2

But a= utyu)=— a when Q is the total heat of melting,
hence also:
2 dy, de ,
cs anata
because w (in the solid Pe: is independent of wv. Hence:
pp Uo
dT da
A tae WOR (4)

7

a . . Oy .
, also aoe and in this way the proposition is
ax

proved. When in the liquid phase there is no excess of one of the
products of dissociation, but instead an indifferent substance, then
there are four kinds of molecules, with molecular quantities resp. :

a iy Pha) ie ie gar.

( 703 )

Instead of (1) we get now:

du du du du
n, = +n, = +n, a +n, ae = U)5 6 oo {(l)
du du ae : :
And as n, ae = 2 remains finite at c= 0, viz. RT, from (1°)
v Lv
d
and (2) z

-=0, when c=0. (n,:2, =?,:?;
x

ryy

is always satisfied in this ¢

du,
| hat x2 — a5 * continues to have a finite value at z—0, follows from this,

du _d RT RT dN
thats — ei. Re log ~ yields Hs + , hence
dx dx a N d«

= RT dN
ON de

| , in which the expression between

di
ay re |
dx

3. We now proceed to derive an expression for the course of the
melting-point curve in the case of increasing excess of one of the
products of dissociation in the liquid phase.

Let us for this purpose suppose, that in this phase there are present
(in Gr. mol.) 1—.a AB and z B, while the 1—~2 AB is disso-
ciated to an amount e«. We have then:

AB A B
(1 — a) (1 — 2) a(1 — 2) a(l1—a)+ a,
together 1 + @ (1 — 2) molecules.

We suppose then, that the compound consists of 1 mol. A and
1 mol. 4, which simplifies the calculations.

The equilibrium between the solid phase and the non-dissociated
molecules in the liquid phase yields:

ois
or (the terms with 7’log T on either side cancel each other)

=

e—ePmy— ol + RP og KE
a(l—@

1) This too is easy to test, when uw’), «',, elc, are considered as constant, so
that e.g. in

uw =u, + RT log

l1—a ou ; 1 6
rat a becomes = R7 --4 ete,

or with e, —e=(e,—~ ~)=(4 +4r—2
row € — pastes.) | ae il Oh! 5 = |)
cas 0 b, a 3 e b,

a .
(«. + kT — *) =q (so that q is the pure latent heat of melting

of the compound, without the heat of dissociation, which is still to
be added), and with c, —c=y:
(toi 2)
ETS)

For the determination of y may serve, that at —=Oand 7’= T
a becomes a,, hence:

g=yTl — RT log

l—a
Cle Td oes *
Hence we finally get:
ih 1 — —
g (= T) SS REG log ee Ese)
l—a 1l+a(i—z)

or

Li. @) (le) egy 1
=o arcs =£(5-z)- eae

0

In this derivation it has also been supposed, that the liquid mixture
is a so-called ideal mixture, i.e. that terms, referring to the influence
of the components inter se, have been left out. It is known that
these terms are of the second degree with respect to «. Equation (5)
represents therefore the course of the “ideal” melting-point curve in
our case.

Further the degree of dissociation @ occurring there is given by
the equation (here too the above mentioned terms are left out, so
that the simple law of mass-action is supposed to hold):

a (1—2) y a (1—2) steno) (l—2)
Ne ae N N

ZG

or

a(a(1 = Oise wy) K
(=e) Gait=2) a =

In this A is now no longer a function of « according to the above
supposition, but it is one of 7’.

Even if we would solve « from this quadratic equation, and sub-
stitute it in (5), we should have gained but little, because A contains
T in a rather intricate way. Therefore the only thing we can do, is
to try and find an approximate expression, which only holds for

(6)

small values of «x.

( 703 )

7

After that a general expression for ae will be given.
&

In order to find the approximate ex-

Tab pression in question for the course of
the curve 7',A, we suppose for the

present, that @ does vary with 2, but

To not with 7. In the result we have then
‘A simply to replace q by the total heat
B of melting at c—0 Q,=q+a, (a is
the heat of dissociation), in order to
introduce the variability of @ with 7.

(see appendix).
ou So From (6) follows now immediately

; the quadratic equation
IG

e (1—zx) + ax — re =

>

By putting 2=0, we see that is then —a,’. According to

1K
the above provisional assumption it is now supposed, that also for
values of 7’, lower than 7,, the value of @, holding for «=O and
T= T., remains unchanged. Therefore in the equation
0 5

a (lL—2) + ae — a,? = 0

«, is no longer a function of 7. So we find for a:
== 2/2 V ieee ay(1—#)
= ee

and hence
eae aa V?/,0?+a,7(1—2a)
l=/7

l1—« =

In consequence of this we get:
d—a)l—a=1—')/,4@-yY
1fe(l—a)=1+/,e4+V
so that we find for the quotient occurring under the sign log in (5):
eas — VV), 2 + a,? ay
1=Yre an V

or also after multiplication of numerator and denominator by
(ep VAR
(L +.4,7) — 2). {/, # — 2.01 — */,#)
Paro a)

( 706 )

5 x?
Let us now approximate Va@,?(1 — 2) +7/,a@7= nf J Sy

for small values of 2.
We shall find:

patees (1 — a,’) (| — 5 a,’)
AF aH a — a, : 7 a

a a,

4
oe

a, VY, multiplied by 2—2, yields then:

1 a, 1 — a,?\?
caf 2 (CUS ah, a al arte...

This, subtracted from (1 + @,”) (1 — a) + '/, 2, gives:

i eer)
(= 4,) @ ay ee

If now finally this formula is divided by (1 — a,*) (1 — a), we get:

24

a,*

a? lta
Wiper ates psa) 5
1—a, | a on
l+a, | ceo ot
Equation (5) changes now into:
| 1, A+) |
) 2 == i 2 ee q l il
—log {1 2. 8 = sepa

a, l—@« Teg te dt.

Notice, that the term with x does not occur, in consequence of

Lv

aT a se ;
which (=) satisfies the condition of becoming 0.
0

If higher powers than a* are neglected, the above becomes:

a’ (1 + 2) pag: 1 1
4a, WTRE ts ;

or also, if we now replace g. by Q, (see above) and 7'7, by 7,?,
which does not bring about a change in the coefficient of 2°, as
T= To G—62")i:

pee Pe RT,’ #’? (1 + a)
Q, 4a,

which approximate expression holds for not too small values of
a (e.g. «@ =1/,) at least up to values of s=01. We see, that
T,—T is not proportional to 2, for small values of x, but pro-
portional to 2’. Hence instead of the usual straight downward course

. . (59)

(207)

of the melting-point curve at the beginning, it presents now an
almost horizontal course.

Observation.
Equation (5) enables us also to compute the melting-point tem-
- perature 7’,, of the unsplit compound (i.e. unsplit in the liquid
phase). (ef. fig. 1).

Then we have namely «=0, «=0, and we get, supposing

Citi = 026
aa gf 1
log = gal "yy ?
oll —— "cee LUNe li ed

0

from which follows:

il 1 R 1+ a,
——s Q
Te Te q a. 1—a,

(7)

Dig

dT
4. We shall now derive the general expression for — all over
ax

the line 7A, in which it is only supposed that we have to do
with ideal mixtures in the liquid phase, so that the terms, referring
to the influence inter se of the different components, are again left
out. But besides on a, @ will now also depend on 7.

In two different ways we can arrive at the correct expression

First of all by total differentiation of the equation (5) with respect
(1 — a) (1 — 2)
SS =e

1+ a (1 — 2) ;

d log c, 4 dloge,\ da q
ar); diy) Daten T

to 7. We get then, calling the fraction

hence
d log ¢,
CNS du
dz q dloge,
RT? dT
Se
ge da 0a dx
/ 1 4 1 oo, ee
Sane Se a ee SE) (92 ee ee cpa ee
l—e# 1+a(l—2) l—a_ l+a(l—2)/) dx
: 2—« da

(l—ea)(1+a(l—2)) (l—a) (1+a(l\—2)) da’

( 708 )

det :
Hence we must calculate =e From (6) follows:
AX

1 da 1 ; a ool +; da
a dx i ata(1—a) =a ees ae

—a dx

1 , da
meriean y

After reduction we find from this:

da a(1— a)
if 1 Ee Te 5 c (@)
His 4 a(1 — e#)
Substitution yields now:
dloge, _ YE 1 4 a (2—2) oe
de — (1—a)(1+a(1—2)) © (1+ a(1—2)) (e-+-2a(1—2))

v

(l—2) (@+2a(1—2)) ©

d log ¢,

For
OF ae.

we find in the same way:

dloge, Ologe, Aloge, da Od loge, da
dl ae on da dT da aT”
because c, is not directly dependent on 7. This gives further (see
above):

dloge, _ 2—wx da
df) 7 7 =e tear
det
So we calculate Aa From (6) follows:
1 da l1—x) da 1 ide l—e da 2
adf ' eas) di i—edr) lots) df = Rigs
0 log K a : ee
ar i To when 2 represents the heat of dissociation.

By solution and reduction we find:

da 2 a(l—a(1 eee

aT RT? 4+ 2@ (=n) a
In consequence of this we get:
dloge, _ 4. a(2—2)(a+a(1—2))
aT RT? «#42a(1—z)

; ; ; F log e, d log c,
If we now substitute the values found for ; and
Ax

( 709 )

a]

the last equation for Ga We set finally :

1 x
Re

aT l—w «42a (1—2a)
2 eae,

v+2a(1—2)

Le.

ey, ee ee a 8
dg: OPN Ogle), a (8)

when for g-+ ete. is written Q, i.e. the total heat of melting.
This formula, combined with (6), indicates therefore the direction
of the melting-point curve throughout its course:
In the second place we could have derived the same expression
from the general equation (4). As namely wu, =u,’ + RT loge,

ea, du, RT dloge,
we have — = FE

, assuming «u,’ to be independent of «, and

hence:
dloge
Rai vf
dT da
de Q

L tS dloge, . . ;
Substitution of the above found value of ae yields immediately
av

(8). But now we have still to prove, that really the total heat Q is

represented by

(2—a) (w+ a (1—a)) ,
a+2a (1—2)

Q=q+e (9)

This takes place in the following way. If a quantity dn of solid
substance passes into the liquid phase, the total quantity of heat
absorbed is evidently :

d
qdn + aadn + (1—a)4 os dn.
In

For qg is the pure latent heat of melting, if only non-dissociated
molecules are formed. But of the dm mols. an amount adn is dis-
sociated; the heat required is @adn.4. Finally the eaisting condition
of dissociation @ of the 1—zv mols. will be changed by the addition

da
of dn new mols., namely to an amount (1—2’) 7, te: For (1—2)a
an

dissociated mols. become (1—.) (a + da).

( 710 )

da dada m

Now —=——. And from 1—r=n, «=m follows «= ;
dn da du m+n
da m _ da da
hence = —- SSO) i
dn (m-+-n) dx da

Dividing by dn, we find therefore for the total quantity of heat,
absorbed per Gr. mol.:
da

Q=qg+ar4— «x (l—x)— 2.

di
2 ee _ da : , : ;
Substitution of ce from (a) yields then after a slight transformation (9).
av

Let us now put «=O, then we find from (8) on account of the

factor 2:
dT 0 2
dzyyr : (67)

If « is very small, this horizontal course does not continue long.
For with small « we may write:
dT Hdl oe ty
de On nanon &

As soon therefore as « becomes so large thet 2a is small with

respect to wv, the fraction

& av
— approaches — = 1, and the normal
+2a a

course is restored. The greater therefore a, the longer the almost
horizontal course will maintain itself in the neighbourhood of 7%.

sf

aX

If @ absolute = 0, then may be replaced by

&
a+2a (1—2) ry
from the beginning, and we have immediately the normal course,
given by
dT I Sagi
da a l—a# q

dT *3 TR
dx a q :

Also JT, and 7, then coincide.

yielding :

5. In fig.1 also the line 7,B has been drawn. This would be the
melting-point line, when instead of an excess of one of the products
of dissociation, an excess of an indifferent substance C was added.

The equation (5) remains then the same. But now (6) becomes
different. We have now namely :

(Galak. )
AB A B C
(l—a) (1—z) a(1—.) a(1—.) x,
together again 1 +- «@(1—.) molecules.
Hence the dissociation isotherm becomes:

a(l—x) a(l1—2) | (l—a) (1-2)
N a“ N : N

— kK,
or
a 1-—z

l—a 1+a(1l—2)

Rare eke = cos GO)

Now a does not decrease with v, but increase. The added indifferent
substance C' may viz. be considered as ‘diluent’, whereas in the
preceding question the addition of one of the products of dissociation
depresses the degree of dissociation «.

If we solve from (10) ee a, we find in this case :

K
1— ar —
sts isk ST eR
B tti O, it in that i that
fa t= VU, c ars aga c pope Sd S ta Te
y putting wv lt appears again 1a 1K a sO lat we

must solve a from
a?(1--x) + a,*aa — a,’ = 0,

in which a@, is again provisionally assumed to be independent of 7.

(Cf. § 3).
Now we find:
a=a, [—-1/,¢,0 + V"/,a,22? + (l—a ]: (1-2),

(1—a) (1—2) = (1—-r) — a, |— */,¢,e + Y]
1+a(1—«)=1-+4 a4, [—'/,¢,7¢ + yY]
The quantity occurring in (5) under the sign dog becomes then:
(l—a) + 1/,a,°4 — ayVv
1 ae a7
Now V1/,c,2? + (1—a) = 1—?/,#—1/,(1—a,”)a?...., so that the
above fraction passes into
1—a,—'/,(1 —a,)(2 + a,)e +7 ree
1+a,—'/,a,(1 + a,)e — aie (le Wars tae

i. e. into
(1—a,)[1—/,2 + ae + */,@,(l +-a,)2".]
(1 + a,)[1 — */,¢,% — */,a,(1 —a,)e*.. ai

or into

Tas)

1—a,
—— [1 —«# —'"/,a,«7...].

1+a,

Owing to this we get:

1 1
— log [i —@ = 9f,a,07 = e es i)
0

RV ZB
or :
2 (2 ea ee Bee
abe OS SNC Ales

RT,
or finally, substituting, Q,=q-+ a,4 for g (cf. §3), and 7; (- 0 *«)

for TL; :

Be : a 2 Ra
fae et %Mi,j1+%/,a,— Q eae a ow)
0 6

which approximate expression will now at least hold for values of
a < 0,26.

1"

. . ¢ . . .
6. <A general expression for fae the case in question may be
ak

most conveniently calculated from (4). (cf. §4). Then we get:
1 @ log cy

dT YS da
dx Oe
where
d log ey _ Ologe, Ologe,da __
ie pn da de
1 2—«a da

(l—a)(l1t+a(l—2)) (1—a)(1+a(1—a)) de”
1

But now = is different. From (6%) we find viz.:
AX

2 da lede 1 1 {
a dx 7 l—adze 1—z 1-+a(1—z2)|
yielding :

da
—e«t (se) = == (I),

da __ a(1—a)
dz (1—a) (2—aw) ’
so no longer negative, but positive as it should be (see above). After
substitution we get:
dlogce, _ 1 a(2—2) oi
de + (l—a)(1tea(I—2)) (—2)(2—aa)(1+e(1—2))
2

~ (1 a)(@ ae)’

(Fa38)
so that we find :
i RT aM 2

dx (O15 VEG

(11)

Ta
In this Q is again = q+ a4 — a2 (1 — 2) = 2. After substitution of
ae

da
the just found value for ae this becomes :

ae
2—a
Q=qtas——4. o- geomeat oe 5 (le)
= o— au
For «=O (11) becomes now:
CEN IE ‘ah
alee Q, ie qtah- ks tol (

So the melting-point curve has now also at 7’= 7, a perfectly
normal course.

For practical purposes we can determine more or less accurately
the value of a, from the approximate equation (5a) (for small values
of «), which according to (7) renders also an estimation of 7%, possible.
The value of Q, must then of course be known. It can, however,
also be calculated from the accurate determination of the initial course
of 7,6 (with indifferent admixture), according to equation (11¢).

yy

If we then determine As once more for that same line for 7 = 0,1
Lv

or 0,2 e.g., we can find Q by means of (11), i.e.
(l—a) «a

9
2—awx

gqtairs—ak

supposing that we may put «=a, by first approximation. We find
then by subtraction of the above found value of g + a,4 the value
(l—a,) «

2—a,u

of aa , so that of 4 separately. Also q is then separately

known.

Appendix. The approximate equations (57) and (5”) might also
have been derived from:

dT OPE Cur
eli — dh & = 2 x? =—S a ry U Sa wien ne,
Cai é \4+ ine (= )4+ Ie (= )+

jy

With (57) we find then easily from the value (8) for =) that
av

( 714 )

dT Rey: TiN: ae 1 3 dT ane! dT Tigh At
day) am " \ dz? ae ap r 2 du ae dau ae Q, 2a, *

(ET 3 (dT oat
and ( = —]. In this it is noteworthy, that on account of

3 : ,
da te dt).

da). dQ
Q=q+] a —x(l—ez) — |4 also 10.
da dx },

ele aS pod
With (5°) we shall find from (11): (: l= —~ = ( =

dx Ay. dx?
se MEE (IN a Oy
= YS S| Jal@Re), 1a! = |} SU,
! Q, dx /, da /,
Chemistry. — “On ocimene and myrcene, a contribution to the

knowledge of the aliphatic terpenes.’ By Dr. C. J. ENkLaar.
(Communicated by Prof. P. van Rompuren).

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

The aliphatic terpene group, discovered in 1890 by SemmiEr’), is
characterised by the absence of closed rings; the terpenes of this
group possess, therefore, three double Iinks in an open chain. The
first aliphatic terpene described was anhydro-geraniol, which SEMMLER
prepared *) from the aliphatic terpene alcohol geraniol by heating
the same with potassium hydrosulphate. This terpene has not yet
been obtained pure, and has been but little investigated. A naturally
occurring terpene of this group was found by Powrr and Kiser *)
in oil of Bay (the ethereal oil of Myrcia acris D. C.); it was ealled
by them myreene. The sp. gr. (0,801 at 15°) was much lower than
that of the eyelic terpenes (0,840—O0,860), the molecular refraction
and the addition of bromine pointed to the presence of three double
links. With permanganate myrcene yielded some succinic acid, on
treatment with mixed sulphuric and glacial acetic acid an alcohol
was obtained, having the odour of oil of bergamotte, which was
taken for linalobl on account of its oxidation to citral. Myrcene oxi-
dised in contact with the air, and polymerised even at the ordinary
temperature. In his studies on caoutchoue Harries *) has for some
time considered these polymerisation products as closely allied or iden-

1) Ber. 28, 2965 (1890) and 24, 201 and 682 (1891).

*) Ber. 24, 682 (1891).

3) Pharm. Rundschau (New-York) 1895, no, 18.

*) Ber. 36, 3256 (1902).

( 715 )

tical with ecaoutchoue, on account of similar nitrites being formed
from them.
On treatment with sodium and alcohol, Semuiur *) obtained from
myrcene a dihydromyrcene, which induced him to suppose the presence
in myrcene of a conjugate system of doubie links. In connection
with the formation of succinic acid from myrcene, and of laevulinic
acid from dihydromyrcene, he constructed the following formulae :

for myrcene: for dihydromyrcene :
C C—C y »—C
< Yo EN a: Var N
CAG C=C C=C C—C
i 3 3 Pes 6 va 2 3 Wz 6
C C=C C C—C
8 7 8 7

For a long time, myreene remained the only known naturally
occurring alliphatic terpene. CHApMAN’) also found it in the oil of hops,
Power and Kuieser*) rendered its presence probable in the ethereal
oil from the leaves of the Sassafras tree, Barpier *) in the ethereal
oil of Lippia citriodora. At Buitenzorg, however, vAN Rompurcu °)
found that the leaves of a variety (sub-variety) of Ocimum Basili-
cum. contained an ethereal oil, in which occurred a still unknown
aliphatic terpene which he called ocimene. Apart from its odour it
was distinguished from myrcene by a greater index of refraction, a
more powerful absorption of oxygen, and by the peculiarity of passing
into an isomer at its boiling point at the ordinary pressure. The
molecular refraction of ocimene gave a considerably higher value
than the calculated value for C,,H,, so that the proof of the aliphatic
character of ocimene was, as yet, wanting.

The further investigation of ocimene was yielded to me by Prof,
van RompureH; the ocimene formed the subject of my dissertation’).
My research, which at first concerned only ocimene, had soon to
be extended to the aliphatic terpenes in general, particularly to
myrcene and the isomer formed from ocimene.

1) Ber. 34, 3122 (1901).

2) Journ. of the Chem. Soc. Trans. 67, 54 (1895) and 83, 505 (1903).

3) Pharm. Review 1896, Chem. Centr. Bl]. 1897, II, 42.

4) Bull. Soc. Chim. [3], 25, 691 (1901).

5) Verslagen van ’sLands Plantentuin te Buitenzorg, 1899, p. 48, and These
Proce. Ill. p. 454.

6) CG. J. Enxtaar, ,Over ocimeen en myrceen, eene bijdrage tot de kennis van
de aliphatische terpenen”. Acad. proefschrift, Utrecht, Dec. 1905. A more extended
communication will appear in the Rec. des Tray. d. chim. des Pays-Bas.

50

Proceedings Royal Acad. Amsterdam. Vol. VIII.

( 716 )

The three aliphatic terpenes investigated by me (the isomer of
ocimene proved later on to be also an aliphatic terpene) form together
a closely related, natural group which, according to my struc-
tural formulae, embraces the dehydratation products of the terpene
alcohol, linalo6l, which occurs so widely in the vegetable kingdom.
The ocimene obtained by distillation in vacuo over metallic sodium
is an optically inactive liquid of an agreeable ester-like odour, which
possesses the following constants :

Sp: gr.,, nd.,, b.p. at 30 mM. b.p. at ordinary pressure.

0,8031 1,4857 Sik 172°,5.

Whilst it boils constantly at diminished pressure, the boiling point
at the ordinary pressure does not remain constant for a minute, and
after 25 minutes the original product is found to be nearly wholly
converted into an isomer, which boils 17° higher than ocimene. By
fractionation in vacuo it may be obtained pure, and it then possesses
the following constants *) :

Spy pie nd. b.p. at 12 mM. b.p. at 750 mM.
0,8182 1,5296 81° 188°.

The myrcene was partly prepared by myself from the oil of Bay,
for another part I used a myrcene, most willingly held at my
disposal by the firm of Scuimmen and Co. In accordance with others,
I found for the myrcene the following constants :

Sp. gt; nd.,, b.p. at 760 mM.
0,8013 1,4700 166°

At the ordinary. temperature these terpenes are stable, except
myrcene, which then undergoes a slow polymerisation; the isomer
of ocimene is pretty soon altered in strong daylight. The chemical
reagents which can be absorbed by unsaturated compounds are
readily taken up by these terpenes. Up to the present, however it
has not been possible to isolate well-defined additive products, with
the exception of the compound formed from myrcene and hydrogen.
Crystallised derivatives of these terpenes are as yet quite unknown,
which very much impedes their detection in ethereal oils and their
investigation. If the chemistry of the aliphatic terpenes is not to
experience the same fate as that of the cyclic terpenes before WALLAcH’s
researches, it now becomes all important, to devise a further charac-
teristic, based, if possible, on crystallised derivatives. The following

1) This change of ocimene into an isomer has been noticed and already communi-
cated by van Rompuren (l.c.). The constants I found agree with those previonsyl
observed by im. I followed his directions for the preparation of ocimene from
the ethereal oil.

(U7)

contains a description of some derivatives obtained from the terpenes
ocimene and myrcene; from ocimene and myrcene a crystallised
dihydrotetrabromide (m. p. 88°), from ocimene a* phenylurethane
(m. p. 72°) from the new terpene alcohol ocimenol, obtained from
ocimene, from myrcene a phenylurethane (m. p. 68°) from the corre-
sponding terpene alcohol, myrcenol, which had not yet been recognised
as a new product.

As will be shown, the preparation of these crystallised derivatives
enabled me to confirm with certainty certain facts already surmised
and to find a few new ones of great importance for the further research.

As regards the additive experiments, it may be mentioned, that in
the bromination the final point is difficult to observe on account of
colorations; in the case of both ocimene and its isomer the quantity
absorbed seemed to point to the presence of three double links.
Of some more importance is the behaviour of these substances
towards oxygen. With some other unsaturated hydrocarbons they
share the property of absorbing oxygen. The isomer of ocimene does
this in'a very striking manner. When a glass plate is moistened
with this liquid, it is found to be changed after half an hour into
a film or resinous crust. Ocimene does this also very strongly,
myreene a little less. The final point of the absorption seems to be
reached after the fixation of two atoms of oxygen’).

I have specially investigated the behaviour of ocimene towards
permanganate. There is no question here of the isolation of glycols
such as Wacner has obtained from many unsaturated substances.
Even should a glycoi be formed with a same number of carbon
atoms as ocimene, this is very rapidly oxidised by the permanganate.
As oxidation products are formed in large quantities carbonic acid,
acetic acid, oxalic acid, acetone and a small portion of higher fatty
acids, also traces of some non-volatile acids, among which is perhaps
pyruvie acid. In a very weak solution of acetone, the oxidation
takes place more moderately. After the absorption of 9 atoms of
oxygen the discoloration of the permanganate ceases; according to
my determinations 25°/, of the ocimene is then oxidised to carbon
dioxide. The greater part of the oxidation products is, however,
volatile with the acetone; a very small quantity of a sirupy glycol
gave on oxidation with hydrogenperoxide a little carbonie acid,
acetone, acetic acid and in addition a fair amount of a non-volatile
acid, which is probably malonic acid. It is remarkable, however,

1) In connection with the researches of Eneter on the oxygen absorption of

the fulvenes and of Wattacu on that of phellandrene, I hope to further investigate
this matter.

50*

(718 )

that in the oxidation in acetone solution no acids higher than acetic
acid were formed and besides not a trace of oxalic acid with hydrogen-
peroxide. I am therefore of opinion that those substances owe their
origin to migration of double links during the oxidation. All
this causes that, I, for one, consider the structural formulae based
on oxidation experiments with such very changeable substances as
very untrustworthy. One should also be very careful in drawing
conclusions from the isolation of smail quantities of the more typical
decomposition products, as these may have been yielded by impurities.

Notwithstanding the beautiful researches of Timmann and SEMMLER *)
on geranial, citral, ete, the structural formulae of none of the mem-
bers of the aliphatic terpene group seems to have been sufficiently
established as was only recently shown by Harrigs oxidations with
ozone *).

Meanwhile I had tried, whether ocimene could be hydrogenised
like myrcene by means of sodium and alcohol. This proved to be
the case. Ocimene, therefore, also contains a conjugate system of
double links. The hydro-product, which I obtained, had the eompo-
sition ©C,,H,,*) (1 will eall it in future dihydro-ocimene) ; it is a
very mobile liquid of an agreeable odour. For its constants I found
the following values :

sp. @V..; nd.,, b.p. at 761 mm.
0,7792 1,4507 166°—168°

whilst SEMMLER *) states for dihydromyrcene :

sp. gr. nd. b.p.
0,7802 1,4501 171°,5—172°,5.

The temperature, at which the specific gravity was determined,
and the barometric pressure at the boiling point are not stated ; at
770 mM., the boiling point of dihydro-ocimene is, however, but
little higher. Owing to this difference of 6° in the boiling point of

1) Ber. 28, 2126 (1895).

2) Ber. 36, 1933, 2998, 3001, 3658; and 387, 612, 839, also Harries und
Scuauwecker, Ber. 34, 2987 (1901) and Harries, Lehrbuch der Org. Chem. by
V. Meyer und P. Jacoxson, II, 754. The above-standing was written before the
latest publication of Harries on this subject appeared (Lieb. Ann. Jan. 1906).

3) In the combustion of these substances with copper oxide in an open tube
the carbon is often found a good deal too low, but in the closed tube with lead
chromate the exact values are always obtained. On the strength of his analyses,
Cuapman also concluded at first to the presence in oil of hops of a hydrocarbon
Cy Hig which by further investigation proved to be myrcene.

4) 1. -¢.

(719)

the said hydrocarbons, these were not considered as identical in the
provisional communication. A second point of difference was the
obtainment of a crystalline bromide from dihydro-ocimene. When,
afterwards, I repeated SremMuLer’s experiments, I found the boiling
point of dihydromyrcene to be the same as that of dihydro-ocimene,
whilst the other constants (as far as could be ascertained) agreed
with those of Semmuer; I found the following values :

Sp. eT.,, nds, b.p. at 761 mm.
0,7852 14,4514 166°—168°

Like Semmiyr, I found that myrcene and its hydro-product have
the same boiling point so that, probably, SemMMLEr’s statement is
based on a mistake; for other investigators also state 166° as being
the boiling point of myrcene.

The now probable identity of the hydro-products became a certainty
by the bromination of dihydromyrcene. As stated, dihydro-ocimene
had given me on bromination a crystalline bromide; from the oil
obtained at tirst, it erystallises to the extent of 12—14°/,. After
repeated crystallisations from methyl alcohol it forms snow-white
erystals which melt, sharply, at 88°. Analysis and determination of
molecular weight pointed to the composition C,,H,,Br,. In most of
the organic solvents, this bromide is readily soluble, but in methyl]
alcohol only to the extent of 1,2°/,; on boiling with sodium hydroxide
and also with silver oxide and water an oil smelling of peppermint
is obtained. From *dihydromyrcene I now obtained the same bromide.
The oil obtained by SemmiEr soon solidifies when, after being purified,
it is put away to crystallise in a cool place; by applying a little
artifice I sueceeded in instantly inducing the crystallisation. The
identity was proved by the fact that this bromide like all its mixtures
with dihydro-ocimenetetrabromide, melted, sharply, at 88° and also
that the solubility of dihydro-ocimenebromide was practically not
affected by addition of this substance.

This now completely proves:

1. that both ocimene and myrcene are aliphatic terpenes.

2. that the dihydroproducts of these terpenes are identical.

From this it follows and this is probably of more importance
still — that we may now deduce the structural formulae of ocimene
and myrcene from the obtained data.

As regards myrcene, owing to its connection with citral and
dipentene*), it was already fairly certain that, like all aliphatic

1) Power and Keser, |. c.

( 720 )

terpenederivatives as yet known, it is a derivative of dimethyl-
2-6 octane :

H, H, H,

C C—C

INA HL! OR
C256 C= CH,

C H,C — CH,

Jal 8 7

On account of the proved identity of the bromides, ocimene should
also possess this carbon skeleton. Now when we accept the correct-
ness of the hydrogenation principle of conjugate systems *):

BS Me A AS ra
C=C—C=C +2H>H—C—C=C—C—H

fi ON a aN Fea a Se 3g:
it is, in the earbon branch formation of dimethyl 2— 6-octane, only
then possible for different triénes to give identical diénes, when these
triénes possess the following conjugate systems :

- Rae era a oO
Se SiN ~ A
C=C C/—C and Jez GC oe C=C
wae 3 < Cae San y ee Ae ¥. 7
C C=C C cate
I Il

and when the third double link occupies the same position in both
formulae, and is not conjugate with the other double links. For
this third double link 1 or 2 is the only possible position; on
account of my experiences as to the oxidation of ocimene, I should
be inclined to accept the position 2, although the position 1 has still
quite as much right of existence’). Perhaps, as SeMMLer believes,
myrcene may contain both forms (ortho and pseudo form).

Dihydro-ocimene and dihydromyrcene then assume the formula of
dimethyl 2—6 octadiéne 2—6:

1) For exact details and the literature of this hydrogenation principle, I must
refer to my dissertation p. 26. I wish only to point out particularly, that my rule is not
based on the theory of Turere, but has been deduced in a purely empirical manner.
1, therefore, make a distinction between the addition of hydrogen and that of other
substances which may prove more complicated.

2) Admitting for this third double link the position I, yet another couple of
formulae seems possible; on account of other facts the latter must however be
rejected. ;

C C—C
aX.
C=C —C€
a 3 U6
C C—C
8 7

which has already been agreed to by SemMLer on other grounds.
Which of the above formulae, however, belongs to ocimene and
which to myreene? A choice is only possible on the strength of other
data. As has been stated, Srmmier had assigned to myrcene for-
mula II on account of the formation of succinic acid in the oxida-
tion. Independently of him and these considerations, I had constructed
for ocimene formula I as the result of my oxidation experiments,
but without attaching any value to this. A closer consideration of
the above formulae, coupled with the peculiar behaviour of ocimene
on heating, as observed by van Rompurcu, led me to the discovery
of a fact, which rendered a choice possible with great certainty.

In one respect formula I differs characteristically from formula II
namely by the presence of the double link 5, which forms an
asymmetric system with the carbon atoms combined thereby and the
groups attached thereto, and so gives an opportunity for the existence
of a geometric isomerism. The transformation of ocimene into its
isomer led me to think that these two substances might be geome-
trically (stereo-) isomeric. Geometrical isomers are often readily con-
verted into each other on warming ; for instance, WisLIceNus noticed the
transformation of the one bromobutylene into the other on distillation.
The hypothesis advanced by me was easy to verify for on hydro-
genation the same dihydro-ocimene ought to be formed from the
isomer as from the ocimene itself. This proved indeed to be the case.
The physical constants of these materials were indeed identical as
is shown from the following table :

sp. gr.., nd.,, b.p. at 761 mM.
dihydro-ocimene 0,7792 1,4507 166°—168°
dihydro-isomere 0,7793 1,4516 167°—168°

whilst the original products exhibit strong differences as is shown
from the subjoined *):

Sp. 2a, nd. b.p. at 760 mM.
ocimene 0,8031 1,4857 172°,5
isomer 0,8133 1,5447 188°

1) The constants of the isomer have been determined with the aid of a purer
preparation than those previously communicated. On heating ocimene some by-
products seem to be formed.

( 722)

With this I consider the identity of these hydro-products and the
geometrical isomerism of the terpenes as proved. The isomer of
ocimene I will call in future allo-ocimene. It is remarkable that
allo-ocimene deviates 6,31 from the theory of Brin; its index of
refraction is also greater than that of the hydrocarbon and it has
also a strong dispersion power. This, as Brinn thinks *), is perhaps
connected with the presence of a conjugate system of double links.
Provisionally, one should be careful in drawing conclusions as other
substances also exhibit such differences. Allo-ocimene is, however,
in this respect a unicum in organic chemistry. Dihydro-ocimene on
the other hand exhibits the correct refraction. The deduced geo-
metrically isomerism was also very much supported by the behaviour
of the isomer towards a mixture of sulphuric and glacial acetic acid.
Whilst ocimene remains for the greater part unchanged and is, to
a small extent, converted into an alcohol, allo-ocimene is for the
greater part converted into a polymerisation product, whilst there
is left a small quantity of terpene, which proved to be nothing else
but ocimene. This typical difference between the two ocimenes is
perhaps connected with the particular tension which the ethylene
link may attain here. Possibly, at the moment this ethylene link
opens, the two connected atoms of three molecules combine to form
a cycle of six atoms; a substituted hexa-hydrobenzene derivative
would then be formed; the polymerisation product would be this
triterpene.

The regeneration of ocimene from allo-ocimene under the influence
of dilute acids renders the analogy complete with the isomerism of
fumaric and maleinic acid. After what has been said, it is no longer
doubtful, that ocimene, which possesses the double link 5, is repre-
sented by formula I, whilst myrcene is represented by formula II,
which has now been deduced independently of the results of the
oxidation. But few instances of geometrical isomerism have been
noticed with hydrocarbons and this is the first known in the terpene
series. It seems to me not impossible that the absence of the cyclic
link has given nature the opportunity of forming a labile geometrical
isomer; it is remarkable, however, that this has taken place without
any admixture of allo-ocimene. I hesitate to pronounce just now an
opinion as to the nature of that geometrical isomerism with ocimene
and allo-ocimene; the following projection formulae seem to me the
most probable.

2) Ber. 38, 761 (1905).

( 723 )

H I am still engaged with this
ote aie geometrical isomerism and the
C Oe Gs other substances described. I
AS om Be \ soon hope to make a further
ocimene: C=C. SOS © communication about — the
Ve “Y aleohols formed from. these

C C=—€ terpenes.
©@ Of late, after this research
Sy had already been _ partly
C finished, Saspatrer and SENDE-
| RENS have made some valu-
C able additions to our methods
\ of research of the unsaturated
allo-ocimene : C compounds. I am engaged in
| applying the same to the
H-..— € —.. aliphatic terpene group and to
NS the sesquiterpenes. Dihydro-
Qua eat ocimene, which eannot be
Sh further hydrogenised by so-
C6 dium and alcohol, eagerly
absorbs hydrogen at 180°
under the influence of reduced nickel; a nearly odourless liquid is
formed which boils at a considerably lower temperature and contains
only traces of the original product. It consists, probably, of dimethy|-
2.6.octane, the as yet unknown foundation of the aliphatic terpene
group. The aliphatie terpene-alcohol, geraniol, also reacts with nickel
and hydrogen; the reaction product is a liquid, possessing a particular
odour; it contains, besides some water, a hydrocarbon, which probably
is identical with the hydrocarbon, obtained from dihydroacimene
and a’ substance of a higher boiling point, which I suppose to be the

saturated aleohol, corresponding with geraniol.

Chemistry. — “On some aliphatic terpene alcohols.” By Dr. C. J.
ENKLAAR. (Communicated by Prof. P. van Rompurau).

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

According to the process of Brrtram and WaA.baAuM ') terpene
alcohols may be obtained from terpenes by digesting their solution
in glacial acetic acid for some hours with dilute sulphuric acid at
50°—60°. The aliphatic terpene ocimene, discovered by van RomBuRGH

1) D. R. Pat. No. 80711, Journ. f. Prakt. Chem. 49. 1. Also compare Watuacu

and Waker, Ann. 271, 285, and Power and Kieper, Pharm. Rundschau (N.-York)
1895, No. 3.

(794 )

and investigated by myself’), was treated by me in this way ’*).
The greater half of the ocimene operated upon was recovered
unaltered while a small portion underwent polymerisation. At the
same time an alcohol was formed, the quantity of which was about
10°/, of the ocimene used. This alcohol was an agreeably smelling
liquid, which gave the following constants :

Spa ete nd,, B.p. at 10 mm. Mol. Refraction (M.R.)
0.901 1.4900 HUE 49.22
(calculated for C,,H,,O|, is: MR = 48.86)

The analysis had given the composition C,,H,,0.

This alcohol, probably an aliphatic terpene alcohol is, therefore,
formed by the addition of the elements of water to ocimene. In
properties it does not correspond with any of the already known
aliphatic terpene alcohols, as is shown by the following table:

Sp. er... nd B.p. at 10 mm.
geraniol : 0,882 1.477 116°
nerol *) : 0,8814 112°
myrcenol (BARBIER) : 0,901 1.477 99°
linalodl : 0,870 1,464 86°

On account of its formation from ocimene, I eall this new alcohol
ocimenol. The investigation of this ocimenol is still of a provisional
character.

The beautifully crystallised phenylurethane, which I could prepare
from it in good yield, renders it possible to characterise and readily
investigate the alcohol. This urethane, when recrystallised from dilute
alcohol, forms white needle-shaped crystals, which melt without
decomposition at 72°, whilst according to the analysis, it has the
composition C,, H,,O,N. I am still oceupied with the regeneration
of ocimenol from its urethane and the closer investigation of these
substances; however from the fair yield of this urethane, and the
absence of oily by-products, it seems that the product obtained from
ocimene is mainly a simple alcohol.

For me, the study of this alcohol was of particular importance
as I wanted to compare ocimene in this respect with myrcene.
Several investigators have been already oecupied with the alcohol,

1) Compare my previous paper and my dissertation.

2) | worked according to the directions of Power and Kuirser. 100 parts of
terpene were heated with 250 parts of glacial acetic acid and 10 parts of 50°/,
sulphuric acid for three hours at 40°.

8) Nerol is distinguished from geraniol by a more delicate odour of roses, by
not combining with calcium chloride and by yielding a diphenylurethane melting at 52°.

( 725 )

which is formed from myreene in the mannev indicated; their state-
ments, however, are often diametrically opposed.

Power and Kiser’), who first prepared it, took it to be linalodél
on account of its odour and the formation of citral on oxidation
with chromic acid. Barsrmr’) declared it to be a new alcohol; on
oxidation, he obtained no citral but another as yet unknown aldehyde.
From the results of the oxidations he deduced for this aleohol, which
he named myrcenol, a structural formula, which had been given
already by Tirmann and Semmuer to linalodl. In a further research
on linalodl, he gave as his opinion’) that it was not a simple
aleohol, but a mixture, and also that its main constituent was not
optically active, a reason why he rejected the formula of T. and 8.
SEMMLER*), however, looked upon myrcenol as a mixture already
partly converted into cyclic products, and upheld his linaloél formula
against Barsier’s objections.

I prepared the myrcenol according to the directions of Power and
Kinser. The greater part of the myrcene was recovered unaltered
(6°/,), a small portion polymerised whilst the alcohol had formed to
the amount of about 20°/,. For this alcohol distinguished from linalodl
also by its intense, agreeable odour, I obtained the constants attributed
to it by Barsigr, who, however, had a much langer quantity of the
aleohol at his disposal :

sp. g0.,, nd,, Bp. at 10 mM. Mol. Refr.
myrcenol (/7): 0,9032 1.4806 97—99° 48,44
rr (B): 0,9012 1.47787 99° 48,34
MR, calculated for C,,H,,O0)> = 48,16

My analyses also pointed to the composition C,, H,,0. I do not
consider this alcohol to be perfectly pure as it has not got a quite
constant boiling point; it seems still to contain a more volatile fraction.

The closer investigation of this substance has, as stated, led to
differences of opinion. It seems to me that these have been caused
by the different methods used. The formation of citral in the oxidation
in acid solution is no reliable test for the presence of linalodl as it
may be yielded also by other alcohols. Barsiser showed, however, that
on oxidation of myrcenol with chromic acid an aldehyde was formed,
having the same formula as citral, but not identical with the same.
He regenerated it, for instance, from its oxime, and obtained a

Dlaics

2) Bull. Soc. Chem. [3], 25, 687 (1901).
5) Bull. Soc. Chem. [3]. 25, 828 (1901).
4) Ber. 34, 3122 (1901).

( 726 )

semicarbazone melting at 197°, whilst citralsemicarbazone melts
at 135°. Here we have a difference in the method of research.
Power and Kuper tested for citral by converting it into citryl-

naphtocinchonie acid; in this way a possibly formed ketone — I
presume myrcenol is a secondary alcohol — must have escaped their

notice, whilst a little citral thus detected may be simply a by-product.
On the other hand, semicarbazone, made use of by Barpigr, is
according to others unfit for testing for citral. Barbier may have
obtained the semicarbazone from the eventually formed ketone, the
main product, whilst a little admixed citral may have given the
aldehyde reactions. Moreover Barsrmr’s oxidations with permanganate
in aqueous solutions cannot be taken as decisive for the differen-
tiation of myrcenol and linalod6l *).

Instead of investigating the oxidation products of myrcenol, I have
prepared from the alcohol itself a crystallised derivative, in the form
of a phenyl-urethane, melting at 68°. The analysis again pointed to
the composition C,, H,, O,N. This urethane has been prepared in the
same manner as Watsaum and Hiruie*) prepared the phenyl-urethane
from linalodl; the latter melts at 65°. By means of the phenyl-urethane
obtained from myrcenol, it could be decided very readily and distinetly,
that the alcohols, myrcenol and linalodl, were totally different. The
mixture of racemic linalodlurethane and myrcenol-urethane melted
at 60°—62°; the depression of the melting point sufficiently proves
the non-identity. The alcohol, which is characterised by the phenyl-
urethane melting at 68°, is also the main product of crude myrcenol.
I obtained from this a yield of nearly 60 pCt. of crystallised urethane ;
besides this alcohol, a little linalodl may possibly be contained in
the myreenol (the hydration product of myrcene); the formation of
some oily urethane in presence of the crystallised substance might
even point to this. The facts mentioned render it possible, however,
to decide the matter. By regenerating myrcenol from its urethane,
the properties of pure myrcenol may be ascertained. I am still engaged
with this. Of this alcohol, myrcenol, it may be stated that it is a
typical derivative of myrceene; its constants differ from those of
ocimenol, in the same manner as those of myrcene do from those
of ocimene; the tendency towards polymerisation of myrcenol is
still larger than that of myrcene.

For ocimenol and myrcenol I devised provisional structural formulae’),
based on their formation from the terpenes ocimene and myrcene.

1) Compare previous communication.

2) Journ. f. prakt. Chem. 67, 323 (1903).
8) Dissertation, p. 73.

(720)

I have not been able to obtain the above racemic urethane of
linalobl by mixing d- and /linalodl and preparing the urethane from
this racemic linalodl ; nothing but an oil was formed, which could
not be brought to crystallise. Still, from each oil separately (d-cori-
androl and /linaloodl, the latter obtained from Scummurn & Co.) I
obtained the arethanes at once crystalline. In order to obtain racemic
urethane, I was obliged to mix these urethanes of d- and /linaloél
in the proportion of their optical activity. The latter, however, had
not been determined; in fact it was doubtful whether they were
optically active at all. Warsaum and Hurnie, who desired to prove
in this manner the identity of linalodl derived from different ethereal
oils, have overlooked the fact, that alcohols of such varying optical
activity as those found with linaloél (from 1° to 35°) could not yield
the same phenyl-urethane.

Racemie urethane has generally quite another melting point than
the pure optically active substance. I was, therefore, obliged to fill
this void in their research. I found that the yield of crystallised
urethane, which only amounts to 15°/,, when one works according to
their directions (time of reaction one week), may be increased to
85°/, increase of the time to three months. The urethanes formed,
which all melt at 65° are optically active in proportion with the
optical activity of the alcohols started from. They consist of mixtures
of racemic urethane (probably a racemic compound) with the opti-
cally active component, which in a pure condition shows a rotation
of 23° 27’ in a 200 mM. tube and has the m.p. 66°. The rotation
of pure optically active linalool under the same conditions may also
be calculated from this; it then becomes 35° 27’, whereas the highest
observed rotation of the natural substance amounts to 35° 14’:
This alcohol appears, therefore, to be very strongly subject to race-
misation, even in nature. By the facts stated it has, therefore, been
proved that linalodl consists of a simple optically active terpene
alcohol; the incorrectness of Barsrer’s formula. for linaloél and
myrcenol has been demonstrated, whilst the linalool formula of
TieMANN and SeEMMLER has received support.

( 728 )

Physics. — “On the propagation of light in a biawial crystal around
a centre of vibration.” By H. B. A. BockwinkeL, (Commu-
nicated by Prof. H. A. Lorentz).

(Communicated in the Meeting of January 1906).

In the electromagnetic theory of light, it is of interest to determine
the electromagnetic field in a crystal due to an action, taking place
in a certain centre QO. In order to fix the ideas, we shall assume,
that in an element of space t at the point O there are certain
periodic electromotive forces (i. M. F.). There will then be a radiation
of energy from © in every direction, the amount of which will
depend on this direction with respect to that of the A IL F.
and to those of the axes of electric symmetry. Our object is to
investigate this dependence, at least for points at a great distance
from O. We might for this purpose use the results of Grinwap‘);
this physicist however takes the equations in the form they assume
for a rigid elastic body and does not operate with an 2. M. F. as
mentioned above; we shall therefore treat the problem independently.
Our method will consist in reducing the question to one of plane
waves, by using a formula, proved: by Prof. Lorentz. In this formula
a continuous function of the coordinates is represented by an integral
over the solid angles of all cones having their vertices in O and
filling the whole space. If the #. M. F. is €e then

078
e=— (Spa [orm

where dn is the element of a line of arbitrary direction within the
cone dw and YW a vector given by

the integral being taken over the plane, passing through the point
considered, perpendicularly to x. Hence, 28 depends on the coordi-
nates, but in such a way as to be constant in every plane perpen-
dicular to n. By (4) the original #. M. F. has now been decom-
posed into a great number of infinitely small vectors, the effect of
which can easily be calculated, each of them being constant in
planes of a certain direction. Thus we determine the field, produced
by each of the elements of the integral (1) and then compose all
the fields obtained in this way into one resulting field, which,

1) J. Grinwatp. Uber die Ausbreitung der Wellenbewegungen in optisch zwei-
achsigen elastischen Medien, Bortzmann Festschrift (1904), p. 518.

( 729 )

according to the principle of superposition, will really be the one
produced by the whole EK. M. F. Each of the separate very small
fields will consist in a propagation of plane waves having the same
direction as the planes in which the corresponding element of the
K. M. F. is constant. The problem will therefore indeed be reduced
to one of plane waves.

§ 2. In order to find the small field, corresponding to a cone of
definite direction, we shall take a system of coordinates O.X', OY’,
OZ', the axis OZ' coinciding with the axis of the chosen cone and
OX', OY" respectively with the two directions of the dielectric
displacement, belonging to plane waves, normal to OZ'. The wave
that has its dielectric displacement along OX' will be called “the
first wave’; the other “the second wave’.

Again we take a system of coordinates OX, OY, OZ, the axes
of which coincide with the axes of electric symmetry. Denoting the

components of the electric force along the first axes by €y , €y , Ey
Qn

ri
: ane ; T
and supposing all quantities to contain the factor e we have to
satisfy the following equations

; 0 4x
Abr g (lie Fea Ee +E) +Ey $Ey) tales +E |

2

0 4;
AE, -— (div. €)=-- aa é,,(Cz +67) +e,.(Ey +) )+e,,(E2 +: ’) (3)
Oy

AG s= Odin GT | al Ce + +e Ey +EP) +eulE +65)

It will not give rise to any misunderstanding that we have denoted
dw 0°28
8x? Oz’?

The quantities #, occurring in these formulae, have particular
properties, because they relate to special directions. These properties
will show themselves in the following development. Since, according
to the preceding considerations, @@ depends only upon z', we shall
find for €-a solution, likewise containing only 2’. By this hypothesis
the equations (8) become

pay

here by €¢ the expression —

oe, 4a? e ‘ e e

eo =~ als (Cy + Ex") + 8, (fy + Cy )+ &,3(€2 + |

07&,, 4x’ 4

Wet ap ene + Ez) + b1, (Ey + Ey) + e13(E2 + EF )| @

0 =, (Ev + Gr) + &, Ey + Gy) +e (Er + &) /

( 730 )

§ 3. The last equation of (4) shows, that there is no dielectric
displacement in the 2’-direction. Further it is evident from these

Cc

equations, that €2 has no share in the disturbance of the state of the

acther at a distant point. Indeed, €; and ©, being zero, the equations
are satisfied by the solution

c= 05 €, = 0, Ev= — &.

At the distant point €% is zero, therefore ©. is so likewise. Electro-
motive forces acting within a layer bounded by two parallel planes
and directed perpendicularly to these planes, do not therefore
produce any disturbance of equilibrium at a distant point.

We eliminate ©. between the first and the third and between the
second and the third equation.

This gives

0°Ey 4x? e
ee alk Ge ~ te eee
PEL oP | (e.— =“) aie ube («. =) (e, +€, |

022 ey 4x? £,.8. . a pe)
— — r= SURE Sy’ ey = 38 &, om .
dz’? Cale (+: E55 ) a ; ) a («. =!) 4 Sy }

According to what has already been said, these equations, if no
E. M. F. are acting, must have one solution in which €, is zero,
and another in which ©, vanishes. This would follow from the
equations themselves, if we knew the above mentioned properties
of the quantities ¢, occurring in them. Conversely, we shall be able
to deduce these properties from the knowledge that the two solutions
must satisfy the equations. Indeed these solutions can only hold if

pice Fi 3&s5 = 0

and 2 e 2 oe

where V, and V, are the velocities of the plane waves in the two
cases. By this the eae take the form

are, be oO
See aes (Gy + €2'), = _ ey + @) - ©)
TV2 T Va

whereas the {third equation of (4) gives €» when €, and €, are

known. We see from (5) that €, depends only on @,, and G,/ only

on &,, further that both equations have the same form. We can

(73i)

therefore confine ourselves to considering only the first; in doing
so we shall write V’ instead of V,. We shall have to remember
however that after having found the result that is due to the X’-com-
ponents of the E. M. F. we have still to add to this a second amount
given by the Y’-component; this amount can be written down at
once by analogy with the first.

§ 4. The general solution of the equation

07E, 4n? 2
Ce ee ie
is given by
Qr2z’ = 2Qrz' nz 2? Qrz
in Ty ip in — ‘tr (ue ' FV
[= Ty" Dies de Ty? 7 fee ated nen (6)
92 n

The lower limit of these integrals:is arbitrary, so that, as could
be expected, two arbitrary constants occur in the solution. It is
easily understood, that in the final result there will likewise be a
certain indefiniteness. Indeed both a propagation towards O and one from
O will be contained in it. It is sufficient for our present purpose to
consider only the first solution and in order to leave aside the second
we have to give completely definite values to the constants, as will
appear in the following manner. We consider the two planes perpen-
dicular to OZ’, tangent to the boundary surface of the space +; let
these planes be determined by the equations

2 —— ie aNd ze) es

Then, since €¢ stands for
1 pee
— — — dw,
8207 Oz!
it will differ from zero between the planes and will be zero in the
space outside them. The first integral of (6) must vanish for
EN lbp

and the second for

This is only possible, if

I< — fh, and
Swe iin
For the rest g, and g, may have any value satisfying these une-
qualities; it is evident that the result of the integrations will always

be the same, if we take into account what has been said about the
i
51

Ia

Proceedings Royal Acad. Amsterdam. Vol. VIII.

values of €¢. We shall therefore put g,= —/h, and g,=h,, so that
.2rz! " _ 2x2! aire cE 2Qrz!
idw ~— ‘py 2 ony idw ‘py (AX, — yy
€,/ = ——_e cia Mea e at dz|' — ——_e ne e PY Ge'(7)
8a TV 0z'? 8a TV 02"
ay he

§ 5. In effecting these integrations we have to distinguish whether
or no the point P, for which we intend to determine the state
of radiation, lies between the two just mentioned tangent planes. First
taking the latter case, the second integral of (7) is zero for positive
values of 2’, whereas in the first case we may take /, instead of
2’ for the upper limit. Integration by parts gives

he One! Qn2! hy he _ Ime!

0728, Ye Ou, ‘ry 2x (08, Rg

Oz? ag a0 | 5
—h, —h, —h,

Now €¢ can only be represented by (1) if it is a continuous
function of the co-ordinates, but we may imagine nevertheless that
at the boundary of the space rt, 3% and 0%8/dz' have arbitrarily
small values. These quantities may therefore be taken zero at the
boundary; as to YW, this has already been done in the considerations
of the preceding paragraph. Hence the first term, given by the
integration by parts, vanishes; the second may again be integrated
by parts, so that finally

hy , 2x2! hy 22m!

Sy “TV ., Ax? Toe

Toss é SS — Ty Sit é dza.
hy —h,

The exponential factor under the sign of integration may be
replaced by 1. Indeed, if a certain length 1, of the same order of
magnitude as the linear dimensions of the space t is very small in
comparison with the wavelength 4 of light, we may omit terms

T se l
containing products of 0 and quantities of the order a Now

Dit = [es do
=!

the integral taken over the portion of a plane 2’ = const. lying
within r. From this we infer

hy
iE dz! ies dt
is

integrated over the volume t. We shall represent this integral by

( 733 )

-€ . =€ . “ y
Gyr, denoting by & a certain mean value of the X’-component of
the E.M.F. within tr. We may now write

hg , 2z2! 5

em, ‘Ty _, An? Eat
an ae ie he RV

=
Similarly

hg , 2x2! é

0728," ‘TV ies An? Cat

02” ¢ 2 Stee T*V2 \

an integral that has to be used for
than — h,.
§ 6. If lastly

negative values of 2’ less

TE h, <i 2" <i h,
the point P lies between the tangent planes and Joth integrals differ
from zero. We find by some transformations

z Qr2' Qr2' Wrz! 2! Qa!
Cee ny Oy. Many | ame em ne A? Be RTA:
—e dz' = ——_e — i—_ We — —— | We Ie!
Oz? Z DAY ay Oke
hy —h,
2" 2n2! Qrz' rz! ‘
PR, “ry, Be -ipy | ee Sty
Se dz' = ——e + {—_%, e =
02” Oz TV
hg
z! Qr2z'
Es — Ty
I, de
Taye |
hy

So that in this case the X’-component of the electric force is
given by

'

Qrz’ z 5 et
idw 4a 4x? = —ipy * 7TV
Ce Se ee Be Tae!
Sal V EV eye
—h,
Qrz' he Qrz!

Since z’ lies between —/, and h,, we may replace the exponential

factors by 1, both before and behind the sign of integration. Then
we find finally

1s, If P lies between the tangent planes
I, in@hr

TE 2ReVE I

( 734 )

2.4, If P lies outside these planes
a. For positive values of 2

+ Ge —i Bie
Ey = eee LY dw
== eee
b. For negative values of 2’
: : 2 a2!
tur Ty
= dw.

2 = oupsiya

The Z’-component of the electric force consists of two parts, one
of which corresponds to €,, the other to €,. Having already omitted
the ’-component, we shall take only the first part, €.,, of the
Z-component and add the second part €.2 to the Y’-component
afterwards. Then by the third equation of (4)

1 0798. & 1 0°,
€.. — — ~— dw = —— 16 — — 2
87? Oz" 55 82? dz!

It appears from this that outside the tangent planes G. and &,-
are connected with each other in the way they always are in the
case of plane waves. We may therefore represent the electric force

ae

by if 9 is the angle between this vector and the corresponding

cos &

dielectrie displacement in a system of plane waves. Finally we have
the following equations for the components of the electric force
along the axes of symmetry

e .2nz \
Oni SUR
as ee hin
2T*V* cosd
E ae . 2rz!
aU COC a ae ra
GS ere TY dos) —*. =e
2T° V* cos}
A «@ 2r2!
MYL gy! T Gen Sra
Ge — io es Ae, TV dw,
2T* V* cos D /

where a, 8 and y are the direction cosines of the electric force with
respect to the axes of symmetry. For negative values of z' the same

'

ot

formulae will apply, provided that z' be replaced by — 2’.

§ 7. In the preceding equations the symbols €:, €, and €. were
used for the (small) electric force, produced by a single element of
the integral (1) in a point P, lying at a given distance 7 from the
origin O. We have seen that the expression for this small electric
force took a different form according to the point P lying or not

( 735 )

lying between the before mentioned tangent planes. Now the direc-
tions for which P lies inside these planes are those excluded by a
certain cone A, which may be defined as the locus of all normals to
another cone, having its vertex in P and tangent to the boundary
of the element of space rt. On the other hand, all directions of wave
normals, for which P lies outside the tangent planes are included
by the cone A. It is clear that this cone will differ infinitely little
from the plane passing through QO perpendicularly to OP.

We may therefore find the total electric force by integrating the
right hand members of the equations (8) with respect to all direc-
tions lying within A’ and then adding to the result the quantity
obtained by integrating the expressions relating to the remaining
directions. In effecting the first integration we must replace 2’ by
—z' for negative values of 2', according to the remark made at the
end of § 6. But we may as well limit the integration to half the
cone A multiplying the result by 2. Again, we may extend this
integration to the plane /’; indeed the right hand members of (8)
contain t as a factor, so that it does not matter, whether or no an
infinitely small solid angle is included in this integration.

It remains to consider the expressions

2 a im Cyt ;
Ee ogsyan on Oeyae
1 028, €,, 0728, é
= = 13 tome 13 ¢¢
EE. = Aan loss ane dw + 5 dw Sey ’
827\ dz &,, 02 En,

which have to be integrated over all directions outside the cone K.

Now, from these expressions we get the components along the
axes of symmetry by multiplying them by finite factors. It is easily
seen that terms already containing the factor t may therefore be
omitted, so that we may write

Ee = spps
g 1/072, £7.10) By
Cp a fice ae ay
a ate ty Bae ( 02? a 05,022 ie

§ 8. We shall resolve this last vector into two other vectors,
the components of the first being

pita é

(Gj rina ; => — = le

and those of the second

Cy — 0, €., ——

U0 E0738," :
Saracen kere Cc me

( 736 )

The first vector has again the same direction as the electric force
in plane waves whose normal coincides with the direction we are
considering; its components along the axes of symmetry are therefore

are,’ _ Wy W,, ;
€. =) oe B COU sg Hee dw.
; 2 T?V? cos 3 : 27° V? cos & 27? V? cos &

Now 2%, , is of the order /? and the integration is to be effected
over a solid angle of the order /. Thus, confining ourselves to
directions in a single plane passing through OP, we may regard as
constants the quantities @,V and cos &, assigning to them the values
they take in the plane P.

We determine an arbitrary direction in the plane passing ouieoess
OP by the angle § which it makes with OP and its azimuth x
with respect to a fixed plane also passing through OP. Then

dw = sin § do dy.

Now we have for the direction considered

W,= fi EF, do

the integral being extended to the portion inside r of a plane G,
passing through P perpendicularly to that direction. If q is the
normal drawn from O towards G, we have

g=recos 6,
|\dq| = r sin § df,
giving
1
dw = — |dq| dy,
7
and
gy eee ase
= Fd vie V? cos nf page

0
Here for each particular value of x, the latter integral is to be
extended to all values that can be given to § or gq. Further

f Be lea = { ial [ee ds = ea

do |dq|

whereas

is the element of volume of an infinitely small cylinder whose upper
and lower base are formed respectively by one of the surface
elements of G and of an infinitely near plane G', the generating
lines of the cylinder being perpendicular to G. It follows from
this that

( 737 )

f Gi do |dq|

is the volume-integral of ©, , taken over the whole volume of rt.

We have already written for this integral G1, denoting by Ge)
certain mean value (§ 5). Hence, the jist part of the components
of the electric force resulting from the integration with respect to
the directions outside the cone K, becomes

2x Phi
1 G,. ‘t 1 GC. t
G; {FES ay, G, = Be. GR? 5 AGS =

273p

V? cos 2T?r, J Vicosd
0 - 0
27
rk yn t
mf Vistas os (9)
0

The second part results from a similar integration of the second
vector

~ 82? \ Oz"? my wee oo

Now it will appear further on, that we can only determine the
exact value of those terms, in. which the denominator contains the
first power of 7. We may therefore confine ourselves to such terms
in the whole course of our calculations. The cone over which we
have to integrate being of the order //r, we may omit terms, which
already contain 7 in the denominator. It will be evident therefore
that instead of

1 (= &,, 078.

O79. 0728.

02"? re 02’?
we may take the values of these quantities, corresponding to that
wave-normal, in the meridian plane passing through OP, which lies
at the same time in the plane /. If dz’ is a line-element of that
wave-normal, we have to consider the integrals

oe, 0218,
—;,— dz and | —— dz!
Oz’? 02’
oe
which evidently are zero, ye being zero at the boundary of tr. It

appears in this way that we need not at all consider the second vector.

§ 9. We now proceed to effect the integration of the right hand

members of the equations (8) so far as is necessary in order to
L,

obtain the terms with —. We shall take the real parts of all expres-
.

( 738 )
sions and represent henceforth by © the whole electric force. Then,

An t
if Ge = b cos os we shall have

G Tat 20 2! s 10
(es, sin =) WwW, . a aes ( )

integrated over all directions on that side of / where 2’ has positive
values. We therefore obtain the resultant luminous vibration in an
arbitrary point P as the sum of small vibrations, belonging to a
great number of systems of plane waves of all possible directions.
These vibrations differ from each other in amplitude and in phase.
The changes of phase are determined by those of the quantity

z

TVA
Since 7V means the wave-length in the crystal for the direction

considered and z' =r cos, the phase will vary very much by small
variations of $, i.e., of the direction of the wave system in question.
There is one direction for which ;

TV
takes a maximum value. This is the direction of the wave-normal
OQ to which OP corresponds as first ray. Indeed, 2'/7’V_ is
proportional to the time in which the vibrations of a certain wave-
system arrive at P and this time is really a maximum for the
system whose normal is OQ. We shall prove, that the resultant
vibration at P is the same as it would be, if we had only to do
with wave systems of this latter direction and of directions in the
immediate vicinity of it. To this effect we shall fix our attention
on an arbitrary normal OWN, making an angle ¢ with OQ, writing
y for the azimuth of the plane NOQ with respect to a fixed plane,
which passes through OQ, and for which we might take the plane
POQ. We shall not however introduce yp and ¢ as variables but » and

Wi

== = cos $,

if V, is the velocity of propagation of the plane wave, having OQ
for its normal. Further we put
2at 2ar

Shs

i TV,

200
= 9, dw =sin?— dudp.
du

Qn 0

J 6.’ 0
G = af lip os ~— sin (qu — h) sin? = quays. », « . (iil)

0 Uo

( 739 )

if w is the value of w for the direction OQ. Indeed the directions for
which w= const. lie on a cone surrounding the line OQ, just because
uw is a maximum for that line. We first integrate with respect to w
and put

Qn

mabyt , . 0D :
—[pepreg in WHS. meow (lta)
0
The result is

0
Ge = fF) sin gu =) du of Oo te ie o (GlS})

0
§ 10. An integral such as (13) has already been considered by
KircnHorr. Kor great values of g it approaches uniformly to zero
and at infinity it may be represented by a development of the form

a, a,
44
Y) 9

It is only the coefficient a, that can be found. Integration by parts
of the integral gives

0
u s (gu, —h)—f(0)cosh a
J 160 sin Cg — 1) oe Od Os DLO . (14)

The first term, taken by itself, gives a sufficiently exact result for
points P, lying at distances r from QO, which are large in comparison
with the wavelength of light; in the following development we have
in view only such points as satisfy this condition. We put therefore

0
Hf F (0) sn (gu —B) dy = OOO LOO (142)

We shall first consider the part
Qn

f(0)cosh TV, or t if wet by t i o¢ ;
RS Et ae s Wn — sin Pi— | dip.
lo 2nr fk; T? V* cos 3 ; Oe an

0 u— 10)

Now
0D es O(cosP) O(coss)
du —- O(cosd) © Ou

Dit, ss 450 | Pos bas ’ 0 V,
O(coss) — O(coss) | V ce | aa pee 0(cosb) = ;

so that for w= 0 or cos¢=0

( 740 )

sin D =| = = dis Ofer) .
Ou |,—0 V, |.0(coss) Ju=o

We may further deduce from the consideration of the spherical
triangle, defined by the directions ON, OQ and OP, that foru— 0

0(cos) a (+
0(cosS) = oe

so that
( ; =) V (dy
sin d — Sy || ==
du ui—i0 V;, dw “u—0
and
f(O)cosh 1 Qat fee
— SS HO || dy.
g Pad hice fly V *cosd

0

The real part of the expression (9), added to this result gives
exactly zero, so that, as we could have expected, there remains in
©, no term with only cos 22'/T. We need hardly add that this is
equally the case with ©, and €..

Finally we have to determine /(u,). Let us denote by 2 the
solid angle of a cone, formed by directions for which w is constant,
then

_ a9
d2 = du | sin d—adw. s,s LO)
Ou
0
Now by (42) we have
Qn
MO, 0,/T _ 00
F (%,) a Sa T3 wee (si ole. dy ’
0

and with a view to (15) we may write for this

ips MA Ort (a8
ars T? V,3cosd, elce

The solid angle d2, of an infinitely small cone with axis OQ may
be found in the following manner. We imagine the wave-surface W,
passing through P, and the polar surface Rk of W with respect to
a sphere of radius unity. Then the point corresponding to P will be
the point of intersection Q of OQ and R&. Further we take a point
P’ on OP prolonged, close to P and describe from P’ the cone
tangent to W. The normals drawn from OQ to this cone will lie on
a second cone and this is the locus of all directions for which w
has the constant value

( 741)
OP

OP' cos va,
The infinitely small cone of normals will intersect R in a curve
lying in a plane, normal to OP; the plane touching F at the point
Q is also normal to OP. Let these last two planes, which are
therefore parallel, cut OP in S’ and S. Then

OS <COP SI" OSS ORs 1;

and
u = OS'. OP cos 3,
du, = — SS'. OP cos 9, .
Further
208 O
dQ, — rd do ,
OQ?

if do is the infinitely small surface of the just mentioned plane curve.
But we have also

do = 21 Ve, 9’, SS’
if @, and g’, are the two principal radii of curvature of R at the
point Q. Combining the obtained equations we find therefore

dQ2 t= SVA6f0')
OND rica = CONOR
, 1
or since OP=r and 0 == OP cos 3, = rics 9,
dQ aes
aise = — 2arV 0, @', cos’ H,,
du u= Ug
so that
: cos(qu,—h A, bor T ia V
T (u,) eatuae — ea — 2ar //o,9',cos?d,. * cos (gu,—h)

LV, coso:, Y 2ar

or by (13) and (14)
Ca) cos (qu, — h) _ 1, bx, tT YO, 0", cos O, De 20 (: ed ‘
Po

9 Tl? Vi ip
if p, is the velocity of propagation of the ray OP, as it is defined
for plane waves. Thus the electric foree appears to have the same
direction as it has for plane waves whose corresponding rays coincide
with OP. Its magnitude is given by
eee Dir T VQ, Qo cos H, ne 2a ¢ r

§ 11. We must add to this a second vibration which may be
obtained by the composition of all wave systems due to the Y’-com-
ponents of the infinitely small vectors into which the original
E. M. F. has been divided. It is this action we have left aside in

( 742 )

§ 3; the total electric force produced by it is given by
Tby'tY 0,0, coed, Ax r
(C= 0 tC

qT? Wee . T
if we distinguish by the index 1 the quantities corresponding to the
second plane wave for which OP is the direction of the ray. The
magnetic force too has in both cases the ordinary direction and may
be derived from the electric force by multiplying respectively by

¢ cicos H, ¢ ccos td,

— = ——— _ and —= ;
Po An Pi Vy
so that the flow of energy is given by
c cos H, c cos }
C= — G,’ 5 —G,',
¢ V, 0 AF ¢ V; 1
or
= Bee, bYaot” Qu g', cos* H, one 20 ee ae
Uses th Po
4, n* ¢? b? t 0, 9, cos® D, ti, 20 A.)
LEY {iY Pr

The mean flow of energy per unit time is therefore
Ge | =" 0, 0', cos* F, “i b3,;, = @; @', cos* |
74 7 6 75 .
27 Vy, V,
The amount of energy travelling outwards in directions lying
within the cone of rays do’, is
r © do’.
We may finally observe that the cone of corresponding wave-
normals has a solid angle

()

do = 9 0' cos* }. 1? do'
so that the total amount of energy radiating from the centre may
be represented by the integral

3:42 b? 1 2 bh? 1T?
rate {( he eas ) ae
FEE I eo Va,
It is only in the case of uniaxial crystals that this integral can
be further calculated.

Geology. — “On brackish and fresh water deposits of the river Silat
in. Western-Borneo.” By Prof. K. Martin.

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

(March 22, 1906).

KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM.

PROCEEDINGS OF THE MEETING
of Saturday March 31, 1906.

—DOCe

(Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige
Afdeeling van Zaterdag 31 Maart 1906, Dl. XIV).

COW HN Ss:

J. BE. Verscwarrerr: “Contributions to the knowledge of van DER Waats’ Y-surface. X. On
the possibility of predicting the properties of mixtures from those of the components”. (Com-
municated by Prof. H. Kameriixcu Onnes), p. 743. (With one plate).

J. E. Versenarrerr: “Appendix to the communications published in the meetings of June 28,
September 1902 and October 31, 1903”, p. 752. ;

P. Hl. Scuovurm: “A particular series of quadratic surfaces with eight common points and
eight common tangential planes”, p. 754.

W. ve Sirrer: “On the orbital planes of Jupiter’s satellites’. (Communicated by Prof.
J. C. Kaprryn), p. 767.

Physics. — ‘Contributions to the knowledge of vax DER WAALs’
w-surface. NX. On the possibility of predicting the properties
of imitures from those of the components.’ By Dr. J. KE,
VpRSCHAFFELT. Supplement no. 11 to the Communications from
the Physical Laboratory at Leiden. (Communicated by Prof.
H. KAMERLINGH ONN#S).

(Communicated in the meeting of January 27, 1906).
1. In the following pages I intend to show that the original
equation of VAN DER WAALS

Sen aes, oe gee ee S(O)

where is put

dz = a,, (1—2)? + 24,, x (1—2a) + a,, x?

by = b,, I—a)? + 2 b,, #(1—a«) + },,
and where also are made the*simplified suppositions *)

1) Comp. Kamertincuh Onnes and Zaxrzewski, Suppl. no. 8, Proc, Sept. 24,
1904, p. 227.

Jt

2

Proceedings Royal Acad. Amsterdam. Vol. VIII,

( 744)

Gig = Vy, Gy, Ons =) (),, aE (Oe) Sk 5 6 (8)

represents pretty well the special properties of mixtures. In cases
where we have no observations of mixtures of two substances,
the formulae given above will probably enable us to predict the
properties of the mixtures of those two substances by means of the
as and 6’s — i.e. the eritical elements — of the components; this
circumstance might be of some use in the choice of substances
whenever one wishes to observe definite phenomena in mixtures.
From formulae (1) and (3) we derive the formulae:

Tae, Dor Tee
eto Gewan ik
V pe V Por V Pik (4)
a ae lo —
b __ Pox Gay % Uk |
Prk Pok Pik

which express how the critical elements 7; and p,, of the mixture
taken as homogeneous depend on the composition ; from these formulae
also. follows, as we know, a linear variation of the critical volume
ve. That the second of the formulae (4) agrees well with the obser-
vations has been shown by van per Waats'). As to the course of
v,~, the curve denoting the variation of that quantity with « not
only deviates considerably from a_ straight line ?) (VeRSCHAFFELT
and Krrsom derive even from their experiments a maximum for
ve) but also the quadratic formula cannot be brought to harmonize
with the observations *).

Not too much importance should be attached to this deviation of a
quantity so closely connected with 6*); of higher import it seemed
to me to investigate in how far the formulae (4) accurately represent
the critical temperatures and pressures, as in connection with the
law of corresponding states, of which the approximate validity may
be considered to be established, these quantities entirely determine
the conduct of a mixture. But also here, of course, we should not
strain our expectations too high.

2. First I have computed from formulae (4) the values of the
quantities : ;

*) Proceedings Nov. 1897.

2) Comp. Kamertincu Onnes and Reweanum. Comm. no, 59), Proc. Sept. 29, 19003
Brinkman, Thesis for the doctorate, Amsterdam 1904, p. 73.

3) Ibidem; comp. also Verscuarretr, Gomm, Suppl. N°. 5.

4) For the possible causes of that deviation comp. Brinkman, loc., cit., p. 75.

( 745.)

EoOveey Be age TORE
(== | | Se oan 2x;
Tor dx 71 Ol: P\k Tok Pik
aes L, (#) a o Pik Was __ Por)

pok \ de Jo Tok Pik Prk

namely for those mixtures for which the @ and @ had been already
derived from the observations by the application of the law of cor-
responding states. The computed values are giyen in the following
table; the values in brackets follow from the observations.

(5)

CO, with CH,Cl «= 0,363 (0,378) 8 = — 0,149 (0,088)
CH,Cl with CO, § —0,270(— 0,221) 0,068 (0,281)
CO, with H, =719:978\(—— dh 29) — 0,439 (— 1,645)
CO, with O, — 0,513 (— 0,6563) — 0,242 (— 1,0871)

Of principal importance are here the signs of the @ and 8, and
in this respect there is a good agreement, when we except the 8 for
CO, with CH,Cl; I must remark, however, that from the experiments
of BriNKMAN a negative 8 is derived for this mixture °).

3. With the derived values of @ and 8 we now, using the formulae
constructed by Kerrsom and me, might compute the quantities

AT’, dpxpl -y, :
=== || and others; but as our principal concern is the
0

da dx

signs of those quantities, it is supertluous to perform these com-
putations. In fact, we immediately obtain a survey of the pro-
perties of the mixtures with a small proportion of one of the
components when we draw the values of @ and 8 as I have done
before’); this time of course in a diagram, based no longer on the
empirical equation of state used then, but on equation (1). It is
still easier, by means of the formulae given in Comm. Suppl. n°. 5
(Proceedings May 30, 1903) to express the @ and @ in KortTEwse’s*)
x and y, and then to use his diagram which, as is known, is
based on the original equation of state. Thus we find that the points
(a, 8} — or (%, y) — lie exactly in the fields which correspond to the
properties of the mixtures; here also, therefore, the investigation has
a favourable result.

4. I shall now communicate some computed values of 77; and pox.
for the mixtures under consideration, and compare them with the
values derived from the observations. (See following table).

1) Loe. cit., p. 73; comp. also the table on the following page.
2) Comm. Suppl. no. 6, Proc. 30 May 1903.
3) Proc. 31 Jan, 1903.

( 746 )

CH,Cl—CO, (BRINKMAN)

| Computed Observed
| Usk pak Tex | pak
«=O (pure CHCl) 4AG.2 65.93 416.16 65.93
i 402.1 66.54 | 398.56 | 65.73
4 388.1 67.20 384.26 | 66.89
} 360.0 68.74 351.88 | 69.19
3 332.2 | 70.70 324.96 | 71.06
x=1 (pure CO,) | 304.2 | 73.40 304.16 73.40

CO,—Hy, (VERSCHAFFELT)

z=O0 (pure CO,) 304.7 | 73.6 304.7 73.6
0.05 989.9 | 72.0 987.8 68.4
0.01 eee) || yl) ® 273.6 | 63.5
0.02 045.7 | 66.4 218.7 | 54.8

x=1 (pure H,) 38.5 20 38.5 20

CO,—O, (KEEsOM)

z=0 (pure CO,) | 304.02 | 72.93 | 304.02 | 7293
04 | 288.6 | 71.17 | 285.68 | 67.70
0.2 | 973.0 | 69.9% | 979.99 | 67.30

z=1 (pure O,) | 154.2 50.7 154.2 | 50.7

HCI—C,H, (Quin)

z=O0 (pure HCl) | 324.3 84.13 304.3 | 84.43
0.1318 | 318.6 | 76.44 | 315.5. | 77.3
0.4035 | 310.9 | 64.89 303.0 | 65.5
0.6167 307.2 57.75 999 4 | 58.6

|
0.7144 306.2 55.47 998.8 | 55.7
2=41 (pure C,I,) 304.9 AS 94 304.9 | 48.94

On the whole the computed course of the Ze and p., agrees
fairly well with the observations. The only important discrepancy is
that from the observations of mixtures of HCl and C,H, there follows
a minimum for the critical temperature, while the computation would
not have predicted this circumstance, This minimum, however, is not

( 747 )

very strongly pronounced; therefore I have investigated whether
perhaps the computation had predicted it for mixtures of N,O and
C,H,, for which the minimum critical temperature is probably much
lower than that of the components. I really find on the side of ethane,
the component with the lowest critical temperature, == —— (0
hence a negative value, which renders the existence of a minimum
critical temperature necessary.

5. After having shown by means of these few instances the
usefulness of formulae (4) for our purpose, I shall now closely
examine the course of the critical lines, in order thence to deduce
which conditions must be fulfilled by the critical elements of the
components, in order that the mixtures may show definite phenomena.

The shapes of these curves in the p7’ diagram have been deduced
by van peR Waats from formulae (1) and (2)') with the single
simplification 4,, = 4 (b,, + 4,,). What we shall find here will there-
fore be a special case of the more general forms found by VAN DER
Waats, namely the transition between the two cases a?,, > 4,, d,,
and a*,, << a,, 4,,; investigated by him.

Pp

T xk Prk :
If we put += a and a2 —=-— and moreover introduce the new

ok Pok
variable z= 2, we may in our case write the equation of the
critical line:
er, (Va,—1) — et (a,—1,) + 1 (7,7, VY a,)= 90. . . (6)

In the zr diagram therefore the critical line is a portion of a
hyperbola (see tig. 1), except when a, —r,, for then it is a portion
ot a parabola (represented in fig. 1 by OAS; a straight line in the
pi diagram), and when x,=1 or Vx, = t,, for then it is a
straight line (CD and OF).

In our drawing (fig. 1) one of the components always lies at the
point A, and we see that the form of the critical line is only deter-

1

: : lk Pik ;
mined by the relations oe and ~—. Besides, when we move the
Ok Pok

second component along one of the critical lines, the shape of that
line remains unchanged. *)

Fig. 1 therefore represents the forms which the critical line can
adopt in our case. In order to show that the observed forms agree
with these in a satisfactory way, I have drawn in the same figure
1 the critical lines derived from the observations. The lines for

1) Versl. Kon, Akad. Nov. 1897.
*) As van pen Waats (Joe, cit.) has remarked in general,

( 748)

mixtures of CO, with CH, Cl and of HCl with C,H, are drawn
twice in if, one time with the one component, the other time with the
other component at A. Of the lines for mixtures of carbon dioxide
with hydrogen or oxygen we could draw only a small portion in
the neighbourhood of carbon dioxide (point A).

These critical lines fit into the system of curves in a satisfactory
way, except the line CO,—O,. Also the beginning of the line
CO,—H, fits well into the diagram, but its further portion, if it is
to terminate at the point H,, cannot but deviate strongly from. it.

6. The drawing of fig. 1 enables us also to determine how we
must choose the pure mixtures in order that the mixtures may
possess detinite properties. VAN per WaAats (loc. cit.) has pointed
out the circumstance that the course of the critical lines (even when

d,,) excludes the existence of a maximum critical tempe-

32/

as ay,
rature or of a Maximum or minimum critical pressure. Yet mixtures
occur which show') a minimum critical temperature, and in our
case we find as conditions for its existence *):

a, +4, >27, Va, and also > 2/x,.

The area, within which the second component must lie, if the
critical temperature is to reach a minimum for one of the mixtures,
is therefore bounded by the two curves

oy}

@—22—2? and r= —

22<—1

y

represented in fig. 1 by OAF and GAH respectively. The first
line is one of the critical lines, namely that which has a vertical
tangent at a; the other contains all the points of the critical lines
where the tangent is vertical. It may be easily seen that the second
component. must lie between those two curves, i.e. in the fields 2
and 38. On the strength of this we may predict that in general a
minimum critical temperature will be observed when the critical

1) The elements of the mixture for which the critical temperature is minimum,
ave here determined by :
san 5) LS aS Vx, wv (V2,—]) (7,—T7, VY %,)
ent == 4 ——— TT? Unt ety 5
LS toa (%,—T,)
9
(2, ate T,— at, Vx,)
a a VA :
(V,—t,) (7,—7,)
®) The general conditions for the existence of a minimum critical temperature
are given in the Molecular Theory of yan pER Waats,

!

( 749 )

temperatures of the components differ little and the critical pressures
differ relatively much. It is known that experience confirms this
conelusion.

7. Now LI shall try to find how the substances must be chosen
in order that one of the mixtures near the critical circumstances may
show a maximum — or a minimum — vapour pressure. At the critical
point (at the same time plaitpoint) of that mixture we then have,

ey qe ak Wik op
along the critical line, — ——= ;
ot k

As we have based our speculations on the original equation of

Ot

the value which follows from this equation, i.e. 4. Thus we find‘)

. ; : Ghy
state of vAN DER WaAats, we must, strictly taken, use for | —
k

that the area within which the second component must lie, is
bounded by the curves :

t=} (82 — 2?) and c= —_.,,”)
represented by OA/ and KAL respectively in fig. 1. KAZ is again
the eritical line which shows at the point A itself the property
; 2 3 ; tdx
mentioned above, while OA/ combines all the points where = == 4
a dt

dz 6
Oe 2. The second component must be situated between these
at

two lines, namely in field 3 or 4.

We may repeat that in order to observe the property under con-
sideration, the pure substances must be chosen so that the critical
temperatures differ little, but the values of the critical pressures
differ relatively much; however, the component with the higher
critical temperature must also have the higher critical pressure *).

') The elements of the mixture, of which the vapour pressure is maximum
or minimum, are given by

Oe 20, Ya,tt, V1,—3 1,2, pea ia Vx,
oP 2 (x,—7,) (Vx,—t,) Naot an TU iment ‘
ak 9 (V2,—1) (7,—17, Vx.)
Trp — 5) LN al (x = )? z .
ifcol a

*) The general conditions for the existence of a maximum or a minimum vapour
pressure have been derived by van pen Waats (Versl. Kon, Ak. 1895/96).

My quotation of van Laar in the Dutch edition (same note) resulted from a
misunderstanding.

5) The latter does not always hold good, as for instance with mixtures of
GO, and CH, (Kuevey, Zeitschr, f, physik, Ghem., 24, 681, 1897),

(750 )

Hence a minimum critical temperature and a maximum vapour
pressure are two phenomena which as a rule occur together, but
not necessarily; this is only the case in field 3.

From fig. 1 it appears that, according to our reasoning, only
a& maximum vapour pressure is possible; yet we know that there
are mixtures which show a minimum vapour pressure, and it has
been proved by Kurnen*) that this phenomenon occurs even under
the critical circumstances. Here it seems that there is a fundamental
deviation from the observation. Nevertheless it is remarkable that
the mixtures which show a minimum vapour pressure are always
of such kind that at least one of the components is an anomalous
substance*); so that there is reason to suppose that with mixtures
of normal substances a minimum vapour pressure never occurs; and
our speculations, which are based on the law of corresponding states,
are applicable to normal substances only.

8. Starting from the same suppositions as set forth here, van Laar*)
has found an accurate expression for the projection of the plaitpoint
line on the va-plane. I have tried to derive from this the equation
of the plaitpoint line in the p7Z’— hence also in the zt — diagram ‘),
but without success. Without therefore occupying myself further with
the general form which the plaitpoint line in our case takes in the 2r-
diagram, | shall investigate a few points, namely the occurrence of
a maximum or a minimum plaitpoint temperature, and that of a maxi-
mum or a minimum plaitpoint pressure.

According to Kuxsom’s °) formulae (2a) and (26) we have

Lal oes T, |
——{ —— | ———_ [x, (38 /a,—1)? — 4a, V2]
GNC Ga oie Aree -
ere)
1 dprpl 1 Pare 4 = ey, fo
— = ==, (0, (8. a, — 1)? 2a (Sy) eel
Pok aa 0 Ty 4
; ¢ Txpl dT zp C
Hence follows that the boundary between ame and = ane <Ois
ev av

formed by the curve:

1) Arch. Néerl., (2), 5, 306, 1900 (Livre jubilaire de H. A. Lorentz).

2) For the bibliography of this ef. Harrman, Thesis for the doctorate, page 84,
Leiden 1899.

5) Proceedings April 22, 1905.

4) By eliminating « and v between the equation of the vx projection of the
plaitpoint line, the equation of the spinodal surface (van Laan’s formula 6, loc.
cit.), and the equation of state,

°y Communication no, 75, Proceedings Dec, 28, 1901,

754 )

434
C= ———,')
(82—1)?
represented in fig. 2 by the branches BAC and OD; to the left of
IT, -&
if <0, to the right > 0. If in consequence we take A as the
Av

component with the lower critical temperature, in other words: if
we put t, >1, there must be a minimum plaitpoint temperature
when the second component lies in the area AGCABDHA; in
general this will again oceur when the critical temperatures differ
little, whereas the critical pressures differ much?). This does not
prove, however, that there may not be other circumstances for which
the plaitpoint temperature reaches a minimum.

If on the other hand we take A as a component with the higher
critical temperature, there must be a maximum plaitpoint temperature
when the second component lies in the area OHKX.’) Neither here
is it proved that the phenomenon is restricted to that area.

il Appl
9. The boundary between ac er 0 and <0 is formed by the
(é da
curve
u? (32—1)? — Qre* (5e—1) 4 424 = 0,
represented in fig. 2 by HAF; ag is negative within that line and

du
positive beyond it. Whence follows that a minimum plaitpoint pres-
sure is impossible, at least little probable, while a maximum _ plait-
point pressure must occur when the second component lies within
the area ALHAF:r OMA; this will therefore in general be the
case when the critical temperatures differ much, which is confirmed by
experiment.

10. Finally, in order to show by means of an example that the
suppositions whence we started in the main represent precisely the

i) Cf. also van Laar, loc. cit., p. 585,

9

2) This is again the same condition as for the existence of a minimum critical

1 (dT zy 1 d Py, ie
temperature; but as —— sii 1 ee a+ — (B—4z)?, aaa may be positive with
Tae dz 0 16 da

negative z, in other words: a minimum plaitpoint temperature requires a minimum
critical temperature, but not reversely. This may also be seen from fig. 2, where
| have once more drawn the line GAH of fig. 1 (dotted line).

*) An instance of this is probably nat to be found,

course of the plaitpoint elements, I shall give here the results of a
computation which | have executed for CO, and H,.

« =O (pure CO,) Tap) == 3044 Ppt = 72,9
04 295,8 90,8
(0,2 287,4 108,7
0,3 274,8 124.8
0,4 260,4 140,0
0,5 244.3 DONG)
0,6 222,10 162,9
0,7 194,0 164,5
0,8 157,0 115255
0,9 108,8 115,2

@ == ts (purest. 38,5 20

The course of the plaitpoint line resulting from this agrees with
fig. 9, plate I of Harrman’s Thesis for the doctorate; in reality,
however, the maximum of the plaitpoint pressure will lie much higher.

Physics. — “Appendix to Communication N°. 81”. (Proceedings

June 28 and September 27, 1902) and Supplement N°. 7
(Proceedings Oct. 31, 1903). By Dr. J. E. Verscaarrent.
Supplement N°. 12 to the Communications of the Physical
Laboratory at Leiden. (Communicated by Prof. H. KamERLINGH

ONNES).

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

In the expression which I have given before (Comm. N°. 81 and

r

also Suppl. N°. 7) for the function y in the neighbourhood of the
plaitpoimt an inaccuracy has remained. I have found that I have

neglected therein more than a mere linear function of w

If we write:
y
eae

where |’ represents a very large volume, then yy is the free energy
in the perfect gaseous state, with the exception of an error which
will be smaller as V itself becomes larger, and which vanishes
when we put V=o.

The first term of w, which depends on v, may be dissolved in the
following way ;:

hayty y

: e

( 753 )

Vr

fo dy 4p 5 Fis f dv.

V
The first part I have developed before, and X = | p dv—RT log V

vA

Lk

(V =o) is the v-function which has then been wrongly left out of
account. This funetion cannot be developed in the same manner as
the first integral, because the series used for that is no longer con-
vergent for large volumes; we must therefore turn to KaMpRLINGH
Ones’ empirical equation of state.

When this equation of state is written in a reduced form, it also
represents the reduced equation of the isothermal of the mixture a,

gal

at the reduced temperature f= —., so that
xk
V
4 Vr}
| pdv = prk rar| pdy =
UTk UTk
Urk

Bie il al iG 1 1

Hence the neglected v-function is:

Bp Beat By © pop v eh

= oF DO Prk V xk Prk V xk

AERP log rps t ee eT a
4 UTk A UTk

and this may be developed again:
Resa Xe XG (a — er) exe (@ — ee) ees
where the co-efficients X,, X,, X, ete. are still functions of tempe-
rature. Fortunately the neglection of that function X has not influenced
the results in first approximation; however in the formulae 4, 5,
12 and 13 of suppl. N°. 7 we have to add 2X, to the factor
RT
we(l—ap)

(754 )

Mathematics. — “A particular series of quadratic surfaces with
eight common points and eight common tangential planes.”

By Prof. P. H. Scnrovurr.

1. “In our space are given a fixed line and four projectively
related plane pencils of rays. To be found the common transversals
of the fixed line and a set of four corresponding rays.”

Notation’). We indicate the fixed line by @°, the vertices and
planes of the pencils of rays by O,, O,, O,,O, and a,, a, @,, @,,
four corresponding rays and_ their two transversals by (,, (4, d,, l,
and 1, the pencils of rays themselves by (/,), (/,), (4), (4) and the
pairs of points of intersection of /,/’ with each of the rays /,, /,,/,, 1, by
(S,, S',), GS;, S45), GS, S'), (S;; S',). Panther the symibols:/,7,.saeeiee
may represent the lines of intersection of the pairs of planes
(CER CAR MCE Ch yrovan o (Gy G2).

2. The order of the locus of the pair of transversals /, /’ is easy
to deduct from its section with @,, which consists of two parts : the
locus {(S,,S,')] of the pair of points (S,,,S,') and some generating
transversals. Each ray /, of the pencil (/,) containing a single pair
of points (S,,/S,'), the locus [(S,,S,')] is an hyperelliptic curve the
order of which exceeds the number of times a transversal passes
through O, by two. Now three transversals pass through O,. By pro-
jecting the pencils (/,), (4,) out of O, on a, we find namely in «, three
projectively related pencils (/,), /',), (/,) and now three times three cor-
responding rays /',,/',,/, pass through one and the same point, the
conics generated by the pairs {(/,), (/,)] and [(/5), (/,)| having besides
(, three more points in common. So the locus {(S,,.S')| is a curve
c,° of order five having in (, a threefold point; its genus is three.
Now that three transversals pass through QO, there must be according
to the principle of duality also three generating transversals in a,.
And indeed, the peneils (/,), (/;), (/,) do deseribe on the lines /,,,, /,,,;
/,,, three projectively related series of points (A,), (A,), (4,) where
three times three corresponding points A,, A,, A, lie on the same
right line a, the conies generated by the pairs [(A,), (A,)] and [(A,), (A,)]
possessing three more common tangents besides /,,,. So the total section
of «@, with the locus of the pair of transversals /,/' is a system of
order eight and this locus itself a scroll O* with a nodal curve of
order eighteen. The order of the nodal curve ensues even from the
fact that the surface U* must correspond in genus to c,° ; moreover

') For notation and reasoning see a former communication,

(755 )

the eighteen points of intersection of the curve with @, are easy to
indicate.

The obtained surface O° is intersected by the given line /’ in eight
points. So in general there are eight lines resting on /’ and on four
corresponding rays /,, (,, (5, /y-

3. In the preceding we have assumed that four corresponding
rays 1, 1, 2, 7, always admit of two common transversals, not
taking into account the possibility that four corresponding rays
have an hyperboloidic position. In the general case this singularity
does not oecur; for the condition that four lines are situated hyper-
boloidically is a threefold one and the number of corresponding
quadruplets of rays is only singly infinite. However, this does not
prevent a proper selection of the data from leading to projectively related
pencils with a quadruplet of corresponding rays lying hyperboloidi-
cally; to this end we have but to assume the points O,, O,, O,, O,
on four hyperboloidie lines /’,,/’,,/’,, l’, and the planes «,, @,, a, a,
through these same lines, and to fix the projective correspondence in
such a way that these four lines correspond.

If the case of four hyperboloidic rays (,, U’,, ’,, ’, really occurs,
the scroll O? of the lines intersecting these four rays belongs to the
locus under consideration; we have thus further to investigate whether
this ©? joins the surface O* of the general case or whether this
surface breaks up in this special case into the surface 0? and a
completing surface V°. At the outset only the first possibility occurred
to me and I contented myself with developing grounds why this
elevation of the order of the locus from eight to ten need not really
clash with the wellkwown principle of the conservation of the number *).

Although at first sight it seems rather absurd that the infinitesimal
small difference between four nearly and four perfectly hyperboloidic
rays should rule the locus obtained by means of the remaining
quadruplets so as to let us find in the first case an V* and in the second
an 0°, yet as will be proved directly the second of the two sup-
positions mentioned above is the right one, not the first; so in that
sense this paper has had to be modified.

The surface (* of the common transversals of any hyperboloidic
quadruplet /’,,/’,,/,,/’, contains these lines and so it must admit
of a transversal through each of the vertices O,, O,, O,, O, and in
each of the planes a,,a,,a,,@,. The deduction of the order of the
locus O° has shown that through each of the four points O three

1) See for this a corresponding case of apparent contradiction in my “Mehrdimen-
sionale Geometrie”’, vol. I, page 263.

(756 )

transversals pass and likewise there are three in each of the four
planes «. Hence the question, whether besides the scroll 0? a surface
O* or a surface O° presents itself, can be decided by the fact whether
the four generators through the points O and the four generators
in the planes @ are common to the two parts of the locus or not.
Now, as a matter of fact, those two parts can have but two gene-
rators in common, viz. those two common transversals of /’,, /’,, ’,, /’,
joining the preceding pairs of transversals and the following of the
adjacent quadruplets. So the eight indicated transversals of 1’,, /’,, 1’,, l’,
are not situated on the other part of the locus and consequently the
latter is cut by each of the planes @; according to a curve c* with
a node in QO; and two right lines; so the remaining part is a surface
0° with a nodal curve of order nine. For the scroll O* of genus
three appears instead a combination of a regulus 0? and ascroll O°
of genus one cutting each other in two right lines and a twisted
curve of order ten.

From the preceding follows immediately what will happen when
the singularity of the hyperboloidic quadruplet presents itself more
than once. If two of those particular quadruplets are at hand O°
breaks up into three parts, two quadratic reguli and a scroll O*
with a twisted cubic as nodal curve; so the latter principal com-
ponent part of the locus is of genus zero and has with each of the
two quadratic surfaces two generating transversals and a twisted
curve of order six in common. If the projectively related pencils
contain three hyperboloidie quadruplets O* breaks up into four
quadratie reguli, three of which answer to these quadruplets whilst
the fourth, really the locus, is supplied by all the remaining qua-
druplets; the latter surface is intersected by each of the others
according to the edges of a skew quadrilateral, whilst these three
intersect each other in. general according to twisted curves of order
four. And if there are four hyperboloidic quadruplets, as will
appear later on, all quadruplets are situated hyperboloidically; then
the case presents itself where the order of the locus, so far always
eight, becomes infinite.

4. The following. simple example will show that it is not diffieult
to choose the data so as to allow each quadruplet of corresponding
rays to lie hyperboloidically.

We imagine the four pencils (/,), (/,), (/s), (/,) situated in the four
sides of a cube (fig. 1), we assume the vertices 0,, V,, O,, O, of
the pencils in the centres of these sides and we allow those rays
l,,/,,1,,/, to correspond which form the same angle g with their

projections ou the plane through the four vertices when one keeps
in the same direction. To each quadruplet of corresponding rays
belongs a hyperboloid of revolution with OZ for axis and circle
O,O,0,0, as minimal circle (‘“cercle de gorge’), whilst the hyper-
boloids of revolution belonging to the various values of gy, touching
each other according to that circle, form a tangential pencil as well
as an ordinary one. Each of those surfaces presents itself twice as
bearer of two reguli corresponding to two supplementary values
of g. In this case is a rule what was an exception above ; here
the number to be found is infinite, as two lines satisfying the con-
ditions pass through each point of /°, the two generators of the surface
of this peculiar pencil passing through this point. Indeed, the case
of an infinite number of solutions makes its appearance even as
soon as there is only one hyperboloidic quadruplet and 7° is at the
same time director ray of the regulus determined by this quadruplet
as director rays; then through each point of /° passes only one
line satisfying the question.

To simplify the representation the preceding particular case has
been taken on purpose as regularly as possible. The principal thing
is what the figure retains after a projective transformation, that
namely the vertices O,, O,,O0,, 0, lie in the same plane, that the
planes «,, @,,4@,,a@, pass through the same point and that all quadratic
surfaces touch those planes in the vertices mentioned above; the
regular situation of the four vertices on the common minimal
circle is of secondary importance.

This leads us to a new question, viz. whether it is impossible to find
four projectively related pencils of rays where each quadruplet of

( 758)

corresponding rays has hyperboloidic position, the vertices not lying
in the same plane, the bearing planes not passing through the same
same point and those planes not being touched in those vertices by
all quadratic surfaces herewith generated. Analytically as well as
geometrically we can convince ourselves in a simple way of the
reverse.

With respect to a rectangular system of coordinates O (X YZ) the
four pairs of equations.
y=perq| —y=per+g ys perq| y= pez
z= rets\)’ —z2z=ret+s\’ —z=— rete 7 z=— ret+s
represent four lines /,, /,,¢,,2, with hyperboloidie position, For the
conditions under which the surface

av? + by? +¢2 = 1
contains one of those lines are
a+ bp? + er? = 0, bpqg+ers=0, bq? + 879 = 1,

any of the four lines being taken. Now these lines /,, /,, /,, 7, hang
together in such a way, that by a rotation of 180°

round the axis OX the lines 4, and /y and likewise the lines /g and J, pass into each other.

ON. ee al a Pel cash oe ne ene
: (WA tal ot eles a8 5 y Mon la = tls z

If now in a plane a, the line (, describes a pencil of rays with
O, as vertex, the lines /,,7/,,/, will describe the pencils obtained by
making the pencil (/,) undergo a rotation of 180° round the axes
OX, OY, OZ, where the four vertices O,, O,, O,, O, will not lie
in the same plane, and the bearing planes will not pass through the
same point. And then is also excluded that the planes @,, @,, a, @, are
touched in O,, O,, O;, O, by the generated quadratic surfaces. For two
quadratic surfaces touching each other in four points not situated in
one and the same plane coincide and the surfaces under consideration
do not.

Let us consider geometrically a more special case connected with
a regular tetrahedron. We start from a cube and take (fig. 2) one
of the two groups of four not adjacent vertices A,A,A,A, as vertices
of this tetrahedron. Then the faces A,A,A,,-A,A,A,, A,A,A,, 4,A,A,
of this tetrahedron are the bearing planes «@,, @,, @,, @,, the centres
of those equilateral triangles are the vertices O,, O,, O,, 0, of those
pencils. And the rays /,,/,,/, corresponding to an arbitrary ray /,
of the first pencil are found again by a rotation of 180° round
the lines OX, OY, OZ through the centre of the cube parallel
to the edges of the cube, which are at the same time the connecting
lines HE', FE', GG' of centres of pairs of opposite edges of the

Fig. 2

tetrahedron. From a simple inspection of the figure appears that
the three points, in which any of the faces of the tetrahedron is cut
by the corresponding three rays lying in the other faces, are situated
on the second asymptote of the hyperbola passing through the three
vertices of that face and having the fourth corresponding ray lying
in that face as an asymptote. So this ensues inter alia for the face
A,A,A, from the three relations:
Ana (OVAL. ABD = DUA, AB = BUA.

So already four lines rest on /,, /,,/,,2,, namely one in each face,

which proves that the lines /,, /,,/,;,/, have hyperboloidic position.

6. We leave our original problem for an other moment in order
to investigate first the series of quadratic surfaces furnished in the
last special case under consideration by the quadruplets of corre-
sponding rays. All these surfaces have eight points in common, the
four vertices U,, U,, O,, OU, of the pencils and the four points O,,
O,, VU, UO, symmetric to these with respect to the common centre
O; so they belong to the net 4, of the quadratic surfaces deter-
mined by seven of those eight base-points O;, forming in their turn
the vertices of a cube. We can likewise point out eight common

ors)

Proceedings Royal Acad. Amsterdam, Vol. VIIL.

¢ 760P)

tangential planes, the four planes «,, @,, @,, @, of the pencils of
rays and the planes a,, @,, @, @, parallel to the former and sym-
metric to these with respect to O; so those quadratic surfaces are
a part of the tangential net MN, determined by seven of those eight
base-planes «;, enclosing together a regular octahedron. So our series
of surfaces being formed by the surfaces common to 1, and wV;, can
be regarded as the intersection of those nets.

The tetrahedron of which the origin ( and the points X,, Y., 4
at infinity of the axes of coordinates are the vertices is common
polar tetrahedron of all surfaces of the two nets VV, and N;,. In
connection with this .V,, has instead of a single infinite number of
cones six pairs of planes, a pair through each of the edges of the
tetrahedron, and NV, contains instead of a single infinite number of
surfaces reduced to conies six pairs of points, a pair on each of the
edges of the tetrahedron. So we find the most general projective
transformation of the series common to V, and .V;,, by starting from
an arbitrary tetrahedron, an arbitrary point and an arbitrary plane
through this point and by then representing to ourselves the surfaces
having the given tetrahedron as polar tetrahedron, passing through
the given point and touching the given plane.

We prepare the deduction of the three characteristic numbers of
our series of surfaces by determining the locus of the points of contact
with one of the eight base-planes, say @,. By considering the indicated
relation

A,C,=C,A, , A,D,==D,A, ; A,B, =B,A,
between the two lines /, and /, (fig. 3), in which @, is cut by the
quadratic surface belonging to /,, we find immediately that B,, C,, D,
on A,A,, A,A,, A,A, describe projective series of points when /,
rotates round QO, and that /', envelops a conie described in triangle

Fig. 3

( 764 )

A,A,A,; by determining for each of the pairs of projective series of
points on each of the bearers the point corresponding to the point
of intersection of the bearers reckoned to belong to the other series it
becomes evident that this conic touches the sides in the centres, so that
it is the inseribed circle c?. The point of contact being the point of
intersection ZL, of 7, and /,, the locus of this point is at the same
time the locus of the point of intersection of the corresponding rays
of the pencil (/,) of order one and the peneil (/',)? of order two
formed by the tangents /', of c?, thus a curve c* of order three with
QO, as node and the tangents from (, to c’, i.e. the isotropic lines
through ©, as nodal lines. This curve represented in fig. 4 touches
the sides of the triangle A,A,A, in the centres and has the points at

Fig. 4

infinity of the sides as inflectional points; the inflectional asymptotes
run parallel to the sides at distanees of four ninths of the height.
In normal coordinates its equation with respect to triangle A,A, A, is
22," (wv, + 2#,) — 22,’ = 4,7,2,,
whilst the Pliicker numbers are
meee ee C8 Os
Pi a ete Oe eee
As is known we mean by the three characterizing numbers of a
simple infinite series of surfaces the numbers u, v, 9 indicating
successively how many surfaces of the series pass through any given
53*

point, touch any given line, touch any given plane. From the following
will be evident that in our case these numbers are 3, 6, 3.

All the surfaces of the net V, with the eight base-points O;, passing
moreover through any ninth point ,, form a pencil with O as
common centre and ON, OY, OZ as common axes. Each surface
of that pencil touching one of the eight base-planes a; touches them
all, so it belongs to the series. A pencil of quadratic surfaces con-
taining three surfaces touching a given plane, we find uw = 3.

All surfaces of the net NV, with eight base-planes a@;, touching
moreover an arbitrary ninth plane a@,, form a tangential pencil with
Q as common centre and ON, OY, OZ as common axes. Each
surface of that tangential pencil passing through one of the eight
base-points QV;, contains all these, so it belongs to the series. So 9 = 3,
as three surfaces of a given tangential pencil pass through a given
point.

The number of surfaces of the series touching an arbitrary line
of the plane A,A,A, is three, because this line cuts the locus of the
points of contact (fig. 4) in three points. As the line is assumed in
a common tangential plane, each of those three cases counts twice;
so » = 6, as is immediately confirmed analytically.

So the indicated series of quadratic surfaces is a series (3, 6, 3).

Indeed, we also obtain a series with eight common points and
eight common tangential planes possessing the same characteristic
numbers (3, 6, 3) when starting from a common polar tetrahedron,
a point and a tangential plane not passing through this point.

7. We have two more points to consider with respect to our
original problem. Firstly, we wish to point out how the case in
which (* breaks up into four quadratic reguli is easily realised ;
secondly, we must show that all quadruplets of corresponding rays
have hyperboloidic position as soon as this is the case with four
of those quadruplets.

When the original part of the locus O* is a regulus QV? the pairs
of transversals of the quadruplets of corresponding rays are the pairs
of generators of this regulus arrayed in a quadratic involution. If
such a quadratic involution of pairs of rays is cut by a plane situated
arbitrarily a quadratic involution of pairs of points is generated on a
conic; this involution is as one knows characterized by the property,
that the connecting lines of the points completing each other to a pair
pass through the same point. To realize the above-mentioned case of
decomposition of the locus O* we start from an arbitrary regulus
*, whose generators we regard as paired off in involution in a

( 763 )

definite way, and from four arbitrary planes @,, @,, @,,«,. If then the
pencils of rays in those planes lying perspectively to the quadratic
involutions of points of the sections are taken as the projectively
related pencils of rays of the problem, then the surface (* is evidently
the integrating part of the corresponding surface O*, so this must
really also be completed to a surface of order eight by three other
reguli (7°. To conform this we allow an analytical treatment to
follow this geometrical consideration.

We suppose the locus proper (* to be decomposed into its genera-
tors by means of the equations

p+4q=9
r + As = 0

and we assume that the generators belonging to 2=Oand toa =
represent the double rays of the quadratic involution on @*, i.e. that
in this involution the rays with the same absolute value of 2 correspond
to each other. Here p,q,7,s are general linear forms in 2, y, Z,
according to the formula

wesueotuytuetu, , (w=p,gqr,s),
whilst the three planes of coordinates « = 0, y = 0, z = 0 and the plane
at infinity will do duty for the planes @,, @,,a,,@, of the pencils of
rays to be found. The tracing of these pencils is simplified by repre-
senting the minors of the determinant

| 8) 84 S, 84

according to the elements pi , qi, 1, si by Pi, Q , Ri, Sp
If we perform the described calculation with respect to the plane
wv =O, there is occasion to represent the equations
(Ps + 49.) ¥ + (Ps 1 293)? + Ps + 49, = 9
(7, = As,) lf alg (", si 4s;) € =f Ms AR as, =0

determining together the point belonging to 4 of the conic of the
section, for shortness’sake by

(p + 49), =0
(r + 23), =0 j
Then

(p + ag ta(r +4), 0. 6 ww ey Gl)

{ 764 )
is an arbitrary line through point. 4 and
Pad QFE (a +48s) + Prt Ags Fes +48) 5 Pep Age (+i s)
| ji! Ts ’ Ps—495 ’ Pa— 24%

ie)

T,—A5S, 5 Tr, —ASs, . TAs)
is the condition expressing that this line (1) through point 4 contains
at the same time the point —2 and is thus the ray of the pencil
looked for, corresponding to the parametervalue 4. If for shortness’
sake we write here

| pi + Agi + w(ri + 48:) |
Dit AiG; | == (I);
| ri — 48; |
direct calculation affords
pi + Agi | i + 48: |
pi — 49: —U my — 48; | = 0,
ri — Asi lb Oe Zh |
or
pe On) |
qi —wu 8; Nise 0,
nm — A138; pi — An
or
Di Pi | ri | | r;
gi | —4) Qi | — wu | $j | + Hu | se — 05
vy CH | | Pi | | qi |
Nee
eae
OERaP,

so the equation of the ray under consideration is
(Q +4P,)(p+49, +6, +42) +48), = 29,
which is reducible to
(Q, p oF S, Deas (en q+ R, s), = 9,
as the coefficient of the first power of A identically vanishes.

If now, for u=p,q,7,s, we understand by wz, Wy, Uz, Uy the forms
into which wv, + uw, y—-+u,v-+u, passes by omitting successively the
term with wv, the term with y, the term with 2 or the constant term
and if & is substituted for 2? the four projectively related pencils are
represented in their planes by the equations

( 765 )
Titten Qh tom aS t= ROP Ge t= Lees, r= 0
IW 5 6 ce (Oy a Si ry + &(P, qy + ie sy) ——(() ]

JE aes OEY Fe aS k (P, gz + fh, sz) = 0 |

TE AER wh ome (Oe Po + Si7% + h: (Uz. Os = iS.) 0
For which values of / have these four rays hyperboloidie position ?
To this end it is necessary and sufficient that the points at infinity of
J, Il, L/T lie on a right line. So for / we find the cubie equation :

0 » Qps FSi rsFk (Pi ga, 89), —L Qh prt Si rep (Pi ge R59)
|—[QopstSars+h (Ps qat-Ross)] , 0 » Qa San$k (Pon Ros) |=
| Qs potPS3 raFk (P3 go Rs 82), —[ Qa~r-PSsn +k Pan + Rss), 0

So in reality three reguli have separated from the surface O*.

8. Finally we have still to indicate that all quadruplets of corre-
sponding rays lie hyperboloidically if four quadraplets do. This proof
we join on to the most general case of four arbitrary planes a,, a,
a,,a@, and four arbitrary points O,, O,, O;, O, in them. If A, A, A,
is the face @, no longer equilateral, then the projective pencils (/,), (4),
(/,) describe on the sides A,A,, A,A,, A,A, the projective series of
points (C}), (D,), (B,) possessing for /—=4 four triplets of points on
the same right line. In that case the conies enveloped by the connect-
ing lines CLD, and 1,2, have five common tangents, i.e. A,A, and
the four lines bearing corresponding triplets of points; then those
conies coincide. So the supposition of four such triplets leads to the
case that there is an infinite number of such triplets. But then in
each of the four planes «,, @,, @,,@, lies a common transversal of
each corresponding quadruplet, ete.

We now conclude by showing that in order to determine four
projectively related pencils of rays with merely hyperboloidie qua-
druplets the four planes @,,@,,@,,a@, and the four vertices O,, O,,
O,, O, in them can be taken arbitrarily by showing that to aray /:
drawn arbitrarily in a, through O, only a single triplet of rays
l,,/,,/, of the remaining pencils corresponds, forming with /, four
lines with hyperboloidic position crossing each other.

If u,¥r,9 represent successively the conditions that a quadratic

surface contains a point, touches a line and touches a plane, then
V, {2v? — 3p? (uw + 0) + v(8u? + 2u0 + 30°) — 2(u* + 0°)

jndicates according to Hurwitz the number of surfaces through an
arbitrary line, which satisfy the sixfold condition V, (Afath, Ann.,

vol, 10, page 854). So the number of quadratic surfaces through /,

( 766 )

containing the points O,, O;, 0, and touching the planes a,, @,, a,
is represented by

1
que {2y* — 3x? (u + 0) + v (8y° + 2n0 + 307) — 2 (u* + 0°),

which in connection with the law of duality can be deduced to

1
3

3,3 3 3 2 2 D,,22
= He? {v* — 3uv? + 3u7v + uve — 2p%}.
_

Out of the wellknown results (H. Scnupert, ‘“Kalkiil der abzihlenden
Geometrie’, Leipzig, Teubner, 1879)

4..2,.3

u*y'o? = 104, u*r’o
we find that there are five quadratic surfaces satisfying the given
conditions.

However it is now easy to see, that only one of those five
solutions furnishes four hyperboloidie lines /,, /,,/,, 7, crossing each
other. We find namely four solutions not to be used for our purpose
(fig. 5) if we determine /,, /,,/, in such a way so as to cut the given

= 08) e von — 429200 134.0 On—alua

( 767 )

line 7,, or when having taken one of these three lines in that manner
we choose the other two in such a way that they both cut this
new line. So there is only one solution, in which the four lines
l,l, ¢,, 1, eross each other.

From the above consideration which can easily be confirmed
analytically ensues that the supposition of four planes «@; given arbi-
trarily and of four vertices 0; given arbitrarily dominates the case
of four projective pencils of rays with merely hyperboloidic qua-
druplets to such an extent that the projective correspondence is fixed
by the condition of the hyperboloidie position. This now again mcludes
that the case of the three quadruplets with hyperboloidic position,
treated above in details, cannot present itself if the planes @; and
the points OU; have been taken arbitrarily. For these three quadruplets
must also put in an appearance if we wish all quadruplets to have
hyperboloidic position, and they determine the projective relation
unequivocally, i.e. three hyperboloidic quadruplets lead here to pure
hyperboloidic quadruplets.

Not to get too redundant, we put aside the examination of the
less remarkable series of quadratic surfaces, answering to this most
general case of four pencils of rays with merely hyperboloidic
quadruplets.

Astronomy. — “On the orbital planes of Jupiter’s satellites”. By
Dr. W. be Sirrer. (Communicated by Prof. J. C. Kaprryy),

The following pages contain a condensed summary of the results
of an investigation, which will soon be published in detail in the
“Annals of the Royal Observatory at the Cape of Good Hope’.
The material on which this investigation is based consists entirely
of observations made at the Cape Observatory, viz. :

1. Heliometer-observations made in 1891 by Git and Finnay,
discussed by me and published in my inaugural dissertation. *)

2. Photographic plates taken at the Cape Observatory in 1891,
measured and discussed by me.

3. Heliometer-observations made in 1901 and 1902 by Cookson,
discussed by himself and published in Monthly Notices, June
1904 p. 728—747.

4. Photographic plates taken in 1903 and 1904, measured and
discussed by me.

1) Discussion of Heliometer-Observations of Jupiter’s Satellites, Groningen J, B,
Worters 1901,

( 768 )

My aim with this investigation was exclusively the determination
of the inclinations and nodes of the orbital planes of the satellites
and of the motions of these nodes. The plates of 1903 and 1904
were taken in order to provide a second epoch from which these
motions could be determined by a comparison with the observations
of 1891.

The fine series of observations, made by Mr. Bryan Cookson in
1901 and 1902 inereases the weight of this determination considerably.

I have already pointed out, in the fourth chapter of my disserta-
tion, that the determination of the other elements, which must be
derived from the observed (jovicentric) longitudes, is probably suffi-
ciently provided for by the observations of eclipses. Moreover from
the observations mentioned above sub 1. and 38. a// elements were
determined,

Eclipse-observations are however not well adapted for the deter-
mination of the inelinations and nodes, which must be derived
from the observed latitudes, as I have shown, l.c. page 77. The
principal interest of the determination of the orbital planes lies in
the comparison with the observations of the large motions of the
nodes, which are demanded by the theory. Since these motions are
produced almost exclusively by the large polar compression of the
planet, the natural fundamental plane to which the latitudes must
be referred is the equator of Jupiter.

If we refer the positions of the satellites to a system of co-ordinate
axes, of which the axis of y is the projection on the sphere of a
line perpendicular to this fundamental plane (i. e. of Jupiter’s axis
of rotation), and the axis of v is the great circle through the centre
of the planet perpendicular to the axis of y, then for the determina-
tion of the inclinations and nodes the 7 co-ordinates of the satellites
are alone important. Only these co-ordinates have therefore been
measured. The plate was, by means of the position-circle with which
the Repsold measuring machine of the Astronomical Laboratory at
Groningen is provided, brought approximately in the position-angle
P+ 90°, where P is the position-angle of Jupiter’s adopted axis
of rotation. The plate then has a motion parallel to a straight line,
which nearly coincides with the axis of , and which is defined by
the axis of the eylinder which guides the plate-holder in its motion.
The co-ordinates perpendicular to this straight line were then measured
by the micrometer screw. These differ from the co-ordinates y only
by small corrections (refraction, orientation and scale-value). In this
method the measured quantities never exceed a few revolutions of
the screw, All errors of résean-lines, division errors of the scales,

( 769 )

error of projection, ete. are avoided. The straightness of the eylinder
was repeatedly tested by comparison with a stretched spiderline.
Its errors are certainly smaller than 0.2 micron. The position-angles
were read off by two microscopes, and the orientation of the plate
was determined from a pair of standard stars, which were for this
purpose photographed on each plate, and from trails of the satellites.
The errors of observation of the measures of the satellites are satis-
factory, distortion of the photographic film cannot be detected, and
the discussion of a dozen plates, which were specially taken for this
purpose, shows that the determination of the orientation from the
trails is always practically free from systematic errors, while the
same can be said of the determination from the standard stars under
certain conditions, which are however not always fulfilled. The
accidental errors of both determinations are very small.

The image of the planet has not been measured, The observed
co-ordinates contain therefore an unknown additive constant (different
for each separate plate), which was eliminated by using in the
subsequent reductions the co-ordinates referred to the mean of all
the satellites occurring on the plate as origin. The equations of con-
dition and normal equations for these relative co-ordinates are very
simple and symmetrical. The limited space at my disposal prevents
me however from entering into more details regarding the measures
and reductions. I will at once state the results.

The unknowns which were determined from each opposition were
the corrections to the adopted values of the elements p and q of
the four satellites which are defined by the formulas:

paisin(— §)

q = cos (— Sr)
where 7 and 9, are the inclination and ascending node of the orbital
plane of the satellite referred to the fundamental plane. The longitude
of the node is counted from the ascending node of the plane of
Jupiter's orbit on the fundamental plane. The quantities referring to
the four satellites are distinguished by the suffixed numerals 1 to 4,

The following table contains the results of the different series of
observations with their probable errors.

The values for 1891 (Heliometer) ave those derived in my disser-
tation with a few unsignificant corrections in the last decimal places.
The results from the heliometer and those from the plates have been
combined with the relative weights 2 and 1.

The results for 1901 and 1902 are quoted from the communication
by Mr, Cookson in the Monthly Notices,

I have howeyer been compelled to reject Ap, and Aq, for 1901

(770 )

[SOL .75 1891.75 1891.75
Heliometer. Plates.
fo] ° °
0050 + 0.0397 + .0038
36 | 4h. 07834 186 | =e Oss 87
= $002 = 49 =) 0030.42 ea
+ .0688 + 12 | + .064 + 9

fe) (oe oO
Ap, | -+ 0.0409 + .0052 | + 0.0372
Ap, | + .0740

A p. — .0033

Ee hk
I+

H+

A ps + 0642

H+
we

Aq — 0.0259 + .0056 — 0.0258 + .0061 — 0.0259 + .0043

ag, | --) 0858 38-| 4 losis 35 |} oean 4 ee

Ags | — ,.0883 20. |e 0748 es 23. || Orb

Ag, | = )018%4p 4a: |) = $0N7 ge hal eee ote
|S agoner. | agon.62% |) 90a roa

COOKSON. COOKSON. | Plates. | Plates.

ap + 00170 + °0077/-+ 0°0137 + °0072'-4 0°o021 + co6o|— 0°0028 + 20078
Ap, |+ 4443+ 56/-+ .0922 + wt 0596 + 338i 015842 (eae
Ap, |— .0148 + 36\— .0072+ 28 .0199+ 221 01044 28
ap, | 0456 ASI: 1085Bi = a5: 0887 ge ASS. Goesaneeemtn
Ag, |— 0.0695 -+ .0084|— 0.0755 + .0065|— 0.0597 + .0048|— 0.0835 + .0077
Ag, |— A770+ 49\— 18604 40— 2190+ sal .om24 48
Aq,|— .09600+ 33\— 033 4 23/— 0494+ 20/— .0464 2%
ag, |— .0880 + 18|— 0070 =) t= ee |= noo ee

and 1902. Cookson found from the reduction of his observations
that the residuals could be much reduced by assuming in the latitude
of satellite IV an inequality of which the period is one half of the
periodic time of the satellite while the coefficient is about 50". I
have searched for this inequality in the observations of 1891, 1903
and 1904 and I can confidently declare that in none of these years
there is even the slightest trace of any inequality of which the argu-
ment should be a multiple of the mean longitude of Sat. LV. Since
also an inequality of this nature cannot be explained by the theory
I cannot but doubt its reality, and since the cause which has produced
this apparent inequality must necessarily also have affected the
determination of p, and q,, the safest course seemed to be to reject
the values of these elements found from the observations of 1901

C7)

and 1902. All other corrections Ap and Aq derived from the obser-
vations are included in the following discussion, with weights in-
versely proportional to the squares of their probable errors and
corresponding to a p.e. of weight unity of + 0.°0050. — *

Before this discussion can be related the theoretical expressions
for p and gq must be developed.

At the time when the analytical theory of the satellites was created
by Lagrancr and Laprace, the eclipses were practically the only
phenomena of the satellites which were observed. For these the
natural fundamental plane is the plane containing the axis of the
shadow-cone, i.e. the plane of Jupiter’s orbit. This was accordingly
used by them. SoumLart, in his theory published in 1880, followed
their example.

The first thing which must be done before the theory can be
compared with modern observations is thus to reduce the expressions
for the latitudes referred to Jupiter’s orbit to latitudes referred to the
equator. This has already been done by Marra, who in 1891
published tables for the computation of the co-ordinates of the satel-
lites, based on Sovmtart’s theory (Monthly Notices, June 1891, pages
505—539).

Let Z/ and JN be the inclination and node *) of the orbital plane
of one of the satellites with reference to the orbit of Jupiter.
SoumLLart’s theory then gives

4
LE; sin N; = = by sin 0; + wi w sin A,
7—
4
UP CINE == >> bj; sin 6; + wi @ sin 6,
—l }
In these formulae @ and @, are the inclination and node of Jupiter’s
equator on its orbit. All longitudes are counted from the first point
of Aries. The quantities 6; are constants, and the angles 6; vary
proportionally with the time. Of the constants 6; four only are
mutually independent. If we put:

shoes GarL)

oe bis = oF Y; ,
then the y; are constants. The multipliers Oi; and w; and the coeffi-
cients of the time in the expressions for 4; are given by the theory
as functions of the masses, the compression of Jupiter and the mean
motions. The constants 6; are small numbers (the largest is 643 = 0.1944)
with the exception, of course, of those in the diagonal, 6, = 1. The
value of uw; differs little from unity. The angles y; and 6; are what
Lapiace calls the “inclinaisons et noeuds propres” of the satellites.

) With node I mean ascending node, unless otherwise stated.

( 7729)

Let now o, and w, be the inclination and the longitude (counted

from the first point of Aries) of the descending node of the plane
which I wish to adopt as the fundamental plane, referred to the
plane of Jupiter’s orbit. Longitudes in the fundumental plane are
counted from the node y, as zero.

Then if ¢ and gs are the inclination and node of the orbit of one
of the satellites referred to the fundamental plane, we have, neglecting
quantities of the third order in 7, Z and o,:
isin §4 = Isin (N—Y,)
icos $4 = Ios (N—y,) + w,

If further we introduce the notations

j=y, —Gi wy, — 6, + 180°
ei yi sin_; vt, —wsiny ee (2)
Yi yi cos T; Y) = W cos P —,

then the expressions for p and g become:

4 \
p= 2 0; j &j_— Wi @, |
JjI=
Wenge wea oo (ce)

é
G== 65 yj; + (Ll — Ui) © — Hi Yo |
yl
Martu has adopted

wo, = the value of w j x :
foe from SovurLart’s theory, *)

YW, ey) 3. Oy 180°
and has computed the values of p and gq by the formulae (3), taking
LY, 0.

The unknowns y;, T;, «, and y, must be determined from the
equations (8). This is, of course, only possible if the coefficients
6, and mw are known. I have adopted these coefficients from
SournLart’s theory, as being the best available. They are very com-
plicated functions of the masses, the compression of Jupiter, and the
mean motions. As a rough approximation, we can say that the
coefficients 6 are proportional to the mass mj. Since the masses are
very imperfectly known, the same thing is true of the coefficients of
the equations (3). Therefore the results of the present discussion cannot
be considered as final, but the discussion will have to be repeated
when better values of the coefficients are available. The results here
derived will however doubtlessly represent a very fair approximation.

It may perhaps be mentioned that the uncertainty of these coeffi-

1) Marru has made one or two mistakes here, which will be duly mentioned
in the detailed publication, but as they have no influence on the result they can
be ignored at present.

(2738)

cients is not due to our ignorance with respect to the masses alone.
The values of these coefficients derived by SoumLart from the same
masses and elements by two different methods of integration show
differences of such amount, that the consequent differences in the
computed values of p and g are of the order of the errors of
observation. It is hardly to be expected that this defect in the theory
will be remedied before the equator is introduced instead of the
orbit as the fundamental plane of the theory. The coefficients adopted
by Marrn and myself are those derived from the second method
of integration, which is also preferred by SovurnLart himself.

In the following discussions these coefficients are treated as absolute
constants. If we denote the corrections to the adopted values of
xv; and y; by dv; and dy;, then the unknowns

Ovi OYi hy
must be determined from the equations

= 6; 02%; — wv, = Ap;
es ae w

= oj Sy; —WY=A4
The term (1 —w;)@, in the second equation (3) must, of course,
be treated as rigorously known.
The solution of the equations (4) is conducted in the following
manner. I define the quantities Aa; and 4y; by the equations
SA ees
Sen | PH. aie (5)
263 Ay; =AQG
These equations are solved once and for all, and the solution is:

A i Da) On A Pj (6)

Ay = Foi 4 Gj
Further, if we put:
Mi = = Oy Us;
then the equations of condition become:

Oe tL a
: - : | Sere saa(T)
Syi — wy, AY;

Next, if we denote the originally adopted values of 2; and y; by
«;, and y;, so that a=2;,+de;, yi=yit+dyi, then the equations
become:

&; — yj 2, = 2;, + Aa,
Yi — Ui Yo = Yin + ia ©)

In these equations 2; and y; are defined by the equations (2), where

dT; Pe
the y; are constants, and Tl; = T;, + 7 (t—t,). The unknowns, which
al

(774 )

must be determined from the solution of the equations (8) are

ae
EMO Pia) rey oe

Eine
The values of ar for the four satellites are however not mutually
¢

independent. The theory gives these differential coefficients as functions
of the masses of the satellites and the compression of Jupiter. The
masses need not be considered here. I have tried to determine a
correction .{0 m,, but this determination had too small a weight to
have any real value. The influence of the other masses is even
smaller.

The compression enters into the formulas through the factor /)7,
where J is the well known constant, which is approximately equal
to o—'/, » (9 = ellipticity of the free surface, g = ratio of centri-
fugal force to gravity at the equator of Jupiter) and 0 is the equatorial
radius ) of the planet.

If we introduce as unknown:

__ db?
arrae

then the true values of the coefficients of ¢ are

a dT;
ee (Sipe

The coefficients a depend practically alone on the mean motions,

RET dT;
and must be treated as absolute constants. They differ little from —~
ai

itself, and consequently the ratios of the motions of the nodes must
be considered as approximately constant. The adopted values accor-
ding to SourLart’s theory are (daily motions) :

dT, dT,
= 0°.14109 —— |} = 0°,007019
dt A
iD IT
an) — 0°.033010 ““+) — 0°.001898
dt 0 dt 0

The 86 equations (8) thus contain the 11 unknowns
View io veoe one sa
These equations must be solved by successive approximations. The
conditions for the application of the method of least squares are far
from being fulfilled.
These approximations have been conducted in the following manner.

1) In the original Dutch ) was erroneously stated to be the diameter, instead
of the radius.

(773)
Let #7. %. be approximate values of «, and y,, thus 7, = .2,, + de,
and ¥, = Yo, + dy, We have then:
Vu — ti Su, =< vty 4 Avi +- fu Loa
yi— pi dy, = yin + Oyit ui y,,
If we suppose that the approximation 7,4 Y) is already so good

ee xO)

that dr, and dy, can be neglected, then these equations become:
yisin Ppa orig + Dat; + 03! 255 )

Yi cos T; => Yio + Ay, + ua; Yoo |
Next I compute the quantities gy; and G; from the equations:

gisin Gi = 4;,, + Aaj + ui e,, ]

gi 008 i = ay + Dye + tel toy |
The other unknowns are then determined from the equations :

Seater ei (0)

SINR aeeiee

Vi — NU

dT:
Vig + ge to) = G: me ee woe ot (EA)

aT,
dt’
which can by an acceptable value of x be made consistent with
the theory, then the appromation is sufficient, if not, then a new

approximation must be made. As a first approximation I have assumed:

If these equations give constant values for y; and values of

Pe aah pt AO)

00
The equations (12) were then formed and solved. In this solution
dT;

I have determined the values of a for the four satellites separately
without introducing the theoretical ratios ab initio. The equations (12)
then consist of two sets for each satellite and each of these 8 sets
is independent of all others. The residuals which remain after the
substitution of the resulting values of the unknowns will be given
below together with those from the other solutions. The probable

error of unit weight was + 0°.0086.
The motions of the nodes in this solution are (Sol. I):

dT. dT, See
a 0801213 aa = 0°.00587
at dt
dr iT
2 — 9°.030266 “+ — 9°,00189.
dt dt

If these are compared with the theoretical values, it appears at
once that their ratios are very different. The node of satellite I,
which according to the theory has a yearly motion of about 50°, in
this solution shows a motion of about 5°, The ratios of the three

54,

Proceedings Royal Acad, Amsterdam. Vol, VIII

( 716)

other motions also differ considerably from their theoretical values.
Moreover the inelinations are far from constant, as will be seen at
once from an inspection of the residuals 4 y.

dT. ?
It must be mentioned that the value of 7 agrees approximately
€

with the value derived by Cookson from the observations of 1891,
1901 and 1902. This could have been expected since Cookson in
this determination also neglected the corrections to the position of

ei teh P!
the equator. The difference between Cookson’s value of = and the
dt

value of Sol. T is not due to a bad agreement of the observations of
1903 and 1904 with those of L9OL and 1902 (which on the contrary
agree extremely well), but to the fact that in Sol. I the corrections
to the elements of the other satellites were eliminated by means of
the transformation from 4p and 4q to 4x and 4y, while Cookson
did not eliminate these corrections but neglected them.

I have now made a number of further solutions, in whieh I started
with approximate values «,, and y,,, and introduced the unknowns
Vi Yar Jw, J 4, 2%,
thus rigorously subjecting the motions of the nodes to the theoretical
condition. The unknowns dy, and x» are badly separated. The
weight of the determination of * is considerably diminished by the
introduction as unknowns of the corrections to the position of the
equator. That this must of necessity be so, is easily seen. If we had
observations of only one satellite at two epochs, it would be impossible
to determine both the motion of the node and the equator. We
would in that case have only four data (the values of p and q at
each of the epochs) for the determination of the five unknowns

’
pee a = x, and y,. Now % is practically determined from sateltite

cd

II alone. The motions of the nodes of III and IV are too slow,
and the inclination of I is too small, to allow a determination of
the motions of the nodes of these satellites to be made, the accuracy
of which would be even remotely comparable to that of sat. IL.
The motions of the nodes of I, HI and IV are derived theoretically
from that of If. If therefore the latter is known, each of the three
others provides a determination of the equator. Then the determina-
tion of x* from II must be repeated with this new position of the
equator, and so on until a satisfactory agreement is reached. *)

") Cooxson has in his discussion of the observations of 1891, 1901 and 1902,
used this method, but he rested content with the first approximation. His corrections

to the equator derived from satellites Ill and IV are in the same direction as the
values found by me.

The solution was not actually made in this way, but all equa-
tions were treated simultaneously. This consideration is only given
here to point out that the position of the equator is ultimately
determined by the condition that it shall be the same for the four
satellites, i.e. that the inclinations shall be constant, and the motions
of the nodes shall be consistent with the theoretical ratios. Since a
small displacement of the equator has a large influence on the
motions of the nodes, in consequence of the small inclinations, it
can be expected that the unknown x and the quantities whieh
determine the position of the equator will mutually diminish each
others weights. (That this decrease of weight is actually much more
marked in the case of y, than for ,, is accidental and depends on
the choice ef the zero of longitudes).

By these considerations I have been led to try whether the value
of x could not be determined from a comparison with other obser-
vations. | have used the values of 6; for 1750 given by Denampre.
A value of « was adopted, such that the value of 6, carried back
to 1750 from the modern observations would be nearly equal to
the value given by Dretampre. The unknowns «,, ¥,, dy; and dT,
were then determined from the modern observations alone.

This gives solution VII. In solution VI on the other hand all
unknowns (inclusive of x) were determined from the modern obser-
vations. I give below the results from these two solutions, which I
consider as the best that can be derived with our present knowledge
of the masses. I do not venture to choose between the two solutions.
Probably an eventual correction of the coefficients oj will tend to
reconcile the two solutions.

Instead of IT; I give at once 6;=y, — T;. The values are given
for 1900 Jan. O Greenwich Mean Noon.

Solution VI Solution VIT Adopted values.
a OVOU72 == ©0028 == (S017 wes= .20.022 0
Y + 0 0427 + .0043 +0 .0489 + . 0022 0
poe 0.032 == .0094 =a QUOM26 0
1, 0°.0259 + °.0032 0°.0248 + .0088 0°.0013
Y, 4696 + 27 .4676 & 24 4694
Vs 1926 + 40 1874 4 26 1789
1. 2540 34 -2D 04) == 25 .2254
6, Gy ae tera) ASO tar ees 99°.8
4, DOS ADT =E 02.35 293 10' 22:0-29 21S?
6, a9) 568) 32) 0) 77 319 .67 + 0.80 330 .59
6 14.40 +0 91 oe 50) SenORo eis)

(778 )

10
From the values of x we find the following values of =

c
dé, WF a aie
a 0°.13664 —- 0°.138952 — 0°.14105
at
dé, o EE aN
aap 0.082105 — 0 .0382638 — 0 .032974
€
dA, Ler oe
Fag 0 006814 — 0. 006916 — 0 .006983
at
dé, ees
retin: 0 .001839 — 0.001854 — 0 .001863
at

From the values of #, and y, we find for the inclination and node
of the equator on Leverrimr’s orbit of Jupiter of 1900-0 :

7) 3°.1107 + °-0048 3°°1169 == *.0022 3°.0680
O (315.727 se. -042 315.7385 2+ -041 315.410

With the exception of x all unknowns in the two solutions agree
within the sum of their probable errors, and with only one excep-
tion (y,) all the corrections to the adopted values are many times
larger than their probable errors.

The residuals of the two solutions VI and VIL are given in the
following table together with those of Sol. I. The probable errors,
which have been added for comparison are somewhat larger than
those of the observed 4p and Aq, because by the transformation
from Lp and 4g to 4w and Ly, the p.e. must be somewhat
increased, even if we consider the coefficients oj as absolutely exact.

The p.e. of weight unity, which was + 0°.0086 for Sol. I, is
+ 0°.0065 for Sol. VI and + 0°.0064 for Sol. Vil. But it is chiefly
in their consistency with the theoretical conditions, that both solutions
are incomparably befter than Sol. 1. The melinations are now constant
within the probable errors. The residuals of the nodes only show a
systematic tendency for Satellite 1 (in Sol. VII, where the motions
of the nodes were not derived from the observations, also for Sat. 11).
Still the agreement with the theoretical motions is much improved.

» )
¢ 7 ° . . .
The value of z derived from Sol. VI irrespective of the theoretical
€

conditions would be 0°.1250, while the value corresponding to the
value of # in this solution is 0°.1866. This is a great improvement
compared with Sol. 1 (0°.01214).

The results for Sat. Hl in 190L and 1902, which in all solutions
eave large residuals, have in the solutions VE and VII been rejected.
This rejection has no appreciable influence on the values of the
unknowns, nor on the other residuals, but it reduces the p.e. of

Sut. I.

Sat. II.

Sat. III.

Sat. IV.

( 975 )
Sol. I Sa VE-, |---: Sok-WE
p. @
Ay sin yAr Ay sin y AT AY sin yOPF
|
1891 | +.0045 | —.0068 —70003 | "0005 +°0128 | 20047 -+-°0093
Poet te os ee) 157. — , 59\|— - 49-401. | — 60 SE) 69
Cert Toe et AS — s8l--. bo = 97 | -g = 400
Cone eCOn ee 1.3704 3k) sR 22 455 = Gt) ae
Gale sO) 997 > AS = 5 90) 5g 78
1891 | +.0030 | 0138 .0002 | +.0017 —.0008 | 4-.0016 +.0045
Hooleec 60) | 187 — 40/4 73 4 0/4 564 8
OBI: atx 50 Peers 0 re 0 63-460: 195, owen
Geeet0e Oh SS 8 ek eg 9
4) + 5 |— 40 4 8 (eas 2 30R 0 ib |e See 0m
4391 | +.0020 | +.0048 +.0007 | — 0014 —.0029 | —.0013 —.0037
Peete Ay) ASL =. 88 I Ans) 29) |[= 1st] 24]
02 | eeoON =e 1g 77 [= A59] — 73) [— 455] [— 67}
Gaeieeeeaon |) 32 67 | 4 ad 56 | 10 + 62
O) + 30 4+ 33 = SON abi a ell |e genes
|
1801 | +.0010 | —.0010 +.0001 | 4.0013 —.0028 | +.0017 —.0031
1901; + 20 |[ DSt]f+ 188]/[— 101)[-+ 205] [— 1101[-+4+ 200]
02) + 20 |[— 86][— 4185]/[— s3][— 466]|{— s5)[— 174]
Pee ise ee 08 4 Sgt Ok a 4
CSE) = 4 = 60 30 — 90)—- 94° 94
weight unity from + 0°.0072 and + 0°.0073 to + 0°.0065. and

+ 0°.0064 for the solutions VI
The values of 6; carried back to 1750 are:

Sol.
151
282
110

I

Tae)

9
3

on Vil

and VII respectively.

Sol. VII Damoiseau Delambre
201-20 282°.0 283ie5
338 .6 SO Sar0 Bry BA)
Te ell 98 .3 “105 .0

( 780 )

In conclusion I must express my deep sense of gratitude towards
Sir Davin Gini, who liberally placed the observations of the Cape
Observatory at my disposal, and was always ready to meet all my
wishes.

(April 24, 1906).

KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM.

PROCEEDINGS OF THE MEETING

of Friday April 27, 1906.

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

Afdeeling van Zaterdag 31 Maart 1906, Dl. XIV).

SO Wy) ey agaN ES:

L. Borx: “On the relation between the teeth-formulas of the platyrrhine and catarrhine
Primates”, p, 781.

F. M. Jaxcer: “A simple geometrical deduction of the relations existing between known and
unknown quantities, mentioned in the method of Vorcr for determining the conduetibility of
heat in crystals”. (Communicated by Prof. P. Zeeman), p. 793.

W. Burcx: “On plants which in the natural state have the character of eversporting varicties
in the sense of the mutation theory”. (Communicated by Prof. J. W. Mor). p. 798.

Hi. Srranx: “The uterus of Erinaceus europaeus L. after parturition”. (Communicated by
Prof. A. A. W. Husprecur), p. 812.

P. Zepman: “Magnetic resolution of spectral lines and magnetic force”, (Ist part) p. 814.
(With one plate).

JAN DE Vries: “Some properties of pencils of algebraic curves’, p. 817.

H. Zwaarpemaker: “On the*strength of the reflex-stimuli as weak as possible”, p. 821.

Anatomy. — “On the relation between the teeth-formulas of the
platyrrhine and catarrhine Primates’. (Communicated by
Prof. lL. Bonk).

(Communicated in the meeting of March 31, 1906),

Among the anatomical characteristics by which the Primates of

the New-World — the platyrrhine apes — are distinguished from
those of the Old-World — the catarrhine apes and man — the com-

position of the set of teeth takes a first place. They are charac-
terized because they possess in the upper and lower jaw one milk-
molar with premolar, which replaces this, more than the latter.
In simplified writing the set of teeth of catarrhine Primates may
be rendered by the following formula:
55
Proceedings Royal Acad. Amsterdam. Vol. VIII.

ye hy kee G5 Ph IP
PAM lk toy 2 ids Be WE,
Oe eile Opn 3 M
7 the WHE 2 IP,

in which the teeth of the permanent set of teeth are written with
a capital letter.

For the majority of the platyrrhine Primates the following formula
holds true :

we Jig AL Gig- Bs IE
Hate LGN eouellls
Oy ae Ih @s 8) Gite a8) JE
Mod& Al C53). 12,

This last formula is only applicable to the family of Cebidae,
whereas the Hapalidae differ from them because they have a molar
less, so that the formula for their set of teeth becomes as follows:

oiled Cretan JE
2%. Lc: 3) mm. 2. iM,
SAMO el (oe ON Gig eA
eR AGP. JP?

The difference however in the set of teeth between Cebidae and
Hapalidae is for the present of less importance, the significance of
it will be shown later on. In the first place the attention should be
fixed on the principal difference between all platyrrhine Primates
on one side and all catarrhine ones on the other, i.e. the occurrence
of only two milkmolars and premolars with these and of three milk-
molars and premolars with those.

It is not doubtful that the set of teeth of catarrhine apes and of
man must be deduced from one that was composed like the set of
teeth of the now living Platyrrhines with three molars, so compared
with the set of teeth of those, the set of teeth of the catarrhine
Primates may be considered as reduced, the total number of teeth
is larger with the former than with the latter. In what way has
this reduction of the set of teeth come about, this is a question
which has been frequently put amd which has been answered in diffe-
rent ways. An obvious conception is certainly this, that a milkmolar
with his replacing tooth, the premolar, has become lost. But which
of the row has disappeared? This question has been answered in
different ways. Whereas the Anthropologists in general are more
of the opinion that the last milkmolar and premolar have been
linked out, zoologists, palaeontologists and anatomists accept the view

(783)

that it has been the first, so that which follows immediately on
the caninetooth. The two opinions have in common that they link
out a milk molar and its replacing tooth from the continuity of the
tooth row. On account of this the two theories may be distinguished
as the excalation theories. I cannot agree with any of these opinions,
it appears to me that the reduction has been brought about in another
way, but this can only be explained more fully, when I shall have
brought forward what pleads for and what against each of the above
mentioned theories.

The Anthropologists look for their proof material, or perhaps more
exactly for the arguing of their theses in the variations in the set
of teeth, which occur with man. Of late DuckwortH among others
has again drawn attention to the fact that rudiments of a tooth, more
or, less developed often appear between the last bicuspid tooth and
the first molar tooth especially in the upper jaw and what is especially
of importance, often on both sides. These rudiments are conical tooth-
points, now occurring single either on the inner or the outer border
of the alveolar margin, then again double on each of the two
borders simultaneously. And Duckworrn does not hesitate to con-
sider these rudiments as the again visible traces of the linked out
third premolar: ‘on the whole we think that it is most reasonable
to adopt the view that they are aborted third premolars, which
constitute a human type of dentition similar to that of the New
World Apes’). From the investigation of Duckwortu the following
must be mentioned. Firstly that the occurrence of these rudiments of
a third premolar is exceedingly different with the different races :
in 3800 old Egyptian skulls he found no single case, on the other
hand in some thirty skulls of Australians he found these rudiments
seven times. The set of teeth of the natives of New-Britain shows
this anomaly exceedingly often. With respect to the set of teeth of
the Anthropoids, DuckwortH mentions that with seven of the thirteen
skulls of gorillas, which he investigated, the rudiments in question
were present whereas on the other hand he found them nota single
time with Hylobates nor with Orang-outans or Chimpanzees.

The reasoning of those who think that the first milkmolar and
premolar, following on the canine tooth have fallen out, in the
passing from the platyrrhine form to the catarrhine form is of quite
a different nature. It is a fact which is generally acknowledged as
being true that originally the number of premolars of the primitive
Primates did not amount to three but to four; so that already the

1) W. H. L. Ducxworrn. Studies in Anthropology. Cambridge 1904. p. 22.

55*

( 784 )

platyrrhines’ also, with their three premolars and milkmolars, possess
a reduced set of teeth, and indeed among the group, of the primitive
Primates which Ossorn puts together under the general name of
-- Mesodonta, forms are found with which both in upper and lower
jaw still four premolars occur (++ Hyopsodus). According to the
investigations of Lecue the number of four premolars has decreased
io three, because the premolar which follows immediately behind the
canine tooth — so the first or front of the row — has become lost.
As most convincing for this opinion of Lecur the set of -+- Micro-
choerus may count, where only three premolars occur in the upper
jaw, and_ still as many as four in the lower jaw, but of these the
first of the row has been reduced to a rudiment without function.
Where it is as good as proved that the reduction of four to three
premolars with the primitive Primates has been brought about by
the disappearance of the premolar following immediately behind the
canine tooth, where moreover we know that with other animals also
reduction of teeth may take place in this spot, there it is quite
comprehensible that the further reduction of four to three premolars
in the same place is localized. The difference between the two
explained opinions is easily made recognizable by writing down the
complete tooth formula of the primitive Primates and that of the
now living ones.

For the primitive Primates we get the following formula in which
the probably original number of /neésivd has not been reckoned with :

Reel Galle ea lis ase, 2
ee lg ts Ore lke vos tls Pe BR dh WAS Be Sh

a Le Ove. all. Snellen 2 3) AS Ceca
(ely PRG aly Je be Aoeee, 24

For the Platyrrhines (Cebidae) the formula becomes :

ee es Ge 5 :
he ln Oe (Gale aoe We Syisk 4h WA lls Bs aks

~
=
~

S)
bo
ce

Us elise recs

lil: 2 Ge

ei ;
aeons OM ches

According to the opinion of Anthropologists this formula becomes
for the ecatarrhine Primates :

EAD e Os Ie ee OG ee se
i ED wes aon OF Qn Ste Ove een aa
i. A. Dic, Le OAS Oren eS
T= Ke ON Gea ee 0 emake

( 785 )

so the milkmolars and premolars of man should be the original 2"¢
and 3'¢,
According to the theory, mentioned in the second place, the for-

mula becomes :

If Pp (Carlier eee WS al)

abe wile ane OR OY

oo

4,
AT MeL 2 oe

oo

te

1
1
52] eo oe ee
Pie Ci lekP> 0.0) 3-4.

so that with man the original 3'¢ and 4 milkmolar and premolar
would still exist.

The last mentioned opinion seemed to me also to be the most
aeceptable. [t pleads for it, that in a phylogenetically older stadium
the first milkmolar and its premolar had already become lost, and
if then one lets the second follow, the reduction-process is localized
and more continuity brought into it. The following can moreover
be said against the opinion of the Anthropologists, that the fourth
milkmolar and premolar with man should have been lost. It may
be justly supposed that only those teeth can reduce, which fulfill
the smallest function. And this now does not apply to the last milk-
molar and premolar. On the contrary. With the Platyrrhines we see,
that just that last molar does not only not stay behind by the others,
but is even the strongest developed of the three. So with those
forms, where we might with some right suppose at least some indi-
cation to a reduction of this tooth, we on the contrary find a pro-
gressive development. No single reason can be given why in the
middle of this toothrow a tooth should suddenly have disappeared,
and why a discontinuity of the set of teeth should have come about
by which the function would have suffered considerably, no single
indication can be found, neither in the ontogenetic nor in the full-
grown set of teeth, in the form of a diastem, that a tooth has really
become lost here, and so the first mode of explanation: that the
last milkmolar and its replacing tooth would have become lost, does
not seem probable to me.

But neither can the theory that at the passing from the platyrrhine
to the catarrhine type, the first milkmolar with the premolar belonging
to it, should have been linked out, satisfy me. The above mentioned
argument about it, is always only an argument per analogiam
without its being possible that a morphological proof for such a
reduction can be given. If the sets of teeth of Platyrrhines are com-
pared in particular with relation to the degree of development of
the first premolar nothing is found that points to a reduction of this

( 786.)

tooth, at least with the now living forms; on the contrary, the first
premolar is often stronger than the second (Cebus, Chrysothrix,
Mycetes, Hapale). Then not a single indication can be observed in
the Ontogenese of the set of teeth of man, which indicates that a
tooth should have become lost behind the canine tooth, the papillae
succeed each other very regularly in their origin and place. Moreover
if it were right that a molar and a premolar became lost behind
the canine tooth the remarkable fact remains still unexplained, that a
rudimentary tooth appears so often between the first molar and the
last premolar.

So I cannot agree with either of the two opinions which prevail
now about the differentiation of the set of teeth of the Primates,
but I am of the opinion that it came about in quite another way.
To be short, my opinion about it is as follows: the set of teeth of
the catarrhine Primates has originated from that of the platyrrhines
by the disappearance of the last or third molar and of the last
or third premolar while the third milkmolar has lost its character
of temporary tooth and has become a permanent tooth.

This opinion is explained by the two following formulas. If we
overlook the original number of four milkmolars and premolars,
and number the elements of the platyrrhine set of teeth according
to their now present amount this set of teeth may be written
according to the following formula:

DAT CEG ceapllct Loe eae
(eee ap eae Uta Ulta rp. Jn shane ites
Pics dann Gn pv iIUe potters Vie
Yell lepen Cne od Caper) epee

The catarrhine set of teeth has originated from this, as P, and M,
have fallen away in the upper and lower jaw, whereas i, becomes
M, in both jaws, by which as a matter of course J, of the
platyrrhines becomes .J/, of the catarrhines, the /, of the former
becomes J/, of the latter. If it had remained the J/, of the platyrrhines
would have become J/, of the catarrhines. Those things are stated
in the following formula in which the reduced teeth are put between

parenthesis.

ue. 2° 1 2 3
dcvisieites ge Tel placa alee OM
i LS ee Gn en 2 1

So the differentiation of the set of teeth of the Primates is accord-

(787)

‘mg to my view more complicated than would have been the case
according to the two above mentioned excalation theories. Two pheno-
mena may be distinguished in this process of development, namely
progressive development of one of the elements: m, loses its character
of temporary element and becomes persistent, and the second pheno-
menon is the reduction of two other elements, These two elements
are at the extremity of each of the two tooth series, P, at the end
of the series of replacing teeth, J7, of the end of the series of
.the teeth of the first generation. Contrary to the two above mentioned
excalation hypotheses I might distinguish the one defended by me
as the hypothesis of the terminal reduction. I shall try to show
the correctness of my Opinion,

If I let m, of the Platyrrhines become a persistent tooth, no
new principle is introduced into odontologie. For it is kwown to us
from other groups of animals that milk teeth may become persistent
teeth; I remind the reader of the Marsupialia, where but for some
exceptions the whole set of milk teeth has become a set of persistent
teeth except a single tooth. Furthermore to Erinacaeus, where according
to the investigations of Lxcnr the so called persistent set of teeth
consists partly of milk teeth partly of permanent teeth. So my
opinion is nothing more than a new example of the tendency also
observed elsewhere of a diphyodont set of teeth to pass into a
monophyodont. So against the principle as such there ean be no
objection.

As a first argument for the correctness of my opinion I state the
morphology of the milk molars in platyrrhines, I had the opportunity
-of studying them from Hapale, Chrysothrix, Cebus, Mycetes, Pithecia
and Ateles. Without going into details it must only be mentioned here
_that m, of the platyrrhines differs a great deal both in the compo-
sition of its crown and in the number of its fangs from im, or m,
and shows much resemblance to J/, of these apes.

It is of much importance with this that im, is functionally a
higher developed tooth than its deciduous tooth /,, so that means
that at the moment that m, is replaced by P,, the set of teeth
becomes to a certain degree functionally inferior. So if m, becomes
persistent, this means a gain for the mechanism of the set of teeth.
This does not hold true for m, and m,, the replacing P, and P,
are functionally higher developed,

A second motive is derived from the development of the set of
teeth of the catarrhine Primates, in particular that of man. So according
to my opinion our first molar has passed from a mitk-tooth into
“a persistent tooth in a relatively recent stadium, with the Platyr-

( 788 )

rhines it still belongs to the milk teeth. May this not be the
explanation of the fact that our first molar still breaks through
in connection with the teeth of the set of milk teeth, and _ still
before the appearance of the first replacing tooth, while the
second tooth appears only after a period of some years? By this
early appearance of our first molar it functionates indeed for
some time together with the complete set of milk teeth and so
according to my opinion the set of teeth of man still possesses in
this period a composition as in the first lifetime of the Platyrrhines.
Still more distinctly than from the time of the eruption this relation
appears when the first forming during the ontogonese is more closely
investigated. I derive the following about this from the wellknown
investigation of Rose '). Between the 9" and 12 week of the faetal
development the papillae of the milk teeth are invaginated in the
dental band (Zahnleiste) which grows on uninterruptedly towards the
back, and already in the 17 week of the development the papilla
of the first molar is invaginated. So with man there is not the least
histogenetical discontinuity between the forming of the milk teeth
and of the first molar. Only after the course of a year, so
four months after the birth the dental band begins to grow on
towards the back and not before the 6% month after birth the
papilla of the second tooth is invaginated. So while J/, is formed
immediately after m, with man, a pause of about a year begins
after this first development. So both from morphology and ontogeny
arguments may be derived for the hypothesis that m, of the Platyr-
rhines is homological to 7, of the Catarrhines.

My hypothesis however still contains another element viz. the
reduction of P,; and M, of the Platyrrhines.

Let us first consider the reduction of /,. From my above men-
tioned deduction of the catarrhine set of teeth given in a formula,
follows that I come in conflict with a rather generally accepted
opinion that the three molars of the Catarrhines should be homo-
logue to the three molars of the Platyrrhines. According to my
opinion J/, of the Platyrrhines is homologue to J/, of the Catarrhines,
M, of those homologue to J/, of these, and in the set of teeth of the
Catarrhines the homologon of J/, of the Platyrrhines is wanting.
If this tooth should also appear by the last mentioned group
of Primates it would become a J/,. Now it is a fact that is
universally known that a more or less developed fourth molar
is not seldom with man and among the Anthropoids, especially

1) G. Rose, Ueber die Entwicklung der Ziihne des Menschen. Arch. f. mikrosk,
Anat. Bnd, XX XVIII.

(789 )

with the Orang and Gorilla. Moreover Zuckerkanpi') has shown
that the epithelial rudiment of a fourth molar of man is formed
with the majority of the individuals. This rudiment of a tooth
and the eventual eruption of the fourth molar were till now
phenomena which were somewhat difficult to interprete. There
was an inclination to keep this fourth molar with man for an
atavism and the set of teeth of man was deduced from a hypo-
thetical primitive form when the set of teeth contained four
molars. Here however.the difficulty offers itself that among the
already numerous well known primitive Primates there has never been
found a form with four molars. ZucKERKANDL also reveals this diffi-
culty where he points to it that four molars should only appear with
the primitive forms of the carnivores. SELENKA®) also, who found from
his rich material that with Orang in 20°/, of the cases appears a
fourth molar feels the mentioned difficulty and interprets the
variation in another way. It should not be atavism but a progressive
phenomenon in that sense, that the set of teeth of Orang is on the
way of bringing into development a fourth molar. It appears to
me that this explanation of Senenka is not correct. If this variation
were only known to us from Orang, no direct difficulties could be
stated against this hypothesis. But such a fourth molar also occurs
as I said before very often with man. And now it is not doubtful
that the extremity of the human set of teeth is in a state of regres-
sion, the third molar is always more or less reduced and even

- according to the investigations of pr Terra *) and others issues no
more with at least 12°/, of the recent Europeans. Where it
is now fixed that our set of teeth reduces at its extremity, the
formation and issue of a fourth molar can hardly be interpreted as
a progressive phenomenon.

The hypothesis brought forward by me gives a simple solution
of the difficulty. The fourth molar of man and of the Anthropoids
is indeed an atavism but does not refer back to a removed primi-
tive form unknown to us, but does not go any farther than to the
nearest past of the history of development of our set of teeth, it
is the homologon of M, of the Platyrrhines. And contemplated in
such a way the relatively frequent occurrence of it can no longer
surprise us.

t) E. Zucxerxanpi, Vierter Mahlzahn beim Menschen. Sitzungsber. der k. Akad.
d. Wiss. Wien Bnd. C.

2) E. Sevenxa. Menschenaffen. Rassen, Schidel und Bezahnung des Orang Utan,
Wiesbaden 1898.

3) M. pe Terra. Beitriige zu einer Odontographie der Menschenrassen. Ziirich 1905,

( 790 )

More direct proofs may however be cited for the conclusion that
M, of the Platyrrhines should be reduced. For if the sets of teeth of
different representatives of this group are investigated, it is undeni-
able that J, is behind in development to M7, and M,.

Not all Platyrrhines are alike in this regard, with some species the
set of teeth is apparently very constant with other it is more varia-
ble. A particularly fixed set of teeth Chrysothrix seems to possess. I
could at least find not a single deviation in the 1380 skulls of Chry-
sothrix sciurea which I possess, no more in 60 skulls of Cebus fatuel-—
lus, although the J/, is already very much reduced with this species.
Ateles on the contrary seems to possess a set of teeth which is
richer in’ variations and Batrson') mentions three cases in which
the J/, which is already reduced in this genus quite fails. The men-
tioned author points to it that in these cases Ateles possessed a
formula for its set of teeth which is typical for the second family
of the Platyrrhines — the Hapalidae. And in connection with this
I may now examine the set of teeth of the Hapalidae in_ the
light of my hypothesis. This hypothesis puts that 1, of the Pla-
tyrrhines became lost in passing to the catarrhine type, that m,
becomes JM, and that P, no longer issues. Where a reduction
of JM, is not seldom found with the Cebidae, and now and then
even it is quite wanting as an individual variation, there M* is
already constantly absent with the Hapalidae. So with these Pla-
tyrrhines one phase of the process has already been run through,
but not yet the second phase, the progression from m, to M,. So
according to my opinion the set of the Hapalidae does not stand
as a deviating form at the side of that of the other Platyrrhines,
but must be considered as an intermediate form, between the original
platyrrhine and the definite catarrhine set of teeth.

So we see that several phenomena plead for my opinion, that the
catarrhine set of teeth has not originated by an exealation but by a
terminal reduction, and I must stop at my assertion that, because
m, has become M, the replacing tooth, which originally belonged
to it, ie. , no longer appears.

By this supposition, the observation of the anthropologists is done
justice to, that a rudimentary tooth does relatively often appear with
man and Gorilla between P, and J/,. When P, has only been sup-
pressed as a normal element of the set of teeth, in a relatively recent
period of the development, than the supposition lies at hand, that this
tooth also like MM, of the Platyrrhines ontogenetically will be formed

1) W..Baveson. Materials for the Study of Variation. London, 1894.

(791 )

still. And it is my opinion that the rudiments of a tooth which so
often occur in the indicated place are indeed traces of the P, which
has got lost.

There could still be mentioned some more anomalies in the set
of teeth of man (the growing together of J/, with a superfluous
tooth, the pushing out of J/, and replacing by a new tooth (so
called third dentition) which would be explained by my hypothesis,
but I will not look more closely into this matter in this place.

By my opinion about the differentation of the set of teeth of the
Primates I come into conflict with a rather universally prevailing
opinion about the morphological significance of the first molar ot
the Placentalia. This molar is universally considered with all Pla-
centalia as-a perfectly homological element of the set of teeth.
Thus says ScHLOsseR') e.g. speaking of the first molar of man:
“Niemand wird sicher die Homologie dieses Zahnes mit dem ersten
Molaren der itibrigen Placentalier bestreiten diirfen”. Where I now
homologise /, of man with m, of the Platyrrhines I come into
conflict with this opinion. If we however try to find motives for the
above mentioned opinion in literature, we seek in vain. And so it
seems to me, that here we have to do with a dogma, which is not
without danger for the comparative anatomy of the set of teeth.
For it lies at hand that as soon as in the whole row of the Pla-
centalia one element of the set of teeth is fixed in its morphological
significance, that then the homologating of the other elements must
join itself to this aprioristical principle. And where such a thing is
possible to a certain degree with a canine tooth, which is sharply
distinguished from the other teeth by its peculiar form, it is absolutely
impossible with a definite molar which possesses no specific mor-
phological qualities.

I cannot finish this communication before having pointed to a
phenomenon, which is immediately related to the here communicated
point of view. If we compare the set of teeth of man with that
of the other catarrhine Primates, it appears that the process, by
which the catarrhine set of the teeth orginated from the platyrrhine
type, is still progressive with man, and that the human set of teeth
is on its way to differentiate from that of the other Catarrhines in
the same way, as these differentiated from the Platyrrhines one time.
I shall try to show this in short. The still active differentiation of
the human set of teeth appears from different facts. First as to the
premolars. In comparison to all other Primates the premolars of men

t) M. Scutosser. Das Milchgebiss der Siiugetiere. Biol. Centralblatt, Bnd. 10, blz. 89,

(792 )

have been reduced considerably, and the 2"¢ premolar more than
the first. Where as the premolars in the upper jaw of all other
Catarrhines possess three and in the lower jaw two fangs, the
premolars of man have normaliter one single fang. That this
has originated from several, appears from the grooves on_ the
surface. Now it is not without significance that the first premolar
shows its origine of a form with several fangs, by a dividing of the
point of the fang. So P, is more reduced than P, with man. If
further the milkmolars, which temporarily precede the premolars
are compared, we state that the milkmolars differentiate progressively
in the group of the catarrhines, and this is especially the case with
the second milkmolar. The progression concerns especially the crown
of the teeth, the number of roots is two in the lower jaw, three in
the upper jaw.

So if we for a moment fix our attention exclusively on m, and
its replacing tooth P, with man, it appears that the first is in
progression, the second in regression, and that with man, the same
relation exists in regard to these two teeth as with m, and P, of
the Platyrrhines. When man namely pushes out his mm, and replaces
it by P,, his set of teeth becomes functionally inferior, for instead
of a tooth with tive or four knobs on the crown and two or three
fangs there comes in its place a tooth with two cusps on the
smaller crown and only one root.

So we see, that the terminal element of the dental band of
the second generation (P,) reduces with man. Still distinetly
may be seen the terminal reduction of the tooth band, of the
first generation, closing with J/,, for as is already mentioned our
M, no longer even issues in + 12°/, of all cases, and is always
behind in development, at least with more highly developed human
races. So the human set of teeth is characterized from the catarrhine
Primates by the following peculiarities ; the last molar is on its way
of reduction, the last premolar is on its way of reduction, the last
milkmolar has developed very progressively. So a trio of phenomena
which are entirely homological to those, by which the eatarrhine set
of teeth has originated from the platyrrhine. Only one phase is still
wanting to the process, namely the remaining persistent of the last
milkmolar and the suppression of the last premolar. And this phase
also is reached now and then individually. This among others
appears from what Magiror says: La persistance des grosses molai-
res temporaires (m,) sobserve trés-souvent, concurremment avec
Vabsence congénitale ou l’atrophie des secondes prémolaires (P,)

( 793 )

Nous en connaissons de nombreux exemples'). If the stated phe-
nomena are connected with each other the conformity to the earlier
process of development of the set of teeth of the Primates as I take
it, immediately strikes the eye; and one would be inclined to this
thesis; In the future set of teeth of man P, will no longer erupt,
m, will have become persistent and functionate as J/,, but by the
simultaneous reduction of J/, the number of molars will not have
become larger than three.

So from this communication appears that the differentiation of the
entire set of teeth of the Primates is from my standpoint more in-
tricate than was supposed till now, but it seems to me that my
principle of the terminal reduction can better be brought into accord-
ance with the function of the set of teeth, and is based on a
larger number of facts than the hypothesis of the excalation.
What from a general point of view, -also seems to me to plead
for my opinion is the fact, that in the exposition given by me the
development of the set of teeth has taken place without a disconti-
nuity in the toothrows at any time.

Physics. — “A simple geometrical deduction of the relations existing
between known and unknown quantities, mentioned in the
method of Voiwr for determining the conductibility of heat
im crystals. By Dr. F. M. Janeur. (Communicated by Prof.
P. ZEEMAN.)

(Communicated in the meeting of March 31, 1906).

It is commonly known that about ten years ago W. Voicr ’*)
indicated a method, based on a recognized principle of Kircunorr,
by which to determine the relative conductibility of heat in crystals
in the different directions. His mode of experimental examination
consists in the determination of the break which two isothermal lines
present at the boundary line of an artificial twin, the principal
directions of which form a given angle g with that line, whilst the
conduction of heat takes place along the line of limit. The isothermal
lines are rendered visible to the eye by the tracings formed by the
fusion of a mixture of elaidie acid and wax with which the plane
of the crystal has previously been covered.

1) K. Maeiror. Traité des Anomalies du Systeme dentaire. Paris 1877. p. 221.
») Voict, Géttinger Nachrichten, 1896, Heft 3.

( 794 )

The method of Voter is far more accurate than that of DE Sinar-
mont’) or even of ROnTGEN *), and, requiring for other purposes to
investigate the relative conductibility of heat in erystals, it was
obvious ‘that I should make use of the method indicated by Voter.

For a erystal, for which oe rotatory coefficients, found in accord-
ance with the theory of G. C. Srokrs*), are = 0, Vorer deducts
the relations required here i constructing the equations of the flow
of heat, conformable to the conditions of limit which are common
to the lateral boundaries of both plates; i.e. that along that line the
loss of temperature must be the same, and moreover that in a
direction normal to that boundary-line the entire flow of heat must
be the same in the two contiguous plates.

Prof. Lormnrz had the kindness to derive the above mentioned
relations in an analogous manner and to note down the conditions
under which the break in the isothermal lines will reach its maximum.

If ¢ be the break, and ¢ the angle, formed in the plates by the
two principal directions, is 45°, the proportion of the two coefficients of

te: Aving rete
the conduction of heat in those directions, consequently — is found
a
as follows:
(4,+4,)

tgs = (4,—A,) 222 .
CSAS IELC

If » differs from 45°, Vorer finds in that case:
(A,—A,) sin 2
(a, +4,)—(4,—2,) cos 2’
which for g equal to 45° passes into the formula of Prof. Lorunrz

tgp =

, é < = : ;
by introducing ty > (= ly8 according to Voiet’s deduction) instead
< =

of tgs.
Instead of the complicated formulae which are required for the
determination of these relations, we here give a simple geometrical

é : ay. :
demonstration, which, besides presenting in a form which is imme-
3
diately available for logarithmic calculations, possesses at the same
time the advantage of being easily discernible.
If, from a given point V in the centrum of a crystal, a flow of heat

can take place without interruption in all directions, the isothermal

1) pe Senanmont, Compt. rend. 25, 459, 707. (1847).
2) Rénreen, Pogg. Ann. 151, 603, (1874).
5) Sroxes, Gambr. and Dublin Math, Journal. 6 215, (1851).

(795 )

surfaces in a similar plane of a crystal are, in most cases, concentric
and equiform three-axial ellipsoids whose half axes stand in the
relation of Va,,Va, and /4,; among these the so-called principal
ellipsoid A, “whose axes are Vi,, V4, and Va, must here be kept
more especially in view.

In the present case we leave unnoticed the rotatory qualities of
the erystal, and suppose an infinitely thin plate, cut parelel to a
plane of thermic symmetry, whose principal directions correspond to
the coordinate axes. Let fig. 1 represent the elliptic intersection
of the plate with the ellipsoid 4; the line traeed by the melted wax
then has the direction of the tangent of the ellipse in the point
P(x’y’), given by the radius vector e, which may enclose the angle
g with the axis Y. The flow of heat may thus proceed along 9,
being the boundary line. In this case the equation for the isothermal
line pq is: ‘

t t
LU yy

=k;

X

Fig. 1.

Thus for the two sections Op and Og cut off on the two axes
the result is:

a 2,
0, SS SS
yo. sin Pp
Og = As — sail
zz OcOsg”
therefore :
O

te cot @p.
q A,

( 796 )

On the other hand however :
& &
90° — G ao >) =r (v == = |h
\ 4 2

é Tees 4 a 2
where z is half of the break of the isothermal lines at the boundary

e— tg d= tg

My

a
line OG.
The immediate conclusion is therefore :

Hau(e +5 )og o 4-25) one
re 2

From this equation the required proportion may be at once dedu-
ced when g represents the direction of the plate and the value of
has been ascertained.

Moreover it will be easy to find the maximum of ¢ — and thus
reduce the errors of investigation to the lowest figures. Suppose

A : : :
A=-—, the above stated formula, after a few goniometrical trans-
2
formations becomes :

, & (A—1) sin 29
ee (A+1) — (A—1) cos 29°
Sue : : 5 GE .
This function will be a maximum for TF ——1() ees
( pp

de 2 {(A*—1) cos 2p — (A—1)}__

= 0.
dp (A?+1) — (A?—1) cos 29

The maximum condition then becomes :
Vea ay eae
cos 2p = — =
ALE ae

and the appertaining maximum break ¢ in the isothermal lines is

?

then expressed by :

raze )

Rh}

In cases where the difference between 2, and {/2, is very small

— and observation teaches that this is usually the case — the
notation may be:

£ 2,—A4,

tg —

= ——_—_ (6!
2 Wee a

For practical purposes therefore, the theoretical maximum g = 45°
may be taken as fairly accurate, so that then the twin plate with

the isothermal lines ete., takes the form of fig. 2. In that case it
follows from A:

(D)

Fig. 2.

. a é . 2 .
By expressing tg as a function of fy 5 from (C’') one obtains the

relation deduced by Prof. Lorenrz;

(4, + 4,)
tg © = (a, — 4,) Tuk :
-_ F| *2

Moreover from the geometrical solution here given the fact is again
brought to light that in general the angle y is not equal to 90°; in
other words in this simple but experimental way is proved by
occular demonstration the truth of the statement already made by
Vorer, that the isothermal lines in crystals do not generally stand
perpendicular to the direction of the flow of heat.

Along the thermic axes however this is the case, because the
tangent lines at the ellipses are there directed perpendicularly to
these axes.

From fig. 1 also follows the form of the break as a result of
ae

I hope soon to communicate the results obtained in the measurement
of crystals by means of this method, together with a few observations
on the differences of these results with those, derived in the same
minerals by the usual methods of DE SENARMONT and RONTGEN.

56

Proceedings Royal Acad. Amsterdam, Vol. VIII.

( 798 )

Botany. — “On plants which in the natural state have the character
of eversporting varieties in the sense of the mutation theory.”
3y Dr. W. Burcx. (Communicated by Prof. J. W. Motz).

(Communicated in the meeting of March 31, 1906).

An investigation of the causes of Cleistogamy') showed that: 1
plants with closed flowers originated by mutation from plants with
chasmogamic flowers and 2 that they occur in the natural state,
partly as constant, partly as ever-sporting varieties.

In the course of this investigation the question arose whether other
wild-growing plants do not also have the character of ever-sporting
varieties.

Especially those plants were thought of that have bisexual and
unisexual flowers in one and the same individual or in which by
the side of bisexual, unisexual individuals are found and also those
plants among the dioecious ones that possess rudimentary stamens or
ovaries, from which may be inferred that they originated from plants
with bisexual flowers.

The agreement between unisexual, cleistogamic and filled flowers
pointed to the same origin, while the resemblance in the manner
in which unisexual flowers occur among the hermaphrodite ones and
closed flowers among the chasmogamic ones, justified the assumption
that in the monoecious and dioecious as well as in the cleistogamic
we have ever-sporting and constant varieties.

This summer I tried to confirm this conception in a twofold manner,
firstly by cultivating the gyno-monoecious Satureja hortensis and
secondly by studying the different forms in which one and the same
andro-monoecious Umbellifer can occur in nature with regard to the
number of male flowers in proportion to that of the bisexual ones
and to the place which the male flowers occupy on the principal
and secondary axes.

To the results of the culture experiments I shall return afterwards
when I shall have had an occasion to repeat these experiments on
a larger scale and with more species. I will here only mention that
they showed that a gyno-monoecious Satureja hortensis begins its
period of flowering with producing bisexual flowers only, that not
until later, when the plant has grown stronger, a few female flowers
appear among the bisexual ones, that their number gradually increases

1) Die Mutation als Ursache der Kleistogamie. Recueil des Travaux Botaniques
Néerlandais Vol. I. 1905.

(799 )

in the following days until a definite maximum is reached, after
which it gradually deereases again until at the end of its flowering-
period the plant again produces bisexual flowers only.

Hence the female tlower follows the law of periodicity established
by pe Vries for the occurrence of anomalies of various nature with
other plants and it may in this respect be put on a line with sueh
anomalies. It may be compared with the increased number of leaflets
of Trifolium pratense quinquefolium, with the filled flowers of
Ranunculus bulbosus semiplenus, with the ramified spikes of Plantago
lanceolata ramosa, ete.

In what follows I shall give the results obtained with the andro-
monoecious Umbelliferae.

The investigations of bBeeErinck '), Scuunz?), Kircuner *), Mac
Leon‘), Lorw'), Warnstorr'), and others on the sexual relations
of the Umbelliferae have shown that by far the most species are
andro-monoecious and that besides in some of them forms occur with
female or with female and asexual flowers. Male flowers appeared
in this family to be as common as bisexual ones. Male individuals
are rare, however. Until now Trinia glauca was considered the
only Umbellifer in Europe, known in the male form. From Scuutz’s
notes it appears, however, that in the environs of Halle a. S. also
male plants of O¢enanthe jistulosa’) and Siwm latifolium *) oceur,
while in this country also Heraclewin Sphondylium can occur in the
male form.

Far less general are female flowers. ScuuLZ only mentions them
for (Lryngium campestre)?*), Trinia glauca, Pimpinella magna,

1) Bewertncx, Gynodioecie bei Daucus Carota L. Nederlandsch Kruidkundig Arch.
Tweede serie 4e Deel 1885, p. 345.

2) Auausr Scnunz, Beitriige zur Kenntniss der Bestiubungseinrichtungen und
Geschlechtsvertheilung bei den Pflanzen. Bibliotheca botanica. Bd. If 1888, Heft 10
und Bd. IIIf 1890, Heft 17.

3) O. Kircuner, Flora von Stuttgart und Umgebung 1888.

4) J. Mac Leop, Over de bevruchting der bloemen in het Kempisch gedeelte van
Vlaanderen. Botanisch Jaarboek Dodonaea 1893 en 1894. :

®) E. Loew, Bliitenbiologische Floristik des mittleren und nérdlichen Europa
sowie Grénlands. 1894.

6) C. Warnstorr Bliitenbiologische Beobachtungen aus der Ruppiner Flora im
Jahre 1895. Verhandlungen des botanischen Vereins der Proving Brandenburg
Bd. XXXVIII. Berlin 1896.

7) Scnutz, Beitr. I p. 47.

8) Scuuuz, Beitr. I p. 48.

®) In his note concerning this plant on page 42 of his first paper, female
flowers are not mentioned. So this is perhaps an error in the general summary
at the end of the second paper.

56*

( 800 )

P. savifraga and Daucus Carota, for which latter plant BetErinck
had already found them before.

In the long list of 66 European Umbelliferae in the Bliitenbiologische
Floristik of Lozw no more than 16 species occur that are only
known as bisexual plants whereas 40 are andromonoecious. It has
appeared since that with three of the plants meutioned as bisexual
also male flowers are found. Of Anethum graveolens, Aethusa
Cynapium and Heracleum Sphondylium namely, Warnstorr found
andromonoecious forms in the environs of Neu-Ruppin; also in this
country they occur in this form. Of the 66 Umbelliferae that were
studied, the following remain of which until now no other than
bisexual plants are known:

Laserpitium pruthenicum, Peucedanum venetum, Crithmum mariti-
mum, Silaus pratensis, Seseli Hippomarathrum, S. annuum, Anthriscus
vulgaris, Bupleurum longifolium, faleatum, tenuissimum and Pleuro-
spermum austriacum, to which list I think must be added: Hryn-
gium maritinum, Berula angustifoha, Conium maculatum and
Helosciadium nodiflorum.

It is probable that of some of these plants andro-monoecious forms
will be found when they are examined over a larger part of their
region of occurrence, especially since it has appeared that the different
forms in which Umbelliferae can occur, are often spread over very
different and widely distant parts, so that, even though the species
mentioned be only known as hermaphrodite plants in a part of
Europe, the possibility must be granted that they occur in other
forms elsewhere.

Of Sium latifolium e.g., no other but the andro-monoecious form
is found in a great part of Middle Europe and until now only in
the environs of Halle a/S accompanied by the male form, evidently
only in a few specimens. Only in our country the bisexual form
is known.

Of Pimpinella magna the bisexual plant is only found in southern
Tyrol and Italy; the andro-monoecious on the other hand in the
whole of Middle Europe, while in southern Tyrol and Italy the
same plant also occurs with female and with female and asexual
flowers.

Of Oenanthe jistulosa the andro-monoecious plant is found every-
where, the male one until now only in the environs of Halle.

Of Aethusa Cynapium the hermaphrodite plant is known in the
whole of Middle Europe, the andro-monoecious one only in the
neighbourhood of Neu-Ruppin and of my residence.

Of Daucus Carota the andro-monoecious form is generally found,

( 801 )

the bisexual one until now only in Flanders!) and in this country *).

So it is not at all unlikely that of those species which until now
are known as bisexual only, later other forms will also be found,
and similarly it may be assumed that of the large number of Um-
belliferae of which now only the monoecious form is known, on
closer examination also the hermaphrodite or unisexual forms will
be found.

Meanwhile it is a very remarkable fact that by far the most
Umbelliferae are andro-monoecious and that exactly these forms are
most generally spread.

Where male individuals are found they only occur in very
limited numbers as rare occurrences among the great majority of
andro-moncecious individuals.

This also holds for the hermaphrodite plants, at any rate for Daucus

‘arota, Sium latifolium and Heracleum Sphondylium. Where these
and andro-monoecious plants occur together the number of bisexuals
is far less than that of the andro-monoecious ones. *)

This general occurrence of andro-monoecious forms gives a very
peculiar character to the family of the Umbelliferae. Nowhere in
the vegetable kingdom these forms are so prominent as here.

In other families with species that are rich in forms, as the
Labiatae, Alsineae, Sileneae and others, where gyno- and andro-
monoecious and female and male forms occur together with bisexual
ones, a similar preponderance of monoecious plants is not found
with a single species.

The rule is there that where the three forms occur together the
monoecious flowers are a minority with respect to the bisexual and
unisexual ones.

Next is conspicuous with the monoecious Umbelliferae the great
variety that may be observed in the occurrence of the male flowers
in the umbels of different order and the many mutually different
forms in which consequently one and the same andro-monoecious
plant may occur.

Sometimes an individual is found which among the large number
of bisexual flowers has a relatively small number of male ones,
another time one in which the number of male flowers is not much

1) J. Srars. De bloemen van Daucus Carota L. Botanisch Jaarboek, Dodonaea
Jaargang I. 1889. p. 182.

2) I shall soon treat elsewhere the different forms in which the Umbelliferae ,
occurring in this country, are met.

3) Male Umbelliferae and exclusively bisexual species are very rare also outside
Hurope. (See Drvpe Umbelliferae. Encuer und Pranti. Die natiirl. Pflanzenfamilien
lil. Teil. Abt. 8. p. 91).

( 802 )

less than that of the bisexual ones, and then again an individual in
which the male flowers are more numerous than the others, and
between these a long series of gradual transitions and intermediate
forms is found.

Not unfrequently the number of male flowers is greatly in excess
of the bisexuals. I met in this country plants of Heracleum Sphon-
dylium in which the inner umbellules of the umbel of the first order
and all other umbels of higher order were exclusively male and
similar plants are also found of Pastinaca sativa and Daucus Carota.
They are found spread among other individuals in which the propor-
tion of male to bisexual flowers is more favourable to the bisexuals
or where the number of males is even very small.

Some Umbelliferae are only known in an almost male form.
“chinophora spinosa e. g. has one bisexual flower in the middle
of the umbel; all other flowers are male. Also with MJewm athaman-
ticum and Myrrhis odorata we may observe in the specimens cul-
tivated in this country in botanical gardens, how also there the
bisexual flower is superseded, so that the umbellules often do not
contain more than one such flower.

An investigation of the andro-monoecious Umbelliferae shows us
at once that there is a certain regularity in the way in which the
male flowers occur. In the first place, when they appear for the
first time in an umbel of a certain order, their number as com-
pared with that of the bisexual flowers increases as we come to
umbels of higher order; and secondly, if in the peripheral umbellules
some male flowers occur among the bisexual ones, their part in
the constitution of the umbellules becomes greater as the umbellules
are more distant from the periphery.

Of Daucus Carota, Pastinaca sativa and Heracleum Sphondylium
whole series of specimens may be collected in the neighbourhood
of my residence, beginning with such which in all the umbels econ-
tain only bisexual flowers up to forms which are almost or entirely
(H. Sphondylium) male. Among these specimens are found in which
the male flowers already appear in the very first umbel of the plant
by the side of other specimens in which the andro-monoecious cha-
racter only appears in the umbels of the second order or later still
in those of the third or fourth order. Now it is a constant rule that
if they appear for the first time in an umbel of a certain order they
will also appear in the umbels that develop later and that their
number in proportion to that of the bisexual flowers in the succes-
sive umbels goes on increasing.

( 803 )

Specimens which in no respect revealed their andro-monoecious
character during the whole summer, which only late in summer
produced male flowers in the umbels of the third or fourth order
or sometimes entire male umbels, are found connected by interme-
diate forms with specimens which already in the very first umbels
contain male flowers.

Concerning the part occupied by male flowers in the constitution
of the peripheral and central umbellules, it must be remarked in
the first place that with all Umbelliferae whose umbels reach a
certain size, the peripheral umbellules consist of a larger number of
flowers than those that occupy the middle part of the umbel. In
some species those central umbellules may be very poor in flowers ;
with Daucus Carota the central umbellules often even consist of
only one flower.

When it was stated that the part occupied in the umbellules by
the male flowers becomes greater the more they are placed near
the centre of the umbel, this must be so understood that as the
umbellules become more distant from the periphery the number of
bisexual flowers decreases and does so much more rapidly than the
number of male flowers. Hence the inner umbellules are often
entirely male while the outer ones bear a number of bisexual
flowers.

This rule is not without exception, however. There are namely
Umbelliferae in the umbels of which the central umbellule occupies
the top of the principal axis of the umbel and may consequently
be distinguished as the top-umbellule.

Such top-umbellules are especially found with Carum Carvi and
Oenanthe jistulosa and occasionally, although not so regularly, also
with Daucus Carota. For such a top-umbellule now the rule does
not hold that the part occupied by the male flowers is greater than
in the surrounding umbellules. Such an umbellule contains a greater
quantity of bisexual flowers. With Carwm Carvi I often found no
male flowers in the top-umbellule when all others, as well the
peripheral as the more inwardly situated umbellules had some of
them. In other specimens the number of male flowers in this top-
umbellule was smaller than in the other.

Of O5nanthe fistulosa the umbels of the second order are in this
country much larger than those of the first order; they consist of
five to eight umbellules and agree in their constitution almost entirely
with that, indicated by Scuunz for the umbellules of the first order.
Here as a rule a top-umbellule can be very easily distinguished ; it
contains only a few (7 to 9) male flowers, but is for the rest entively

( 804 )

hermaphrodite, while the side-umbellules are generally exclusively
male.

With Daucus Carota, where the umbellule as was remarked
above, often consists of no more than one flower, this latter is very
often hermaphrodite, also when the surrounding umbellules consist
entirely of male flowers.

It must still be remarked for the andro-monoecious Umbelliferae
that both sorts of flowers as a rule occupy a fixed place in the
umbellule.

In by far the most Umbelliferae the bisexual flowers are found
near the edge and the male ones in the middle.

Only a few make an exception to this rule; with Oenanthe jistulosa
and Sanicula europaea the opposite is found and with Astrantia the
bisexual flowers as a rule occupy a definite zone between the peri-
pheral and central male flowers. Advancing from the circumference
to the centre we find there first one or two whorls of male flowers,
then a whorl of bisexual ones and finally at the centre male flo-
wers again.

But although it may be the rule for all other Umbelliferae that
in all the umbellules, containing the two forms of flowers, the her-
aphrodite ones are placed at the edge and the male ones in the
middle, an exception must be made for those Umbelliferae which
in the middle of the umbellules develop a top-flower, for this latter
is as a rule bisexual.

Such top-flowers are e.g. regularly found with Chaerophyllum and
with Meum; in each umbellule of Chaerophyllum temulum and Meum
athamanticum bisexual marginal flowers and a bisexual top-flower
are found and for the rest male flowers.

Also with Aegopodium Podograria, Carum Carvi and Daucus
Carota bisexual top-flowers are found in the umbellules, but in these
species this top-flower is not always found in all umbellules.

No extensive argument will be needed to understand that the two
forms of flowers, found in the same individual of the plants men-
tioned, may be considered, like the two flowers of a cleistogamic
plant, as two antagonistic characters which mutually exclude each
other and that consequently these plants may be compared with
ever-sporting varieties, originated by mutation, the existence of which
was shown by DE Vrins.

Every andro-monoecious Umbellifera of which we compare a
number of individuals among themselves, affords an opportunity for
noticing that the two antagonistic characters evidently fight for

( 805 )

supremacy, in which combat now one, then the other gains an
advantage.

But if of a species which is rich in forms we mutually compare
a fairly complete series of andro-monoecious forms, we are struck
by the circumstance that between these and the ever-sporting varieties
known until now, there is this important difference that while
with other ever-sporting varieties the original specific character is
always more conspicuous than the racial character, here very often
the opposite takes place.

We met in what precedes plants like Myrrhis odorata, Meum
athamanticum or forms of Pastinaca sativa, Heracleum Sphondylium
and Daucus Carota, where the specitic character had been entirely
superseded by the racial character, and this raises the question whether
the andro-monoecious Umbelliferae, looked upon as races originated
by mutation, must be placed on a line with the above-mentioned
gyno-monoecious Satureja hortensis and other ever-sporting varieties.

We know from the theory of mutation that the interaction of two
antagonistic characters may show itself in more than one way and
that a character originated by mutation may be inherited in a different
degree in various plant-species, by which process various races are
formed.

To a race in which the anomaly comes only little to the front,
much less than the normal character, and which consequently is
hereditary in a small degree only, pn Vrins has given the name of
a half-race, and the abnormal character he has called sem?-latent.
That, however, among these half-races important differences may
occur in the measure in which the character is semi-latent, clearly
appeared from the statistical investigation of the half-races, e.g. of
Trifolium incarnatum quadrifolium and Trifolium pratense quinque-
folium.

It may be imagined that there exist races in which the two antago-
nistic characters possess nearly the same degree of heredity so that
then it is often difficult, under favourable circumstances, to settle
whether the specific or the racial character is more prominent and
sometimes even, when the conditions of life are very favourable,
the anomaly gets the upper hand. In such a race as well the specific
character as the anomaly are then to be considered as semd-active.
The statistical investigation of the anomalies has not yet revealed
that such races really exist.

But it may be further imagined that between these latter races
which pr Vries called middle-races and the constant varieties, in
which the specific character is latent and the anomaly active, there

( 806 )

exist still other races in which the normal character is semi-latent
to a different degree.

Dr Vrins thinks such cases possible, but until now they have not
yet been noticed’). Now the question arose to me whether in the
andro-monoecious Umbelliferae we may not have such races in which
the specific character has become semi-latent ? *)

Let us start our speculations with one of those Umbelliferae of
which besides andro-monoecious ones also hermaphrodite and male
forms are known, e.g. Heraclewm Sphondylium.

As was remarked above, Heraclewm-Sphondylium appears in a
ereat part of Middle Kurope as a hermaphrodite plant. In the
environs of Neu-Ruppin at the same time forms are however found
which are only bisexual in the umbels of the first order, whose
umbels of the second order are composed on half bisexual and half
male umbellules and whose umbels of the third order are exclusively
male, and which in consequence may be considered to produce about
as many male as bisexual flowers.

In this country now I found besides the hermaphrodite and the
Neu-Ruppin middle forms a great variety of forms which may be
considered either as gradual transitions of those middle forms to
perfeetly hermaphrodite ones or as gradual transitions of those middle
forms to perfectly male individuals, which latter occur also in this
country.

If we uow for the present consider this andro-monoecious plant
which is so rich in forms as an ever-sporting variety, and if we
compare its properties with those of Trifolium pratense quinquefolium,
which has first been extensively dealt with by Dr Vries, and later
has been investigated in all its details by Miss Tames‘), so that of
this race the properties are most completely known, then we begin
with asking what peculiarities Heraclewm should present if its mo-
noecious form represented an ever-sporting variety.

Then we should observe :

1. that a strongly developed specimen, e.g. a plant with umbels
of the first to the fourth order, produces more male flowers than
an individual which has not succeeded in getting beyond the formation
of umbels of the first and second order.

1) De Vries, Mutationstheorie, I, p. 424.

2) In my article on cleistogamic plants I already briefly raised the question
whether Ruellia tuberosa, Impatiens noli tangere, Impatiens fulva, Amphicarpaea
monoica, Viola spec. div. are not in this condition.

3) Bot. Zeit. Iste. Abt., Heft XI, 1904.

(2307)

2. that plants on fertile soil produce on the whole more male
flowers in proportion to the bisexual ones than plants on less fertile
soil.

3. that the male flowers only appear at a stage in which the
plant has grown stronger, that they gradually increase in number
as the individual grows stronger and gradually decrease in number
again when the plant has passed its highest point of development.

4. that in each umbel as well as in each umbellule which contains
both forms of flowers, the male flowers are preferably found in
those places which are most favourable with respect to nutrition.

It is not difficult to show that observation does not confirm these
four points.

Let us in the first place consider point 4.

There can be no doubt that (excepting the just mentioned terminal
umbellules and terminal flowers) the peripheral umbellules are more
favourably placed with regard to nutrition than the more inwardly
situated umbellules, and that in each umbellule the flowers at the
circumference also occupy a more favourable position than those in
the middle. This is seen not only by the inner umbellules being
less rich in flowers but also in the flowers becoming smaller the
further they are distant from the periphery; often the central flowers
do not reach their normal development or the setting of the fruit
does not take place. We see here the same with the umbels as with
long-drawn inflorescences like those of Capsella Bursa pastoris ox
Pisum sativum, that namely the last-formed flowers, at the top of
the inflorescence, no longer reach their normal development on
account of insufficient nutrition. Further every umbellule (not only
a mixed one but also a purely hermaphrodite one) allows us to
notice that the peripheral flowers are ahead of the central ones in
their development.

And now we see with all Umbelliferae without exception:

that the peripheral umbellules retain their bisexual character longest,

that the male flowers always occur first at the centre of the umbel,

that where the umbellules are mixed, the number of bisexual
flowers always decreases from the periphery to the centre,

that the inner umbellules often are already entirely male when
the outer ones still contain bisexual flowers, and

that everywhere, except with Oenanthe sistulosa, Sanicula europaea
and Astrantia the marginal flowers in the umbellules are bisexual
and the central flowers male’),

1) I think an explanation may be found for the anomalous behaviour of these
three genera. { cannot dwell on this point, however, in this short communica-

( 808.)

In other words, we may say that as well in the umbel as in the
umbellule. the biseaual flowers always occupy the place which is most
favourable with respect to nutrition.

That terminal umbellules and flowers are placed most favourably
is evident; it can be readily explained why a top-umbellule is often
richer in bisexual flowers than other umbellules from the centre
and why as a rule the top-flower of the umbellule is hermaphrodite.

That this position is by far the most advantageous can also be
inferred from the fuct that often the top-flower is the only bisexual
one of the whole umbellule. So with Mewm athamanticum e.g. it is
very often found that in the umbels of the second order, the 6—8
inner umbellules possess no bisexual flowers at all; the only bisexual
flower of these umbellules is the top-flower. *)

So we see exactly the opposite from what we should observe if
the andro-monoecious plant represented an ever-sporting variety like
Trifolium pratense quinquefolium. It is not the male flower — the
anomaly —- which is preferably found in the best places, but the
bisexual flower, and on further examination of the above points
1, 2 and 3 we shall again see how it is this latter that depends
on the nutritive conditions and in all respects behaves like a character
in a semi-latent condition opposed to the active condition of the
anomaly.

I pointed out already that with all andro-monoecious Umbelliferae
the umbel of the first order shows the anomaly least.

With very many forms the male flower appears first in the umbels
of the second order, with others in those of the third order, and
sometimes it is the umbel of the fourth order in which the male
flower appears first.

But where these flowers are already observed in the umbels of
the first order their number is there always less than in the umbels
of the second and higher orders.

The umbel of the first order consequently retains, in all andro-
monoecious Umbelliferae, the pure racial character longest.

If we remember that the umbel of the first order is at the same
time the terminal umbel of the plant and is extremely favourably
placed at the end of the principal axis with regard nutrition, we
cannot wonder at this, bearing in mind what was said when

tion. I shall return to it elsewhere when exposing the differences between the
forms occurring in this country and those that have been observed in other parts
of Europe.

1) This reminds us of what may be noticed with Hchinophora spinosa. Vide
supra.

ef

( 809 )

diseussing point 4. We find the already stated conception confirmed
that the bisexual flower, being in a latent condition with respect to
the anomaly, preferably occurs in the most favourable places.

We may also assume that the plant during the flowering of its
top-umbel, which only occurs after it has reached its full vegetative
development, is also in the strongest stage of its growth, in a stage
in which a good part of its nutritive material may be spent on the
development of its top-umbel, while all umbels that bud forth later,
are in less favourable conditions, first on account of their being
placed on lateral axes of the second or higher order and secondly
because a very great part of the nutritive material is spent on the
ripening of the fruit of the first umbel during the development of
the umbels of the second or at any rate higher orders. This would
explain why in the umbel of the second order the semi-latent bisexual
flower is no longer prominent in the same degree as in the terminal
umbel, and why in the umbels of the third and fourth order it more
and more gives way before the racial character.

This also explains why in very strong specimens the male flowers
first appear in the umbels of the third order, and why often with
Sium latifolium, Daucus Carota and others, not until late in summer,
when the plant has already passed its highest point of development,
male flowers and even male umbels appear in plants which in their
umbels of the first and second or first, second and third order have
exclusively produced bisexual flowers.

That in fact strongly developed specimens produce more bisexual
flowers than weak specimens was already noticed by Mac Lxop,

With strong specimens — he says in his note on Aeyopodium
Podagraria — the umbels of the first order and with very strong

specimens also those of the second order consist almost exclusively
of hermaphrodite flowers, while with ordinary specimens the
umbellules in the umbels of the first order consist partly and in those
of the second order exclusively of male flowers. Also Scuuz made
the same remark with Vorilis Anthriscus and Pimpinella savifraga
and personally I found the justness of his remark repeatedly confir-
med with Pimpinella magna, Aeyopodium Podagraria, Aethusa
Cynapium, Astrantia major ete.

If now finally the numerical relations of the two flower-forms are
examined in umbels of such species as are found in large numbers
on soils of different constitution and fertility, the examination at once
shows that the number of bisexual flowers in a fertile place is
considerably greater than in a less fertile one. Anthriscus silvestris
and Chaerophyllum temulum are plants which in our country are

( 810 )

very general as well on sandy soil (at the edge of the dunes) as on
fertile claygrounds. Both plants can be best judged by the constitution
of the umbels of the second order.

Of Anthriscus silvestris the average constitution is :

on sandy soil on clay ground
of the six outer umbellules 4-58111-137 7-103+-3-4¢
of the seven inner umbellules 2-4$-++ 8-11¢ 6-73 +4-7h

And of Chaerophyllum temulum :
of the outer umbellules 1584100418 208+-7¢+1%

while the 2 or 3 innermost umbellules of the plants on sandy soil
are entirely male.

So the results are in perfect agreement with my observations on
the influence of the fertility of the soil on the appearance of chas-
mogamic flowers with Ruellia tuberosa at Batavia and with those
of GorBeL on the chasmogamic flowers with /mpatiens noli tangere
in places of different fertility near Ambach *).

From what has been communicated here it appears that the andro-
monoecious Umbelliferae in the natural state have the character of
ever-sporting varieties in which the racial character, the bisexual
flower, is in a semi-latent condition.

By assuming this it becomes clear why the anomaly shows itself
least in the terminal umbel, why, after it has once appeared, it
increases in number in the umbels of higher order, why in each
umbel the number of hermaphrodite flowers decreases from the
periphery to the centre, why in each umbellule the bisexual flowers
are placed at the cireumference and the male ones at the centre and
why with those species in which the umbels have a top-umbellule,
this latter often has again relatively more bisexual flowers than the
surrounding umbellules and finally why, where in the umbellules
a top-flower is found, this is as a rule bisexual and holds out longest
when the umbellules grow more and more male, so that it often
still occurs in such umbellules where the bisexual marginal flowers
have already had to give way to the male ones.

Although I am of opinion that many things plead for my conception,
yet I am perfectly aware that certainty about the true nature of the
race, about the influence of fluctuating variability on the numerical
relations between bisexual and male flowers, about the question
whether perhaps locally different varieties or ever-sporting varieties

1) Gorsex. Die kleistogamen Bliiten und die Anpassungstheorién, Biol, Centralbl.
Bd. XXIV. No. 24, p. 770.

( Sit )

may exist of one and the same Umbellifer and other related questions
can only be obtained by culture experiments and statistical investi-
gation.

Yet I thought it worth while to communicate these observations
although they must only be considered as an exposition of the
grounds why culture experiments were undertaken. If may be useful
to indicate these grounds, first because they support my conception
about the racial character of many cleistogamic plants, and further
because in my opinion we may certainly expect that besides monoe-
cious and cleistogamic plants, other plants in the natural state will
turn out to have the character of races originated by mutation, so
that this communication may to some extent draw attention to this
point.

The culture experiments will from the nature of the case oceupy
a few years.

In the Erganzungsband of Flora 1905, Heft I, p, 214, Goxprn
communicates as a sequel to his paper ‘Die kleistogamen Bliiten
und die Anpassungstheorien” the results of his continued culture
experiments with cleistogamic species of Viola. The results of his
experiments confirm his formerly pronounced opinion that the appea-
rance of a cleistogamic or chasmogamic flower depends entirely
on nutritive conditions. If these are favourable the chasmogamie
flower is seen to appear; in the opposite case the cleistogamie one
appears.

I communicated in my former article my objections to this con-
ception. I will now only remark that the influence of the nutritive
conditions shows itself in such a way that with favourable conditions
the semé-latent character is developed, and with unfavourable is
suppressed.

Now if in Goxrsen’s experiments the chasmogamie flowering is
suppressed when the plant is under unfavourable conditions, this is
because Viola is an ever-sporting variety in which the chasmogamic
flower is in a semi-latent condition. If the cleistogamie Viola be-
longed to one of the other ever-sporting varieties, if e.g. it were
an ever-sporting variety like the gyno-monoecious form of Satureja
hortensis or Trifolium pratense quinquefolium in which the anomaly:
(the female flower and the composite leaf) is in a semi-latent con-
dition, then under favourable nutritive conditions the anomaly, the
cleistogamic flower and under less favourable conditions the chas-
mogamic flower would be fostered.

( 812 )

Zoology. — “The uterus of Erinaceus europaeus L. after parturi-
tion”. By Prof. H. Srrann, of Giessen. (Communicated by
Prof. A. A. W. Husrecut).

(Communicated in the meeting of March 31, 1906).

Through the obliging kindness of my colleague Prof. Huprecut,
to whom I owe my sincere thanks, I was enabled to continue my
researches on the involution of the uterus post partum with a species
which, as far as I know, had not yet been studied in this respect.
The examination of a larger number of uteri of Erinaceus europaeus L.
made it possible sufficiently to investigate the regressive development
in question.

In the pregnant uterus of the hedgehog shortly before parturition,
pretty large foetal chambers are found, as was shown by Huprecut’s
extensive investigations. These chambers are entirely lined with
epithelium which extends a little under the edges of the discoid
placenta, the relative size of which is not very large. This placenta
consequently belongs to the stalked ones, although the stalk is a
very broad one. ;

The wall of the uterus of a hedgehog which was killed immediately
after parturition is accordingly almost entirely covered with an
epithelium which proved to consist of high, cylindrical cells. A
layer of epithelium is only wanting in a small antimesometral region
which is characterised as the site of the placenta by the large
vascular stumps.

Excepting the specimen just mentioned the time post partum could
not be determined in my preparations. So I had to arrange them in
a series according to the thickness of the uteri, beginning with such
as were still very thick and admitted of a determination of the number
of former foetal chambers by swellings corresponding to the placental
places and ending with others the appearance of which did not
reveal any traces of pregnancy. The sections obtained from such
uteri were in good agreement with each other and gave a sufficient
idea of the various stages of involution.

I will not give here a detailed description of the phases of the
retrogade development but only remark that the essential changes
occur in the connective tissue of the uterine mucous membrane and
in the glandular apparatus. The surface epithelium which with many
animals (e.g. with Putorius furo) undergoes considerable changes of
form, here shows these to a relatively smaller extent. They are
limited to the casting off of superfluous parts and to the change of

larger cells into smaller ones.

( 813 )

The epithelial defect of the placental spot is covered by epithelium
advancing from the edges by a similar process as has become known
of late years for a number of other mammals. Since a spot without
epithelium is found in several stages, it must be assumed that the
covering of the gap does not take place so rapidly as e.g. in many
Rodents.

Characteristic for the connective tissue is the great abundance of
liquid in it; after parturition it appears to be of a loose irregular
texture and contains a considerable number of large blood- and
especially lymph-vessels, the former especially in the placental spot, the
latter spread over the whole mucous membrane. In this connective
tissue during the first period following parturition only small and
irregularly shaped glands are found, with a low epithelium. These
glands oceupy little place in the pretty thick mucous membrane.

In the completely retrograde uterus I find a mucous-connective
tissue which is not particularly strong and is rich in cells; in this
long glands reach in a very graceful and regular arrangement from
the inside of the uterus to the musculature, while larger blood- and
lymph-vessels are lacking in it. (see fig. 1 in Husrecut’s Studies in
Mammalian Embryology. Quart. journ. of micr. sc. vol. XXX. new
ser.). A comparison of these two stages, representing the beginning
and the end of the involution, shows the direction of the involution,
It consists, not to speak of the just mentioned minor changes in the
epithelium, in the connective tissue becoming more compact, the
total calibre becoming considerably less, and in a re-arrangement of
the glandular apparatus which is probably accompanied by a new-
formation, but certainly with a re-arrangement and considerable
lengthening of the single glandular tubes.

In the connective tissue it is not so much the single cells which
change (as is e.g. conspicuously the case with the female dog post
partum) as there is a clear indication that intercellular substances
diminish, which finally leads to a consolidation of the whole tissue,

At the same time the swollen lymph-vessels become smaller and
narrower as well as the stumps of the torn blood-vessels in the
placental spot, the trombi of which organise themselves. The retro-
gression at the placental spot takes place distinctly more slowly
than in the remaining mucous membrane so that the placental spot
is still recognised as something particular when the gap in the
epithelium has become completely covered.

The return of the glands to their regular form takes still more
time than that of the connective tissue, perhaps its last phase only
sets in with a new rut.

57

Proceedings Royal Acad. Amsterdam. Vol. VIII.

( 814)

Comparing the puerperal involution of the uterus of the hedgehog
with the same process as it occurs in other mammals, hitherto studied,
we may state that in this respect the hedgehog occupies an intermediate
position between Rodents and Carnivora. It stands near the former
in the way in which the epithelium regresses, near some of the latter
in the regression of the layer of connective tissue, although in this
respect the analogy is not complete.

The more accurate details of the involutional processes of which
a short sketeh is given here, will be published elsewhere.

Physics. — “Magnetic resolution of spectral lines and magnete
force’. By Prof. P. Zerman. (First part).

The intensity of a magnetic field may be defined by the amount
of splitting up of a given spectral line emitted by a source placed
in the field.

The distance of the outer components of a triplet can be measured
with great accuracy. The components of a line resolved by the
action of magnetism are of the same width as the original line and
the high degree of accuracy obtainable in the measurement of spee-
trum photographs is generally known.

We may call two magnetic intensities equal, when producing
equal amounts of separation of a spectral line, and we may eall
two differences of magnetic intensilies equal, when the changes of
the distances of the components are the same. In this way we
obtain a scale of magnetie forces, the zero point and the magnitude
of the units can still however be chosen arbitrarily. All conditions
necessary for the indirect comparison of different intensities of a
quantity are fulfilled. ')

In this method of measuring magnetic forces we adopt a natural
unit of magnetic force.

In applying the specified method we need not know the functional
relation between magnetic force and magnetic separation of the
spectral lines. It is sufficient to know that this funetion is one-
valued. The most accurate measurements of the present time ”)
and also theory render it extremely probable that the separation
of the spectral lines is proportional to the intensity of the field
wherein the source of light is placed. If this simple relation be

1) Comp. Runer, Maass und Messen. Encyclopiidie der mathematischen Wissensch.
Bd. V. I. 1903:

2) See specially: A. Farper, Uber das Zeeman-Phinomen. Ann. d. Phys. 9
p. 886. 1902,

( 815 )

the true one, then our scale of magnetic forces is identical with the
one commonly used.

We may then deduce from a given separation of a well-defined
spectral line the strength of a field in absolute measure, the constant
of reduction being once for all determined.

In the measurements of FARBER") relating to the lines 4678 Cd
and 4680 Zn (produced by a spark between zine-cadmium electrodes)
the constant of reduction could be determined with a probable error
of far less then */,,,.

This method and all methods used till now for measuring
magnetic fields, give the intensity in a point. Or rather the mean
value in a small area (often rather extensive) or in a small space
is considered to be the intensity in a point of that area or of that
space,

The magnetic separation of the spectral lines enables us to measure
simultaneously the magnetic force in all points belonging to a
straight line.

In my experiments vacuum tubes charged with some mercury and
excited by a coil were used. The tubes had capillaries of 8 em.
length, the interior diameters varying between ‘/, and */, mm.
The shape of the tubes was that given by PascuEn *), also used by
Ronen and Pascuen in their investigation concerning the radiation of
mercury in the magnetic field.

Avery moderate heating is required for the passage of the discharge,
the light in the capillary is then fairly intense, it becomes very
brilliant as soon as the tube is placed in the magnetic field.

It was noticed that for a given vapour density there exists a
definite intensity of field for which the luminosity is a maximum.
This is easily seen when putting on the current of a pu Bots half ring
electromagnet. Owing to the large inductance (relaxation time 50")
the intensity of the field rises gradually. If the vapour density in
the tube is not too high, there is clearly one moment of maximum
luminosity.

If with a given field the density of the vapour is well chosen, then
only a very moderate heating of the tube is sufficient for keeping
it luminous.

When the tube is placed between the conical poles of a pu Bots
electromagnet and in a plane perpendicular to the line joining the
poles, there is of course a different field intensity in every point of

1) Farser. 1. c.
2) Pascuen, Eine Geisslersche Réhre zum Studium des Zeeman-effectes, Physik,
Zeitschr. p. 478. I. 1900.
57*

( 816 )

the tube. Analysing the light of the different points of the tube
with a spectroscope, we find of course a different magnetic separation
for every point. .

We can however spectroseopically analyse simultaneously the light
of all points of the tube.

We have only to focus an image of the tube upon the slit of the
spectroscope. This spectroscope must satisfy one condition. This econ-
dition is that to every point of the slit there corresponds one point
of the spectral image. In the case of a prism spectroscope, of an
echelon spectroscope, and of a plane grating spectroscope, this condition
is clearly fulfilled, but the concave grating mounted in Row nanp’s
manner forms an exception. The use of the concave grating necessitates
in our case the employment of the method proposed by Runer and
PASCHEN 1),

My experiments were made in the above manner.

To illustrate this method I shall take the blue line of mercury (4859),
which divides into a sextet.

The distribution of the magnetic force in a plane perpendicular to
the axis of a pu Bors electromagnet with a distance of 4 mm. between
the poles is mapped out in a spindle-shaped magnetogram, of which
a part is reproduced in Fig. 1. This figure is from a negative enlarged
9 times. We may extinguish by means of a Nicol the light of the
inner components. At both sides two narrow lines remain, Fig. 2
is a natural size reproduction of a magnetogram taken under the
specified conditions. The duplication of the outer components is lost
in the reproduction. The extension of the field, mapped out by this
magnetogram, may be better understood if l observe that 1 mm. in the
focal plane of the spectroscope corresponds to 1.80 mm. in the plane
between the poles or 1 mm. in the latter plane to 0,556 mm. of the
negative. Hence in Fig. 1 5 mm. corresponds to 1mm. between the
poles. The complete magnetogram gives the magnetic force ina line,
30 mm. in length. Using a lens of shorter focus we can represent,
of course, a greater part of the field. In the middle of the field the
magnetic force is about 24,000 C.G.5. A comparison of field strengths
ean be made with a decidedly higher degree of accuracy than that
which is given above for an absolute measurement.

The method set forth above will be applied, of course, only in diftieult
eases. As long as our spectroscopes of great resolving power are
rather cumbersome, no practical application of the method is possible.

In many cases there will be great advantage in selecting a spectral
line which is tripled in the field.

4) Kayser. Handbuch Ba. I, p. 482,

P. ZEEMAN. “Magnetic resolution of spectral lines and magnetic force.’

Bigvea:

Proceedings Royal Acad. Amsterdam. Vol. VIII.

(S17)

The magnetisation of the spectral lines enables us to determine the
maximum value of the force with phenomena varying rapidly with
the time, and with non-uniform fields.

In some cases it is of great importance to follow the behaviour of
a spectral phenomenon with different strengths of field. The above
described method might then be called the method of the non-uniform
field.

In a future communication I hope to study in this manner the
asymmetry of the separation of spectral lines in weak magnetic
fields, predicted from theory by Vorer. On a former occasion | have
communicated some experiments giving rather convincing evidence of
the existence of this asymmetry ').

In the mean time, I think that the developments lately given by
Lorentz *) make it desirable to corroborate the reasons for accepting
the existence of this extremely small asymmetry.

Mathematics. — “Some properties of pencils of algebraic curves’.
By Prof. Jan pe Vrigs.

§ 1. Let A be one of the »’ basepoints of a pencil (c%) of curves
c” of order n, B one of the remaining basepoints. If we make to
correspond to each c” the right line c’ touching c” in A, then we
get as product of the projective pencils (c") and (c’), a curve 7’, of
order (2 -+ 1) forming the locus of the tangential points of A, i.e.
of the points which are determined by each ec” on its tangent cl.
This tangential curve has in A a threefold point where it is touched
by the inflectional tangents of three c” having in A an inflection;
it has been considered for the first time by Emm Weryr (Sitz. Ber.
Akad. in Wien, LXI, 82).

I shall now consider more in general the locus 7, of the mtb
tangential points of A. The order of this curve is to be represented
by t(m), whilst a(m) and Bim) are to indicate the number of branches
which 7, has in A and B.

Prof. P. H. Scnourm has drawn my attention to a paper inserted
by him in the Comptes Rendus de Académie des sciences, tome CI,
736, where the corresponding curve is treated for a cubic pencil.
I found that the numbers obtained there for m= 8 appear from the
results to be deduced here.

1) Zeeman. These Proceedings, December 1899.

2) Lorentz. These Proceedings, December 1905,

( 818 )

To determine the functions zm), ag) and pon) I shall make use
of an auxiliary curve already used by Wryr, which might be called
the antitangential curve of A. It contains the groups of (m—1)—2
points A), having A as tangential point; so it passes three times
through A and once through all points BL. So it has (2n? — n)

points in common with any ec”, from which it is evident that it is of

order (27 — 1).

§ 2. The (a—41)" tangential curve (A"—!) of A is cut by the
antitangential curve (A~') of A, save in the base points, in the points
Av—!) having A as tangential point. Their number amounts to three
less than the number of tangents which 7, has in A, so a(m)—8;
for, on the three c” which have in A an inflection A coincides with
one of its m' tangential points.

The three inflectional tangents being also tangents of the curve
(A—'), the tangential curve (A”—!) and the antitangential curve
(A—') have 3a(m—1)-+3 points in common in A. In each base-
point 6 lie B(m—1) points of intersection. So

(2n — 1) tr (m — 1) =a (m) 4: 8a (m — 1) + (nr? — 1) B(m— 1)... (1)

A second relation is found by noticing that (A™~!) has with the
antitangential curve of , save the basis, the Bim) points in common
for which & is an m'*tangential point. In 6 lie 33 (m— 1) points
of intersection, «@(m—1) points of intersection lie in A, B(m— 1)
in each of the other basepoints. So

(2n — 1) t(m — 1) = B(m) + a(m — 1) 4+ (nv? 4- 1) B(m — 1) - (2)

With any c” the locus 7, has, save the basis, only the (m— 2)”

pomts A in common; so

n t(m) == a(m) + (rn? — 1) B(m) + (n— 2)" 2. (8)

§ 3. To find a homogeneous equation of finite differences for the
determination of t(m) I eliminate from the three obtained relations
the quantities em) and pi), and I find
ne(m) = n?(2n—1)r(m —1) — (n? + 2)a(m—1) + (rn? —1)B(m—1)} + (n— 2)".

Here the expression within braces can be replaced on account
of (3) by nt(m—1) — (n— 2)". Then

t(m) = (n? — n — 2) e(m — 1) 4+ (rn 4+ 1)(n— 2)m 1. . A)

So

t(m — 1) = (n* — n — 2) t(m — 2) + (n+ 1)(n — 2)". (5)
Equations (4) and (5) finally furnish

t(m) — (n — 2)(m + 2) t(m — 1) + (wn — 2)?(m 4+ 1) t(m — 2) = 0 (6)

(819 )

To determine a particular solution t(m) = « we have

x? — (n — 2)(n + 2)x + (rn — 2)? (r+ 1)=0,
therefore

en? —n — 2 or c= 1 —

9
Consequently the general solution is
t(m) = ¢,(n® — n — 2)m + 6,(mn — 2)m,
To determine the constants c, and ce, I substitute in (4) the known
values (2 + 1) of r(1) and (m+ 1) (n? —4) of 1(2).
Now
n+ 1=c, (n? — n — 2) + ¢, (n — 2),
(n? — 4)(n + 1) = ¢, (n? — n — 2)? + oc, (n — 2)?.

Finally we find by elimination of c, and ce,

t (m) = (n + 1) (n — 2)r—1 a (7)
From (1) and (2) ensues
a(m) — B(m) = — 2 fa (m — 1) — B(m— 1),

50
a (m) — B (m) = (— 2)" fa (1) — BCR = — (— 2) ~~ (8)
Making use of (3) and (7) we now find
n* a (m) = (n — 2)™—l' h(n + 1)m+l — 2n + 1} — (n? — 1) (— 2)". (9)
n? 8 (m) = (n — 2)™—1{(n + 1)mt1 — 2n 4+ 134+ (—2)m . 2. . . 6 (10)

§ 4. For m=2 we find a(2) =n? + »—9; as A is inflection
for three curves c,, there are therefore (m? + 2— 12) curves on
which A coincides with its second tangential point. From this ensues
the wellknown result that A is point of contact of (m + 4) (n— 3)
double tangents.

In a former paper?) I have brought into connection the locus of
the points of contact DY of the double tangents with the locus of the
points W in which a c” is cut by its double tangents. To determine,
how often a point ) coincides with one of its tangential points WW
I consider the correspondence of the rays d= OD and w= OW
which the correspondence (/), }I’) forms in a pencil with vertex 0.

As the curves (2) and (JW) are of orders (2 — 3) (2n? + 5n— 6)
and 4 (nm — 4) (n — 3) (5n? + 5n —6), to each ray d correspond
(n — 4) (n — 3) (2n? + 5n — 6) rays w and to each ray w correspond
(n — 4) (n — 38) (5n? + 5n — 6) rays d.

1) “On linear systems of algebraic plane curves.” Proc. April 22 1905, Vol. VII
(@) repeal:

( 820 )

3ecause each of the 2n(m— 2)(n— 3) double tangents out of O
represents 2 (n — 4) coincidences d= w, the number of coincidences
D= W is Pere by

(n — 4) (n — 8) (2n? + 5n — 6) + (nm — 4) (nm — 8) (5n? 4- dn — 6) —
— 4 (n — 4) (n — 3) (n — 2) n = 8 (n — 4) (n — 3) (n? + 6n — 4).

In a pencil (c") * + 6n — 4) curves
have an inflection, of which the tangent touches the curve in one other

point more.

In the paper quoted above I thought 1 was able to determine this
number out of the points of intersection of the curves (D) and (IW);
here 1 overlooked the fact that a point of contact of a double tangent
can be tangential point JV of another double tangent.

§ 5. To find the number of threefold tangents I consider the
correspondence between the rays projecting out of O two points
W and W’ lying en the same double tangent. The characterizing
number of this symmetric correspondence is evidently equal to
3 (n — 4) (n — 8) (5n? + Sn — 6) (n — 5), whilst each double tangent
borne by O replaces 27 (2% — 2) (7 — 3) (n — 4) (x —5) coincidences.
The number of coincidences }/’== W’ amounts thus to

(n — 5) (nm — 4) (n — 3) (5n? ++ 5n — 6 — 2n* + 4n).

As each threefold tangent bears three of these coincidences we
have the property :

In a pencil (c") we find that (n — 5) (n — 4) (n — 8) (nv? + 38n — 2)

curves have a threefold tangent.

§ 6. In my paper indicated above I have tried to determine the
number of undulation-points out of the points of intersection of the
inflectional curve (/) with the locus of the points (V) which e
determines on its inflectional tangents. As each inflection which is
also tangential point of another inflection is common to (/) and
(V’), the number found elsewhere is too large. The exact number I
can determine by means of the correspondence between the rays O/
and OV.

As the orders of (J) and (V) are 6(n—1) and 3(n—38)(n?+2n—2)
and each of the 32 (nm — 2) inflectional tangents drawn from O replaces
(1 — 3) rays of coincidence, we get for the number of coincidences
[=V
6(n—1) (n—3) 4+ 3(n—38) (n? 4+ 2n—2)—3n(n—2) (n—3)= 6(n—38) (8n—2).

In a pencil (c) we find that 6 (n — 3) (3 n— 2) curves have a
Jour-point tangent.

( 821 )

§ 7. The curve of inflections (J) and the bitangential curve (D)
have in each of the 3 (2 —1)* nodes of (c”) in common a number
of 2 (nm — 3) (m+ 2) points.

For, out of a node we can draw to the c” to which it belongs
(n® —n—6) tangents, to be regarded as double tangents, whilst each
node of a c” is at the same time node of (/).

In each basepoint lie moreover 3 (7m + 4)(7 — 8) points of inter-
section (§ 4). The remaining points common to (2) and (/) are
the inflections of which the tangent touches the c™ once more (§ 4)
and the undulation-points (§ 6) where the two curves touch each
other.

Indeed, we have

6(n—1)? (n— 8) (n4-2) + 38n? (n+-4) (n—8) + 8 (n— 4) (n—3) (n? + 6n—4) +
+ 12(n—3)(3n—2) = 6(n —1)?(n—3)(n42) + 3(n—3)(2n'46n?— 16048) =
= 6(n—1) (n—3S) (2n? + 5n—D),

and this is the product of the orders of (2) and (D).

Physiology. “On the strength of reflex stunuli as weak as possible.”
By Prof. H. Zwaarpemaker. (Report of a research made by
D. J. A. vAN REEKUM).

(Communicated in the meeting of March 31, 1906).

Investigated were chemical, thermal, mechanical and electrical
stimuli, which partly acted upon the skin partly on the sensible nerves
of the animals, which were experimented on.

§ 1. The chemical stimuli were applied by immerging the hind-

leg of a winterfrog in a little bowl with a solution of sulphuric
n n .

acid varying from */, to */,,°/, Cs to = The spinal cord system
was withdrawn in the usual way from the influence of the cere-
brum. After the experiment the legs were washed with distilled
water and the experiment repeated after a pause of 5 minutes.
Neglecting the preliminary reflex, only a complete reflex was consi-
dered as a positive result. After-reflexes and general movements did
only show themselves when rather strong concentrations were used.

as a rule a */,,°/, ie a solution of sulphuric acid may be accepted

as the minimum stimulus which still produces reflexes. The retlex-

( 822 )

time at an immerging of the two legs was 10 seconds, at an immerging
of one leg 22 seconds.

It was calculated how much sulphuric acid disappeared in the
skin of the frog, when */,,°/, sulphuric acid (=) was used, respec-
tively how much was fixed by the excretion-products. This occurred
by titrating the immerging liquid with caustic soda (methylene orange
as indicator) before and after a series of 20 singular reflexes.

Then it appears that about '/,, of the total quantity of the
used sulphuric acid has been bound. Supposing the heat of reaction
of 2 aequivalents natron and 1 aequivalent sulphuric acid to be
31,4 great calories and supposing that our sulphuric acid has been
bound in a reaction of this kind then the heat of reaction of
the chemical process pro singular reflex, reckoned over the whole
immerged surface of the skin, amounts to 1,37 gram-calorie. It is evident
that only a small part of this supposed reaction can have taken
place in or near the terminations of the nerves and that this value
of 1,37 gram-calorie must be also a limit under which is situated
the heat of reaction.

This amount may surpass the real value of the reflex-stimulus
perhaps a million of times. By measuring the electrical conductivity
of the stimulating solution before and after the reflexes it was
controlled if anything else had passed into the immerging liquid
in the place of the disappeared sulphuric acid. This proved to
be the case for the increase of resistance of the liquid experi-
mented with, was greater than would follow from the decrease of

the sulphuric acid.

§ 2. As a thermal stimulus served immersion in cold or warm
water. The most favourable result was obtained by a decreasing dif-
ference of temperature between animal and water of 10° C. and
by an inereasing difference of temperature of 15° C. The reservoir,
isolated by an asbestos envelope, in which the immersion of the frog
took place contained 50 cem. The immersion was performed once
and after that the reflex was waited for. Then it could be stated
that the temperature of the water increased on an average of 8
centigrades by the immersion of the heated frog and decreased on an
average of 22 centigrades by the immersion of the leg of a frog
which was cooled down. Some experiments already gave a reflex
before it had come to this. <A sufficient quantity of refleses large
enough to avoid casualties, were accompanied by an increase of
temperature of 7 centigrades resp. a decrease of temperature of 19

( 823 )

centigrades. Consequently at these last experiments a quantity of
heat of 3,5 gr. calorie must have been withdrawn from the leg of
the frog, and 9,5 calorie have been added. This heat divided itself
during a reflex-time of average 7'/, sec. resp. 9 sec. over the whole
immerged part of the skin. Only a very small part will have come
to the benefit of the terminations of the nerves and what appears
as a reflex-stimulus may very well be millions of times smaller than
the tatal quantity of the heat which is given or taken up. The
above mentioned values have again only the significance of limit
values beneath which the heat resp. cold stimulus, which causes a
reflex movement, must be necessarily situated.

§ 3. To produce mechanical retlex-stimuli first falling mercury
drops were made use of*), afterwards a little ball of resin fastened
to a pig’s-bristle, which by an electrically moved tuning fork of
16 double vibrations was kept in a forced vibration of fixed ampli-
tude. In both cases as much as possible the lateral side of the
foot, where the corpuscula tactus are situated, was taken.

The mercury drops were all of the same size (average 100 mer.)
and were used to the number of 1 to 15, trickling down one after
the other. The height from which the drops were falling varied
from 1 to 20 em. At each experiment the vis viva was calculated
with which the drop came down on the skin of the animal. It was
obvious that for causing a reflex the vis viva had to be in minimo
686 ergs which amount was obtained by dropping 7 drops one after
the other from a falling height of 1 em. Once it was possible to
obtain a reflex by the fall of one drop from a height of 7 em.
which shows the same quantity of energy now contained in one
single stimulus without any summation.

The smallest results according to vis viva which still produce
a reflex were obtained with a little ball of resin of 7 milligram
which vibrated with an excursion of 5 mm. After a retlextime of
on an average 3 sec. the reflex movement was obtained. The
quantity of energy which was added to the skin in this way in
summing contains 212 ergs.

The result of the mechanical stimulation is quantitatively consi-
derably lower than the above mentioned chemical and caloric
stimulation. It leads to a minimum, which however put together
in a restrict spot still possesses the peculiarity of having been
communicated to a part of the skin which probably is considerably

') E. A. Scuirern, Proc. Physiol. Soc. 26 Jan. ‘901.

( 824 )

larger than the surface of a corpusculum tactus. The divergency
between the quantity of energy applied and that which is used for
reflex-stimulus is in this- last case not by far so great as in the
thermal forms of reflex stimulus. The simplest relation might be
expected in the very favourable case already mentioned, in which
only one drop of mercury falling from a height of 7 em. was used.
Meanwhile, with the ball of resin, still smaller values were obtained,
notwithstanding the summation was taken into the account, so that
we may accept, this most simple case has not at all been a most
favourable one.

§ 4. The electrical stimulation brought about by discharges of a con-
densator which was immediately before charged with a voltage varying
between 0 and 2 volts. The capacity of the condensators, which were
constructed in the laboratory from mica of different thickness and cover-
ings of tinfoil different in surface varied from 15 >< 10~5 to 4X
10-3m. F. They were wholly closed in by paraffine and verified by com-
paring with an air-condensator. The following stimuli were used:
firstly on the skin of the leg of the frog by means of little catches
of steel which surround the leg: secondly on the posterior roots of the
lumbal-cord, by means of platinum-electrodes set in pavraffine, thirdly
on the nervus vagus of a rabbit by means of platinum-electrodes set
in ebonite. The stimuli were for the greater part supplied in series
with an interval of ‘/, sec. in a number varying between 1 and 15.
All those regulations took place automatically by properly isolated
swings and keys. The best results gave a condensator of 59><10—° m. F.

Skinreflexes (not ordened series)

(with condensator of 59.40—5 m.F.)

1 2 3 1 5 ES i) | 9 | 10 | 11 | 12 | 13 | 14 | 15 |number of stimuli
121 | 103 | 158 | 98 76 BA ol | 40 | 9 20 6 | 6 2 5 | 10 Jnumber of obseryat.
0.87 | 0.81 lo. 83 10. 77 | 0.77 | 0.75 | 0.74 | 0.79 lo. 94) 0.86) 0. 95 |0.7 75 | 0.65, 0.62 0.67l/average voltage
99:99 /19.85| 39|20.32/17.49) lee 59}16. 15118. rk 26° rapa 81/21.% mateeea| 59)12.4611.3413.24energy in 10—4

The above mentioned experiments were taken without a system.
Observing a more judicious succession of the stimuli more favourable
conditions of stimulation were obtained in the following series.

From this table it distinetly appears that the stimulus is limited
to the smallest quantity of energy when a condensator 0,00035 m. F.
is used. Then 1,4 < 10-4 ergs is sufficient on condition that
the stimulus is repeated three times with an interval of */, sec.
Consulting the experiments about reflexes which are not mentioned

( 825. )

Skinreflexes (ordened_ series)

(the average for the different condensators).

ner el

Sheil r
Capacity | | Number Energy
‘ Voltage of of each stimulus
in mF. | stimuli in 10—4 ergs.

|

0.00025 0.40 2 2.0
0.00035 0.28 3 1.4
0.00059 0.24 8 eT
0.0013 0.294 3 3.7
0.004 0.34 15 23).4

in the tables a minimum value is obtained which is only slightly
larger, namely an amount of 5 & 10~4 ergs.

The result got at the last root of the lumbal region with frogs
cannot be given in one table as the individual experiments differed
too much and have not been numerous enough to fix the average.
In a very sensitive preparation when the above mentioned condensator
of 0,000385 m. F. was used, a distinet reflex was obtained with a single
discharge of only 8,6 10-6 ergs, a result which shows clearly that
in the experiments of Mr. van Reekum the reflex sensitiveness has
been considerably greater from the root than that from the skin. In a
single case there was even found a value still three times smaller.
The above stated number however was not obtained accidentally
but represents a whole series of observations (12 in number).

By central stimulation of the cervical part of the nervus vagus
of a rabbit reflex-changes of the breathing were caused, which could
be registered by means of the aerodromograph ‘). The said reflex
consists according to the intensity of the stimulus 1. if stimulating
with very weak discharges in a slight increase of frequency of
breathing and in an increase of the rapidity of the current of air in
in- and expiration 2. if stimulating with somewhat greater discharges,
an increase of the rapidity of the stream of air notwithstanding
decrease of frequency 3. stimulating with sufficient great discharges
a distinct decrease in rapidity of the stream of air and frequency
both. If we only examine. the result mentioned in the third
case as the reflex on which we want to base our measurements,
the results of the experiments may be taken together as follows:

1) H. ZwaarpemakerR und C. D, Ouwenanp. Arch. f. Physiol. 1904. p, 241,

( 826 )

Breath-reflexes.

——
capacity 15 successive discharges | 1 discharge
Auer energy of the | energy
pe voltage stimulus voltage in 10-1 ergs
in 10—4 ergs |
0.00015 Oni 0,24 0.23 0.40
0.00025 0.13 0.21 0.21 | 0.55
0.000385 0.410 0.47 0.17 0.54
0.0059 0.09 0.24 0.16 0.76
0.0013 O44 0.79 0.19 2.35
0.004 0.42 2.88 0.18 6.48

CONCLUSION.

The reflex stimuli of different kinds used as weak as possible on
cold- resp. warmblooded animals have in minimo very different
value. Thus one and the same effect was brought about by applica-
ting on the skin of a frog of an electric stimulus of 3,15 > 10~ ergs
by a mechanical stimulus of 212 ergs, by a thermal stimulus of
11,5 mega-ergs and by a chemical stimulus of 57 mega-ergs. So
of all these forms of stimulus the electrical is the most favourable.
It may be still more favourable when we let the stimulus act not
on the skin but on a posterior lumbal root of the frog. Then 8 >< 10 &
ergs is sufficient to cause a typical reflex and so the amount ap-
proaches to that which occurs with weak sensorial stimuli (light stimuli
vary in general between 1 < 10—!° as lowest and 6 X 10° as highest
value; acoustical stimuli between 0,3 < 3-® as lowest and 1 > 10°
as highest value’). What holds true for frogs, as a rule holds
true for mammals. From the nervus vagus there can be brought
about by central stimulation with an electrical stimulus of 0,17
10~4ergs a very marked change of breathing, whereas a few times
smaller value causes an indistinct but yet an unmistakable accele-
raion of breathing. Here also the minimum reflex stimuli have a
limit value of the order 1 <x 10-® ergs.

1) Die physiol. wahrnehmbaren Energiewariderungen, Ergebnisse der Physiologie
Bd. IV. 1906. p. 423.

(May 25, 1906).

CONDE Nas:

ABSORPTION and emission lines (The) of gaseous bodies. 591.

AciD (The amides of g- and 8-aminopropionic). 475.

actps (On colorimetry and a colorimetric method for determining the dissociation
constant of). 166.

— (Ester anhydrides of dibasic). 336.

ACRALDEHYDE (The reduction of) and some derivatives of s. divinylglyeol (3.4 dihy-
droxy 1.5 hexadiene), 541.

ALGEBRAIC CURVES (Some properties of pencils of). 817.

ALGEBRAIC SURFACES (On pencils of). 29.

— (On the rank of the section of two). 52.

ALLYL FORMATE (On the action of ammonia and amines on). 188.

AMIDES (The) of g¢- and @-aminopropionic acid. 475,

AMINES (On the action of ammonia and) on allylformate. 138.

— (On the action of ammonia and) on formic esters of glycols and glycerol. 339.
AMMONIA (On the action of) and amines on allylformate. 138.

— (On the action of) and amines on formic esters of glycols and glycerol. 339.
AMYRINE-ACETATE (The occurrence of B-) in some varieties of gutta-percha, 544.
Anatomy. L. Bouk: ‘On the development of the cerebellum in man”. 1st part. 1.

2nd part. 85.

— A. J. P. vAN DEN Broek : “On the sympathetic nervous system in Monotremes”. 91.

— D.J. Hunsnorr Pou: “Bor1k’s centra in the cerebellum of the mammalia’’. 298.

— LL. J. J. Muskens: “Anatomical research about cerebellar connections”. 563.

— LL. Bo.x: ‘On the relation between the teeth-formulas of the platyrrhine and

catarrhine primates”. 751.

apocamy (On a case of) observed with Dasylirion acrotrichum Zuce. 684,

Astronomy. J. Weeper: ‘Approximate formulae of a high degree of accuracy for the
ratio of the triangles in the determination of an elliptic orbit from three obser-
vations.” If. 104.

— J. A. C. Oupremans: “Supplement to the account of the determination of the

longitude of St. Denis (Island of Reunion), executed in 1874, containing also
a general account of the observation of the transit of Venus”. 110.
— H. G. van pe Sanve Bakuuyzen: “Preliminary Report on the Dutch expedition
to Burgos for the observation of the total solar eclipse of August 30, 1905”. 501.
58

Il CAO NDE SN Eoes:

Astronomy. H. J. Zwiers: “Researches on the orbit of the periodic comet Homes and

on the perturbations of its elleptic motion”. 642.
— J.C. Kapreyn: “On the parallax of the nebulae”. 691.
— W. ve Sirrer: “On the orbital planes of Jupiter’s satellites”. 767.

atoms (The introduction of halogen) into the benzene core in the reduction of aromatic
nitro-compounds. 680.

AZOBENZENE, Stilbene and Dibenzyl (On Diphenylhydrazine, Hydrazobenzene and
Benzylaniline, and on the miscibility of the last two with) in the solid aggregate
condition, 466.

BacTERIA (Methan as carbon-food and source of energy for). 327.

BAKHUIS ROOZEBOOM (u. w.). The different branches of the three-phaselines
for solid, liquid, vapour in binary systems in which a compound occurs. 455.
— presents a paper of Dr. F. M. Jazcer: “On Diphenylhydrazine, Hydrazobenzene
and Benzylaniline, and on the miscibility of the last two with Azobenzene, Stil-
bene and Dibenzyl in the solid aggregate condition”. 466.

— The boiling points of saturated solutions in binary systems in which a compound
oceurs. 536.

— presents a paper of Dr, A. Smrrs: “On the hidden equilibria in the y, -sections
below the eutectic point.” 568.

— presents a paper of Dr. A. Smits: “On the phenomena which occur when the
plaitpoint curve meets the three phase line of a dissociating binary compound”. 571.

— presents a paper of J. J. van Laar: “On the course of melting-point curves
for compounds which are partially dissociated in the liquid phase, the proportion
of the products of dissociation being arbitrary’’, 699.

— and J. Outre gr. The solubilities of the isomeric chromic chlorides. 66.

BAKHUYSEN (#. G. VAN DE SANDE). v. SANDE BakauyseENn (H. G. VAN DF).

BAROMETRIC HEIGHTS (On frequency curves of). 549.

BENZENE CORE (The introduction of halogen atoms into the) in the reduction of aromatic
nitro-compounds. 680.

BENZYLANILINE (On Diphenylhydrazine, Hydrazobenzene and), and on the miscibility
of the last two with Azobenzene, Stilbene and Dibenzyl in the solid aggregate
condition. 466.

BESSEL FuNCTIONS (The quotient of two successive). I. 547. If. 640.

BEIJERINCK (M. W.) presents a paper of N. L. Sounaun: “Methan as carbon-
food and source of energy for bacteria’. 327.

BINARY COMPOUND (On the phenomena which occur when the plaitpoint curve meets
the three phase line of a dissociating). 571.

BINARY MIXTURE (The molecular rise of the lower critical temperature of a) of normal
components. 144.

— (The properties of the sections of the surface of saturation of a) on the side of
the components. 280.
BINARY MIXTURES (On the course of the spinodal and the plaitpoint lines for) of normal

substances. 3rd communication, 578.

CO NT EON Ts: Tit

BINARY SYsTEM (On the hidden equilibria in the p, v-diagram of a) in consequence of
the appearance of solid substances, 196.

BINARY SYSTEMS (The different branches of the three-phaselines for solid, liquid,
vapour in) in which a compound occurs. 455.

— (The boiling points of saturated solutions in) in which a compound occurs. 536,

BLAAUW (A. H.) and F, A. F. C. Went. On a case of apogamy with Dasylirion
acrotrichum Zuce. 684.

BLANKSMA (J. J.), Nitration of symmetric nitrometaxylene. 70.

— The introduction of halogen atoms into the benzene core in the reduction of
aromatic nitro-compounds. 680.

— and F. M. Jancrr, On the six isomeric tribromoxylenes. 153.

BLOOD (On catalases of the). 623.

BOCK WINCKEL (H. B. A). On the propagation of light in a biaxial erystal around
a centre of vibration. 728.

BOILING pornis (The) of saturated solutions in binary systems in which a compound
occurs. 536.

BOLK’s CENTRA in the cerebellum of the mammalia. 298.

BOLK (L.). On the development of the cerebellum in man, Ist part. 1. 2nd part. 85.

— presents a paper of Dr. A. J. P. van DEN Broek: “On the sympathetic nervous
system in Monotremes”. 91.

— On the relation between the teeth-formulas of the platyrrhine and catharrine
primates. 781.

BOLTZMANN'S Vorlesungen iiber Gastheorie (Some remarks on the quantity Hin). 630.
Botany. K. VerscuarreLt: ‘Some observations on the longitudinal growth of stems
and flower-stalks”. 8.

— F. A. F. C, Went: “Some remarks on the work of Dr. A. A. Pubte: “An
enumeration of the vascular plants known from Surinam, together with their
distribution and synonymy”. 639.

— F. A. F.C, Went and A. H. Buaauw: “On a case of apogamy observed with
Dasylirion acrotrichum Zuce.” 684.

— W. Burck: “On plants which in the natural state have the character of ever-
sporting varieties in the sense of the mutation theory”. 798.

BOUMAN (% P.). An article on the knowledge of the tetrahedral complex. 358.

BROEK (A. J. P, VAN DEN). On the sympathetic nervous system in Monotremes, 91,

BROMINATION (The) of toluene. 512.

BROMINE and jodine (Contribution to the knowledge of the isomorphous substitution
of the elements fluorine, chlorine,) in organic molecules. 614.

BU RCK (w.). On plants which in the natural state have the character of eversporting
varieties in the sense of the mutation theory. 798.

CARBON-FOoD (Methan as) and source of energy for bacteria, 327.

CARDINAAL (J.) presents a paper of Dr. H. pp Vries: “Central projection in the
space of LopatscHErsky”. 1st part. 359,

caTALAsEs (On) of the blood, 623,

IV ClO NET AE eNA LS

caucuy’s THEORY (Derivation of the fundamental equations of metallic reflection
from). 486.

CENTRAL PROJECTION in the space of LoBatscunrsky. 1st part. 389.

a

CEREBELLAR connections (Anatomical research about). 563.

CEREBELLUM in man (On the development of the). 1st part. 1. 2nd part. 85.
— of the mammalia (Bonk’s centra in the). 298. ;

Chemistry. J. J. van Laan: ‘On the shape of the plaitpoint curves for mixtures of

normal substances”. 2nd communication. 33.

— H. W. Bakuuts Roozenoom and J. Ori sr: “The solubilities of the isomeric
chromic chlorides’. 66.

— J. J. BuanksMa: ‘“Nitration of symmetric nitrometaxylene”. 70.

— F. M. Jancrr: “On some derivatives of phenylearbamic acid’. 127.

— P. van Rompureu : “On the presence of Jupeol in some kinds of gutta-percha”. 137.

— P. van Rompurcu; “On the action of ammonia and amines on allyl formate”. 138.

— A. J. Uuren: “On the action of hydrocyanic acid on ketones”. 141.

—J. J. van Laar: “The imolecular rise of the lower critical temperature of a
binary mixture of normal components’. 144.

— I. M. Janaer and J. J. Buanxsma: “On the six isomeric tribromoxylenes”. 153.

— F. H. Eypman gr: “On colorimetry and a colorimetric method for determining
the dissociation constant of acids’. 166.

— D. Mor: “Ester anhydrides of dibasic acids”. 336,

— L. van Lraviie: “Thalictrum aquilegifolium, a hydrogen eyanide- yielding plant”. 337.

— P. van Rompureu: “On the action of ammonia aud amines on formic esters of
glycols and glycerol”. II. 339.

— W. A. van Dorp and G, C. A, van Dorr: “On the chlorides of maleic acid
and of fumarie acid and on some of their derivatives”. 387.

— H. W. Bakuuts Roozesoom : “The different branches of the three-phaselines for
solid, liquid, vapour in binary systems in which a compound occurs”. 455.

— A. P. N. Franemmont and H. YrrpmMann: “The amides of g- and B-amino-
propionic acid’. 475.

— A. F, Hotneman and F. H. van per Laan: “The bromination of toluene”. 512.

— H. W. Bakuvis Roozrsoom: “The boiling points of saturated solutions in
binary systems in which a compound occurs”. 536,

— P. van Rompurau and W. van Dorssrn: “The reduction of acraldehyde and
some derivatives of s. divinylglycol (2.4 dihydroxy 1.5 hexadiene). 54).

— P. van Rompurcu and N. H. Conen: “The occurrence of B-amyrineacetate in
some varieties of gutta-percha”. 544.

-- P. van Rompurcu and W. van Dorssen: “On the simplest hydrocarbon with
two conjugated systems of double bonds, 1.3.5. hexatriene”. 565.

— A. Smits: “On the hidden equilibria in the p, 2-sections below the eutectic
point”. 568.

— A. Smits: “On the phenomena which oceur when the plaitpoint curve meets

the three phaseline of a dissociating binary compound”. 571,

COOP Ne LSE ENS Des: v

Chemistry. J. J. van Laan: “On the course of the spinodal and the plaitpoint lines
for binary mixtures of normal substances”. 3rd communication. 578.
— A. F. Honieman: “On the nitration of ortho- and metadibromobenzene”. 678.
— J. J. Buayxsma: “The introduction of halogen atoms into the benzene core in
the reduction of aromatic nitro-compounds’’. 680.
— J. J. van Laar: “On the course of melting-point curves for compounds which
are partially dissociated in the liquid phase, the proportion of the products of
dissociating being arbitrary”. 699.
— C. J. Eyxnaar: “On Ocimene and Myrcene, a contribution to the knowledge of
the aliphatic terpenes”. 714.
— C. J. Enkiaar: “On some aliphatic terpene alcohols’. 723.
CHLORIDES (The solubilities of the isomeric chromic). 66.
— (On the) of maleic acid and of fumaric acid and on some of their derivatives. 387.
CHLORINE, biomine and iodine (Contribution to the knowledge of the isomorphous
substitution of the elements fluorine). 614.
CLIMATE (Oscillations of the solar activity and the). 20d communication. 155.
cocks (HuyGEns’ sympathic) and related phenomena in connection with the principal
and the compound oscillations presenting themselves when two pendulums are
suspended to a mechanism with one degree of freedom. 436.
COHEN (N. H.) and P. van RomsBurew. The occurrence of @-amyrine acetate in
some varieties of gutta percha. 544.
COLORIMETRY (On) and a colorimetric method for determining the dissociation constant
of acids. 166.
COMET HOLMES (Researches on the orbit of the periodic) and on the perturbations of
its elliptic motion. 642.
COMPLEX (An article on the knowledge of the tetrahedral). 358.
COMPLEXES of rays (A group of) whose singular surface consists of a scroll and a
number of planes. 662.
*COMPONENTS (The molecular rise of the lower critical temperature of a binary mixture
of normal). 144.
— (Properties of the critical line (plaitpoint line) on the side of the). 271.
— (The properties of the sections of the surface of saturation of a binary mixture
on the side of the). 280.
— (The exact numerical values for the properties of the plaitpoint line on the side
of the). 289.
— (On the possibility of predicting the properties of mixtures from those of the). 743.
compounbs (On the course of meltingpoint curves for) which are partially dissociated
in the liquid phase, the proportion of the products of dissociation being arbitrary. 699.
CONDUCTIBILITY of heat in crystals (A simple geometrical deduction of the relations
existing between known and unknown quantities, mentioned in the method of
Voter for deterniining the). 793.

CORALLIUM from Timor (On a new species of). 268.

corONA (Measurement of the heat produced by the integral radiation of the) and of
the solar disk. 503.

CORTI’s ORGAN (On the pressure of sound in). 60.

vI co N TE NLS.

cRrANIUM (The primordial) of Tarsius spectrum. 397.
CREATININ (On the excretion of) in man. 368.
CRITICAL LINE (plaitpoint line) (Properties of the) on the side of the components. 271.
CRYOGENIC LABORATORY (Methods and apparatus used in the). VIL. 77. VILL 79. IX. 82.
cryostat (A modified). 77.
— with liquid oxygen for temperatures below — 210°C. 79.
crystaL (On the propagation of light in a biaxial) around a centre of vibration. 728.
Crystallography. F. M. Jazcer: “On Diphenylhydrazine, Hydrazobenzene and Benzyl-
aniline, and on the miscibility of the last two with Azobenzene, Stilbene and
Dibenzyl in the solid aggregate condition”. 466.

— F. M. Jazcur: “Contribution to the knowledge of the isomorphous substitution -

of the elements fluorine, chlorine, bromine and iodine, in organic molecules”. 614.

crystals (A simple geometrical deduction of the relations existing between known and
unknown quantities, mentioned in the method of Vorer for determining the con-
ductibility of heat in). 793.

curVE (The PLicker equivalents of a cyclic point of a twisted). 498.

cycLic Pornt (The PLicker equivalents of a) of a twisted curve. 498,

DASYLIRION ACROTRICHUM zZuUcc. (On a case of apogamy observed with). 684,

pEposits (On brackish and fresh water) of the river Silat in Western-Borneo. 742.

DIBENZYL (On Diphenylhydrazine, Hydrazobenzene and Benzylaniline, and on the
miscibility of the last two with Azobenzene, Stilbene and) in the solid aggregate
condition. 466.

piLuviuM of Holland (The geographical and geclogical signification of the Hondsrug,
and the examination of the erratics in the northern). 427.

— of the Netherlands North of the Rhine (On fragments of rocks from the Ardennes
found in the). 518.

DIPHENYLHYDRAZINE (On), Hydrazobenzene and Benzylaniline, and on the miscibility
of the last two with Azobenzene, Stilbene and Dibenzyl in the solid aggregate
condition, 466.

DISSOCIATION CONSTANT of acids (On colorimetry and a colorimetric method for deter-
mining the). 166.

DIVINYLGLycoL (The reduction of acraldehyde and some derivatives of s.) (3.4 dihy-
droxy 1.5 hexadiene). 541,

DORP (w. A. and Gc, A. VAN). On the chlorides of maleic acid and of fumaric
acid and on some of their derivatives. 387.

DORSSEN (W. VAN) and P. van Rompurcu. The reduction of acraldehyde and
some derivatives of s. divinylglycol (3.4 dihydroxy 1.5 hexadiene). 541,

— On the simplest hydrocarbon with two conjugated systems of double bonds
1.3.5 hexatriene. 565.

DUBOIS (EUG.). The geographical and geological signification of the Hondsrug,
and the examination of the erratics in the Northern diluvium of Holland, 427.

pyYNamics of the electron (Remarks concerning the). 477.

EASTON (c.). Oscillations of the solar activity and the climate. 2nd communication.
155,

———=—s > |

CeO NT EVNSDs, VII

EINTHOVEN (w.). Analysis of the curves obtained with the string galvanometer.
Mass and tension of the quartz wire and resistance to the motion of the string. 210.

ELECTRON (Remarks concerning the dynamics of the). 477.
ELLIPTIC MOTION (Researches on the orbit of the periodic comet Hoxmes and on the
perturbations of its). 642.

ELLIPTIC ORBIT (Approximate formulae of a high degree of accuracy for the ratio of
the triangles in the determination of an) from three observations. Il. 104.

EMISSION LINES (‘The absorption and) of gaseous bodies. 591.

ENKLAAR (Cc. J.). On Ocimene and Myrcene, a contribution to the knowledge of
the aliphatic terpenes. 714.

— On some aliphatic terpene alcohols. 723.
EQuatTrons (Derivation of fundamental) of metallic reflection from Caucuy’s theory, 486.

EQquitisria (‘ihe Tx-) of solid and fluid phases for variable values of the pressure. 193.
— (On the hidden) in the pa-diagram of a binary system in consequence of the
appearance of solid substances. 196.
— (On the hidden) in the p,-sections below the eutectic point. 568.
ERINACEUS EUROPAEUS L. (The uterus of) after parturition. 812.
eRRaTIcs (The geographical and geological signification of the Hondsrug, and the
examination of the) in the Northern Diluvium of Holland. 427.
ERRATUM. 426.
ESTER ANHYDRIDES of dibasic acids. 336.
gsTERS (On the action of ammonia and amines on formic) of glycols and glycerol. 339.
EUTECTIC POINT (On the hidden equilibria in the p,a-sections below the). 568.
EYDMAN JR. (F. H.). On colorimetry and a colorimetric method for determining
the dissociation constant of acids. 166.
EXCRETION (On the) of creatinin in man, 363.
FISCHER (EUGEN). On the primordial cranium of Tarsius spectrum. 397,
FLOWER-STALKS (Some observations on the longitudinal growth of stems and). 8.
FLUIDs of the body (On the differentiation of), containing proteid. 628.
FLUORINE, chlorine, bromine and iodine (Contribution to the knowledge of the isomor-
phous substitution of the elements), in organic molecules. 614,
FRANCHIMONT (aA. P. N.). presents a paper of Dr. D. Mou: “Ester anhydrides of
dibasie acids.” 336.
— presents a paper of Dr. F. M. Jatcer: “Contribution to the knowledge of the
isomorphous substitution of the elements fluorine, chlorine, bromine and iodine,
in organic molecules.” 614.
— and H. FrigpmMann. The amides of g- and B-aminopropioniec acid. 475.
FREQUENCY CURVES (On) of meteorological elements. 314.
— (On) of barometric heights. 549.
FRIEDMANN (u.) and A. P. N. Francuimonv. The amides of g- and 6-aminopro-
pionic acid. 475.
FuMaRic actb (On the chlorides of maleic acid and of) and on some of their derivatives, 387.
FUNCTIONS (The quotient of two successive Besse). [. 547. Il. 640.

ViIr C10 NeT EN 2s:

GALVANOMETER (Analysis of the curves obtained with the string). Mass and tension
of the quartz wire and resistance to the motion of the string. 210.

GAs (Improvement to the open mercury manometer of reduced height with transference
of pressure by means of compressed). 75.

— (Improvement in the transference of pressure by compressed) especially for the
determination of isothermals. 76.

GAsEous BopIEs (The absorption and emission lines of). 591.

caseEs (The purifying of) by cooling combined with compression especially the preparing
of pure hydrogen. 82.

GASPHASE (Contribution to the knowledge of the pa- and the p7Z-lines for the case
that two substances enter into a combination which is dissociated in the liquid
and the). 200.

GASTHEORIE (Some remarks on the quantity Hin BoLTzmann’s Vorlesungen iiber). 630.

Geology. H. G. Jonker: “Some observations on the geological structure and origin
of the Hondsrug’’. 96.

— Eve. Dusots: “The geographical and geological signification of the Hondsrug,
and the examination of the erratics in the Northern Diluvium of Holland”. 427.

— C. BE. A. WicumMann: “On fragments of rocks from the Ardennes found in the
diluvium of the Netherlands North of the Rhine”. 518.

— K. Marri: “On brackish and fresh water deposits of the river Silat in Western-
Borneo.” 742.

GLycoLs and glycerol (On the action of ammonia and amines on formic esters of). 339,

GRowTH of stems and flower-stalks (Some observations on the longitudinal). 8.

GUTTA-PERCHA (On the presence of lupeol in some kinds of). 137.

— The occurrence of §-amyrine acetate in some varieties of). 544.

HAGA (H.) presents a paper of Dr. C. Scuoure: “Determination of the Thomson-ettect
in mercury”, 33).

HAMBURGER (H. J.). A method for determining the osmotic pressure of very small
quantities of liquid. 394.

HEAT (On the radiation of) in a system of bodies having a uniform temperature. 401.

— (Measurement of the) produced by the integral radiation of the corona and of
the solar disk. 503.

— in crystals (A simple geometrical deduction of the relations existing between
known and unknown quantities, mentioned in the method of Vorer for deter-
mining the conductibility of). 793.

HPXATRIENE (On the simplest hydrocarbon with two conjugated systems of double bonds,
13505): Dba.

HICKSON (SYDNEY J.). On a new species of Corallium from Timor, 268.

HOEK (p. P. c.). On the polyandry of Scalpellum Stearnsi. 659.

HOLLEMAN (A. F.) presents a paper of Dr. J.J. Buanksma: “Nitration of symmetric
nitrometaxylene.” 70.

— presents a paper of Dr. F. M. Jancer and Dr. J. J. Buanksma: “On the six

isomeric tribromoxylenes,” 153.

CON DEN 2s: IX

HOLLEMAN (a. F.). On the nitration of ortho- and metadibromobenzene. 678.
— presents a paper of Dr. J. J. Buanxsma: “The introduction of halogen atoms
into the benzene core in the reduction of aromatic nitro-compounds.” 680.
— and F. H, van per Laan. The bromination of toluene. 512.

HOLMES (Researches on the orbit of the periodic comet) and on the perturbations
of its elliptic motion. 642.

HONDskuG (Some observations on the geological structure and origin of the). 96.

— (The geographical and geological signification of the), and the examination cf
the erratics in the Northern Diluvium of Holland. 427.

HOOGEWERFF (s.) presents a paper of #. H. Eypman Jx.: “On colorimetry and
a colorimetric method for determining the dissociation constant of acids.” 166.

HUBRECHT (a. a. W.) presents a paper of Prof. Kugrn Fiscurr: “On the primordial
cranium of Tarsius spectrum.” 397.

— presents a paper of Prof. H. Srrau: “The uterus of Erinaceus europaeus L.
after parturition.” 812.

HULSHOFF POL (Db. J.). Bouk’s centra in the cerebellum of the mammalia. 298.

HUYGENS’ sympathic clocks and related phenomena in connection with the principal
and the compound oscillations presenting themselves when two pendulums are
suspended to a mechanism with one degree of freedom. 436.

HYDRAZOBENZENE (On Diphenylhydrazine), and Benzylaniline, and on the miscibility
of the last two with Azobenzene, Stilbene and Dibenzyl in the solid aggregate
condition. 466.

HYDROCARBON (On the simplest) with two conjugated systems of double bonds 1, 3,5
hexatriene. 565.

HYDROCYANIC ACID (On the action of) on ketones, 141.

HYDROGEN (The purifying of gases by cooling combined with compression, especially
the preparing of pure). 82,

HYDROGEN CYANIDE-yielding-plant (Thalictrum aquilegifolium, a), 337.

IcE (On the motion of a metal wire through a lump of). 653.

INTEGRAL of Kummer (A definite). 350.

INTENSITIES of tones (On the ability of distinguishing). 421.

IODINE (Contribution to the knowledge of the isomorphous substitution of the elements
fluorine, chlorine, bromine and) in organic molecules. 614.

ISOMORPHOUS SUBSTITUTION (Contribution to the knowledge of the) of the elements
fluorine, chlorine, bromine and iodine, in organic molecules. 614.

ISOTHERMALS (Improvement in the transference of pressure by compressed gas especially
for the determination of). 76.

ITALLIE (L, VAN). Thalictrum aquilegifolium, a hydrogen cyanide-yielding plant. 337.

— On catalases of the blood. 623.
— On the differentiation of fluids of the body, containing proteid. 628.

JAEGER (fF, M.). On some derivatives of Phenylearbamic acid. 127.

— On Diphenylhydrazine, Hydrazobenzene and Benzylaniline, and on the miscibility
of the last two with Azobenzene, Stilbene and Dibenzyl in the solid aggregate
condition. 466.

x CiOUN DaEENGESs:

JAEGHR (tv. M.). Contributions to the knowledge of the isomorphous substitution of
the elements fluorine, chlorine, bromine and iodine in organic molecules. 614.

— A simple geometrical deduction of the relations existing between known and
unknown quantities, mentioned in the method of Vorer for determining the
conductibility of heat in erystals. 793.

— and J. J. Branksma. On the six isomeric tribromoxylenes. 153.

JONKER (UH. G.). Some observations on the geological structure and origin of the
Hondsrug. 96.

suLrus (w. .). Measurement of the heat produced by the integral radiation of
the corona and of the solar disk. 503.

— A new method for determining the rate of decrease of the radiating power

from the center toward the limb of the solar disk. 668.

yuprrer’s satellites (On the orbital planes of). 767.

KAMERLINGH ONNES (H.). Improvement to the open mercury manometer of
reduced height with transference of pressure by means of compressed gas. 75.

— Improvement in the transference of pressure by compressed gas especially for
the determination of isothermals. 76.

— Methods and apparatus used in the cryogenic laboratory. VII. A modified
eryostat. 77. VIIL Cryostat with liquid oxygen for temperatures below —210°C.
79. 1X. The purifying of gases by cooling combined with compression, especially
the preparing of pure hydrogen. 82. I

— presents a paper of Dr. J. E. Verscuarrenr: “Contributions to the knowledge
of vaN DER Waats’ y-surface. X. On the possibility of predicting the properties
of mixtures from those of the components”. 743. :

— presents a paper of Dr. J. KE. Verscoarrenr: “Appendix to the communications
published in the meetings of June 28, September 27, 1902 and October 31,
1903”. 752.

KAPTEYN (J. c.). On the parallax of the nebulae. 691.

— presents a paper of Dr. W. pe Srrrer: “On the orbital planes of Jupiter’s

satellites”. 767.

KAPTEYN (w.). A definite integral of KumMMER. 350.

— The quotient of two successive Brssen functions. I. 547. IL 640.

KETONES (On the action of hydrocyanic acid on), 141.

KLUYVER (J, ©.). A local probability problem. 341.

Kk OH NSTAMM (PH.) — Some remarks on Dr.— last papers (on the osmotic pressure). 49.

KORTE WEG (D. J.) presents a paper of Dr. W.-A. Verstuys: “On the number of
common tangents of a curve and a surface”. 176.

-- Huyeens’ sympathic clocks and related phenomena in connection with the
principal and the compound oscillations presenting themselves when two pendu-
lums are suspended to a mechanism with one degree of freedom. 436.

KUMMER (A definite integral of). 350.
LAAN (f H. VAN DER) and A. F, Hotzeman. The bomination of toluene. 512.
LAAR (J. J. VAN). On the shape of the plaitpoint curves for mixtures of normal

substances. 22d communication. 33.

CONTENTS. XI

LAAR (J. J. VAN). Some remarks on Dr, Po. Kounstamm’s last papers. 49.

— The molecular rise of the lower critical temperature of a binary mixture of
normal components. 144,

— On the course of the spinodal and the plaitpoint lines for binary mixtures of
normal substances. 3rd communication. 578.

— On the course of melting-point curves for compounds which are partially dissociated
in the liquid phase, the proportion of the products of dissociation being arbitrary. 699.

LIGHT (On the theory of reflection of) by imperfectly transparent bodies. 377.

— (On the propagation of) in a biaxial crystal around a centre of vibration. 728.

LINES (Contribution to the knowledge of the pa- and pT’) for the case that two
substances enter into a combination which is dissociated in the liquid and the
gasphase, 200.

LIQUID (A method for determining the osmotic pressure of very small quantities of). 394.

— and the gasphase (Contribution to the knowledge of the pz- and the pZ-lines for the
case that two substances enter into a combination which is dissociated in the). 200.

LIQUID PHASE (On the course of melting-point curves for compounds which are
partially dissociated in the) the proportion of the products of dissociation being
arbitrary. 699.

LOBATSCHEFSKY (Central projection in the space of). 1st part. 389.

LONGITUDE of St. Denis (Island of Réunion) (Supplement to the account of the deter-
mination of the), executed in 1874, containing also a general account of the
observation of the transit of Venus. 110.

LORENTZ (H. a.) presents a paper of J. J. van Laar: “On the shape of the
plaitpoint curves for mixtures of normal substances”, 2nd communication. 33.

— presents a paper of J. J. van Laar: “Some remarks on Dr. Pu. Kounstamm’s
last papers”, 49.

— presents a paper of J. J. van Laar: “The molecular rise of the lower critical
temperature of a binary mixture of normal components”, 144.

— presents a paper of Prof. R. Stssineu: “On the theory of reflection of light by
imperfectly transparent bodies”. 377.

— On the radiation of heat in a system of bodies having « uniform temperature. 401.

— presents a paper of Prof. R. Sissincu: “Derivation of the fundamental equations
of metallic reflection from Caucuy’s theory”. 486.

— presents a paper of J. J. van Laan: “On the course of the spinodal and the
plaitpoint lines for binary mixtures of normal substances. (34 communication). 578.

— The absorption and emission lines of gaseous bodies. 591.

— presents a paper of O. Posrma; ‘Some remarks on the quantity H in Bourz-
MANN’s: Vorlesungen iiber Gastheorie’”’. 630. :
— presents a paper of L. 8. Ornstern: “On the motion of a metal wire through a

lump of ice”. 653.
— presents a paper of H. B. A. BockwinkeL: “On the propagation of light in a
biaxial crystal around a centre of vibration”. 728.

LUPEOL (On the presence of) in some kinds of gutta percha. 137.
MAGNETIC FORCE (Magnetic resolution of spectral lines and), 1st part. 814.

XII 0 O° °N of EN es;

MALPIC acibD (On the chlorides of) and of fumaric acid and on some of their dert-
vatives. 387.

MAMMALIA (BonK’s centra in the cerebellum of the). 298.

MAN (On the development of the cerebellum in), Ist part. 1. 2nd part. 85.

— (On the excretion of creatinin in), 363.

MANOMETER (Improvement to the open mercury) of reduced height with transference
of pressure by means of compressed gas. 75.

MARTIN (K.) presents a paper of Dr. H. G. Jonker: “Some observations on the
geological structure and origin of the Hondsrug.” 96.

— presents a paper of Prof. Kue. Dusots: “The geographical and geological signi-
fication of the Hondsrug, and the examination of the erratics in the Northern
Diluvium of Holland.” 427.

— On brackish and fresh water deposits of the river Silat in Western-Borneo. 742.

Mathematics. Jan pe Vries: “On pencils of algebraic surfaces.” 29.

—- W. A. Verstuys: “On the rank of the section of two alvebraic surfaces.” 62.

— W. A.Verstuys: “On the number of common tangents of a curve and a surface.” 176.

— J. C. Kivuyver: “A local probability problem.” 341.

— W. Kapreyn: “A definite integral of Kummer.” 350.

— Z. P. Bouman: “An article on the knowledge of the tetrahedral complex.” 358.

— H. pe Vries: “Central projection in the space of Lobatschefsky.” (1st part), 389.

— D. J. Korrewec: ‘HuyeEns’ sympathic clocks and related phenomena in
connection with the principal and the compound oscillations presenting them-
selves when two pendulums are suspended to a mechanism with one degree of
freedom.” 436.

— P. H. Scuours: “A tortuous surface of order six and of genus zero in space
Sp, of four dimensions.” 489.

— W.A.Versuvys: “The PLicker equivalents of a cyclic point of a twisted curve.” 498,

— W. Kapreyn: “The quotient of two successive Bessel functions.” I. 547. IL 640.

— Jan ve Vries: “A group of complexes of rays whose singular surface consists
of a scroll and a number of planes.” 662.

— P. H. Scnourn: “A particular series of quadratic surfaces with eight common
points and eight common tangential planes.” 754.

— Jan pe Vries: “Some properties of pencils of algebraic curves.” 817.

MELTING POINT cURVES (On the course of) for compounds which are partially disso-
ciated in the liquid phase, the proportion of the products of dissociation being
arbitrary. 699.

mERcuRY (Determination of the Thomson-effect in). 331.

METADIBROMOBENZENE (On the nitration of ortho and). 678.

METAL WIRE (Qn the motion of a) through a lump of ice. 658.

Meteorology. C, Easton: “Oscillations of the solar activity and the climate”. 2nd com-
munication. 155.

— J. P. van pur Srox: “On frequency curves of meteorological elements”. 314.

— J. P. van per Stok: “On frequency curves of barometric heights”. 549.

—— es

CO Nee NOES: XTIT

METHAN as carbon-food and source of energy for bacteria. 327.

METHOD (A) for determining the osmotic pressure of very small quantities of liquid. 394,

METHOD of VorcT (A simple geometrical deduction of the relations existing between
known and unknown quantities, mentioned in the) for determining the conduc-
tibility of heat in erystals, 793.

METHODS and apparatus used in the Cryogenic Laboratory. VIL. A modified Cryostat.
77. VILL. Cryostat with liquid oxygen for temperatures below — 210° C. 79. IX.
The purifying of gases by cooling combined with compression, especially the
preparing of pure hydrogen, 83.

Microbiology. N. LL. Sounern: ‘“Methan as carbon-food and source of energy for
bacteria”. 527.

mixtures (On the possibility of predicting the properties of) from those of the com-
ponents. 743.

MIXTURES of normal substances (On the shape of the plaitpoint curves for). 2nd com-
munication. 33.

MOL (D.). [ister anhydrides of dibasic acids. 336.

MOLECULAR RiSE (The) of the lower critical temperature of a binary mixture of normal
components. 144.

MOLL (J. w.) presents a paper of Dr. W. Burck: “On plants which in the natural
state have the character of eversporting varieties in the sense of the mutation
theory”. 798.

MONOTREMES (On the sympathetic nervous system in). 91.

MUSKENS (1. J. J.). Anatomical research about cerebellar connections. 563.

MUTATION THEORY (On plants which in the natural state have the character of ever-
sporting varieties in the sense of the). 798.

MYRCENE (On Ocimene and), a contribution to the knowledge of the aliphatic terpenes.
714.

NEBULAE (On the parallax of the). 691.

NERvoUS system (On the sympathetic) in Monotremes. 91.

NITRATION (On the) of ortho- and metadibromobenzene. 678.

— of symmetric nitrometaxylene. 70.

NITRO-cCOMPOUNDS (The introduction of halogen atoms into the benzene core in the
reduction of aromatic). 680.

NITROMETAXYLENE (Nitration of symmetric). 70.

NYLAND (A, A.). The prismatic camera. 505.

OCIMENE (On) and Myrcene, a contribution to the knowledge of the aliphatic terpenes.
714.

OLIE sr. (3.) and H. W. Baxuurts RoozeBoom. The solubilities of the isomeric chromic
chlorides. 66.

ONNES (H. KAMERUINGH). v. KAMERLINGH Onnes (I1.).

orbit of the periodic comet Hotmrs (Researches on the) and on the perturbations
of its elliptic motion. 642.

ORNSTEIN (1. s.). On the motion of a metal wire through a lump of ice. 653.

oriHO- and metadibromobenzene (On the nitration of). 678.

XIV COON DPE NN:

oscILLAwions (lluyGEns’ sympathie clocks and related phenomena in connection with
the principal and the compound) presenting themselves when two pendulums
are suspended to a mechanism with one degree of freedom. 436.

orpit of the periodic comet Hormes (Researches on the) and on the perturbations
of its elliptic motion. 642.

ORBITAL PLANEs (On the) of Jupiter’s satellites. 767.
OsMOTIC PRESSURE (Some remarks on Dr. Pu. Konnstamm’s last papers on the). 49.
— (A method for determining the) of very small quantities of liquid. 394.
OUDEMANS (J. 4. C.). Supplement to the account of the determination of the
longitude of St. Denis (Island of Réunion), executed in 1874, containing also a
general account of the observation of the transit of Venus. 110.
PARALLAX (On the) of the nebulae. 691.
PEKELHARING (C. A.). On the excretion of creatinin in man. 363.
— presents a paper of Dr. L. van Ivaunte: “On catalases of the blood,” 623.

— presents a paper of Dr. L. van Irautie: “On the differentiation of fluids of the
body, containing proteid.” 628.

pencits (On) of algebraic surfaces, 29.

— (Some properties of) of algebraic curves. 817.

PHASE LINE (On the phenomena which occur when the plaitpoint curve meets the
three) of a dissociating binary compound. 571.

puasp LInEs (The different branches of the three-) for solid, liquid, vapour in binary
systems in which a compound occurs, 455.

PHASE PRESSURE (The shape of the sections of the surface of saturation normal to the
x axis, in case of a three) between two temperatures. 184.

puasns (The Zv-equilibria of solid and fluid) for variable values of the pressure. 193,

PHENYLCARBAMIC AcID (On some derivatives of). 127.

Physics. J. J. van Laar: “Some remarks on Dr. Px, Konxsramm’s last papers”. 49.

— H. Kameriinca Onnes: “Improvement to the open mercury manometer of
reduced height with transference of pressure by means of compressed gas” 15.

— H. Kameruincu Onnes: “Improvement in the transference of pressure by com-
pressed gas especially for the determination of isothermals”. 76.

— H. Kameriineu Onnes: “Methods and apparatus used in the eryogenic laboratory.
VII. A modified eryostat. 77. VIL. Cryostat with liquid oxygen for temperatures
below — 210° C. 79. IX. The purifying of gases by cooling combined with com-
pression, especially the preparing of pure hydrogen”. 82.

J. D. van per Waats: “The shape of the sections of the surface of saturation
normal to the a-axis, in case of a three phase pressure between two tempera-

tures”. 184.
_— J. D. van per Waats: “The Z,2- equilibria of solid and fluid phases for variable

values of the pressure”. 193.

— A, Smrrs: “On the hidden equilibria in the p,2- diagram of a binary system in
consequence of the appearance of solid substances”. 196.

— A. Smits: “Contribution to the knowledge of the p.2- and the p7- lines for the
case that two substances enter into a combination which is dissociated in the
liquid and the gasphase”. 200.

~~

ClO N TE Nese ‘ xv

Physics. J. D. van per Waats: “Properties of the critical line (plaitpoint line) on
the side of the components.” 271.
— J. D. van per Waats: “The properties of the sections of the surface of
saturation of a binary mixture on the side of the components.” 280.
— J. D. van per Waais: “The exact numerical values for the properties of the
plaitpoint line on the side of the components.” 289.
— C. Scuours: Determination of the Thomson-eflect in mercury’. 381.

— R. Stsstncu: ‘“ On the theory of reflection of light by imperfectly transparent
bodies”. 377.

— H. A. Lorentz: “On the radiation of heat in a system of bodies having a
uniform temperature”. 401.

— J. D. van per Waats Jr.: “Remarks concerning the dynamics of the
electron”. 477.

— R. Sissinew: “Derivation of the fundamental equations of metallic reflection
from Caucny’s theory”. 486.
— H. A. Lorenz: “The absorption and emission lines of gaseous bodies”, 591.

— O. Postma: “Some remarks on the quantity Z in Bontmann’s “Vorlesungen
iiber Gastheorie.” ” 630.

— L. 8. Ornstein: “On the motion of a metal wire through a lump of ice”. 653.
— W. H. Junius: “A new method for determining the rate of decrease of the
radiating power from the center toward the limb of the solar disk”. 668.

— H. B. A. Bockxwinkes: “On the propagation of light in a biaxial crystal around
a centre of vibration’’. 728.

— J. E. Verscuarrect: “Contributions to the knowledge of vaN per WAats’ y-
surface. X. On the possibility of predicting the properties of mixtures from those
of the components.” 743.

— J. E. Verscuarreit: “Appendix to the communications pablished in the mee-
tings of June 28, September 27, 1902 and October 3], 1903”. 752.

— F. M. Jarcer: “A simple geometrical deduction of the relations between
known and unknown quantities, mentioned in the method of Voter for deter-
mining the conductibility of heat in crystals”. 793.

— P. Zneman: “Magnetic resolution of spectral lines and magnetic force”. 1st part.
814.

Physiology. H. Zwaarpemaker: “On the pressure of sound in Corti’s organ”. 60.

— W. Erytuoven: “Analysis of the curves obtained with the string galvano-
meter. Mass and tension of the quartz wire and resistance in the motion of the
string’, 210.

— G. van Rynserx: “The designs on the skin of the vertebrates, considered in
their connection with the theory of segmentation”. 307.

— C. A. PEkeLuarine: “On the excretion of creatinin in man”, 363.

— H. J. Hampourcer: “A method for determining the osmotic pressure of very
small quantities of liquid”. 394.

— H. ZwaarpeMaker: “On the ability of distinguishing intensities of tones”. 421.

— IL van Irauuie: “On catalases of the blood”. 628.

— L. van Itatiin: “On the differentiation of fluids of the body, containing pro-
teid”. 628.

XVI CONTENTS,

Physiology. H. Zwaarppmaker: “On the strength of the reflex-stimuli as weak as
possible”. 821.
PLAYTPOINT CURVE (On the phenomena which occur when the) meets the three phase
line of a dissociating binary compound. 571.
PLAITPOINT CURVES (On the shape of the) for mixtures of normal substances. 2nd com-
munication. 33.
PLAITPOINT LINE (Properties of the critical line) on the side of the components. 271.
— (The exact numerical values for the properties of the) on the side of the com-
ponents. 289.
PLAITPOINT LINES (On the course of the spinodal and the) for binary mixtures of
normal substances. 3rd communication. 578.
pian? (Thalictrum aquilegifolium, a hydrogen cyanide-yielding). 337.
-pranrs (An enumeration of the vascular) known from Surinam, together with their
distribution and synonymy. 639.
— (On) which in the natural state have the character of eversporting varieties in
the sense of the mutation theory. 798.
pLucKkeER equivalents (The) of a cyclic point of a twisted curve. 498.
POL (D. J. HULSHOFE). v. Hutsuorr Pon (D. J.)
POLYANDRY of Scalpellum Stearnsi (On the). 659.
postMa (0.). Some remarks on the quantity H in Boltzmann’s “Vorlesungen iiber
Gastheorie.” 630.
pressurB (Improvement to the open mercury manometer of reduced height with trans-
ference of) by means of compressed gas. 75.
— (Improvement in the transference of) by compressed gas especially for the deter-
mination of isothermals. 76.
— (The Tx-equilibria of solid and fluid phases for variable values of the). 193.
— of sound (On the) in Corti’s organ. 60.
PRIMATES (On the relation between the teeth-formulas of the platyrrhine and catarrhine). 781.
PRISMATIC CAMERA (The). 505.
PROBABILITY PROBLEM (A local). 341.
prorerp (On the differentiation of fluids of the body, containing). 628.
PULLE (A. A.). An enumeration of the vascular plants known from Surinam, together
with their distribution and synonymy. 639.
QUADRATIC suRFACES (A particular series of) with eight common points and eight
common tangential planes. 754.
quantity IZ (Some remarks on the) in Bourzmann’s “Vorlesungen tiber Gastheorie.” 630,
Quartz wien (Mass and tension of the) and resistance to the motion of the string. 210.
RADIATING POWER (A new method for determining the rate of decrease of the) from
the center toward the limb of the solar disk. 668.
RADIATION (Measurement of the heat produced by the integral) of the corona and the
solar disk. 503. -
— of heat (On the) in a system of bodies having a uniform temperature. 401.
RATE OF DECREASE (A new method for determining the) of the radiating power from

the center toward the limb of the solar disk. 668.

CONTENTS, xvit

RaTIo of the triangles (Approximate formulae of a high degree of accuracy for the)
in the determination of an elliptic orbit from three observations. IL. 104.
rays (A group of complexes of) whose singular surface consists of a scroll and a
number of planes. 662.
REFLECTION (Derivation of the fundamental equations of metallic) from Cavucay’s
theory. 486,
— of light (On the theory of) by ‘imperfectly transparent bodies, 377.
REFLEX-STIMULI (On the strength of the) as weak as possible. 821.
rocks (On fragments of) from the Ardennes found in the diluvium of the Netherlands
North of the Rhine. 518.
ROMBURGA (P. VAN) presents a paper of Dr. F. M. Jancer: “On some derivatives
of Phenylearbamic acid.” 127.
— On the presence of lupeol in some kinds of gutta-percha. 137.
— On the action of ammonia and amines on allyl formate. 138.
— presents a paper of Dr. A. J. Unren: “On the action of hydrocyanie acid on
ketones”. 141.
— presents a paper of Dr. L. van [vaniie: “Thalictrum aquilegifolium, a hydrogen
eyanide-yielding plant”. 337.
— On the action of ammonia and amines on formic esters of glycols and glycerol. 339.
— presents a paper of Dr. C. J, EnkLaar : “On Ocimene and Myrcene, a contribution
to the knowledge of the aliphatic terpenes.” 714.
— presents a paper of Dr. C.J. Enkuaar: “On some aliphatic terpene alcohols.” 723.
— and N. H. Counn: “The occurrence of @-amyrine acetate in some varieties of
gutta percha”. 544.
— and W. van DoxssEn: “The reduction of acraldehyde and some derivatives of
s, divinylglycol (3.4 (lihydroxy 1.5 hexadiene). 541.
— On the simplest hydrocarbon with two conjugated systems of double bonds,
1.3.5 hexatriene. 565.
ROOZEBOOM (H, W. BAKHUTS). vy. BakHuts RoozeBoom (H. W.).
RIJNBERK (G, vAN). The designs on the skin of the vertebrates, considered in
their connection with the theory of segmentation. 307.
SANDE BAKHUYZEN (H, G. VAN DE) presents a paper of J, WEEDER: ‘“‘Ap-
proximate formulae of a high degree of accuracy for the ratio of the triangles
in the determination of an elliptic orbit from three observations”, II, 104,
~ Preliminary Report on the Dutch expedition to Burgos for the observation of
the total solar eclipse of August 30, 1905. 501.
— presents a paper of Dr. H. J. Zwirrs: “Researches on the orbit of the periodic
comet Holmes and on the perturbations of its elliptic motion”, 642.
SATELLITES (On the orbital planes of Jupiter’s). 167.
SCALPELLUM STEARNSI (On the polyandry of). 659.
scHOUTE (c.). Determination of the Thomson-effect in mercury. 331,
SCHOUTE (P. H.) presents a communication Dr, W. A, VersLuys: “On the rank of
the section of two algebraic surfaces”, 52. :

XViIl ClOUN LE NTs:

scHoure (v. u.). A tortuous surface of order six and of genus zero in space Sp,

of four dimensions. 489.

— presents a paper of Dr. W. A. Verstuys: “The Piicker equivalents of a cyclic
point of a twisted curve.” 498.

— A particular series of quadratic surfaces with eight common points and eight
common tangential planes. 754.

section of two algebraic surfaces (On the rank of the). 52.

sections (On the hidden equilibria in the p,x-) below the eutectic point. 568.

SEGMENTATION (The designs on the skin of the vertebrates, considered in their con-
nection with the theory of). 307.

stLav in Western-Borneo (On brackish and fresh water deposits of the river). 742.

SISsINGH (R.). On the theory of reflection of light. by imperfectly transparent
bodies. 377. 2

—- Derivation of fundamental equations of metallic reflection from CavcHy’s

theory. 486.

SITTER (W. DE). On the orbital planes of Jupiter’s satellites. 767.

sktn (The designs on the) of the vertebrates, considered in their connection with the
theory of segmentation. 307.

smM1vrs (A.). On the hidden equilibria in the p,e-diagram of a binary system in
consequence of the appearance of solid substances. 196.

— Contribution to the knowledge of the pa- and the p7Z-lines for the case that two
substances enter into a combination which is dissociated in the liquid and the
gasphase. 200.

— On the hidden equilibria in the p,2-sections below the eutectic point. 568.

— On the phenomena which occur when the plaitpoint curve meets the three phase
line of a dissociating binary compound. 571.

SOHNGEN (N. L.). Methan as carbon-food and source of energy for bacteria. 327.

SOLAR activity (Oscillations of the) and the climate. 2nd communication. 155.

SOLAR DISK (Measurement of the heat produced by the integral radiation of the corona
and of the). 503.

— (A new method for determining the rate of decrease of the radiating power
from the center toward the limb of the). 668.

SOLAR ECLIPSE (Preliminary Report on the Dutch expedition to Burgos for the obser=
vation of the) of August 30, 1905. 501.

— (Report on the operations with the two slit-spectrographs for the) of August

30, 1905. 506.

SOLUBILITIES (The) of the isomeric chromic chlorides. 66.

soLutions (The boiling points of saturated) in binary systems in which a compound
oceurs. 536,

souND (On the pressure of) in Corti’s organ. 60.

space Sp, (A tortuous surface of order six and of genus zero in) of four dimensions. 489,

— of LosatscuErsky (Central projection in the). 1st part. 389.
SPECTRAL LINES (Magnetic resolution of) and magnetic force. 1st part. 814.

CONTENTS. XIX

SPECTROGRAPHS (Report on the operations with the two slit-) for the solar eclipse of
August 30, 1905. 506.

SPICULES of sponges (On the structure of some siliceous). I. The styli of Tethya
lyneurium, 15.

SPINoDAL and the plaitpoint lines (On the course of the) for binary mixtures of
normal substances. 8rd communication. 578.
SPONGEs (On the structure of some siliceous spicules of). I. The styli of Tethya
lyneurium. 15.
st. DENIS (Island of Réunion) (Supplement to the account of the determination of
the longitude of), executed in 1874, containing also a general account of the
observation of the transit of Venus. 110.
sTEMs and flower-stalks (Some observations on the longitudinal growth of). 8.
STILBENE and Dibenzyl (On Diphenylhydrazine, Hydrazobenzene and Benzylaniline,
and on the miscibility of the last two with Azobenzene) in the solid aggregate
condition, 466.
STOK (J. P, VAN DER). On frequency curves of meteorological elements. 314.
— On frequency curves of barometric heights. 549.
STRAHL (u.). The uterus of Erinaceus europaeus L. after parturition. $12.
sTRING (Mass and tension of the quartz wire and resistance to the motion of the). 210,
suBsTances (On the hidden equilibria in the p,v-diagram of a binary system in conse-
quence of the appearence of solid). 196.
— (Contribution to the knowledge of the p,v-and the p,7- lines for the case that
two) enter into a combination which is dissociated in the liquid and the gasphase. 200.
surface (A tortuous) of order six and of genus zero in space Sp, of four dimensions. 498.
y-surFace (Contributions to the knowledge of van per Waats’). X. On the possibi-
lity of predicting the properties of mixtures from those of the components. 743,
SURFACE OF saTURATION (The shape of the sections of the) normal to the a-axis, in case
of a three phase pressure between two temperatures. 184.
SURFACE OF SATURATION (The properties ot the sections of the) of a binary mixture
on the side of the components. 280.
surracrs (A particular series of quadratic) with eight common points and eight
common tangential planes. 754.
sURINAM (An enumeration of the vascular plants known from) together with their
distribution and synonymy. 639.
TANGENTS (On the number of common) of a curve and a surface. 176.
TARSIUs sPecTRUM (On the primordial cranium of). 397.
TEETH-FORMULAS (On the relation between the) of the platyrrhine and catarrhine
primates. 781.
TEMPERATURE (The molecular rise of the lower critical) of a binary mixture of normal
components, 144.
— (On the radiation of heat in a system of bodies having a uniform). 401,
TEMPERATURES (The shape of the sections of the surface of saturation normal to the
x-axis, in case of a three phase pressure between two). L84.
— below — 210°C. (Cryostat with liquid oxygen for). 79.

Xx CONTENTS.

TERPENE ALCOHOLS (On some aliphatic). 723. j

TERPENES (On Ocimene and Myrcene, a contribution to the knowledge of the aliphatic). 714.
TETHYA LYNcURIUM (The styli of). 15.

TETRAHEDRAL COMPLEX (An article on the knowledge of the). 358.

THALICTRUM AQUILEGIFOLIUM, a hydrogen cyanide-yielding plant. 337.

pHEoRY of reflection of light (On the) by imperfectly transparent bodies. 377.
THOMSON-EFFECT (Determination of the) in mercury. 331.

Timor (On a new species of Corallium from). 268,

TOLUENE (The bromination of). 512.

tones (On the ability of distinguishing intensities of). 421.

pransit of Venus (Supplement to the account of the determination of the longitude
of St. Denis (Island of Réunion), executed in 1874, containing also a general
account of the observation of the). 110.

TRIANGLES (Approximate formulae of a high degree of accuracy for the ratio of the)
in the determination of an elliptic orbit from three observations. I]. 104.

TRIBROMOXYLENES (On the six isomeric). 153.

ULTEE (A. J.). On the action of hydrocyanic acid on ketones. 141.

urerus (The) of Erinaceus europaeus L. after parturition. 812.

VENUS (Supplement to the account of the determination of the longitude of St. Denis
(Island of Réunion), executed in 1874, containing also a general account of the
observation of the transit of). 110.

VERSCHAFFELT (&.). Some observations on the longitudinal growth of stems and
flower-stalks. 8.

VERSCHAFFELT (J. £). Contributions to the knowledge of van DER Waaxs’
p-surface. X. On the possibility of predicting the properties of mixtures from
those of the components. 743.

— Appendix to the communications published in the meetings of June 28, Sep-
tember 27, 1902 and October 31, 1903. 752.
VERSLUYS (w. A.). On the rank of the section of two algebraic surfaces. 52.
— On the number of common tangents of a curve and a surface. 176.
— The Priicker equivalents of a cyclic point of a twisted curve. 498.

VERTEBRATES (The designs on the skin of the), considered in their connection with
the theory of segmentation. 307.

VIBRATION (On the propagation of light in a biaxial crystal around a centre of). 728.

vorerT (A simple geometrical deduction ot ‘the relations existing between known and
unknown quantities, mentioned in the method of) for determining the conduc-
tibility of heat in crystals. 793.

vosSMAER (G.c. J.) and H. P. Wissman. On the structure of some siliceous
spicules of sponges. I. The styli of Tethya lyncurium, 15.

VRIES (H. Dz). Central projection in the space of LoBarscHErsky. Ist part. 389.

VRIES (HUGO DE) presents a communication of Prof. E. VerscuarreLt: “Some
observations on the longitudinal growth of stems and flower-stalks”. 8.

VRIES (JAN DE). On pencils of algebraic surfaces, 29,

CO NaTTE Nets. XXI

VRIES (JAN De) presents’ a paper of Z. P. Bouman: “An article on the know-
ledge of the tetrahedral complex.” 358. Se
— A group of complexes of rays whose singular surface consists of a scroll and a
number of planes. 662. ;
— Some properties of pencils of algebraic curves. 817.

WAALS (VAN DER) g-surface (Contributions to the knowledge of). X. On the
possibility of predicting the properties of mixtures from those of the components. 743.
WAALS (J. D. VAN DpR). The shape of the sections of the surface of saturation
normal to the axis, in case of a three phase pressure between two temperatures. 184.
— The 7,x-equilibria of solid and fluid phases for variable values of the pressure. 193,
— presents a paper of Dr. A. Smrrs: “On the hidden equilibria in the p,w-diagram
of a binary system in consequence of the appearance of solid substances.” 196,
— presents a paper of Dr. A. Smrrs: “Contribution to the knowledge of the pa-
and tne pZ+lines for the case that two substances enter into a combination which
is dissociated in the liquid and the gasphase.” 200. ;
— Properties of the critical line (plaitpoint line) on the side of the components. 271.
— The properties of the sections of the surface of saturation of a binary mixture
on the side of the components, 280.
— The exact numerical values for the properties of the plaitpoint line on the side
of the components, 289.
— presents a paper of Prof. J. D, van per Waats JR.: “Remarks concerning the
dynamics of the electron.” 477.
WAALS JR. (J. D VAN DER). Remarks concerning the dynamics of the electron. 477.
WEBER (MAX) presents a paper of Prof. Sypney J. Hickson: “On a new species
of Corallium from Timor.” 268.
WEEDER (J.). Approximate formulae of a high degree of accuracy for the ratio of the
triangles in the determination of an elliptic orbit from three observations. II. 104.
WENT (fF. A. F. C.). Some remarks on the work of Mr. A. A. PuLLE, entitled: “An
enumeration of the vascular plants known from Surinam, together with their
distribution and synonymy.” 639.
— and A. H. Buaauw. On a case of apogamy observed with Dasylirion acrotrichum
Zuce. 684,
WICHMANN (c. E, A.). On fragments of rocks from the Ardennes found in the.
diluvium of the Netherlands North of the Rhine. 518.
WILTERDINK (J. H.), Report on the operations with the two slit-spectrographs for
the solar eclipse of August 30, 1905. 506.
WIND (c. H.) presents a paper of C. Easton: “Oscillations of the solar activity and
the climate”. 2nd communication. 155.
WINKLER (c.) presents a paper of D. J. Hutsuorr Pou: “BoLk’s centra in the
cerebellum of the mammalia’’. 298.
— presents a paper of Dr. G. van Risnperx: “The designs on the skin of the
vertebrates, considered in their connection with the theory of segmentation”. 307.
— presents a paper of Dr. L. J. J. Muskens: ‘‘Anatomical research about cerebellar
connections,” 563.

XXII REGISTER.

WigSMAN (H. P.) and G. CU. J. Vosmaer. On the Structure of some siliceous spi-
cules of sponges. I. The styli of Tethya lyncurium. 15.

ZEEMAN (v.) presents a paper of Dr. F. M. Jagcer: “A simple geometrical
deduction of the relations existing between known and unknown quantities,
mentioned in the method of VoreT for determining the conductibility of heat in
crystals.” 793.

— Magnetic resolution of spectral lines and magnetic force. Ist part. 814.

Loology. G. C. J. Vosmagr and H. P. Wrssman: “On the structure of some siliceous

spicules of sponges. J. The styli of Tethya lyncurium.” 15.

— Sypney J. Hickson: “On a uew species of Corallium from Timor.” 268. ,

— Evaen Fiscuer: “On the primordial cranium of Tarsius spectrum.” 397. ;

— P. P. C. Hoek: “On the polyandry of Scalpellum Stearnsi.” 659. be

— H. Srraui: “The uterus of Erinaceus europaeus L. after parturition.” 812. .
ZWAARDEMAKER (H.). On the pressure of sound in Corti’s organ, $0.

— On the ability of distinguishing intensities of tones. 421,

— On the strength of the reflex-stimuli as weak as possible. 821. j
ZWIERS (H. J.). Researches on the orbit ot the periodic comet Hotmes and on the 7

perturbations of its elliptic motion. 642.

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