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KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN -- TE AMSTERDAM -:-
PROCEEDINGS OF THE SECTION OF SCIENCES
NG) al ME XI 257 PART — )
JOHANNES MULLER :—: AMSTERDAM JULY 1909
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- en Natuur
(Translated from: Verslagen van de Gewone Vergaderingen der Wis Afdeeling van 24 December 1908 tot 23 April 1909. DI XVII
Uy MI Aberrn7
‘
_ Proceedings of the Meeting of
December 24
January 30
February 27
March 27
April 23
LR RD a a
KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN TE AMSTERDAM.
PROCEEDINGS OF THE MEETING of Thursday December 24, 1908.
DCC
(Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige Afdeeling van Donderdag 24 December 1908, Dl XVII).
EE EE NT TE S.
W. Borek: “On the biological significance of the secretion of nectar in the flower”, p. 445,
W. Kapreyn: “On a theorem of PAINLEvú”’, p. 459.
P. ZEEMAN: “The law of shift of the central component of a triplet in a magnetic field”, p. 473
J. D. van per Waars: “Contribution to the theory of binary mixtures”. XII, (Continued), p. 477.
A.J P. van DEN BROEK: “About the development of the urogenital canal (urethra) in man”. (Communicated by Prof. L. Bork), p. 494. (With one plate).
Jan DE Vries: “On bicuspidal curves of order four”, p. 499.
E. H. BücnNer and Miss B. J. KarsreN: “On thesystem hydrogen bromide and bromine”. (Communicated by Prof. A. F. HorreMan), p. 504,
Miss T. Tammes: “Dipsacan and Dipsacotin, a new chromogen and a new colouring- matter of Dipsaceae”. (Communicated by Prof. J. W. Morr), p. 509.
A. F. HorLEMAN and J. J. Porak: 1. “On the bromation of toluol” 2. “On the sulfo- nisation of benzol sulfonic acid”, p. 511.
Botany. — “On the biological significance of the secretion of nectar in the flower.’ By Dr. W. Burck. (Communicated in the meeting of November 28, 1908.)
In an article in the Recneil des travaux botaniques Néerlandais vol. IV.*) I have explained in detail, that Darwin in 1859 put forward the hypothesis, that a cross with another individual is indispensable for (he species and that, at the time, he considered the structure of flowers to be generally such as to ensure, or at least to favour, eross-fertilisation, but that in later years he, however, left this stand-point. I showed from his later writings, that the observa- tions and experiments of many years had brought him more and more to the conclusion, that a much greater significance should be attached to self-fertilisation, than he had at first imagined; I also showed that, at the close of his studies, he was not very far from giving a negative answer to the question whether floral structure favours cross-fertilisation. Since then, observations have been made on a number of tropical plants, the flowers of which are always closed, so that in such plants the possibility of cross-fertilisation is
1) An abstract of this may be found in Biolog. Centralblatt. Bd. XXVIII. N°. 6. 1908. 30 Proceedings Royal Acad. Amsterdam. Vol, »i.
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excluded and I further pointed out, that from these results there can be no question of a natural law in the sense imagined by Darwin. Furthermore, the view that cross-fertilisation might be advantageous to the species, has been rendered untenable by our present knowledge of the structure of the nucleus, and its function in the life of the plant, and also by our modern ideas concerning the nature of fertilisation.
Suppose we now give up this view, and fall back on the funda- mental hypothesis, which was put forward by GArrner in 1849, that only by self-fertilisation, vigour and fertility of the species are pre- served, since a cross may lead to hybrid-formation, which diminishes the fertility of the plant. It then follows, that floral biology, which has started in its considerations from the opposite view, has lost its basis and must be built up anew. We have been led astray by our ideas regarding the significance of the properties of the perianth
— its shape and dimensions, its colour and odour — and regarding the various mechanisms of the flower — dioecism, monoecism, hete- rogamy, dichogamy, hercogamy and self-sterility — all of which we
thought we could explain as useful adaptations for visiting insects in order to ensure cross-fertilisation; it must be possible to explain them in another way. I have already sbown in my previous paper that diclinism and hereogamy can be explained by mutation and that protandry and protogyny must be considered as characters of organisation, and not of adaptation.
With regard to the phenomenon of self-sterility I limited myself to pointing out, that this should be considered primarily as the result of hybridisation, rather than as a special adaptation.
In order that we may now obtain a better conception of the qualities of the floral envelopes, we must again adopt the view of the older biologists, who regarded these envelopes as organs for the protection of the sexual apparatus.
We must therefore consider to what extent the sexual organs require the protection of the perianth, not only when they originate and develop, but also during the flowering period. Hitherto we have been accustomed to look for a connexion between the various pro- perties of the perianth and its significance in the attraction of insects. Now we shall have to test these same properties, especially of shape, dimension, position and the distribution of fragrant vapours, by the question, how they may be considered to be of importance to an organ, which is intended to protect the sexual organs from unfavour- able external influences. More than has hitherto been the custom in floral biology, we shall have to pay attention to the anatomical structure and the physical and chemical properties of the floral
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envelopes, to the way in which the floral leaves are arranged in the bud, with reference to each other, (the aestivation) to mutual coalescence, io the presence of scales, hairs and glands, to the secretion of water, honey, mucilage, etc. Also, if we give promi- nence to the protection of the generative organs as the basis of our considerations, we shall have to investigate whether the secretion of nectar is not connected with this protective function.
This connexion may be inferred all the more readily on account of the unmistakable correspondence of the secretion of nectar during the flowering period to that of water or of a mucilaginous fluid in the so-called water-calyces, the latter secretion being considered a means of protection of the sexual organs.
With regard to this secretion of water I beg to recall, that 20 years ago Treus first drew attention to the remarkable phenomenon that the floral buds of Spathodea campanulata Bravv., a tropical Bignoniacea, are filled with a watery liquid, secreted by a large number of glands, which cover the inner surface of the calyx, so that the petals, stamens and ovaries develop under the protection of this fluid. The liquid contains traces of the hydrochlorides, carbonates, nitrates and sulphates of potassium, sodium and calcium, has an alkaline reaction and contains traces of ammonia; sugar was not found in it.
A similar secretion of water in the closed flower-bud was after- wards also observed in other plants. We may mention the papers of LAGERHEIM, GREGOR Kraus, HALLIER, KOORDERS, SHIBATA and SvEDELIUS, to whose investigations [ do not propose to refer here in further detail, as I intend to publish my own observations on this subject before long. From these it will be evident, that the phenomenon is not limited to the tropics, but can also be studied here.
I only wish to emphasize, however, that all naturalists, who have occupied themselves with the subject, have accepted the opinion of TrevuB, that the secretion of water is a means of protecting the sexual organs against the unfavourable consequences of too strong transpira- tion, and that my personal observations, especially in this country, have shown me the connexion between the secretion of water and of nectar, and have gradually confirmed me in the conviction, that by the nectar-secretion the sexual organs are protected.
I wish briefly to explain the train of thought, from which I started my investigation.
The observations on plants with water-calyx and especially the detailed investigations of Koorpers have taught us, that already long before the corolla and the sexual organs are laid down, the very
30*
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young calyx (the development of which in all these plants is burried on, before that of the other parts of the flower) is protected against the dangers of exposure to the atmosphere ; sometimes the calyx is protected ‘by glands, which may or may not be active, sometimes by a thick covering of hairs, which retains air, sometimes by both these means. It is intelligible, that this young organ, the vascular bundles of which are as yet unperfectly developed and therefore unable to compensate adequately for the loss of water through tran- spiration, should require a special, temporary protection, in so far as it is not surrounded by bracts.
Now we see at a later stage, in which the calyx has already acquired certain dimensions and in which its anatomical structure is nearing completion, that the same glands appear on the inner surface, and by their activity more or less fill the cavity of the calyx with water; this secretion of water supplements the protective function of the calyx towards the other parts of the flower, which are now beginning their development.
Later, in older buds, when the stamens and ovaries have already made considerable progress, the same glands appear on the outer- and on the inner surface of the corolla. The former, the outer glands, are especially active in protecting the petals against excessive trans- piration in the short period, between the bursting open of the calyx and the development of the petals to their full size — temporarily therefore ©. The significance of the hairs on the ner surface would then be, that in the same period they keep the sexual organs in a moist space.
When we see therefore, that the flower is carefully protected against transpiration from the first stages of its development to the moment of opening, the question naturally arises, whether at the opening and during the flowering-period, the sexual organs are under such especially favourable conditions, that they require no protection ?
This is certainly not the case; during the flowering period the ovary is not placed in favourable conditions.
When opening, the flower enters upon a period, in which the stamens and the ovaries — exceptions apart — have reached their highest stage of development, do not require food for further growth and are in a state of rest; the ovaries are awaiting fertilisation in order to be called to new life by that stimulus; the stamens in a state of maturity, await the evaporation of the superfluous water from the anthers.
1) This opinion will later be supported by examples; I propose to show, that at this stage, when the corolla is still completely closed, water or nectar is found in many flowers.
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At the opening of the flower the perianth, and especially the corolla, are in very different condition, the latter generally not having reached anything like its full size, when the sepals move apart. In a very short time, the corolla grows out to its normal dimensions, to be afterwards during the whole flowering period the seat of various physiological processes, consisting partly, in the transformation to its own use of material laid down in its tissues in the bud, and partly in the continuous production of fragrant vapours, which the flower gives off in that period, very often also in the production of nectar etc. If we further remember, that the considerable quantity of water which the corolla gives off to the atmosphere by transpiration, is con- tinually replenished by fresh supplies, while the stamens on the other hand receive less water from the thalamus than they give off, it becomes clear, that the nutrition-stream moves principally in the corolla.
The consideration suggests the following questions: What means are at the disposal of the ovary for escaping the harmful consequences of too strong transpiration? Is the secretion of nectar perhaps to be regarded as one of these means?
I venture to think that I have obtained an affirmative answer to the last question and hope that I may succeed in obtaining acceptance of my opinion.
I wish to preface a description of the ponies and of nectar-secretion in Fritillaria Amper ialis.
Fritillaria imperials bears large, bell-shaped flowers turned with the opening downwards, and consisting of a perianth of two trimerous whorls, a superior ovary with a long style and tripartite stigma, and 6 long stamens, with filaments entirely enclosed in the bell, but with anthers protruding outside. Generally the style is somewhat longer, so that the stigmas are under the anthers and outside the flower. The cylindrical ovary escapes observation, as it is wholly surrounded by the fleshy filaments of the stamens, which form a close-fitting tube around it.
Not until fertilisation has taken place and the perianth has withered, do the flowers become erect; the fruits afterwards are also erect.
Each perianth-leaf bears close to its base a large saucer-shaped, shiny, white nectary, which is surrounded by an elevated border, and secretes heavy drops of fluid during the flowering-period.
The whole of the perianth is very rich in glucose, not only at the time of flowering, but already much earlier. A section through the middle of an adult perianth leaf, about half-way between base and top, shows, that the mesophyll, which is here 13—14 cells thick,
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consists of thin-walled cells, which leave large intercellular spaces between them. The vascular bundles are strongly developed and take up almost the whole thickness of the leaf. In no part of the transverse section can starch be detected, but the whole of the mesophyll is very rich in glucose; starch occurs only in the nectary. In a section through the nectary, 4 different parts can be made out even at low magnification. First there is the honey-secreting tissue proper, consisting of 3—4 layers of small clost-fitting cells, densely filled with protoplasm and containing large nuclei. Under this there is a tissue, 8 or more cell-layers thick, composed of larger cells with very distinct intercellular spaces; these cells are crowded with numerous small starch-grains. Outwards or downwards there follows the region of the vascular bundles, where the mesophyll still contains starch. Finally the latter tissue gradually passes into that containing chromatophores, which again consists of considerably smaller cells and is closed off on the outside by an epidermis, consisting of pina- coid cells, the outer wall of which, in this region, is much thicker than in any other part of the perianth.
The starch which collects under the secretory layer, is already found in sections of very young nectaries, for instance in buds about 2.5 em. long.
That it’ is from this material that nectar is afterwards formed, becomes evident on the examination of nectaries, which have already been forming honey-drops for some days; a distinct diminution of starch may then be observed, and at the end of the flowering no starch whatsoever is found.
A section through a stamen shows, that the latter is traversed by a comparatively thin vascular bundle, and that for the rest the tissue consists of large cells, which give a very strong reaction with Frurine’s test-solution for glucose. Externally the tissue is enclosed by a small-celled epidermis with a comparatively thick outer wall, which presents a granular cuticle. It may be, that by being enclosed by stamens, which are rich in glucose, the ovary is not so completely protected against the harmful consequences of exposure to the atmos- phere as an inferior ovary is by the thalamus, but nevertheless the two kinds of protection are comparable; in any case the ovary thus receives considerable protection during development.
It may be of interest to note, that the stamens continue to enclose the ovary, when the anthers have fallen off. The filaments remain fresh and in their original position, as long as flowering continues.
The secretion of nectar begins soon after the perianth-leaves separate, and the tips of the anthers protrude out of the flower.
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The secretion is very abundant. Generally large drops hang down from the nectaries in plants in the open. If a cut flowering specimen
be placed in a glass of water, under a high bell-jar — in a fairly moist space therefore, where evaporation is limited — drops may
be seen to fall down from time to time. When plants which have been grown in pots, are placed in a dark room some time before the opening of the flowers, it is found that the secretion of nectar is quite independent of light and continues day and night. If the nectar be removed by means of a pipette, the drops are renewed as well and as quickly as in the light. The nectar can be removed for several days; each time new drops appear again. From this we may deduce, that the evaporation of nectar in plants in the open air is fairly considerable, and that the nectaries continue to act as long as the flowering-period lasts.
Fritillaria imperialis is one of those plants, in which the dehiscence of the anthers depends on loss of water by transpiration. Although in many orders, such as the Papilionaceae, Antirrhineae, Rhinanthaceae, Malvaceae the dehiscence of the anthers is independent of the hygros- copie condition of the atmosphere, and the pollen is equally well liberated in a moist flower as in dry air, this is not the case in Fritillaria. As has been said above, the tissue of the filament indeed contains a considerable quantity of glucose, but nevertheless the osmotic action, which the sugar exerts in abstracting water from the anthers, is evidently not enough to make them dehisce. If a young flower be enclosed in a moist glass box, or a cut plant be placed under a high bell-jar in surroundings, which are only moderately damp, the anthers remain closed during the whole of the flowering period, whereas in the open air they often dehisce on the first day in bright, dry, spring weather, after having lost 90°/, of water. It follows from this experiment, that the anthers can dehisce, because they protrude from under the flower. If this were not the case, if the filaments were a few centimetres shorter, the moist air, inside the flower, would prevent the dehiscence of the anthers. That during the flowering-period there is a strong current of water through the vascular bundles of the perianth-leaves, which continually supplies the latter with water to compensate for the loss by transpiration, needs as little proof as the fact, that this watercurrent has been turned away from the stamens. If this were not so, there could be no question of the dehiscence of the anthers.
I now come to the conclusion, that the Mritillaria-flower is to be regarded as a cup in which the air is continually kept moist during the flowerlng period by the evaporation of 6 large drops of fluid,
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secreted in its upper parts by as many nectaries, the transpiration- loss of which is made good by fresh supplies of fluid, day and night, as long as flowering continues.
Inside this moist cup there are the ovary and the stamens, which remain in a state of rest during the flowering period, and receive only a small supply of water from the thalamus. For to the extent that they are enclosed in the cup (the ovary for its full length, and the stamens with the exception of the anthers) they are protected against dessication by damp surroundings, whereas the anthers, hanging out of the cup, are exposed to evaporation.
According to the analysis of Bonnier the nectar is very rich in water and contains at most 5—-7 °/, of sugar. If there were no sugar at all in the fluid, one would not hesitate to call the nectaries of Fritillarta perianth-hydathodes, and to consider them quite similar to the calyx-hydathodes of Spathodea campanulata and similar plants.
In fritidlaria the nectar does not come into direct contact with the ovary, but is found outside the sexual organs. This method of nectar-secretion, which I purpose to call, for the sake of brevity, a peripheral one, is not the most general. A number of plants may indeed be cited, which agree with Mritillaria in this respect, suchas Trollius, Abutilon, Liliwn and Helleborus, but in most plants the nectar is secreted in such a way, that the ovary is directly moistened by it, as in Labiatae, Boraginaceae, Solanaceae and other orders. In contradistinetion to the peripheral, | wish to call this a central secretion of nectar. Very often the nectar is secreted in more than one part of the flower; in such cases there is a combination of the peripheral with the central method.
In numerous plants the moistening of the ovary is greatly increased by a thick covering of soft hairs or by a thick felt, which covering is saturated with nectar in various ways. Sometimes the nectar is secreted by the ovary-wall, and ascends between the hairs, as is for instance, the case in most species of Verbascum and in Heli- anthemum vulgare, which are wrongly called nectarless plants. In other cases the covering itself consists of hairs which secrete glucose ; this occurs for instance in the species of Paeonia, another genus which is wrongly considered to be devoid of nectar. Often, however, the nectar which saturates the ovary-covering, is brought up from the thalamus, as for instance in Pulsatilla and other Ranunculaceae, which will be considered below. Especially when such covered ova- ries are close together (e.g. in Pulsatilla each flower has about 100 Ovaries) it may be readily imagined, that by evaporation of the nectar
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the ovaries are always in a moist atmosphere. By this I mean, that one may assume, not only that the nectar is continually replenished by fresh secretion (this can indeed be observed in many plants) but also that on increased concentration, the nectar never dries up, be it, that it absorbs aqueous vapour from the air, or abstracts water from the ovary itself. This moistening of the ovary reminds us vividly of certain well-known mechanisms for protecting an organ against excessive transpiration, such as a covering of wax, or of mucilage- secreting glands. In this connexion | may point out that among plants without nectar, there are indeed some, in which the ovary is protected by wax, as in Papaver, Eschscholtzia, and Glaucium or by mucilage, as in species of Lysimachia, Ononis spinosa, and Verbascum Blattaria. It thus becomes intelligible, that these plants can do without nectar. In Verbascum Blattarta the ovary, which is fairly deeply hidden, is covered from top to bottom with compound glands, which correspond in structure with lupulin- and /zbes glands, continually pouring out a layer of mucilage over the ovary.
This is the more remarkable and important, since, as was men- tioned above, the ovary of all other Verbascum-species is covered with a felt, rich in glucose. We find therefore in different species of the same genus two different means of protection, to which the same biological significance must be attached.
I now wish to explain further, by some notes on Ranunculaceae and Malvaceae, what was said above with reference to the secretion of nectar in different parts of the flower.
Let us consider first of all the flower of T'rollius europaeus L.
In Prollius the 11 or 13 large, hemispherical. sepals with over- lapping edges, form an approximately ball-shaped envelope round the sexual organs. The petals, generally 10 in number, are yellow and spatulate, and secrete honey on the middle of their inner surfaces. The stamens numbering about 160 and placed in numerous whorls, surround about 30 ovaries. Except for a small opening, facing upwards, the flowers are closed; only the stigmas come wholly or partially into view.
At the beginning of the flowering period the anthers are at about the same height as the stigmas, and the ovaries are surrounded and protected by the column of stamens.
Later this is not the case to the same extent, although a few whorls of the inner stamens, the anthers of which do not come to complete development, retain their places.
As in Fritillaria, the ovaries of Trollus are in a moist space, and are furthermore protected laterally by the stamens. Whereas, however,
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the liumidity of the flower in Fritillaria does not interfere: with the dehiscence of the anthers, because these are outside the flower, this is not so in Trollius, where the dehiscence of the anthers is equally dependent on the evaporation of superfluous water into the air, for in Trollius the stamens are enclosed within the calyx.
This is the explanation of the remarkable phenomenon, that the stamens, beginning with those of the outer whorl and then grad- ually from the periphery to the centre, become elongated soon after the opening of the flower and bend inwards, until their anthers are near the opening; tle anthers of the inner staminal whorl then come to lie immediately above the stigmas. If one places a young flower in a closed glass box, the phenomenon may be followed step by step, and one observes at the same time, that as long as the flower re- mains in the glass box, the anthers remain closed. In an open box on the other hand, the anthers are seen to dehisce as soon as they have come under the opening of the flower, and their pollen is seen to be seattered on the stigmas. Observation in the field likewise proves, that the anthers remain closed in damp weather.
Honey is not secreted in any place other than the petals. In the main the arrangement of the flower is quite like that of Fritillaria. The closed condition of the corolla can hardly be explained other- wise than as a device to prevent the rapid evaporation of the nectar into the air and is connected with the erect position of the flower’).
As a second example of the methods of nectar-secretion in Ranunculaceae, | now choose the flowers of Clematis and of Anemone, which do not possess petals, but where the calyx takes the place of the corolla, and where no nectar is observed on the periphery of the flower. This is the reason, why they are referred to as nectarless plants in the literature on the biology of the flower. That this is by no means correct, is at once evident when we wash the ovaries, which are thickly covered with silky hairs, for a moment with a drop of distilled water on a slide, and then warm the water with a drop of Frarine’s solution; we then obtain a strong glucose-reaction, proving that the hairy covering of the ovary is saturated with nectar. Further investigation shows, that this nectar is derived from the interstaminal portion of the thalamus.
The droplets of nectar, which are secreted here, are sucked -up between the stamens and the ovaries and are retained, especially by the hairy covering of the latter.
I must now recall that many years ago, Bonnier already drew
1) | believe that this is also the explanation of the closed flowers of Calceolaria, Fumariaceac, Antirrhineae, Rhinanthaceae etc.
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attention to the interstaminal secretion of nectar in Anemone nemorosa. He stated that the thalamus contains much sugar, and that its inter- staminal portion is covered with numerous thin walled papillae, from which, under favourable conditions, minute drops of nectar are seen to exude. My own investigations have shown me that what BONNIER ') observed, may be called a pretty general phenomenon in the order of Ranunculaceae i.e. in many genera, nectar is secreted from this portion of the thalamus.
The flowers of Anemone and of Clematis may therefore be con- trasted with those of 7vol/ius, as regards secretion of nectar. Here the nectar comes into direct contact with the ovaries and it is evident, that the numerous drops of honey, which are found every- where between the stamens, and which are constantly renewed, contribute not a little to the maintenance of a certain degree of humidity in the neighbourhood of the ovaries.
It is remarkable, that in many other Ranunculaceae the nectar is secreted in the flower in two places, so that a peripheral and a central secretion may be distinguished. It should be noted, that in some genera the two methods of secretion are of about equal im- portance to the plant, but that in other genera the peripheral one is much the least important.
The flower of Aconitum may serve as an example of a plant in which both secretions are of importance for the protection of the sexual organs.
At the beginning of the flowering-period the 3—5 quite glabrous ovaries have not yet reached their full development. They can scarcely be discerned, as they are enclosed by the numerous stamens. These stamens are distinguished by broad filaments, which are very rich in glucose, and which, being closely pressed against the ovaries, protect the latter against external influences. The sexual organs are kept moist by a secretion of nectar from the interstaminal portion of the thalamus.?) The sepals and petals are also rich in glucose.
The two superior petals are metamorphosed to nectaries with long stalks and during the time of flowering these secrete a copious supply of honey. The two superior, dark blue sepals have coalesced to form a helmet-shaped hood, which, as long as the flower is still in bud, encloses it for the most part and further, during the
1) Bonnier, G,. Les nectaires. Annales des sciences naturelles. Botanique. Tome
VIIL. 1879. p. 141.
2) Not unfrequently the nectar-drops can be detected on the stamens with a simple lens; the presence of nectar between the stamens may moreover be easily demonstrated chemically, by depriving a young flower of its calyx and corolla, and washing it with water.
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flowering-period, acts as a protective roof to the two nectaries and the sexual organs below them, while the latter are surrounded by the remaining sepals and petals. The secretion of nectar has once more rendered the flower a moist chamber, in which the sexual organs are protected against the dangers of dessication. At first the stamens, with anthers bent downwards and closed, lie turned away from the entrance of the moist chamber. Later they become erect; afterwards they become elongated, and so bring the anthers to the entrance of the flower, where they can give up their excess of moisture to the air, at least when the latter is not too damp. As they dehisce, the stamens again bend downwards with empty anthers. The broadened parts of the filaments do not, however, bend in this way, but retain their original position and protect the ovaries throughout the whole of the flowering period. It is not until this stage that the stigmas, which are now fully developed, come to the entrance of the flower.
Although the corollar-nectaries of Aconitum are not much less important than the thalamus, as regards secretion of nectar, this is not so in all genera of Ranunculaceae, as has already been pointed out. In Ranunculus, Batrachium, and Ficaria the corollar-secretion is of much less significance and that of the thalamus certainly much more important. In Pulsatilla the corollar-secretion is still further reduced and in the genera Paeonia, Caltha, Anemone, and Clematis the corollar-nectaries no longer occur; here the honey-secretion of the thalamus has become of primary importance.
In Caltha palustris secretion of nectar can be observed in the flower in three places: first at the periphery of the thalamus, where in the allied Helleboreae the stalked corollar-nectaries are placed ; secondly at the interstaminal part of the thalamus; thirdly on the wall of each ovary. The ovaries of Caltha are glabrous, but on both sides of each ovary there is a spot, covered by hundreds of delicate papillae with very thin walls. Each of the latter secretes a minute droplet of nectar, and the large drop, which is formed by the fusion of the droplets, can easily be detected with a lens between any two adjacent ovaries. The parietal papillae here replace the hairs of other genera,
The extent of the reduction in the peripheral nectar-secretion of other genera is best observed in Ranunculus and in Pulsatilla.
The flower of Ranunculus acer for instance, agrees with that of Trollius both as regards the position of the stamens relative to the ovaries and the elongation and inward-movement of the stamens. The nectar-secretion at the base of the petals cannot contribute to the protection of the sexual organs by keeping the flower moist,
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except possibly on the first day of flowering, when the corolla is still cup-shaped. In no case can this secretion be of importance during subsequent stages, when the corolla is spread out. If there were here no nectar-secretion at the interstaminal portion of the thalamus, the ovaries would be in danger of rapid destruction owing to dessication.
In Ranunculus auricomus the peripheral secretion is still much less important. Here often one or two and sometimes all petals ure wanting, and with them the nectaries; frequently, moreover, the nectaries are rudimentary.
In the genus Pulsatilla the peripheral nectar-secretion is likewise insignificant (its seat is in the metamorphosed anthers of the outer whorl). In Pulsatilla vulgaris, P. pratensis and P. vernalis it has been observed, that the nectaries frequently do not secrete any nectar; here nectar-containing and nectarless plants are found; P. alpina is quite free from nectar, according to Scnurz. The nectar-secretion from the thalamus is therefore, also in this genus, of primary importance; during the flowering period the numerous ovaries are each, as it were, covered by a mantle saturated with glucose.
In the natural order of Malvaceae the true significance of nectar- secretion is not less clear than among Ranunculaceae.
I shall not be able to consider this subject in detail in the present communication, but may recall, that BeHrens showed in 1879, that in Abutilon, Althaea, and Malva the bottom of the calyx bears a nectary, consisting of a large number of closely crowded multicellular “Sekretions-Papillen”, which together form a large secreting surface. Each “Papille” consists of a large number of cells, placed in a row, e.g. in Abutilon insigne 12—14. What Benrens thus describes pro- bably applies, as far as my own investigation extends, to all Malvaceae. I found these nectaries also in the genera Hibiscus, Kitaibelia, Malope, Anoda and Sidalcea.
Whether in general, bowever, secretion is a constant phenomenon in these calyx-nectaries, is doubted by various authors. Of many species it is not known whether they ever contain nectar, and of other species the accounts are contradictory; in the case of some, it might be assumed, that the individuals of the same species differ among themselves. Thus, for instance, Kircaner could not find any nectaries in Abutilon Avicennae, whereas in this country the same plant is so rich in nectar, that the latter can be seen with the naked eye. As regards Hibiscus, those species, which are best known in Europe, namely #H. syriacus, H. Trionum, and H. esculentus are regarded as nectarless. The large flowers of Abutilon are however very
( 458 )
rich in nectar, so much so, that the nectar is removed by honey-birds.
Being peripheral, the secretion of the calyx-nectaries may be compared with that of the corollar-nectaries of Manunculaceae. My investigations have now shown me, that in the order of Malvaceae a central secretion of nectar may also be observed, which in most genera gives the impression of being the more important — perhaps in all genera except Abutilon.
As is well known, the stamens in Malvaceae are united to form a tube. This staminal cylinder, which extends upwards round the ovary, is, at its base, joined to the corolla in such a way that their common tissue encloses the ovary and hides it from view. If the ovary be now liberated from its little “house”, its wall, in almost all Malvaceae, is found to be thickly covered with nectar-secreting trichomes of the same structure as those, which constitute the calyx- nectary (Sekretionspapillen of Brnrens) and these trichomes conti- nually pour a layer of glucose on the ovary. In Hibiscus esculentus and in H. Trionum these ovarial trichomes are even larger than those of the calyx-nectary, and consist of 28 cells. The ovaries are therefore not only enclosed in the staminal tube, but are always confined in a space, kept moist by nectar-secretion.
I hope afterwards to return to a detailed study of this order, which is so extremely interesting as regards nectar-production.
Before closing this communication, I still wish to call attention to two important matters. In the first place to the secretion, which takes place in many flowers, while they are stil in bud. We are accus- tomed to assume, that secretion only begins at or after the opening of the flower, but I have found many exceptions to this rule. The phenomenon may be observed in Ranunculaceae especially. The ovaries of Clematis Viticella, covered with silken hairs, the ovaries of Paeonia, Pulsatilla and of Aconitum are bathed in nectar, long before the opening of the bud, and it may probably be assumed with safety, that the secretion of nectar, which already takes place in the bud, serves here to protect the sexual organs, and is therefore comparable to the secretion of water in flowers with a water-calyx. In the flowers of Aconitum I found that indeed the central, but not the peripheral, secretion may be observed before the opening; this suggested to me that the latter secretion serves more especially to keep the flower moist during the flowering period. Further investigation will be required to show, whether this difference can also be traced in other plants with a double secretion of nectar.
Before there is any question of the flower’s opening, a copious secretion of nectar may also be observed in other plants, such
( 459 )
as Melandrium album (Lychnis vespertina), Hyoscyamus niger, Galanthus nivalis, many Papilionaceae and Epilobium angustifolium.
In the second place I think it may be useful to refer briefly to the so-called nectarless plants, because it might be argued that these do not support the truth or general validity of the hypothesis, put forward above.
I have already had an opportunity of pointing out, that some plants, which do not contain nectar, have their ovarian-wall covered with war, and others with glands secreting mucilage; to these secretions the same biological significance is attached as that, which I think should be attributed to nectar-secretion. Furthermore, I have already mentioned a number of plants, which are recorded as nectar- less, but which, nevertheless, must certainly be reckoned among those containing nectar, namely species of Anemone, Clematis, Pulsatilla, and Paeonia in the order of Ranunculaceae, also Helian- themum vulgare and the various species of Verbascum and Hibiscus. I will only add, that it can be easily shown by chemical means, that the so-called nectarless Rosaceae: Rosa, Poterium, Agrimonia, Aruncus and Spiraea have been wrongly included in this class. Here indeed the nectar is often difficult to observe, but it is none the less present, as in other Rosaceae. If the flowers are extracted with water, so that the nectar, which has been thickened by evapo- ration, passes into solution, the presence of glucose may readily he demonstrated in all these plants. Finally it may be pointed out in this connexion, that very many plants do not require a special protection by nectar, either because the ovary continues its growth without interruption, (on account of early fertilisation, which often already takes place in the bud) or because it is not exposed to the air during the flowering period.
The latter case occurs especially in the genera Plantago and Luzula, in Nymphaea alba and Erythraea Centaureum, in Luncus, in most Grasses and in other anemophilous plants.
Mathematics. — “On a theorem of Painrevé’s.” By Prof. W. Kaprnyn.
1. Parninve, in his well-known memoirs on differential equations of the first order, investigated the question when the integrals possess a definite number of values or branches if the independent variable turns round the critical parametric (not the fixed) points.
For differential equations of the first degree
BY — ul 1259 ) : (1) TE EEC NEN
( 460 )
where Pand Q represent polynomials in y, he has proved that if the integrals possess m branches, there always exists a substitution me Oe: + Lit git = eee = Ly
My ae
by which the equation (1) may be reduced to an equation of Riccati
u
(2)
.
du Gat ee KE ne ec da the coefficients L, WM, G, H, K being functions of x.
Our object in this paper is to prove this proposition in another way, starting from the form of the integral
oe an YP + An gee -. thy +4, fg Sn ee ee Seat AY oe eg where C represents an arbitrary constant and 2 and u functions of z.
The treatment of the two cases n=? and „=3 will be sufficient to show that the proposition holds good generally.
(4)
2. If n=2, it is evident from the integral
Ay? + Ay + Aa C= a gie os nst on + i 1 y? > BY > Bo
that the differential equation must be of the form
dy ay’ + ay’ Hay + ay +4, de by? + 2b,y + b,
(6)
the coefficients « and 5 representing functions of z. Differentiating the equation (5), we find between a, 0,4, u, the following relations 6 being an indefinite factor, Ce =e Oa, = wa,’ + a,’ — Au, Ga, = ud, Hud +4,’ — Au — At Oa, = ud, Hud, — Jol — Allo Oa == Uik =d 0b, = 4, — ud, Ob == A = ba, Gb, == 1,4, — ots -
From the three latter equations (7) may be induced
( 461 )
hie bb, = 0 bu, Ek bu, me b, = 0 and from the five preceding
ee OF 0 Bos OF. 0 0 Gert
Gt. PE 0 ==; fi, 1 40 —a, 0 te) rane B Bh, A, “| 2 pp, 1 Al —A,
co eg | let Age =d, 0 uw —A, —A,
Gone OVE Nae 0 Oy) OV raar 0 -A,
This equation may be easily reduced to an equation of Rriccarr. For adding up, in the first determinant the third column multiplied by 4, to the fifth and in the second determinant the second and third columns each multiplied by 4, to the fourth and last, we get
Ge ty OF eo BO Oee Oe: as ns Me OO wb OO. 6 oo has Both kb, . u. U, 1 b, 0 a, 0 u, u, 6, O uu, bb, a0 Ou, 0 || OF Oe 006, If now we substitute ee Oey ae eae ie
1
in the denominator, and subtract the fourth and fifth columns each
multiplied by = from the second and third, .we find
ER m1 0 0 0 Beb ee bh, OE ED. Go 6 0 &,
1 If we in the same way subtract the fifth column multiplied by Ss from the third, the numerator takes the form
dl
Proceedings Royal Acad. Amsterdam. Vol. XI.
1 0 o +0
a, b +b Pipe i Me ne I, b ve ° | 5 < a, Wo De 1 b, | == A fit, aia Bu, = C b,+6 a, 0 0 bt allo b, b, BA 0 0 it 20
where the coefficients have to be determined still. If we put u, = 0, the coefficient C is found to be
oe |
| b, mo
1
a C=a, => (0, — 5,65 1
Dividing further both members by u,? and supposing afterwards u, = 0, we get
a,0 00 0 i la, 2 1 0 0 : EEN b, hi 4 3 Ae eee |p Gh) 1 ee 1 b, a 0055, yA A 0 dea
Differentiating both members with respect to u,, and substituting u, — 0, we get for B the form
a, 0-0 One ged Oe 20 be b a= Leas oe 1 WD b, b, b b Bi Sal = Re) = 0 5 0 dy 1 1 : b ; 2 a. 0 (=o aor B 1 b, 1 pk b, 1 a, 0. 0), Qe coo 0° 1 0
The first of these determinants is identically zero; the second developed, gives
( 463 )
B bj B == — a a, + boa, — b,a, + bya, co ay. 1 1 Hence u, satisfies the following equation of Riccati a 1 Ho ET A (6,>—, by) BT = (6,’a,—b,b,a, + b,?a,— 6, bya, tT bra) — 1 1 a, : Sets nn ee 8) 1
We now proceed to find the substitution of PAINLEvÉ. From the general integral erste ad 1s | y+ hy + Uo it is evident that u, is that particular solution of the equation (9) which satisfies the equation by + bake
ner rd nt 1
if we attribute to y that particular integral of (6) which corresponds to the value C= ox. Therefore
a by’ + boy f boy zl b, is the substitution which reduces the differential equation (6) to (9).
3. From the preceding we may also deduce the conditions which must be satisfied by the given differential equation. For the three last equations (7) give
d bz nee cms: Eed b, _ d Wido Hod de \ b, de 2,-—U,d, dx \b, de 2,—U,Az
6 (5,b,'— 6, b,') mT Doda + b,a,'—b9A,'— 6, Azu, Horde,
or and 6 (6,5,'—4,6,') = bokod2 — bm yA,’ 55 bood ER bilt, —baAoldo- Combining each of these with the five first equations (7) and eliminating 2, 4,’ 2, u, u, we may write the conditions
a, i, 9 0 0 0 | ds pt 0 —A, 0 | a, B 1 —à, —A, fee! a, wn, > Be, A, —A, | a | 4% 0 0 u, 0 —A, | | (b,5,,,—6, b, —b, —b,A, ba, |
31*
and a, 1 0 0 0 0 | a, uy 1 0 —i, 0 | Ge? 25 he ee en a oe " a, 0 by gate PE. hi a, 0 0 He 0 —A,
| (b,5,) bo — ie, bu, b,A, oA,
where (,6,') and (b,b,) mean 6,6,’ — bb, and bb, — bb, respectively. Reducing these determinants in the same way as before, we have immediately
ye ORD. Div] a. pd 0 BO
6, a, B, 140, 0
b, | | : [SO ee a, 0 0 == b, b,
1
a, “D007 0, sb. C08 Be ON os
the latter row representing the following values
a= (.6.) §=—b, y=, db e= sl, 11) a=(bb,) B=bu, y=O d=0 e=—b 5=bb, 6, tb,
~~ the determinant (10) takes the form Au, + B.
If we write u,— 1 By differentiation with respect to u,, A is determined by
3 | LS ae du, | b Ed 0 | b, A= — a,b, | 5 eeh, Bit | b, | 6 a0 op, b,
or
( 465 )
In both cases this expression vanishes. Therefore both conditions are found by writing u, =O in the equation (10). In this way the conditions we looked for, are the following
where the last row
4. When n= 3,
1 Orr Or Ops 0 b, — Bee cle 70 b, b, 0 — 1 5, 0 b, ih den Ed ret ab) b, 0 0 En Pega
0 0 0 @Duz=oy Td e §
is given by the relations (11).
the general integral
NL RE | = nd a = sale Pec agit, yv Huy +wy+4,
shows, that the differential equation must be of the form
dy ay Hay Hay Hay’ + ay’? + ay + a,
= 14 da b,y* + 4b,y° + 6b,y° + 4b,y + b, VE with the following relations between the coefficients a, 6, A, u: Ga, =d, \ Oa, = ud, + 4,' — Al, Oa, = ud, Hud, +4 — Au, — As Ga, = ud, EE ud, ae 4,2, ' = Jo ns hu, FE do, eS Ao Oa, = ud, Hud, Hud, — Agu,’ — Al — Alo 6a, = Hod, oF ud. nie Aobty’ aes Alo (15)
Ga, = Udo i Au.
6b, =À, — Ay,
40b, = 22, —
24,u,
60d, = 34, +5 Au, En Ast, Ee 3À, U, 40b, = 2d,u, — 2A,u,
8b, DE Jo, ig du,
( 466 ) Eliminating alternately the ws and 4’s from the five last equations (15) we have ' Bb ae bbl eha) 6b,4,? — 6b,4,A, + 26,4,? + b, (84,4, — AA) = 0 | (Bu, A (U4) b, sE 2u, 'b, a 6u,b, =f 6d, = 0 ub, 8 (Bu, a uu) b, = 6u,5, a 36, = y
(16)
The two latter equations (16) enable us to express u, and mw, in function of u,. For multiplying the first of these by 25,, the second by 4,, and adding up, we find the following quadratic equation
(u,b, — 2u,6,)? + 6d, (u,b, — 2u,6,) + 3 (40,5, — 5,6,) = 0 so u,b, — 2u,b, = — 3b, + V8, where the square root stands for both values, and 2, represents the expression i, = 3b,? — 4b,b, + bb This result, in connexion with
u, (4,5, a 2u,b, + 65.) = 36, = bu), gives ant 3b, st bu), ris 6b, a Sub, REVE ae
Now the first seven equations (15) lead up to
Hy
a 2).0 0 0 0 SOR iO a DT 6 OD gee 0e Dar AD " En MOE Oak OE a, Ui u, l 0 =d, —A, u, u, 1 0 —Ay —A, 0
| i eee
a, =o a, Wo u, u, 1 —A, ay >= Uy U Wz 1 ja 0 De Pa fh, — 4% —A, 0 Lo Lh, fly eG oe —Ag| 0 Wo u, 0 —A, 0 0 U, u, 0 aks 4;
lo 0 0 O pp 0 Of HO 0:0 pp, OO
which reduces to an equation of Rrccarr. For adding up in the numerator 4, times the third column to the sixth and 4, times the fourth to the seventh, and in the denominator 2, times the second, the third, and the fourth columns respectively to the fifth, sixth, and seventh, we find
ape ROTO 0 0 MA Oey CO ORE 0 Gere ee 00 0 0 fom U Or On ELO Taare te ba 0 oe rr 0 0 Pee el 20 Dt le gy kel 2, B, O| Goe OR a er ús U 2D, | B hy Up u 2d, B, | a, Oren Me te XO 2% | OP np, Ow. 4 Gee Oe OO. me Oe 0 ae en Os ees OO |
where w is determined by the relation
6Ob54-A,us—A 6b,+4,b,—2ub, O 6 bug Es, St Bh) = om.
op If we substract in the numerator — times the sixth and seventh me
3 columns from the third and fourth and in the denominator ee me times the fifth, sixth, and seventh from the second, third and fourth
columns, the value of uw’, reduces to
NIR Og SUE GRUT lg fe AO.) or = ge See “ 0 e, EEN 0 BORE) A Oa. . O oe 6b, 66, a, eet) On" 0 ee tol Oeh Oe 0 me me 3b, 6b, I 38, 65: BI Wo — — 1 26, ble u,— — i 2b, b, 0 m m m m 3b go. 60. om Goa Gi EE 5 “A Oy ee ee b ay Ea Ug 3 3 Te a 2b, 4 m BY) ey dees Oi en rn ee <0 5 | OL ose 0 = 0 a 2b, m OT EO Od 0a TOO a
Here the denominator N is evidently independent of u, and may be written
dede ped 65, En Seb aby (OBD Ae.) on I N=— ee La Be gnl 27 3 AF mm 3 Hie thy SP! — 120,055, 3h te 200, -- 1 U TRP ROE
( 468 )
This takes a simpler form if we eliminate all the powers of m except the first. The definition of mm gives
m* — (36b,? — 3b,5, — 12b,b,) m + 18b,b3b, — 72b,bob 1 8by — Big m 3(45,5, — bb)
hence 3b,7b,? — 12b,b,b,b,
m
— — bob, (6b, — m).
With these values, and putting i, = b,bob, + ‚bob, — ba — b,b,? — b,°b, we obtain finally
Aig rion, @ 4 4 N= = m — = GH) = 5 HV BiA, = | (4/3i—9%,). (18)
Introducing now the values of u, and u, in function of uw, in the numerator, we may reduce this to Au,’ + Bu, + C, where the coefficients are to be determined still.
If we put u, == 0, C is immediately found
1 0 oS De 6b, —' l 25, . 6, m 3 tr den yl a mem 3 3b, m 0 — 0 — | m 3
If we divide further the second and third columns by u, and substi- tute afterwards w=, the equation is reducible to (3b |
— Le 0 ee B Aa DD m 6 6b 6d eer SR fell (a Wa m m m m 3 AE b, 6b > RNN Tes abe he Eee SS | m m 3 om m | 3b, m 3b, m | 0 Un (Sees ao == m 3 | | m 3 |
Differentiating the numerator with respect to u, and putting u, = 0 afterwards, we find the value of B. This value consists of two determinants; the first of these is identically zero, therefore
Daa 4 0 0 0 0 0 a) Be a, — 1 0 0 0 0
m
36, 65 a, yt ed On: Dyer 0
m m
36, 66 == od = a reg aay m m 36, 36 Ort Oe ae ees oop. m 3 6b POE bon Ux eek m 3 ER SN RIA AS or 12b,—2m ‘ , B=-a4, cad Say yo EE (46,° + 6, 5, — Bb.b0) |
2 Re (bom + 66,7? — 95,3)
2
=i ean (b,m — 3b,6,) 2
oe (bam — 2b,b, — bb) 2
a (bm — 3b,b,)
2 5 (bam + 6b, — 9555,)
12b,—2m
rk en «| Lea ear ay
(45,?-+ b,6,2— Bhsbd) |
With these values the differential of Riccati takes the form
3 Se 3 8 = — — a — — ——— 5 5 pe e 5 He Sate + ye Sots (19)
and the same reasoning as before shows that if the necessary condi- tions are satisfied the substitution which reduces the given differen- tial equation (14) to the equation (19) may be inferred from y + Moy? + ey + Wo =O. Substituting the values (17) we conclude finally that
( 470 )
my’ + 6b,y* + Shy 3b,y° + 6b,y + m
— (20) reduces (14) to (19).
5. To determine in this case the conditions, we differentiate the
b ° expressed in 4 and u by (15). This gives
il J ] 5 ‘ 5 four values ET
GOB) = (BB apty—BB,,)2,'—(B,p1, + GBy)Ag! + batt! HBA | + (DA, + 6b3,)u9 —b,A.u,'—36,a,u,' 64(b,by') = (3b ,4, —3b,u,)A,' — bq) + (b,42—3b9)A,' + 36,2.) + + b,2,u2' + (8624,—b,49)u,' —36,2,u,. 64(bb9'J=—3b eds + (Bbout,—b,U, )ag + 6, My4,' H(3D,— Bbz), + if + (6,2, —3b A, ta —b Agu,’ + (8b242—3b,a,) yo 64(b,by')=— 3b, ,A,,—b uy hg + (b fa 6bou)d, +(3b,—6bou)d, + + Did,’ —(b Ag+ 6b94,)u,' + (6624,—3b,2,)u,.
(-1)
Combining each of these equations with the seven former equations (15) and eliminating the quantities 2, 25 2,'à, u u, #,… we obtain
a es ee 0 0 0
|% Wz i oD. 0 A 0
| dB Po derd hoes 0
| Ce, ig Oh ea eee wig a O Uy Bi Uy —4, Ah Ag | a, © B dE 0 —A, —A, aq 0 Dk: 0 0 —aA, ellis We Wa en CG: C,
where the last row is formed by the coefficients of each of the four equations (21). Hence for the first of these Ge == 6(b,52). C, == BboUt2 = 36 Uy, etc. If we reduce this determinant in the same way as before, the last row becomes in the first place CHC, ACC, CdC, 6 lo, 4 6
Cor Cyr Car Csr Cos
and secondly
that is for the four cases successively
6 (6,6,'), 6 b.ug—3b,uU,, med Da zi
6 (bebo). 3 bou, —3b,u,, —
6 (b,62'), a blos Det
m
: ' ; 36,b, 6b,° 6 (b, 55 ), i Duo EPE ij
’
m m 3b,? 6b,b, m : m
3b,b, 6b,b,
ft, 4,C, 4 C, Su, a,C, + C, m 0 ek m 0 CHC, 4,C,+C, CdC, i) ’ i) ’ 4 9 36,b, 66, SS Abr 0 m m — 3b, 3b,, — 2b,°, b, b,, 0 m 185,53 , 3b, — ——., bym-2b,b,, b,b,, -3b3b, m 185,53 ’ OR ee See met m — 2b,b,, 5,6, + 2bym, — 12b3b,
which may be represented for a moment by D, D, ), D,D,D,D,D,,. After these reductions it is evident that only the second column contains the quantities u, mu, u,. Hence, with regard to the relations (17), this determinant may be written in the form Au,+B, where the value of A is found by differentiating with respect to u, and B by substituting u, = 0.
In this way A takes the form of a determinant of the eighth order which immediately leads to the following of the sixth order.
3b EN | 0 Oy Bhs m 3b, 65, re pee mm A= —a,u 3b, 0 0 m 0 0 0 (Dy D, D, dD where D= —
du,
c= 0 0 2b, b, m 5 2%, m rie 3 D, D,
m — jp) 3
b
4
ve. Ber sie m eer 3
m dps
3 b, 2b Ones 0 U
m — D 3
4 DD. D, qr. 0 b, 0 2b, b, m 3 26,
mL 0 ; 3
( 472 )
Developing this determinant, and putting m* 4b,b,+2b,b, , 4b,b,°+4b,°b, ann. ae ss a ae “tm + b,b,(6,6,—46,),) =P we have oy.) o =. "ome 2b,m m Axau(=)P| 52, Dit OT 1D}.
If we introduce now the values of the quantities D in the last factor, this leads in the four different cases to
2b x = m? + 4byb,m + 2b,(b,b,—40,),)
2b — a m? —- A4bybym En 2b,(b,b,—46,5,)
25 as 5 m’ =o 4b, bom <7 2b,(6,b,—45,),)
26 — =z! m* + beg + 2B4(046,—48,5,)- If we observe that we have by definition m*?—6bym bb. —4bb, — tae ca
it is evident that in all cases A = 0. The conditions are therefore determined by 5 = 0, and this may be written, after a slight reduction
foie 0 - 0 >. OC eae 3 m a De ee ee ee 3 m a, be 2b, 3 0 b, 0 0 a, 0 b, OW = WB 0 =— 0 a, 0 0. sbr 5 2b, b, m a, 0 VW wee 3 >a a, 0 0 0 DEES | m m m m D, z Pe Peak go ale D, D, D, |
( 473 )
where the elements of the last row are respectively in the four cases : 6(b,b>'), 12b,b9, —(b,b,+2bym), Wb, bam, —2b,b,, be
3 4) AS 0 6(b,b9'), Sheba, —b,b,, Wibe bom, bom, —2b,7, b,d,, 0
34?
6(5,b,'), 0, —b,b,, 20,2, b,(m—6b,), bom —2),b,, b,b,, —3b3b, 6(b,b,'), 0, —b,?, 2b,b,, b,(m—6b,), —2b,b,, b,b,-+2bjm, —12b,b,
OE AS
6. Following the same way in the general case, we obtain for Ht, the quotient of two determinants each of order 2n-+1. If we reduce these as before, the denominator will be seen to be independent of 4 and u; and the numerator will only contain the quantities Unie Un—2-- By, Hy in two columns. Now pa, Un—2,-- U, may be expressed as linear functions of u,, and this proves at once that the numerator must be a polynomial of the second degree in u. If, therefore the necessary conditions are satisfied, the quantity u, is an integral of an equation of Rrccarr. The substitution which reduces the given differential equation to this equation of Rriccarr will then be
found from
gr Frit IH =9 by determining wn—1,---U, in function of u, and expressing u, in function of y.
Physics. — “The law of shift of the central component of a triplet in a magnetic field.” By Prof. P. Zeeman.
In two communications to this Academy *) on “Change of wave- length of the middle line of triplets’ I gave conclusive evidence obtained by means of MrcnersoN's echelon-spectroscope that the central line of some triplets is shifted. The fact of this displacement was established simultaneously with my own observations by GMELIN?) and JAcK*). GMELIN first gave the law of shift in the case of the mercury line 5791. According to him the change of wavelength under consideration is proportional to the square of the magnetic force.
In the second part of a former paper on “Magnetic resolution of spectral lines and magnetic force” measurements concerning the asymmetrical resolution of the mercury line 5791 are given“).
1) P. Zeeman. These Procedings February 1908, April 1908.
2) Guus. Physikalische Zeitschrift. 9. Jahrgang S. 212—214, 1908, 8) Jack see Voter. Magneto-optik. S. 178.
4) ZEEMAN. These Proceedings November 1907,
( 474 )
Supposing that the asymmetry of the separation is entirely due to the shift of the central line towards the red, one should conclude from the communicated numbers that the displacement increases nearly linearly with the strength of field. This investigation was made with Rowranp’s grating, the principal object in view being to prove the existence of asymmetrical separations. I succeeded in this respect, but I think now I have overrated the accuracy of the extremely difficult determinations of the amount of the asymmetry. In fields of the order of 20000 gauss the asymmetry is 35 thousandth parts of an Angstrom unit, while the RowLanp grating used permits in the chosen, first order to resolve lines, the difference of whose wavelengths is 0.12 AU. hence with the field intensities mentioned we have to do with a quantity which is already four times smaller than the limit imposed by the resolving power.
It is only because we have to do in determining the asymmetry with a difference of two quantities which are above the limit set by the resolving power, that there may be question of measurement. _ When we reach however the utmost limits of the method used then sources of error come to the front, which partly are caused by our mode of appreciation of the distance of two adjacent lines, partly are connected with particularities in the formation of images by gratings, not yet sufficiently understood.
It is therefore undoubtedly to be preferred to use for the further investigation of the shift of the central line a method warranting greater resolving power. GMELIN in his investigation has used MICHELSON’s echelon grating, and it seems that he has largely succeeded by syste- matic procedure to interprete quantitatively the results given by this instrument. His result therefore possesses high probability and more- over is now supported by the theory given by Vorer *) in order to explain the large asymmetrical separations, a theory which assumes the existence of couplings between the electrons.
I thought it however to be worth while to investigate the matter by a method independent of RowLanp’s and MiIcHELson’s apparatus. Fapry and Prrot’s method seemed most appropriate. The greater part of the measurements communicated in this paper have been obtained with a 5 m.m. étalon, already used on a former occasion. Some determinations were made with an étalon with distance-pieces of mvar as suggested by Fasry and Perrot in order to diminish the dependence upon temperature. It was constructed for me by JoBIN.
1) Vorer. Magneto-optik. S. 261.
( 475 )
The thickness of the air-layer in this étalon was nearly 25 m.m. With this distance and using the light of the mereury line 5790 in the magnetic field the limit of the method is being rapidly approached. Hence the accuracy of the results obtained with the 25 m.m. étalon is in our case hardly superior to that to be reached with the 5 m.m. apparatus.
The arrangement of the apparatus was described with sufficient detail on a former occasion *). For the purpose now in view it was desirable to investigate exclusively the vibrations parallel to the magnetic force. A calespar-rhomb therefore was placed between the source of light and the first lens. Two images of the radiating vacuum-tube are now obtained near together on the étalon, the non-desired one being screened off. A photograph was taken with the field on, and before and afterwards one with the field off.
Besides the inner ring, always also the second ring, in some cases also the third and fourth one, was measured and the result used in the wave-length calculation.
The formula for the calculation is the one first given by Fasry and Perrot, still remarkably simplified in our case ’).
In the following table the results are given relating to the mercury line 5791. The first column contains the number of the experiment, the second one the reference-number of the spectrogram; A2, is the change of wavelength of the central component. The field intensities are given in the last column. Their relative values, which are only necessary for establishing the law connecting displacement and strength of field, are exact. These numbers must be increased with 1 or 2°/, in order to reduce them to gausses.
Experiment Plate n°. AP in Ase. H. 1 208¢ 0.0085 12700 2 2095 0.0088 12700 3 211 0.0169 20700 4 212¢ 0.0074 13950 5 214° 0.0201 20600 6 218% 0.0367 28250 7 218¢ 0.0358 28250 8 219% 0.0360 28250 9 220% 0.0353 29170 10 2204 0.0406 29780
1) Zeeman. These Procce dings December 1907. *) See These Proceedings December 1907, February 1908.
( 476 )
The experiments 4 and 5 are made with the 25 m.m. étalon, the other ones with the 5 mm. apparatus. In the figure the results are graphed. The smallness of the displacements may be illustrated by the statement, that the outer components of the triplet 5791 are separated 0.500 A.U. from the unmodified position in a field of 29750 Gauss. The ordinate measuring 0.500 A.U. would be 75 em. in the figure. ‘
The results 1, 2 and 4; 3 and 5; 6, 7, 8, 9, 10 were combined in each case by assigning simply to each mean displacement the mean magnetic intensity. The three principal values, thus obtained are indicated by crosses. These points and the origin lie very approxi- mately on a parabola.
Inspection of the figure or a simple calculation easily shows that the quadratic law is obeyed within the limits of the errors of obser- vation of the measured displacements. The magnitude of the dis- placement has been measured in the average in each of the ten points to within 0.002 or 0.003 A.U.
In order to show how the values of A2, were obtained, I will give the calculation of one case in full.
. A, Amie SR a, = 5791 A. E. H=12700 Etalon 2d=10m.m. R=120m.m.
(z,° — Am)
>
( 477.)
v,,¢m diameters of the rings in m.m. x, mean of 2 diameters on plates taken before and after ain.
First ring:
# = 9.662 @> 245.410 0.160 ty, — 3.640 On == 13.200 Second ring: 0.171 en 4608 Pe == 6.802 0.182 an BAS Ws a 6/620 0.171A, A A 14, = ——— = 0.0086 A. E. ee i
In the case of the triplet of the mercury line 5770 no displacement of the central line could be found. In a field of 28250 the following values of the diameters were obtained with the 5 m.m. étalon:
First ring Second ring
2.199 3.409 field off. 2.193 3.408 field on. 2.199 3.394 field off.
Kr
Hence the central line of 5770 remains within the limits of experi- mental error exactly in the position of the unmodified one.
Physics. — “Contribution to the theory of binary mixtures,’ XII. (Continued). By Prof. J. D. vaN per WAALS.
In the discussion in the preceding contribution on the question whether there is any possibility that values of v >>b, might occur in the case that the locus of the points of intersection of the curves
dy dp —- == 0 and =O is a closed curve, we have also discussed dx? dv? (p. 433) the case that (g") or: dA dA n—1—nyY {\A—a@—( s=YVy jA4+(1—2)—| = 0 dz da
would be imaginary over the full width from #=0 to e=1. We have reduced this equation there to the following form:
n—1—ne pe ee mn = (1 — 2) pa oe 0 a a a a
and shown that if „>> 2, the value of a,—cx* may become negative for the high values of rv. The limiting value of x is then 32
Proceedings Royal Acad. Amsterdam. Vol. XI.
( 478 )
a al Fi equal to [42 so that we have «, = |/*. We then observed C C
(p. 485) that if such a limiting value for x exists, our conclusion that g!'=0 must possess a minimum value which is negative, can no longer be considered as proved; but we omitted the observation that the thesis that » would have to be < 6,, may not be considered as proved any longer either. If viz. the substitution of «=, should make the first member of (g'”) negative, whereas, as we saw before, the substitution of «=O makes the first member of (g'") positive, then a value of 2 must exist which makes (g"’)—O both on the branch of (g"’) with the negative sign for the third term as on that with the positive sign. Then it is therefore unnecessary, that (@'’) possesses a minimum value, and there is no reason for the positive sign for the third term, and so no necessity for v being smaller than 5. Let us seek the condition for:
VY ja,—c (1—a,)*}
n— 1— ney di 0 : a or a, i; je LA aM (1 —ay)’ AE Rr n a Cc - Let us write; a a, alg en 1» A it (la) HSS — ay (le) or a det ~ = 2g (Lary) + 4 i 2, (1—2,) or a a, = — — (1l—a,)?}. Cc “9 c ( 9) The condition put above, becomes then : n—1 1 im < 7 a, (1 )? (lS A g or
n°
a — (1—2,)’ << (n—1)
or
( 479 )
“aay U »
a: — (Le) <0 Ka
Ie 15;
EEE
——, we obtain
And taking into account that «2, = Vase
as condition :
Est
nl> VIe He,
I have given it in this form in the “Erratum” accompanying the preceding Contribution.
Before discufssing the signification of this condition I will remark that we might, indeed, have obtained this result in a less intricate way.
Let us directly put the value »v = b, in the equation for the closed curve, and let us examine what value of « then satisfies the equation. If v= b,, then v—b = (6,—4,) (1— 2), and v? = 6,?. Equation (a) of Contribution X p. 318 becomes then:
wal dens cx (l—e)
n a
or (a) OREN EE 1 Lo ee) c c or 1 1
We ole ila Ee) oz (len)
then we find as condition for the ay eee ae of z for which v= bb:
1+-e, if ae a Ge
—x(l—z)=0
or OLS aes alice Ea RENT ae (x—1)?
As 1+ «, must certainly be positive, because a negative value of a, is inconceivable, we see that if the above equation has real roots, it must have two for positive values of z in all possible cases, also if «, and «, should be negative. The condition for the roots being real is:
32*
( 480 )
lt¢,—n’e,_ 2V1+s,
i; (n—1)? ee ack) or V+) Ve n—l n—l
So the same condition as had been found above.
If we represent the condition for the possibility of v >> b, again graphically, it is given by a parabola, and that the same as occurs in fig. 86 p. 321, but shifted downward in the direction of the e,-axis by an amount —1. We need not draw it, but we shall think the points of contact with the ¢,-axis and with a line ¢,=—1 indicated by the letters Q' and P". To satisfy the circumstance v>b,, the point (¢,, &,) must lie inside the space which | shall call O"P"Q". But for the possibility of the closed figure the point (e,, &) must lie inside the space OPQ — in both cases below the corresponding parabola. This can only occur when the two areas mentioned cover each other or as least overlap. This requires (n—1)? >1 or n> 2. So the points (¢,, €,) giving a closed curve, for which the value v > 6, occurs between two values of x, are confined to a smaller space, again bounded by the axes and a parabola. In this case the parabola touches the ¢,-axis at a distance n(n -—2) from the origin, but intersects the ¢,-axis at a distance
n(n—2 n—2 tet ane ( apes from the origin. The condition that the two values
n? n of x for which v= b,, coincide, and that the closed curve touch a line y=), is this: that the point (¢,, ¢,) shall lie on this parabola. ie nwe : an Land 1 —a«= ——. If we compare this value (n— 1) n—l of zw with that which we have called z, above, z, appears to be
Then r=
dv besides highest value of a for which = is equal to 0 for the points a“
of the closed curve, also the value of z for the point in which the closed curve touches the line v=8,. If volumes occur which are larger than 5,, then the greatest volume lies at a value of z < ay.
Let us now more closely examine the space which OPQ and 0" P'Q" have in common, and inside which the points (¢,, €) must lie for the condition v > 6, to be satistied. For very large this space will be very large in the direction of the ¢,-axis, but in
the direction of the ¢,-axis it remains limited to an amount 1—— Nn
and so below unity. Also by simple construction we can now indicate
(ABL)
a rule for the place of the points (e,‚e,) which satisfy the require- ment that the portion cut off by the closed curve from the line v= 0,, have a given value.
From equation (8) of a 479 follows:
1e — a —nte, le Rn et ee ee 1 Oak ed Bere ay | If we represent the highest value of « by «,, and the smallest by z,, then :
eee] 4 le,
(n—1)' or 1e, (v,—2,) Br Bee: (a —#,) oh (n —1)? 4 | ET DL PO ER ret ET 1 An? |
Ie krk) Sa _& (ee) ae =e eee jo ee An” |
So the points for which z,—z, has an equal value, lie again on a parabola, and one of the same shape as that of fig. 36; but now it has undergone two shiftings.
The first shifting is that in which all the points of the parabola have descended by an amount—=1 in the direction of the ¢,-axis which makes it the upper limit of the space now under discussion. But the second shifting is one which takes place in the direction of the diameter or the axis of the parabola. The amount of the second shifting must be such that it can be considered as the resultant of a displacement in the direction of the negative ¢, by an amount
(enal
equal to " (1 —1)* and a displacement in the direction of the (age)? (n—1)? 4 dek ing as ”‚—®, is greater, this second shifting is more considerable — but as soon as the shifting would proceed so far that the parabola would have no more points inside the original space VPQ we have exceeded the possible value of z,—,. The extreme limits of z,—, 1 n—2 are then on one side O, and on the other side 1 — a ae This greatest value of x, —v,, which is equal to 0 for n= 2 itself approaches 1 with increasing value of 7. We may also express the
Accord-
negative ¢,-axis by an amount equal to
( 482 )
above as follows. When we have a point (¢,, ¢,) in the space which OPQ and O"P'Q" have in common, the closed curve will possess volumes which are greater than 4, — and by shifting this point in the direction of the axis of the parabola till it meets the first-
mentioned shifted parabola, we find the value Be) (n—1)’, in the
projection of this displacement on the ¢,-axis, or the value of (@,—2,)? (n—1)? 4 n° So the length of the line drawn through the given point in the direction of the axis of the parabola till it meets the second parabola teaches us the value of (z,—a,)* ; to which we may add that the same line prolonged to the other side so below the given point, shows us also at what value of a the middle of x, and x, lies. If the continuation of this line passes through the point ¢, — O ande, = — 1,
in the projection of this displacement on the ¢,-axis.
1 ete the middle of 2, and 2, lies exactly at Gc If this line intersects
; ete 1 the ¢«,-axis below e, = — 1, then a ee and the other way
about. We have viz. from (9):
il = pst #,+2@#,=14 ple ne (n—1)? bP re ae ae or putting aan Lm 8 1+¢,—n’e, 1 — 22 — ree we .
For given value of x, this represents a straight line, the direction é : : : : : of which is given by — =n’. This straight line intersects the €,-axis é, in a point «, + 1 = — (n—1)’ (1—2z,,); from this formula the given rule appears.
Such rules may also be given for the dimension and the place of the closed curve itself — and for the accurate knowledge of the properties of this curve the knowledge of such rules is not devoid of importance. Thus the equation (3) of p. 319 Contribution X leads to:
(z,—«,)* ==
when the values of z between which the curve exists, are represented by a, and z,. If we derive from this :
( 483 ) €, Ph met it &, (, - A (n—1)? 4 (n—1)? An? it appears that the locus of the points ¢, ande, for which the closed curve has the same width, is again the same parabola OPQ, but shifted in opposite direction of the axis by an amount of such a value that the projection on the e-axis is equal to (n—1)’
i cad . For the points of OPQ itself the width is, therefore, equal
be
= ny
to 0, and for the origin, in which ¢, and e, is equal to 0, 2,—7,=1, and the curve occupies the whole width. The decrease of the values of «, and «, obtained by shifting in the opposite direction of the
2
te Lv
axis of the parabola, promotes therefore the intersection of
2
and ce = 0, and so furthers the non-miscibility. In the same way ve E oe, we find, representing the value of Sn Lm: &,—n’é 1 a Dare = = ee . (n—1)?
So if we trace a line parallel to the axis of the parabola through
if the origin, this line is the boundary for the points for which z,, re
1 For the points for which ¢, >n*e,, tm > ca and the other way
about.
And finally this property. We may also write the equation (@’) of p. 319 Contribution X indicating the limiting value of & which cor- responds to given value of e‚ and «, as follows:
Eerd nen TES ae
Let «=z, for one of these limiting values, then this equation
becomes :
nete A at ee ee (ale, (elle,
And for constant value of z,, this Jast formula represents a straight line for the points (,, &,). On this straight line also the point must lie for which not only the one limiting value of z==,, but also the second, and for which the two values of x therefore coincide.
as
V In this case 7, = and 1 —z, = n—l n—
ne,
. Hence we get back again
( 484 )
the limiting relation between ¢, and «, or in other words the equation of the parabola by this substitution in the equation of the straight line. So this straight line is a tangent to the parabola, and one touching in the point in which also the second limiting value of x, or v, coincides with z,. From this follows then this rule. If we draw a tangent to the parabola in the area OPQ, then all the points («,,¢,) for which one of the limiting values is equal to the value for a of the point of contact, lie on this tangent. If we draw a second tangent to the parabola, the point of intersection with the first tangent has _the property that the values of v of the two points of contact belong to it for 2, and a,. If we have drawn one tangent, tangents may be drawn from all the points of this line lying on the lefthand side of the point of contact, so from all the points for which e, is smaller, and ¢, larger than that of the point of contact, to the points for which e‚ is larger, and so 2, >,, and the other way about. If we wish to indicate in what part of the space OPQ below the parabola the points lie for which the values of ¢, and «, are such that the whole
; ; 1 closed curve remains restricted either to values of rn or to 1 | Eten values of # << me must begin with finding the point on the para-
l zetel . bola “for ‘which ‘x, = 2, EEN This isthe pomt for which‘ =e.
and which therefore lies on the line which is drawn from the origin
in the direction of the axis of the parabola. In this point we must
trace the tangent to the parabola. From the ¢,-axis this tangent ents
(n —1)? (n—1)?
off a portion =-——— and from the ¢,-axis a portion =-—_—__. 2 2n?
So it is a line parallel to the straight line PQ of fig. 36, and it
OP GU a En cuts off from the axes parts equal to 7 ane oe This tangent divides
the space OPQ below the parabola into three parts, viz. the part below this tangent, and the two other parts above this tangent and further bounded by the parabola and one of the axes. The righthand one of these two parts contains the points, for which the closed curve
1 remains confined to values of Ht For the lefthand part the
reverse applies. So according to this result either of these cases would be possible
1 either that the closed curve remains restricted to values of ae ,
( 485 )
it or to values of BE sat But if it is asked whether it is probable
that both cases occur, this probability depends on the value which [* must assume in these two cases. The point in which these spaces
n—1)? touch, is the point where ¢, = n° ¢, = omen For this point to be
possible the following equation must hold: (Qn He, +n’? ey —=407 (14+ &,)n’?(1 + &,) We find from this by substitution of the values ¢, and g, : fol (n +1) (n+ DH An — I
So in any case a value of £ <1. It becomes smaller as 7 increases,
i?
and the limiting value for =o amounts to En Such a small value,
however, / will most likely never assume. And if we now take into consideration that for the points of the lefthand part, for the
ji points of which ee the value of / will have to be still smaller, we arrive at the conclusion that if is large, the case that the closed
; 1 curve remains restricted to values of «> rs will not have much
chance of occurring. For the point in which the two spaces touch 81 4 ? is equal to aE for n = 2, and this value is equal to 5 tor md;
and we may consider these values of / as probably possible. So that we arrive at the conclusion that for not great values of n, e.g.
f 1 n = 3, the closed curve, if it exists, can occur atx > ah but that for higher values of mn, and also if / should be >1, the other case, 1 Bo is possible.
Let us now proceed to derive some results on the miscibility or non-miscibility in the liquid state from what has been observed on
the intersection of EM Sn and ee 4, for the case that the locus dx? dv?
of the points of intersection is a closed curve, and to compare these
results with the observed facts. All the properties discussed of the
closed curve are perhaps no longer necessary if we could have
anticipated this result. They have, however, been necessary for me
to come to this conclusion. And if we do not content ourselves with
( 486 )
more or less vague indications, but want to give clearly defined statements, the knowledge of most of the properties discussed is necessary.
I already treated one of the meanings of the closed curve, p. 331
2 2
ae d d Contribution X. In this case contact of e= 0 and —" =0 occurs TL v
for the first time at low temperature 7; at rising temperature there is intersection of these two curves. But with further rise of 7’ the two points of intersection draw nearer together, and at 7’= 7’, there
2
dy is again contact. For the case mentioned —- — 0 had again to lie at
dp in the region where = = 0 is negative above 7'= 7,. But a second v
ease is possible. With constantly rising temperature the intersection of the two curves may always proceed in the same sense, and then there can
bl
d also be contact at 7'= 7. Then the curve nl = 0 must disappear at
a? in the region where er is positive. In Contribution HI I gave the av
See dp equation which is to decide whether =e is to disappear in the Ai one region or in the other, viz. :
cx (1 — #4) > Ay? a a +49
d? ; : d? If the sign > holds, = = 0 disappears in the region where — : Ax Vv
is positive, and the other way about. And now, to answer the question whether the first mentioned case takes place or the second, we must examine this equation, bearing in mind that «, and e, is positive, and that the points (¢,,¢,) lie below the parabola OPQ.
The values of x, and Yg are dependent on mn, and quite determined by this quantity; and according to the list of calculated values occurring in the beginning of Contribution III, z, can only vary between */, and */,, and y, between */, and 0. So the second member of the inequality to be investigated is entirely determined by the ratio of the size of the molecules, but the first member depends moreover on ¢, and &,.
Let us write this first member, omitting the index to z,:
( 487 )
cx( 1 — x) ca(1—.w) eee (sa) aa En 1 ae 1 ACE a eee ce cl—x (n--l)P? ew = (nm - 1)7(1 — 2) or call — x) L tae Pie ee 1 +| F, 1, we, i! zij (n—1)?# = (n—1)?1l—«a (n—1)?w © (n—1)? le , Now there is a series of values of ¢, and e, (see p. 483) for which En el n'e,
1 ei, ——— = Fis d : g =i Co is equal to 0. All these values
are given by a line which touches the parabola in a point for which We, ates mined by the value of n, and lies on the line which passes through
the value of
==, so a point which, as the parabola itself, is entirely deter-
ae deld ieee Era VEN
the origin in a direction ——=7? (=) This direction approaches to Es ei
Re ies :
on for very great values of n, and to »° itself for values of # which
are but little greater than 3. All the values of ¢, and e, occurring below the parabola are reached when lines are traced parallel to the said tangent. Thus:
i n's 1
eps cx ee SE 4 (n—l?e = (n—1)? l—ez
represents all the points below this tangent, when « is given the negative sign; and then the second member can descend to — 1, in which case the origin itself might occur. All the points above the said tangent are reached, when « is given the positive sign, and
I then made to ascend till 1 + a@—=-—, in which case the point Q is Hi
1 reached. For « such that 1 + « = me the point P is reached. it
So we have for points below the tangent: er (le) l a eal 1 n° 1 — + —a (n—1)? (n—1)? le in which a lies between O and 1, and is =O on the tangent itself. For points above the tangent we have:
( 488 )
ce (la) 1
a 1 l n° 1 (ale (n—1)? le ne
1 in which « lies between O and — — 1; whereas to reach the points x
lying above the tangent on the side of P we need not go further
— 1. Of course in the same way as illustrated in an
[ian ae —
example above we have again to consider whether all these points points probably occur by investigating the value of 7°.
cv (1—2) ; — has been given, now consists of two a
The form in which 1 n° 1
(n—1)?a | (n—1)? l-«
depends only on n, but the second part « depends also on e, and é,,
and as the second member of the inequality which is to be in- vestigated, does depend only on n, we cannot expect the circumstance
parts in the denominator. The first part
d et? on . whether eae = 0, when disappearing, lies in the positive region of Ls dp . | — or in the negative one, only to depend on the ratio of the size
of the molecules. But this we may at once consider as a result obtained that as the parailel line is farther from the origin, and so the values of s, and e, are larger, the value of the first member of the inequality becomes smaller, and so there is a greater chance that the second member exceeds the first. For greater values of ¢, and e,
Pw
dz’
there is a greater chance that the disappearance of == '0 takes
dt place in the region where el < 0, and the degree of the non- av
miscibility will be limited. Or rather, a phenomenon that attends non-miscibility, will be checked by this. Thus for 7 == %, for which
iL 1 n “v——, and y= ee and —=— — 1, the first member of the inequa- 3 f 2 n—
2 lity will be equal to 2 for the origin, to = for the points of the
1 = tangent mentioned, and 5 for the point P if we include also the
lefthand part above the tangent in our calculation ; the second member
2 d* : ‘ is equal to En Then ST disappears just on the verge of the la
( 489 )
9
: yw ae ; : . region of 2 positive or negative for the points of the tangent. For Vv
dp the points above the tangent, however, ar 0 disappears where av dp. zond negative, and the reverse for the points below the tangent. ae 2
d But let us try to answer the question where en Ak
= 0 disappears
for arbitrary value of x. The reiation between n, 2, and y (Contri- bution III) is, indeed, a very intricate one — but to my astonishment it proved to be possible to find an answer by a comparatively simple reduction. If we start from equation (4) of Contribution III, we may write:
1 a xv (le) n—l je agens 4H} and n x (le) Be ear el mn oe If we take the square of the first of these equations, and then divide by « — and the square of the second of these equations and then divide by 1 —, the sum of the two values obtained yields: ee t n° e(l—e) ra TE Sl maha wv (n—1) 1—z(n—1) (1— 22) yee CE js For the second member may also be written 1 sand the y
2
d EEP condition whether — is positive or negative for the point in which Vv
d : ne =0 disappears, becomes then for the points below the tangent: av” 1 PENS 1 (1-4) SI eg derd
In this equation we have a—=1 for the origin and a =0 for the tangent itself. With a—1 we find as condition:
v +3) 2 —y)?
1 For OE which belongs to mo, the first member of the
3 1 inequality is 7 and the second member ris So, as we found above,
(490)
d? el >0. But for y=0, which would belong to n=41, the first (dj ER. dp member =O and the second =1. So for this limiting case —, <0 v
So there is a transition value of n, namely for that which belongs 1 o-r 1 or = AD According to Contribution III the value of z
is about 0.41 and of n about 3.4 for this value of y. For the points of the tangent for which a= 0, the condition is:
1 = 4y? 1 (l—y)? < 1+ ae y or 02 4y? — By 4 1 or
OZ =: Bayt a): So this inequality can never be satisfied by the sign >; only for
1 Us there is equality, as we saw already above. We conclude from
= 0 disappears in
d this that however great the value of 7 be, 5
d er is negative for all the points of the tangents. v
the region where So this is a fortiori the case for all the points above the tangent.
1 When y lies between = and and so n > 3.4, a line is to be
9’ ol
indicated parallel to the tangent on which the points (¢, , ¢,) must p) d?
hie) tor ED to disappear, just on the verge of me But da? dv?
1 for values of 4 = and n < 3.4 the disappearance will take place
dp
where dv?
is negative for all the points below the parabola, and so
3 2 VS 0 will he inde ten Dig dv?
perature below 7, so before the first contact, and at a temperature above 7,, so after the second contact. The place of the straight line which contains the points at which the transition of the sign of
d the curve == 0 both at-a tem-
( 491 )
d 3y—1 nh takes place is determined by the value of «= 1— : dv dy?
ee) a -— wa: ee .
1 and as it cannot be greater than 1, y must always be greater than 7
or
So the quantity « has always the same sign,
So the equation of this line is: Eat nie, Enten (n—1)?@ (n—ljl—e 4y?
Now we have also the means to decide whether the temperature at d* ye a which = Q disappears, is higher or lower than the critical tem- wv
perature of the mixture of the value of z=, — in other words
d? whether 7, en Teelt T= Te) then Ee 0 has left the region where dp Tw
daz" Dn <0 on the side of the branch of the small volumes of ae Vv Vv
and this branch is still found even at the temperature 7,. For the other case we have a representation of the relative position of the two curves after they had left each other in fig. 10, Contribution
—0,
Ill. The condition (Ce Ty, (see Contribution III) may be written : 2 ly > 8a — PH == nnn —_ — pea man Se or 27 ca (1—2) AL ge A BR (1—y)
If we write further ss — == Et pr the condition becomes: la + OE 27 1 > (1-4) ‘ l—a + En = me
For a=1, or for the origin O, this condition becomes : 27 y ZL Hy) (1 —g).
i For y= Ss 2 the first member of the inequality becomes
27 9 equal to re and the second member to a which means that
( 492 )
T,=3T;. But for y=0 or n=1 the first member = 0, and the second =—1. So there is a value of y, for which 7, = 7; and of course this value must be larger than that which we found above,
2
dy .. — disappears de
when we determined for what value of y the curve
2
dp 1 a= 0. So if we put cs the first member D
on the boundary of
52 is equal to 1, and the second to re: The equality of the two members
requires y about 0,36, to which n= 3.7 corresponds, which is but little greater than we found above for the smallest value of 7 for which d? d*
x = 0 goes beyond mt a dx? dv?
For the tangent for which a == 0, the condition becomes:
a Sd Se
ee ai (1—y)
We cannot expect another ise for the points of the tangent than
1 ’ : Ie The last inequality may also be written :
0 2 (1—2y)* (1 + 4y + 10y? + 9)
( 493 ) If we call the value of a required to change the inequality into equality for given value of y, « — then the relation: 27 1— 1—y)’ ee ee at y) 4 (l+y)° Ay?
holds for this quantity. For the preceding problem, viz. the determination of the relation
3 dw
d between « and y causing —=0 to disappear on the curve T= 0, & v~
dy—l Ay?
1 —a=
held. For a@'—a we find then: : lty 27 1—y a'—a= pat ee == 4y? 4 (1+y)’
or (ten (veer (EN orice 4y* (149) uP (ty
From this it appears, what had been clear beforehand, that a’ is
eo —
1 always greater than «,‚ except for y= oe when they are both equal
to 0, and so for the points of the tangent. A case, however, which we can only think as a limiting case, because it would require n=o. The adjoined figure 38 gives the relation between «a and x for the two problems graphically. For the origin «= 1, and for the
points of the tangent ¢=0O. For the first problem vz for the origin, and for the second y= 0,36 — whereas for a=0O the two values of y are =>. For the second problem the line y= f(«) always lies above that of the first problem. Hence for equal value
of y the point P’ lies at higher value of a than the point P.
(To be continued).
33 Proceedings Royal Acad. Amsterdam. Vol. XI.
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Anatomy. — “About the development of the urogenital canal (urethra. in man.” By A. J. P. v. p. BroEK. (Communicated by Brof. 1. Borg).
In the following communication I am going to give a deseription of the way in which ontogenetically the closure of the urogenital canal comes about in man; next I intend trying to throw some light upon the composition of this canal from a comparative point of view.
The youngest stage that 1 examined was a male (?) embryo of a length of 30 m.m. from crown to coccyx; a stage which is a little younger than the oldest female embryo (l.c. Embryo Lo) described by Keren *).
The urodaeum (entodermal cloaca) is divided into rectum and sinus urogenitalis; there is a primitive perinaeum. The anal mem- brane no longer lies near the surface of the body, but forms the bottom of a short proctodaeum. Sinus urogenitalis and proc- todaeum combine into a short (200 u) ectodaeum (ectodermal cloaca), in whose walls the two component parts are easy-to recognize. If we follow the part of the wall proceeding from the sinus urogeni- talis, it appears that this at the basis of the penis contributes to the limitation of the short genital groove (“Geschlechtsrinne”) ; before this it continues in the beginning of the penis as an epithelial double lamella, phallusframe (‘Urogenitalplatte”, ‘Urethralplatte”’, “lame cloacale” etc.). There is not yet a fossa navicularis.
In an embryo of 4 cm. the apertures of proctodaeum (anus) and sinus urogenitalis are separated by a definitive perinaeum.
The sinus urogenitalis mouths on the perineal penis-surface with an aperture about lozenge-shaped, situated immediately behind a circular furrow on the penis. This furrow denotes the limit between the glans and the corpus of the penis.
Following the transverse sections, starting from the apex of the penis, it appears how in the part before the navicular aperture (fossa navicularis) the phallus-frame as double-lamella penetrates into the tissue of the penis (fig. 1 a). In the sphere of the fossa navicularis the lamellae of the phallus-frame partly deviate (fig. 1.b.), by which on the perineal surface a groove becomes visible. The angle between the two leaves becomes gradually larger, till at last, in the widest part of the aperture, one is the continua-
1) Keiser (F.). Zur Entwickelungsgeschichte des menschlichen Urogenitalapparates. Archiv f. Anatomie und Physiologie. Anat. Abth. 1896. pag. 55
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tion of the other (figure 1c. and d.). The upper part of the phallus frame stands like a crest upon the cornerplace of the deviating lamellae. (fig. 1 b-d.).
If we look more closely at the wall of the fossa navicularis, it appears that it is only partially formed by the lamellae of the phallus-frame; the rest originates from the penisectoderm, which by the side of the phallus-frame bends like a fold over its edge (marked in fig. 1 b and ec. with g.p.). If this fold is to be called sexual- fold, it must be borne in mind that it does not represent the tran- sition-edge of the phallus-frame into the penisectoderm, but entirely originates from this ectoderm. In figure 1 b the two sexual folds are situated close to each other, in figure 1 ¢, corresponding to the middle of the fossa navicularis they are farther distant.
Towards the base of the penis the two lamellae of the phallus- frame remain each other’s continuation; likewise the median crest remains present; the two sexual folds, on the other hand, keep bending to one another till they reach each other in the median line and close the urogenital canal. Accordingly the wall of this canal consists of two parts, originating from the phallus-frame and from the sexual folds (penisectoderm) (fig. 1 d.). At the nature of the epithelium they are to be recognized microscopically.
In the discussion of the older embryos I shall restrict myself to that place, where comes about the closure of the urogenital canal. I mention in passing that the part already closed, grows in length during the following time of development and contributes to the growth of the perinaeum.
In an embryo of 5 em. the place where the two sexual folds meet in the median line, is situated somewhat behind the broadest part of the fossa navicularis. Here, too, the two wall-parts of the urogenital canal, originating from the phallus-frame and from the sexual folds are clearly to be distinguished from each other. The part of the phallus-frame not separated lies like a crest on the ventral wall of the urogenital canal; before the fossa navicularis the phallus-frame forms an epithelial double-lamella. In this embryo a praeputium has appeared which has not yet entirely grown about the penis. The closure of the urogenital canal now goes on in apical direction, so that the orificum externum urethrae is removed to the point of the penis. This removal runs almost parallel to the growing of the praeputium round the glans penis.
In the closed part of the urogenital canal the wall every time consists of the two parts described higher up, which are microscopi- cally sharply to be distinguished. Differences appear only in the
33*
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proportions in which the two epithelia contribute in the formation of the wall. In an embryo of a length of 8.5 em the praeputium has grown round the whole glans. The orificium externum urethrae finds itself not far behind the apex of the penis on the perineal surface of the glans.
The first sections, beginning at the apex of the penis, still show the solid phallus-frame (fig. 2 a). The aperture of the urogenital canal is to be seen in fig. 2 a as a groove in the thick mass of epithelium, having arisen by the meeting of the two edges of the praeputium. Through this the urogenital canal runs in an oblique direction and after some sections it reaches the surface of the glans. In that place the two lamellae of the phallus-frame have partly deviated a little from each other (fig. 2 b.). The adjoining penisecto- derm forms at the edges of the phallus-frame two small sexual folds (marked in fig. 2 b with g p). By the meeting of these two folds, some sections further on, the closure of the urogenital canal is brought about (fig. 2 ¢). In contradistinction to what we saw in the sphere of the fossa navicularis, the phallus-frame has by far the greatest part in the formation of the wall of che urethra; only a very small part proceeds from the sexual folds (penisectoderm).
That here, also, the two wall-parts are easy to distinguish from each other, is taught by fig. 3, in which a part of fig. 2c under high power is sketched.
The epithelium of the phallus-frame is to be recognized in a very distinct stratum germinativum of high cylindrical cells; between the stratum germinativum on either side there are a number of big, little coloured, polygonal cells with large round nuclei. The cell- boundaries are very clear. The groove between the deviated parts of the phallus-frame possesses a smooth surface.
The epithelium proceeding from the penisectoderm and covering the foremost part of the canal, has quite a different appearance.
It has a much darker colour, probably partially a consequence of the much closer arrangement of the nuclei. A clear stratum germi- nativum is not to be recognized, no more are the cell-boundaries visible; the limitation of the lumen is not so smooth and sharp as in the phallus-frame.
If we follow the urethra towards the fossa navicularis, we see two kinds of changes taking place. First in the wall-formation a place getting larger and larger is given to the penisectoderm; secondly the two lamellae of the phallus-frame deviate more and more, only a small part remaining in the shape of a crest on the urethra (fig. 2e). The epithelium of the phallus-frame is gradually replaced by an
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epithelium having the character of the penisectoderm. In the section from which fig. 2g has been borrowed, the two components which are ontogenetically contained in it, are no more to be recognized. I cannot omit directing attention in this figure to the epithelial knob lying dorsally with respect to the urethra. It represents the “Anlage” of one of the so-called para-urethralpassages and is to be considered as a separated part of the phallus-frame or as a cell-cord grown inside from this frame.
Finally I give in fig. 4 a series of sections through the urethra of an embryo 13cm. long (+ at the end of the 5" month) in which embryo the state of the full-grown man has been reached.
The urethra mouths at the end of the penis with a vertical aper- ture. Where the urethra is vertical, accordingly before the fossa navicularis, its wall, as is shown in fig. 4a, consists principally of the epithelium of the phallus-frame; only an exceedingly small part proceeds from the penisectoderm, resp. the sexual folds. The lamellae of the phallus-frame are almost entirely separated, not because they are deviated, but because the central mass has disappeared.
In the direction to the fossa navicularis also here the composition of the wall changes and the part proceeding from the phallus-frame becomes smaller, the part originating from the sexual folds becomes larger. In fig. 4c the vertical part of the canal certainly answers to the phallus-frame, the rest is for the greater part a production of the sexual folds. Also in this preparation the difference between the two kinds of epithelium disappears in the sphere of the fossa navi- cularis; in the sections from which fig. 4d-g has been borrowed the boundaries between the two components are no more to be seen.
In different places separated cell-cords and tubes are present which must be considered as the ‘“Anlages” of paraurethralpassages; the tube in fig. 4f marked s.g. is the “Anlage” of the sinus of Guérin.
The series fig. 4, like fig. 2, shows the cause of the change in the position of the urethra, which, as is well-known, stands vertical before the fossa navicularis, behind it mostly horizontal. The diffe- rence is based upon the difference in composition. For before the fossa navicularis it is the phallus-frame, which has a vertical position, that forms the greatest part in the wall-formation of the urethra, only a small part proceeds from the sexual folds. Behind this fossa, on the other hand, the wall of the urethra is for the greater part the production of the united sexual folds, only a small part proceeding ontogenetically from the phallus-frame. The deviation of the two lamellae of the phallus-frame is in this transformation an important factor.
( 498 )
Considering the ontogenetical processes which contribute to the closure of the urogenital canal, as they have been described before, I have to join the group of investigators (Rerrerer, ReicHeL, HeRzOG) who assume a closure in consequence of the combination of two folds (sexual folds) in the median line. I deviate from their opinion as to the origin of the sexual folds, which are not the edges of the phallus-frame, but which represent folds of the penisectoderm.
In connection with the processes described above I finally wish to give some ideas about the value and the importance of the urethra from a comparative ontogenetical point of view. For this purpose I have to remind of the state, as it occurs in Echidna, one of the Monotremata. In this animal, as Keren’s') investigation taught us, a couple of tubes, the so-called “Samenurethra’” and the “Harn- urethra’ are developed caudally from the glands of Cowper. The former runs like a canal through the penis and is a production of the phallus-frame; the latter goes from the urogenital canal oblique caudally to the ectodaeum (ectodermal cloaca). Genetically this tube is formed, because the original single ectodaeum is divided by means of two folds which come together and unite, into two halves, the proctodaeum and the “Harnurethra”. For the group of the Marsupialia [*) have proved that the urogenital canal must not be considered as a homologon to the ‘“Samenurethra” of Echidna (as is generally done for the urethra of placental mammals on the ground of its topography with respect to the corpus cavernosum), but that it must be considered as a combination-product of “Samen- urethra” and “Harnurethra’, which placed themselves against each other and formed one canal. In Perameles there exists a transition between Echidna and placental mammals (man).
Applying the explanation given for the marsupialia about the genetical composition of the urogenital canal to the urethra of man, l come to the conclusion that here, too, a real “Samenharnurethra’’ exists, homologous to the ‘‘Samenurethra” + “Harnurethra” of Echidna. To be compared with the “Samenurethra”’ is that part of the urethra which owes its origin to the phallus-frame. The homologa of the two folds of the ectodaeum are the two folds which I described as sexual folds, by whose meeting the closure of the urogenital canal is brought about. The part bounded by these folds thereby becomes homologous to the “Harnurethra.”
1) Keren (F.). Zur Entwickelungsgeschichte des Urogenitalapparates von Echidna aculeata var. typica. Semon. zoöl. Forschungsreisen. Lieferung 22. pg. 153—206.
2) v. p. Broek (A. J. P.) Zur Entwickelungsgeschichte des Urogenitalkanales bei Beutlern. Verhandl. der Anat. Gesellschaft. 22. Berlin 1908, pg. 104—120.
A. J. P. VAN
Fig. 4.
Proceedings Royal
A. J.P. VAN DEN BROEK, “About the development of the urogenital canal (urethra) in man.”
Fig. 2.
In the figures 1, 2 and 4 the phallus-frame is black, the penisectoderm marked
with transverse lines. g. p. sexual fold.
Proceedings Royal Acad. Amsterdam. Vol. XI
( 499 )
From a comparative ontogenetical point of view, therefore, also the value of the urethra before and behind the fossa navicularis is different. For, whereas behind the fossa navicularis only a very small portion of the wall can be considered as a production of the phallus- frame, perhaps the vertical part of the lumen as it is found in the urethra of man, this changes before the fossa navicularis in such a way that there the greater part of the wall originates from that frame; therefore behind this fossa the urethra is principally homolo- gous to the “Harnurethra”’, before it to the “Samenurethra”.
Mathematics. — “On bicuspidal curves of order four.” By Prof. JAN DE VRIES.
1. It is easy to see, that each curve of order four, C,, with two
cusps can be represented by the equation Be, + 22,0,0,’ + 26,0,2,° + 2b,2,2,° + ew, = 0.
The triangle of reference has then the cusps O,, 0, and the point of intersection 0, of the cuspidal tangents as vertices.
From the equation
(zie, + «,°)? + 2(6,2, + b,c, + bewo), = 0, where 26, —c— lI, is evident that b, = b,#, + 6,4, + 6,7, = 0 represents the double tangent d of C, and that the conic tbe)
passes through the tangential points D,, D, of d and osculates C, in the cusps QO, and 0,
By combining the equations
i aie =O oand w= 26,0, we understand that the conics A, through O, , O, , D, and D, generate a system of pairs of points on C,, which are lying in pairs on the rays 2x, + Abr — 0
of the pencil, having the point of intersection H of k= 0,0, andd as vertex.
As this system of points with the curve is given we shall denote it as the fundamental involution F,.
If we put 2? = u, it follows from
oy. fo, = 0, 0? = pd,*4,*,
that C, can be generated by a pencil of conics (0,0, D,D,) arranged in the pairs of an involution and a pencil of lines (47) between which
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such a projective relation exists that the rays d and & through H correspond to the double-elements of the involution, the first of which is composed of the right lines d and &. The locus of the points of intersection of corresponding elements thus consists of the line d and a C, with cusps O,, O,. The polar line k of point A (6,,— 6,, 0) with respect to the conic A,, ut, Ha — Abe, + bz, + bez) oo has as equation b,(w,—àb,r,) — b,(x7,—Ab,x,) = 0 or be =br
On the line A lie the points Q,, Q,, which are connected with the pair of points P,, P, of F, generated by A, in such a way that we have
Q, = (O,P,; OF) and Q, (O,2;; O,P,).
The fundamental involution #, is thus projected out of O, and out of O, in the same involutory system of points (Q,, Q,). Now Q, is the projection of two points P, and P,’ of C,, so it is conjugate to two points, Q, and Q,’, by means of /,. Therefore the pairs Q,, Q, form on kh an involutory correspondence (2,2).
2. The points of C, are projected out of O, and O, by two pencils in correspondence (2,2); the line # is for both systems a branch-ray, because it is conjugate to the two cuspidal tangents k, and k,; the remaining branch-rays are the tangents out of O,
and” O10 1: These tangents are represented by 2b,e,* + 26,272, — Abr, — be, =, abe, en 2b,2° 2, => 26,0,0," EN bin — 0.
Through the points of intersection of these two three-rays passes the figure, represented by (b°z*—b re’) + ber, (6,7x,*—b,?2,7)—},b,2,’ (0,4, —6,2,) = 0. It is composed of the line h, bri =O and the conic (b,c, + 6,a,) br — b,b, (‚rs 4+ #,°) = 0.
The tangents 7,, 8, t, out of O, can thus be conjugated to the tangents 1,,8,,t, out of O, in such a way that the points of inter- section R=r,r,, S=s,s,, T=t,t, le with the point of intersection of the cuspidal tangents on a right line h.
At the same time a new proof has been given for the well-known
( 501 )
property *), according to which the singular elements (branch-elements and double-elements conjugate to them) of a correspondence (2,2) can be arranged in such a way that the singular elements of the first system correspond projectively to those of the second one.
For, if two pencils are connected by a (2,2) we have but to rotate them around their vertices until a branch-ray of the first pencil coincides with a branch-ray of the second; in the new position they then generate a C, with two cusps. From this is evident that there are four projectivities between the singular elements *).
The (2,2) between the pencils «,—= Ar, and «,—=wuea, has as equation
| Au + 2u + 26,44 2bu+c=— 0.
By the points of 4 these pencils are arranged in the projectivity 6,4 = bi!
By eliminating 2 we find out of these two relations the equation of the correspondence (2,2) between the points which conjugate rays of the pencils (O,) and (O,) generate on h. And now it is evident from
bu? aw? + 2b, by uw + 2b,°b, (ut uw) + 6% c=0 that this correspondence is involutory.
This result is in accordance with the well-known property *), according to which a (2, 2) between two collocal systems is involutory when the two systems have the same branch-elements.
3. Evidently the involutory (2,2) on A does not differ from the (2,2) which was deduced from the fundamental involution F,. Its coincidences arise from the four tangents which one can draw from H to C,. Indeed, the polarcurve of H consists of the line h and the conic u (passing through the points of contact of d).
If the branch-point R=vr,r, is conjugate to the double-point 2’, then A’ must be the point of intersection of the rays which the points of contact R, and R, of r, and-r, project out of O, and O,.
We conclude from this that the tangential points R,, S, T, of the
1) Emm Weyr, Beiträge zur Curvenlehre, Vienna 1880, Alfred Hölder, p. 32, or Annali di Matematica, 1871, IV, p. 272.
3) In my paper “Over vlakke krommen van de vierde orde met twee dubbel- punten’ (N. Archief voor Wiskunde, 1888, XIV, p. 193) I have applied the pro- perties of the (2,2) correspondence to those curves.
3) Emm Weyr „Ueber einen Correspondenzsatz’, Sitz. ber. der K. Akad. in Wien, 1883, LXXXVII, p. 595, or my paper under the same title in N. Archief voor Wiskunde, 1907, VII, p. 469.
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tangents 7,,8,,t, are projected out of the point H into the tangential moms Res Ss de 07 whestangentsnns sat.
If S’ corresponds as a double-point of the (2,2) to S=s,s,, then it follows from O(RR'SS') = O,(RR’SS’), that we have O,(RR’SS’) = O,(R’ RS'S).
From this follows that the points R,, R,, S,,.S, are connected with ORO mby a. conte. Also “the (groups 05, O.R, Roden OPO Tar hie on Jconies:
If K=hk we find out of 5
OO int ONO MC REN OF CK OF RA) that through A, and A, passes a conic which is touched in O, and 0, by the cuspidal tangents. The pairs of points SS, and 7’, 77. procure two analogous conics.
If two arbitrary points X and Y of h are projected out of O, and O,, then the points (O,X, O,Y) and (O,Y, O,X) lie in a right line through #.
From this follows that A bears three right lines which contain successively the pairs of points
aa | o's i. |
> =r
En
Lik
Above we found that these six points lie on a conic and form
two hexagons having 0, and QO, as point of BRIANCHON: it is now
evident that they determine a third hexagon, having A as point of BRIANCHON.
d. From (Ar, st) = (4r,5,7,) follows (Ar,s,t,) = (r‚kt.s,) == (s,t,47,) = (245,74).
So we can bring through O, and O, three conics Q,, 6,,T, with respect to which the line & has as poles the points &, S, T, whilst containing successively the pairs of points 3,6; 2,5 and 1,4.
On these three conics the pencils (O,) and (O,), arranged in (2,2) determine, just as on A, involutory correspondences (2,2); for, the two systems of points generated on them have again the branch- points in common.
If M,, M, is a pair of the (2,2) determined on g,, then the points (O,M,, O,M,) and (O,M,, O,M,) lie on C, and in one line with the point R, namely on the polar line of the point (M/‚M,, O,O,) with respect to @,.
The pencils with vertices A, S and 7 generate therefore on C, three more fundamental involutions of pairs of pomts where again each
( 503 )
ray contains two pairs. They differ from PF, in this, that unlike the former they do not contain the tangential points of the double tangent as a pair.
For M,=WM, we have a coincidence of the (2,2). From this is evident that the tangential points of the four tangents which can still be drawn from Mè, S or 7’ to C,, are every time connected with OQ, and O, by a conic (Q,, 6,, 7,). |
In an analogous way as for /, we find by paying attention to the singular elements of the (2,2) on @,, 5, and r,, that the lines ST, and S,T, concur in R, the lines R,T, and R,T, in S the lines R,S, and R,S, in T.
5. The polarcurve of the point (y,,7,,0) has as equation p I Yr Ya q yy (o,o? sl Laity” = b,@,°) zi (e‚°z, BE ORR zi b,x,°) =; or (ye, + Y, 24) (rara + 2,7) + (Oy, + by.) %,° = 0. By combination with the equation (wr, + #,7)? + Aber, = 0
of the C, is evident that the points of intersection of the two curves lie on 7,2, J- 2,27 == 0 and on the curve syst + Yot.) by = (b,y, + bay.) (e,2, + #,?).
Therefore the tangential points of the tangents out of a point of O,O, lie on a conic 1,
For y,:y, = 6,:6,, ie. the point K=hk, we find the conic (biz, + bw) bc = 6,6, (a,x, + 2,7) through the points 1,2,3,4,5,6.
Out of the equation
YO, (@ 1%, aac © len 2.056, ti bale, Log) = 2a be} = 0 is evident that the conics 4, form a pencil having as basis the points of intersection of z,7,-++2,2=0O with 6,—O0 (the points D,, D,) and two points of 6,7, = b,«, (the line A).
One of the pairs of lines consists of the lines d and /; it contains the tangential points of the tangents out of H, two of which are united in d.
The other two pairs of lines belong to two points of O,0,, for which the six tangential points lie every time on two lines passing
through D, and D,.
6. If (yx) is a point of d, thus 6,=0, then its polar curve with respect to C, is represented by
2 (Yoe, + Yat, + 2y,2,) (t,2, + Lo) + Oyst Or = 0.
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The points of intersection of these curve with C, which are not situated at the same time on ‚rt, + 27,?— 0 lie on the conic &, 3Ys (23 ie w,*) =2 (voe, sin gia a 243%) Eat
So the tangential points of the four tangents out of any point of the double tangent lie on a come through the cusps.
For y,=0, so y,:¥y,=0,:—6, (the point H) we find as it ought to be (Mer bya.) 2, = 0.
The conies § form evidently a pencil of which two basepoints lie on h, the remaining two in O, and Qj.
7. The curve of Hrssr of C, has as equation.
6wv,°z,* + 18 (bz, + bee) 2,727,247, + (18e + 32) wv ww, + + 60 (bie, + bw) 7, a,2,*° + (86b,6, + 24e — 8) x,2,2,* + + 9 (bie, + bew) #4 + 18 (bw, + bew) 10° + (185,5, + ¢) 7° = 0.
By combination with the equation of C, we find that the points of intersection of the two curves not lying in the cusps are situated on the curve
12 (b,4,+6,a,)a,0, + (185,6, — 18e —30)r, m,r, — 27 (b‚e, Hb) er, —
— (54e+22) (b,v,+b,0,)a,2 + (18b,b,—190— 18e’), =O.
So the eight points of inflerion of the C, are situated on a cubic curve passing through the cusps and the point H.
The polarcurve of the point O,=4,, consists of #,=0 and the conic .
22,0, + 3 be, + 3 been, + 2cx,? = 0, passing through the cusps and through the points of contact of the four tangents which meet in the point of concurrence of the cuspidal tangents.
It is easy to see that U, and H are the only points for which the polarcurve degenerates.
Chemistry. — “On the system hydrogen bromide and bromine.” By Dr. E. H. Bicuner and Dr. B.J. Karsren. (Communicated by Prof. A. F. HOLLEMAN).
The research, a report of which is given here,’ was undertaken in connection with a remark from Prof. HOLLEMAN, that the exis- tence of compounds of the type HBr, has been assumed several times in order to explain the mechanism of reactions in organic chemistry. In order to test the validity of this assuiaption it was thought desi- rable to ascertain, in the first place, whether pure bromine and
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hydrogen bromide are capable of forming a compound. As in binary systems the safest conclusions as to the existence or non-existence of a compound may be drawn from the course of the melting point curves we have attempted to determine the melting point figure of the system HBr-Br.
It soon became evident that, at atmospheric pressure, the hydrogen bromide instantly escaped from the mixtures so that we were com- pelled to use sealed tubes. The experiments were now carried out as follows: A quantity of specially purified bromine was weighed in a glass tube a part of which was drawn out; the tube was now connected to a HBr-generating apparatus and placed in a bath of solid carbon dioxide and alcohol. As soon as a sufficient quantity of HBr had condensed the tube was sealed and reweighed. The hydrogen bromide which was prepared from bromine, phosphorus and water was dried by passing it through two U-tubes containing P,O, whilst care was also taken that no moisture could enter the tube during the condensation. The tube was now fixed in a frame of copper wire and suspended in a rectangular wooden case, the long sides of which consisted of glass panes; in order to get a better isolation a second pane was fixed to each of these. Inside this case was placed a mixture of calcium chloride and ice for the higher temperatures whilst for the lower ones down to — 50° solid carbon dioxide and alcohol were used. For still lower temperatures this apparatus is unsuitable and the ordinary vacuum vessels were used; these, how- ever, suffer from the disadvantage that, unlike in the other apparatus, the tubes cannot be shaken properly without lifting them out. Any- how, in all cases we allowed the temperature of both to rise very slowly and the reading of the thermometer was taken at the moment that the last crystals fused. If only care be taken that the bath is kept constant at a trifling lower temperature for some time and that the tube and the bath are well stirred we may assume that the temperature of the mixture is practically the same as that of the bath. The observations were made with an “Anschütz” thermometer down to — 40° and a BaupiN toluene thermometer for the lower temperatures ; each determination was repeated a few times and the subjoined figures represent the mean result.
Before stating our results we just wish to explain, that, strictly speaking, we do not determine a melting point curve by means of the method described, for a vapour phase is also existent in the tubes which deviates considerably in composition from the liquid, and exists perhaps under a relatively high pressure. And from the weighings we know only the total concentration, and not that of
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the liquid phase alone. From some calculations, however, it appears that the composition of the liquid corresponds fairly well with the total-composition '), so that the curve representing our results graphi- cally does not differ much from the projection on the ¢, v-plane of the liquid-branch of the three-phase line, when we call to mind the p,t,@ model in space of BaKnuis RoozrBoom. In any case, the con- clusions as to the existence of compounds which we can draw from the course of the curve, remain unaltered.
In the subjoined table our figures are united whilst in the annexed drawing they are represented graphically ; it should be observed that the composition is expressed in mol. percentage of Br.
en ka melting point. Initial melting point
0.0 — 87.39 42, — 88 — 940 3.3 — 91 — 95 9.6 — 73.5 — 95.5 17.4 — 61.5 — 96 31.6 — 48 | — 93 41.0 — 41.5 — 95
50.5 — 355 55.8 — 32.5 69.0 — 24.5 77.6 — 19.6 87.7 — 13.4
The drawing, as will be noticed, does not leave the least doubt; bromine and hydrogen bromide do not form a single solid compound. It has not yet been decided whether the solid phases which are deposited, consist of pure bromine and pure hydrogen bromide or of mixed crystals; in the latter case there is a discontinuous series as at about —95° a eutectic point was observed.
Some experiments on the composition of the liquid and vapour phases at a pressure of one atm. render it highly probable that a compound of the type HBr, does not occur in the liquid or the
1) Only in the case of one tube — 77.6"/, Br, — the deviation might amount to about 2°, at least under a pressure of 5 Atm.; with the others it amounts to at most 1/3 %o.
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vapour. Moreover, the fact that in our tubes the pressure exceeded 1 atm. showed that at 1 atm. solid bromine (or the mixed erystals) would be in equilibrium with a gas-phase which contains much more HBr; from this we deduced that the liquid- and the vapour branches of the f,z-curve for constant pressure (the boiling point line) are much diverged. We tried to prove this by passing gaseous hydrogen bromide through bromine at 0° and analysing both the liquid and
40 - 40 x x x x a ses oe eee Rie ee fou fa) ele eal, 0 ier sba VA sy bo 0 £0 0 120
the gas. The bromine was placed in a tube furnished at the bottom with a tap by means of which the solution saturated with HBr could be removed. The hydrogen bromide which had bubbled through the bromine was passed through a tube furnished with stopeocks at both ends, from which it finally emerged in a flask over water. After the gas had passed for some time so that it might be taken for granted that the bromine was saturated and the tube completely
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filled with the vapour which was in equilibrium with the liquid the two stopcocks were closed. After introducing aqueous sodium hydroxide by gently opening one of the stopcocks until all HBr and Br had been absorbed, the solution was introduced into a measuring flask and diluted to the mark. An aliquot portion was then titrated at once with KJ and Na,S,O,, and in another portion the bromine was all converted into bromide by means of H,O, and then titrated according to VoLHarD with AgNO, and NH,CNS. In this way we found the free bromine and the total bromine from which the relation HBr: Br, may be calculated. In a similar manner the composition of the liquid was determined. At O° we found for the liquid 8 mol.°/, of HBr and 92°/, of Br,; for the vapour 87 °/, of HBr and. 13°), “ofelare2):
This result renders the existence of a compound in the vapour highly improbable, for if a compound occurs in a binary system in the fluid phases an inward bend is noticed in the p, 2- or ¢, a-curves; the liquid- and the vapour branch approach each other more or less according to the degree of dissociation of the compound. Judging from our observations there can be no question of something of the kind taking place in our case.
We beg to say just a few words as to the significance of these results in connection with the supposition mentioned above. Although we have proved that HBr and Br, in a pure state do not form a compound it cannot be denied that facts may be disclosed which plead for the existence of such compounds in solvents. But those facts only relate to solu- tions which possess electrical conductivity power and in which we must assume a powerful action of the solvent on the dissolved matters: in our case splitting into H’- and Br'-ions. One might cer- tainly imagine that the Br'-ion has a tendency to take up Br, and to pass into Br'’,-ion without this necessitating the existence of a compound HBr,, but in non-conductive solutions the idea of the existence of compounds HBr, should, in our opinion, be rejected.
Amsterdam, December 1908. Inorg. chem. labor. University.
1) These experiments are being continued.
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Botany. — “Dipsacan and Dipsacotin, a new chromogen and a new colouring-matter of Dipsaceae’. By Miss T. Tamars. (Com- municated by Prof. J. W. Morr).
If leaves of Dipsacus sylvestris are heated for a few hours in a moist -space to a temperature of 60° C., they acquire a fine dark- blue coloration. I have more closely investigated this phenomenon, which once accidentally came to my notice, and have studied the conditions of the formation of the blue colouring-matter dipsacotin, its properties and those of the chromogen dipsacan, the localisation of the latter and its distribution in the vegetable kingdom. At the same time I have traced the occurrence of dipsacase, the enzyme which splits the chromogen.
Here I wish briefly to communicate the chief results of the in- vestigation; a more detailed paper on this subject will be published in Recueil des Trav. bot. Néerl. Vol. V, 1908.
The investigation, which was chiefly carried out with radical leaves of Dipsacus sylvestris and fullonum, has shown that for the formation of the blue colouring-matter a temperature of at least 35° C. and the presence of water and oxygen are necessary.
Between 35° and 100° C. the rate of formation of dipsacotin in- creases with the temperature. It is only formed after the death of the leaf. No blue colouring-matter is formed in the living plant, even when exposed for several days to a temperature of 35°— 40° C. ; the pigment only appears in the dead leaves, when the plant is dying off.
If leaves are dried very rapidly at a temperature above 30° C., no dipsacotin is formed, or only a very small quantity ; if, however, during the warming, the leaves are in a moist atmosphere,t hey are coloured blue.
Neither does the blue coloration occur when oxygen is absent. Since it is extremely difficult to free the leaves completely from air, I have proved in another way, that oxygen is necessary. The chromogen can be extracted by warm water, and if the extract is warmed in a space completely shut off from the air, no dipsacotin is formed, even on heating for days together. As soon as the extract is warmed in contact with the air, the blue colour rapidly appears.
The formation of dipsacus-blue is therefore accompanied by an oxidation. Experiments have shown, however, that the colouring- matter does not result directly from dipsacan by oxidation. An inter- mediate product is first formed, as is shown by the fact that the light yellow extract becomes yellowish red on being heated in a
34
Proceedings Royal Acad. Amsterdam Vol. XI.
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space shut off from the air, and that the yellowish red solution has acquired the property of turning blue even without being heated. In the formation of dipsacotin from dipsacan a chemical transformation, which can only occur on warming, evidently takes place first; the subsequent oxidation can also proceed at the ordinary temperature, although it is greatly accelerated by warming.
Of the properties of dipsacotin I only propose to mention, that this colouring matter is soluble in water, that it is decomposed by sulphuric acid with the formation of a yellowish red product, and that it is decomposed by light; three points in which it differs from indigo.
The chromogen dipsacan is decomposed by acids and by alkalies, and can only exist in a feebly acid solution, such as that of the extract. Acids and alkalies do not, however, ever on heating, produce the transformation-product which by oxidation forms dipsacotin. This is formed from dipsacan, not only by warming above 35°C., but also at the ordinary temperature, through the agency of dipsacase, the enzyme occurring in the plant. This perhaps explains an observation made long ago by pr Vries *), that the press-juice of Dipsacus fullonum becomes black after a few days’ exposure to the air. Probably the juice contains both the chromogen and the enzyme, and the former is decomposed by the latter. That the colour, after oxidation, is black and not blue, may perhaps be attributed to the presence of other substances, or to other chemical reactions taking place simultaneously.
Dipsacan occurs in all organs, even including the flower and the seed, and all tissues, except the pith of the stem, contain it. The cellwall is probably free from dipsacan, as it does not become coloured blue.
The quantity of the chromogen, present in the various organs, depends on internal and external causes. Young parts growing vigorously, contain most. Under favourable conditions of life the quantity is larger than under unfavourable; af temperatures which approach the limits of life of the plant, the quantity of dipsacan is less. Light exercises no direct influence on the presence of the chromogen. In the dark the dipsacan does not disappear from the leaves, but it is formed in new, completely etiolated ones. Dipsacan is therefore not directly related to carbon-assimilation. More probably the chromogen takes part in metabolism, and as it occurs in the plant in such large
1) Hueco pe Vries, Een middel tegen het bruin worden van plantendeelen by het vervaardigen van praeparaten op spiritus. Maandbl. v. Naluurw. 1886, No. 1.
bower)
quantity, and especially in parts growing vigorously, it must indeed be an important substance to the plant. I imagine that dipsacan is continually formed and continually decomposed in the plant, and that the product of transformation, most probably that product which yields dipsacotin on oxidation, is used in various vital processes. In those places, where it is required, it is formed by the enzyme from the dipsacan present, and since it is not oxidized in the living plant to dipsacus-blue, we must conclude, that it is used up at once. Probably therefore dipsacan is the form under which the product used in metabolism, is stored up by the plant. This view not only explains the presence of the enzyme, but also the fact, that no dipsacus-blue is formed during life.
Besides in Dipsacus sylvestris and fullonum, | have been able to demonstrate dipsacan in several other species of Dipsacus, and in various species of the genera Succisa, Scabiosa, Knautia, Astero- cephalus, Pterocephalus, Trichera and Cephalaria. It is not wanting in any of the members of the order Dipsaceae which I have examined, so that I conclude, that it is characteristic of this order. It does not occur in other plants, as was shown by an examination of about 80 species, belonging to widely different orders. Only in the three species of the genus Scaevola of the order Goodeniaceae, which were at my disposal, I found, after warming parts of the plants in a moist space, that a blue colouring-matter occurs which is doubtless dipsacotin. The occurrence of dipsacan is therefore limited to two closely related natural orders, and a certain systematic value must undoubtedly be attached to it.
Groningen, Botanical Laboratory, Nov. 23:d, 1908.
Chemistry. — “On the bromation of toluol” and “On the sulfoni- sation of benzol sulfonic acid.’ By Prof. A. F. Hotieman and Dr. J. J. Porak.
(These communications will not be published in this Proceedings).
(January 27, 1909).
—
KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN TE AMSTERDAM. PROCEEDINGS OF THE MEETING of Saturday January 30, 1909.
—————— ES CO
(Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige Afdeeling van Zaterdag 30 Januari 1909, Dl. XVII).
e-O inl EN I AES:
N. L. Sénncen: “The splitting up of ureum in the absence of albumen”. (Communicated by Prof. 8. HOOGEWERFF), p. 518.
L. S. Ornstein: “Statistical Theory of Capillarity’. (Communicated by Prof. H. A, Lorentz), p. 526.
P. H. ScHourr: “On fourdimensional nets and their sections by spaces”, (4th part), p. 543. (With 3 plates).
J. P. VAN DER STOK: “On the duration of showers at Batavia”, p. 555.
JAN DE Vrims: “On curves of order four with two fleenodal points or with two biflecnodal- points”, p. 568.
Jan DE Vries: “On curves which can be generated by projective involutions of rays”, p. 576.
J. D. vaN DER Waats Jr.: “On the law of the partition of energy in electrical systems”. (Communicated by Prof. J. D. van per Waats), p. 580.
F. A. F. C. Went: “Some remarks on Sciaphila nana Br”, p. 590.
A. BRESTER Jz.: “The Solar Vortices of Hale”. (Communicated by Prof. W. H. Juus), p. 592,
Pu. Konnstamm: “On the course of the isobars of binary mixtures”. (Communicated by Prof J. D. vAN DER WAALS), p. 599.
Erratum, p. 614.
Microbiology. — “The splitting up of ureum in the absence of albumen.” By Dr. N. L. SÖHNGEN. (Communicated by Prof. S. HoOGEWERFF).
(Communicated in the meeting of October 31, 1908).
§ 1. General considerations. Ureum as a source of energy.
Ureum, secreted as a product of the katabolism in the higher organized animal world, leaves the body, dissolved in urine.
As such this nitrogen-compound cannot be assimilated by the higher vegetable world, and hence it would be of no practical impor- tance for us, if there were no fungi, especially certain microbes everywhere in the ground, which changed it into assimilable compounds,
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Proceedings Royal Acad. Amsterdam. Vol. XI.
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It is for this reason that we have to consider urine, more parti- cularly ureum, as one of the most valuable sources of nitrogen for arable land.
Millions of kilogrammes of the indispensable nitrate-nitrogen are annually in a biological way formed from urine in the ground and are of the greatest use to vegetation.
Nitrogen taken by man and animal as vegetable albumen, leaves the body again for the greater part in the form of ureum, and in this way describes a cycle.
A rough caleulation of the quantity of ureum, which in our country is produced by the population and the cattle, gives an idea of the enormous quantity of nitrogen describing this cycle.
The data for the amount of cattle have been taken from Verslagen en Mededeelingen van de Directie van den Landbouw. (Reports and Communications of the Board of Agriculture).
The quantity of ureum, daily secreted by the population, amounts to + 125000 K.G.; by the cattle = 225000 K.G., making + 350000 K.G., or + 350 tons a day.
By biological oxidation, a quantity of nitrate-nitrogen would be produced equal to that found in + 900 tons of saltpetre.
Annually from the + 125000 tons of ureum formed, + 350000 tons of saltpetre could be produced, representing a value of + 3.5 millions of £. sterling and this, distributed over the 2155000 acres of arable land, would yield + 160 K.G. of saltpetre pro acre.
That only a trifling part of this enormous mass is utilized for agricultural purposes, need not be proved here. Especially in large towns for hygienic reasons almost all ureum is lost to any useful purpose; on the other hand it would decidedly be of great value for farms in the country, to be more careful about collecting urine.
The above mentioned considerations may serve to draw once more attention to the enormous value represented by ureum as a manure.
In the experiments about microbes decomposing ureum the culture media generally were characterized by the presence of albumen and peptones.
It is true that von JAKscH *) and BEIJERINCK *) have made experiments with salts of organic acids as a source of carbon, but a systematic investigation in this direction has not been made as yet.
Von JAKSCH’s investigation in 1881 was especially of importance for
1) Zeitschrift fiir Phys. chem. 1881.
2) Centralbl. f. Bakt. Il, Abt. VII, Bd, 1901.
4
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the study of the conditions of nutriment of ureum-bacteria. It taught us that carbo-hydrates, salts of fatty acids and of organic multibasic acids can be assimilated.
The so highly interesting studies of BeiserINcK about the decomposition of ureum by microbes principally treat of the ureum-decomposing organisms which in cultures, on application of his accumulation-method in bouillon 10 °/, ureum make themselves conspicuous. In the course of the investigation some experiments have been made with culture- liquids, composed of water, 5 °/, ureum, 0.025 K,HPO, and 1 °/, ammoniumoxalate, natriumacetate, seignette-salt, ammoniumcitrate and ammoniummalate. In these culture-media a strong decomposition of ureum takes place after infection with mould.
The 5 °/, ureum added, however, are not entirely decomposed. The easily assimilable compounds, such as ammoniummalate and citrate, give rise to a greater ureum-decomposition, respect. 4°/, and 3°/,, than those which are not so easily assimilated, such as ammoniumoxalate and natriumacetate, in whose presence only 2 °/, ureum is decomposed.
The study of the microbes which are found in these cultures, was not continued at that time.
The purpose of this investigation is therefore principally to prove that the life of numbers of ureumbacteria is by no means dependent on the presence of albumen, but that for these ferments the large quantities of carbo-hydrates and salts of organic acids, which for microbial life are available in mould are extremely fit as a source of carbon, whilst at the same time the ureum can serve as a source of energy as well as as a source of nitrogen.
Some preliminary experiments led to the conviction that the most different sources of carbon, in culture-liquids containing these com- pounds and ureum, dissolved in water, 0,05 °/, K,HPO, are excellently fit for the growth of weak as well as for very strong ureum- splitting microbes.
Cultivated in a thin layer of liquid in Erlenmeyer-recipients at + 33°, being the optimumtemperature of the growth, strong species, especially those producing spores appear; at a low temperature, 15°—23°, less strong splitting ferments, especially micrococci are produced.
The exclusion of other groups and the privilege of the ureum- bacteria in these culture-media is so complete, that the latter mixed with raw materials, such as mould, sewer or ditchmud, after some days contain only ureumsplitting organism.
If one of these cultures, infected with raw material, is put into sterilised liquids of the same composition, the ureum-fermentation also progresses very well there.
35*
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Which ureum-splitting species will appear depends upon the com- position of the souree of carbon added and the degree of alkalinity of the culfure-medium.
In $ 2 and $ 3 we shall revert to this more in detail.
Ureum as a source of energy.
Ureum gives to the ureum-splitters exclusively energy; in no circumstances whatever it is fit to serve at the same time as a source of carbon.
Different experiments which I have made about this, corroborate the truth of former investigations; neither can ureum serve as oxidizable material in the sulphate-reduction; denitrification with ureum is also impossible.
The part that ureum plays in the growth of microbes, is therefore sharply determined. Always the presence of some suitable source of carbon is necessary ; this carbon-compound is partly oxidated and there- fore also this part serves for energy, partly it is assimilated.
For the above-mentioned oxidation of the source of carbon atmos- pheric oxygen is used; the quantity necessary is very small, which can be proved by cultivation in bottles with a stopper, which are entirely filled with the culture-liquid.
Only the oxygen dissolved in the culture-liquid is then available tor the microbes and nevertheless ‘the ureum-splitting then takes place just as well as when the supply of oxygen is abundant.
If, however, the culture-liquid has previously been made free from oxygen by boiling, after infection no ureum-splitting takes place in a bottle completely filled.
From these experiments follows that a good ureum-splitting is possible, while only very little organic matter is oxidated.
Now it is a fact that on the whole strong splitting ferments show in the cultures a very slight growth and from this it follows that also the quantity of carbon, necessary for the structure of bacterial bodies is very small.
These facts prove that a small quantity of a suitable organic compound, in the presence of ureum, must be sufficient for a complete development of the organisms and a normal ureum-decomposition.
Now, in order to ascertain what part of the sum of energy developed in the culture, is developed in the splitting up of the ureum the minimum quantity of carbon-compound, sufficient for a normal ureum-decomposition and growth, was determined. For this purpose experiments have been made with the afterwards described Bacillus
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erythrogenes and Urobacillus jakschii in series of culture-liquids, which, besides ureum, contain a diminishing quantity of asparagina or ammoniummalate.
Indeed a very trifling quantity of these materials proves to be sufficient for a normal ureum-decomposition.
From the results of the investigations, laid down in the subjoined table, it follows that the Bacillus erythrogenes at a normal growth splits 500 mG. of ureum with 20 mG. of carboncompound whilst the Urobacillus jakschu splits 1800 mG. of ureum with 10 mG. of carbon-compound.
With smaller quantities of carbon-compound the growth of both microbes is considerably less than above.
The quantity of energy, which in the erythrogenes- and jakschi- cultures was developed by the splitting of the ureum, amounts respec- tively to + 96°/, and 99°, of the total sum of energy developed in these cultures.
At the same time it appears from these numbers that the less splitting species want a larger quantity of carbon-compound for the decomposition of a certain quantity of ureum than the very strong splitters.
The figures in the subjoined table denote the number of c.c. N H,SO,, necessary for neutralizing 50 e.c. culture-liquid after five days of culture at a temperature of 30°. |
The 50 ce. eulture-liquid inoeculated with the Bacillus erythro- genes, consist of water, in which 0.05 ®/, K,HPO, 2°/, ureum and the carbon-source are dissolved.
The 50 ce. culture-liquid infected with Urobacillus jakschi has, besides 5°/, ureum instead of 2°/, ureum, the same composition as the above mentioned.
Decomposition by Bacillus erythrogenes. Quantity of carbon-source in milligrammes 50 40 30 20 10
5 Decomposition if the latter is asparagine 18.5 175 17 a 13 8 Decomposilion if the la'ter is amm. malate 19.8 17.9 18.5 180 142 9.5
Decomposition by Urobacillus jakschu. Decomposition if the latter is asparagine 61.5 60 59 60 54 42 Decomposition if the latter is amm. malate 60 58 60 59 56.5 39
§ 2. Caleiumsalts of organic acids as a carbonsource for ureum-splitting microbes. The organie acids proceeding from plants or produced by fermen- tation thereof are principally neutralized in arable soil by the frequently occurring calciumearbonate.
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The general occurrence, therefore, of these salts in the soil caused, for the following investigations, caleium-compounds of organic acids to be chosen in the first place as a source of carbon for ureum- splitting bacteria.
If in a eulture-liquid, containing these salts and ureum, dissolved in water to which 0.05 K,HPO, is added, ureum is split, the ammonium-carbonate formed will not directly bring about a consi- derable increase of alkalinity of the medium, but in the first place it will be neutralized by the calciumsalt and that according to the formula:
Ca R + (NH), CO, = CaCO, + (NH,), R
Therefore the calecium-compounds of the organic acids exercise a retarding influence on the alkalinity; for not until all the calcium is united with the carbonic acid formed, the alkalinity will advance rapidly; then the culture-liquid is like one that contains an ammo- niumsalt of an organic acid as source of carbon. This is treated of in § 3.
But it is especially because of the existence of this period of rest before the increase of alkalinity, that cultures with calciumsalts of organic acids are so particularly fit for the accumulation of less strong splitting organisms, by which means every opportunity is afforded for study of these kinds, which are otherwise so rapidly supplanted.
The cause that during this first period also the ureum-splitters have the advantage of all the microbes contained in the raw infection-material, so that the latter are already then entirely supplanted is that their specific source of energy, the urenm, is at their disposal.
So if we want to get an insight into the numerous kinds of weak ureum-splitters, we have to make a plate-culture before all calcium is united with carbonic acid.
As a rule a good result is obtained when after 2 or 3 days the plate-making takes place on meat-gelatine or on a culture-medium, composed of water, 10°/, gelatine, 0.05°/, K,HPO,, 0.05 NH,Cl and 0.5 °/, calc. malate.
These experiments give a fair idea of the great number of ureum- splitting microbes which the soil contains; they follow each other as the process proceeds and as the alkalinity increases, till at last the strongest, the most powerful hydrolysing kinds are left.
To give a detailed description of the many weak ureum-splitting kinds that exist, would be of hardly any use.
During my experiments in October, November, and December 1904 in the Microbiological Laboratory, under the guidance of Prof.
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Beijerinck at Delft, there regularly occurred in these cultures a microbe which drew the attention by the formation of a red and yellow colouring-matter on meat-agar and meat-gelatine. The colonies are of a bright yellow colour, whilst a red colouring-matter diffuses in the culture-plate.
This bacterium shows itself more especially in cultures with citrate and tartrate in large numbers, so that in using these salts the above mentioned species can be obtained in great numbers.
If in the same culture-liquid inocculation is repeated twice or three times at a temperature of 23° and a titre of + 35 ec. N. per 100 ec.e. culture-liquid, this bacterium is often accumulated almost to pure culture.
Description of Bacillus erythrogenes.
Bacillus erythrogenes are among the very strong oxidating fer- ments; both sugars and salts of organic acids and also albumen are assimilated. In tap water 0.05°/, K, HPO, a fair growth takes place in the presence of ureum, if one of the following compounds as a carbon-source is added: glucose, maltose, cane-sugar, asparagine, caletum- and natriumsalts of the volatile fatty acids (except of formic acid, which gives a slight growth) and the multibasie acids, such as apple acid, lactic acid, lemon acid, argol acid and amber acid (except oxalic acid).
Milk appeared to be a very suitable culture-medium. The develop- ment herein is attended by the appearance of a disagreeable sweet smell.
Even calciumhumate added as a carbon-source causes growth and therefore ureum-splitting.
Amylum, however, is not affected, so that evidently no diastatic enzym is formed.
The yellow colouring-matter belongs to the body of the bacteria and arises independent of the composition of the medium; however for its formation the influence of light is necessary.
The red, diffusing colouring-matter arises only in case the feeding takes place with albumen and the light is excluded. Arisen in the dark, this colouring matter will soon be decomposed when exposed to the light.
Cultivated on meat-agar while light is excluded, the white colony shows itself on a fine red diffusion-field ; cultivated in the presence of light, there arises a bright yellow colony on a colourless field.
What influence the two colouring matters have on the vitalfunctions of the microbe, could not be stated.
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Gelatine is melted by the strong splitting varieties, not by the weak ones. |
The length of the bacterium amounts to 2-—4u
Breadth 1—1.5 u
The bacterium is endowed with the power of motion, and in liquids mostly occurs as a double bar; on solid media it sometimes forms strings.
No formation of spores takes place.
The optimum of the growth lies near + 30°.
The optimum of its urease near + 51°.
Ureum-splitting by the strongest species is found in the subjoined table.
The figures denote the number of ee. '',, N.H,SO,, which are necessary for the neutralization of 10 ¢.c. culture-liquid.
The culture has taken place at 43°.
In bouillon with ureum.
After days 1 2 3 4. 2° , ureum 13.5 30. 45 44. 6°/, ureum 13, 45 68 68.
If we compare the species described here with those isolated by Lohnis'), they prove to agree in size and formation of a double colouring-matter ; striking is the difference in the power of splitting ureum.
In his experiments a bacillus erythrogenes split in bouillon 2°/, and 5°/, ureum resp. */,, °/, and 1°/, ureum, whilst the one described here splits in bouillon ©°/, and 6°/, ureum resp. 1'/,°/, and 2°/,.
The less strong species, isolated here, still split in the culture- liquids named '/, °/, and 1 °/, ureum respectively.
So it is clear that the species Bacillus erythrogenes includes varieties of very different ureum-splitting power.
The powerful splitters are at the same time characterized by the possession of tryptic enzymes.
§ 3. Ammoniumsalts of organe acids as a carbon-source for ureum-splitting microbes.
Ammoniumsalts of organic acids are in media, which at the same time contain ureum and anorganic salts, superior to any other com- pounds for the development of strong ureum-spliting microbes.
Both the split ureum and the ammoniumearbonate of the oxidated
1) Centr. bl. f. Bakt. Abt. XIV Bd 1905.
( 521 )
ammoniumsalt that has become free contribute to the rapid rise of the alkalinity of the culture-liquid.
Provisional experiments proved that with a ureum-quantity of + 5 °/, in these cultures the best results could be obtained ; with this ureum concentration growth is still very good.
In a way guite analogical with the ammoniumsalts behave diffe- rent sugars as carbon-sources for ureumsplitters; the species which are most remarkable generally agree with the powerful species isolated by Mriqurr.
A culture-liquid consisting of :
100 water
0.05 K,HPO,
1 ammoniummalate
5 ureum
infected with + two gr. of mould or sewage and cultivated at + 33° contains after 48 hours, sometimes even after 36 hours only ureum-splitting ferments. A total supplanting of all other organisms has taken place in that short time.
The decomposition of the ureum takes place in presence of easily assimilated carbonsources, such as malate and lactate, more rapidly than with compounds which are not so easily assimilated such as tartrate or acetate.
Malate is also for these organisms an exceedingly easily assimilable compound, as is generally the case; lactate, citrate, succinate, tartrate, butyrate and acetate follow next.
When, however, in a culture with one of these salts the final titre has been reached, the same powerful species are on the whole observed in malate as well as in tartrate and acetate cultures.
Now if we examine the sorts succeeding each other in these culture- liquids, it appears that, when sown upon meat-gelatine '/, °/, ureum or ammoniummalate-gelatine */,°/, ureum, already after two days, when the titre is + 60 cc. N. per 100 ec. culture-liquid, the num- ber of micrococcis and melting bacteria rapidly diminishes; whilst the alkalinity increases, bacteria forming spores together with a ureum-splitter not forming spores take their place.
The many weak splitting organisms observed in the cultures with calciumsalts rapidly die off.
After 3 or 4 days only very strong hydrolysing microbes are left, whilst micrococcis and melting species have disappeared.
The growth on neutral meat-gelatine of the species found in strongly alkaline liquids is very slight or does not sueceed at all.
( 522 )
In general the colonies on meat-gelatine 1 °/, ureum or ammo- nium malate-gelatine 1°/, ureum are characterized by their small dimensions, whilst a field of calciumphosphate- and calciumearbonate- crystals surrounds them.
After 5 or 6.days the titre has risen toa maximum of + 125 c.c.N. per 100 cc. of culture-liquid, so that + 4°/, ureum has been split.
The 4 or 5 species present are the Urobacillus leubii (Brtserinck) and the most powerful species described by Mique, the Urobacillus maddoat, freudenreichit and duclauxii together with a species not yet described and not forming spores, which will be called wroba- cillus jakschit.
After infection of cultures with ammoniummalate it is especially the Urobacillus maddori and wurobacillus duclauxi together with the Urobacillus jakschii which predominate. Sometimes the Urobacillus jakschu supplants the two other species almost entirely and is almost accumulated to pure cultivation.
If we start from pasteurized material, it is especially the Urobacillus maddoati and Urobacillus duclauxii which make themselves con- spicuous.
In these culture-liquid the Urobacillus pasteuri BrimrincK did not occur, so that the latter may be said to belong to the ureum- bacteria which positively want albumen for their growth.
For the description of the Urobacillus leubit, freudenreichit, maddox and duclauxu it is sufficient to mention the chief characteristics.
The Urobacellus jakschii, however, will be described more in details.
Urobacillus lewbit (BrIJERINK).
Urobacillus leubii, which generally occurs in the Vorflora of BEIJERINCK’s accumulation-experiments, is a little moving bar which can get oblong spores.
On meat-gelatine with ammoniumearbonate it is difficult to distinguish this species from wrobacillus pasteurii. Inocculated from this medium on neutral meat-gelatine it grows into two species of colonies: viz. into yellow, troubled, thin colonies forming spores and into glassy, transparent colonies free from spores.
The growth is, however, upon meat-gelatine with ammoniumcar- bonate much better than upon neutral meat-gelatine. -
The spores can bear boiling heat and can be dried.
Gelatine is not melted.
In bouillon 6°/, ureum 2'/,°/, ureum is split in 4—5 days
( 523 ) Urobacillus freudenreichii Miqver,
Urobacillus freudenreich is a little moving bar, 5—6 u in length, 1 u broad; on a firm medium it grows into long threads.
Elliptic glittering endospores are formed, which can stand a heat of 94° for two hours.
Neutral gelatine is slowly melted by the irregularly formed colo- nies, whilst- gelatine to which ureum has been added, is not melted and the colonies on it assume the characteristic globular form.
2°/, ureum in bouillon are decomposed within 4 days at 30°—35°.
MrqurL isolated this species out of air, riverwater, soil and from the excrements of ruminants.
Urobacillus maddoxti Miqver.
A little moving bar, 3—6 « long, 1 u broad, forming oval endo- spores, which are able to bear a heat of 94° for two hours.
On neutral meat-gelatine it does often not develop, on ammoniacal gelatine the growth is rather good.
Within 3 days 2°’, ureum in bouillon is split.
The bacterium has been isolated from sewage and river-water.
Urobacillus duclauxii Mrquer.
Like the two preceding species moving; length 2—10 4, breadth 0,6—0,8 u.
The bacterium forms small elliptic endospores which are able to bear a heat of 95° for 2 hours.
In a neutral medium no growth arises, on ammoniacal meat- gelatine or on meat-gelatine provided with ureum there arise very small hardly observable colonies which are surrounded by crystals.
The gelatine does not melt, but it becomes like viscous after 40 —50 days.
2°/, ureum in bouillon are decomposed within 24 hours.
Urobacillus jakschit.
Urobacillus jakschti is a small quickly moving bar in a culture- medium that is not too alkaline; if some percents of the ureum in it have been split, the motion stops.
Length of the bacterium 5—7 u; breadth 1—1.5 u. Spores are not formed.
( 524 )
On neutral meat-gelatine growth is seldom obtained ; if, however ammoniumearbonate or 1°/, to 2°/, ureum is added, there arise small coli-like colonies, surrounded by a wreath of caleiumphosphate crystals.
The gelatine is not melted, but after a month, it is viscous.
2 °/, ureum in bouillon are split in 24 hours. Of 10°/, ureum in bouillon 6—7 °/, are changed into ammoniumearbonate.
In culture-media containing the necessary anorganic salts together with ureum, a good growth is obtained with the following compounds after infection : pepton, asparagine, glucose, cane-sugar, maltose, citrates, lactates, tartrates, and salts of volatile fatty acids (except salts of formic acid).
§ 4. Lrsating cultureplates.
The faculty in bacteria of splitting ureum ean according to the method of BrErinck by means of the yeast-water-gelatineplate 2 or 3°/, ureum, be proved in a very elegant way by the Jris- phenomenon formed on this culture-medium by those bacteria.
It is supposed that the ammoniumearbonate getting free at the decomposition of ureum causes the phenomenon, in consequence of the precipitation of calciumearbonate and -phosphate.
An explication of the origin of the irisphenomenon on the yeast- water-ureum-gelatineplate, has, however, its difficulties, the culture- plate being so complicated that it is not easy to get an exact idea of the process.
In the experiments with ureumbacteria on plates composed of water, 0.5 °/, calcium salt of an organic acid, 1°/, ureum, 0.05 °/, K,HPO,, 10°/, gelatine or, 1.5°/, agar, the iris-phenomenon often produced itself.
The possibility of composing a simple culture-plate, if possible coagulated by agar, which produces the iris-phenomenon in a beautiful way, seemed not to be excluded, when the above facts were taken into consideration.
In this way corresponding phenomena on the yeast-water-ureum-gela- tineplate and the irisating of more complicated culture-plates might be generally explained.
After some trials I succeeded in the following manner in composing a plate which entirely answers the requirements.
In pure water agar + 0,5°/, calciummalate or -lactate and 0,05 °/, ammoniumcitrate are dissolved; the melted agar is cooled down to the still just liquid state, after which a K,HPO, solution is
(525 )
added, till a slight opalizing is observed; now the culture-plate is formed of this material.
This culture-plate is, if made with care, almost clear. The calcium- phosphate that has been formed remains dissolved with the ammonium- citrate. A drop of ammoniumearbonatesolution on this medium causes the irisphenomenon, while after some moments produces itself a precipitate round the drop.
This phenomenon shows itself in quite the same way, if, instead of agar, gelatine is taken.
The irisating field and the precipitate are microscopically and chemically identical to those which are produced on the yeast-water- ureum-gelatineplate.
If the culture-medium contains no phosphate, ammoniumearbonate put on it gives a very slight field of CaCO,; a drop of ammonia produces no irisating field at all. |
If, however, only calciumphosphate, dissolved in ammonium- citrate, is present as the only calciumeompound, ammoniumearbonate and also ammonia on such a plate cause an extremely fine irisating field.
If + 2°/, ureum is added to this plate ureum-splitting microbes cause thereon the “trisphenomenon’”’.
From these experiments it appears that the calciumphosphate-preci- pitation has to be considered as the real cause of the irisating of the culture-medium, whilst the calcium-carbonate formed at the same time plays a subordinate part. .
Accordingly the irisating of culture-plates by certain bacteria growing on them and the irzsphenomenon of BriJRRINCK have to be regarded as a consequence of the precipitation of calciumphosphate in the first and of calciumearbonate in the second place.
§ 6. Results obtained.
1. Decomposition of ureum, in the absence of albumen, may take place by different microbes, if some suitable carbon-source is present.
2. In cultures in which ureum-splitting takes place, + 98°/, of the total energy is developed by the decomposition of the ureum.
3. Cultures with calciumsalts of organic acids as a carbon-source, are extremely fit for getting weak splitting ureumbacteria. The bacillus erythrogenes occurring herein has been described more in detail.
4. Cultures with ammoniumsalts of organic acids or sugars as
( 526 )
a carbon-source, rapidly lead to the accumulation of strong ureum- splitting bacteria forming spores and the urobacillus jakschit forming no spores.
5. The irisating of culture-plates and the “irisphenomenon” on the yeast-water-gelatineplate are the consequence of the precipitation of calciumphosphate, whilst calciumcarbonate formed at the same time plays a subordinate part in it.
At the end of this investigation I beg to express my sincere thanks to Professor M. W. Brwrinck for advising and “supporting me in these experiments wherever and whenever he could.
Physics. — “Statistical Theory of Capillarity.” By Dr. L. 5. ORNSTEIN. (Communicated by Prof. H. A. Lorentz).
(Communicated in the meeting of December 24 1903).
In a paper *) published in 1893 VAN DER Waars has developed a theory of capillarity, which leads to results agreeing sufficiently with observation, as has been shown by the experiments of Dr. E. O. DE VRIES.
The methods used in the above mentioned paper have been repro- duced with only a slight change in the lectures of van per WaaLs recently published by Prof. Pr. KOHNSTAMM.
Both in the paper and in the treatise the hypothesis *) is introduced, that the entropy of an element of volume is a function only of the number of molecules it contains and of that of their collisions.
By the statistical method of G1BBs we can deduce the condition of equilibrium for the capillary layer without using a hypothesis of this kind and we can easily show that it must be true when certain condi- tions are fulfilled. This is the object of the present paper in which I shall also determine some quantities that play a part in the theory of capillary action.
§ 1. Let us suppose that m spherical molecules of diameter o, per- fectly rigid and elastic, are enclosed in a vertical cylinder of height Z, and of unit of horizontal section, closed at the top and the bottom by horizontal walls. Let the axis of 2 be drawn upward and let us further suppose that the molecules exert attractive forces on
Dele. Vier Day VAALS, Thermodynamische theorie der capillariteit in de onder- stelling van continue dichtheidsverandering. Verh. d. K. A. v. W. Deel I. 8. 1893. 2) Compare |. c‚ p. 16.
( 527)
each other up to distances which are large in comparison with the diameter o and with the distance of neighbouring molecules. I shall denote by —g(/) the potential energy of this attraction for a pair of molecules whose centres are at a distance fand I shall suppose that - g(f)=0 for values of f which are large compared with o (and the distance between neighbouring molecules) but small compared with finite lengths, the same being also true of the function yw (/) determined by the equation
NN =d Gale ite ar ots Dau GD
Let us now consider a canonical ensemble with modulus @ built up of N systems of the above kind.
We divide the volume of the cylinder by horizontal planes into a great number & of elements of a height dz, this height being large compared with o and small compared with the distance at which the molecules sensibly attract each other. I shall further suppose that the potential energy of attraction changes but little over a dis- tance of the order of magnitude dz, *).
We shall determine the number, or, let us say, the “frequency” 5 of those systems in the ensemble in which there aren, ... 7, ... Np molecules respectively in the elements dz,... dz, ... der. I shall suppose that the numbers », are very large; their sum being 72 we
have the relation k
ym ar at eee Pr Rie ree sce HT
1
The number of molecules per unit of volume in the element dz, (the molecular density) will be represented by nz,.
I shall consider the mutual energy of a pair of molecules as belonging for one half to the first and for the other half to the second of the molecules. The energy determined in this way is the same for all the particles of the layer dz,. I shall represent this energy per molecule by €,.
The total potential energy can therefore be represented by
Kk > Ny Be. 1
The frequency ?) in question is given by the formula
1) For the sake of simplicity I shall take the elements dz, of equal magnitude ;
. Ny : ; F : our result will be that —~ => n, (the molecular density) is a function of 2, showing dz,
that we do not lose in generality by this simplification. 2) In explanation of the formula (II) the following may be observed, Let us consider a system constituted of 2 molecules of the kind above described enclosed.
( 528 ) oi PAE: NE, Ny
ie ae my 7 2 @ 1 0 S= N(2amO) e lS @. de) e oen VAE) 1 fe
§ 2. The properties of an observed system are identical with those
in a vessel of the volume V. And let us imagine a canonical ensemble built up of N systems.
In this ensemble the number of systems — having the coordinates of the centres of the molecules between x, and #, +dx)... 2n and zn + den and the components of the velocities of these points between a, anda, and Ly ie cat En and zn + den — amounts to
We G ‘ 6 N m3" e Ul ae den ADs a UB ken ge
Here, the energy of the system is expressed by «, and ¥ is a constant for the ensemble depending on © and V. The value of ¥ can be found by integrat- ing (d) with respect to the coordinates and the velocities. The result of this integration must be N, which yields a relation for w. The number of the systems in which the velocities have any values, but whose coordinates are lying between the specified limits is obtained by integrating (a) over the velocity components from — ooto + oc.
n
| : : : The energy € is given by the relation € = @, + ) = m (a?,-+ y?, + 27,) 1
in which e, is the total potential energy and m the mass of a molecule. Therefore the result of the integration is
—n
2 g . N(2xOm) e dens dan. : (b)
Let us now divide the volume V into k elements dV,.. dVy .. dVk. If nz molecules are situated in an element of volume dVx the 277 coordinates of their centres may still vary between certain limits; in other terms, a certain extension is left open to the point representing these coordinates in a 37,-dimensional space. l shall represent the magnitude of this extension by
4 (n,, dV;).
The repulsive forces between the molecules are accounted for by excluding from the 3n,-dimensional space (JV) all those parts in which there exists a relation of the form :
(a, — ap)? + (yo yr) H (eve) <0 ... + (©)
between the ordinary coordinates of the centres of two molecules. We can
represent 7 (nx, dVz) by RENEE. dee eN
where the integration has to be extended over the whole space (dV), with the exception of the parts determined by (c). By a simple reasoning we can show
( 529 ) of the system of maximum frequency in an ensemble (whose modulus
that with a fair approximation y (%,d@V») can be represented by
Ny, (w, dV,) . ° ° . ; 5 . (e) where @, is a function of nz. I have calculated for w the approximate value.
; Me en fay = — n{ — — — n’/[— 76 og @ 5% ren
(Cf my dissertation and also these Proceedings 1908 p. 116). The extension of the 37 dimensional space covered by the systems containing Ny... M.. Nk definite molecules in the elements dV,..dVx..dVk can now be
represented by k
IT (i dV,). 1
The extension covered by all possible systems of this kind amounts to
“1 (My dV,) ni eae nn
In the potential energy we may neglect the repulsive forces, these forces having been already taken into account by the exclusions (c). Supposing that the energy is the same for all the molecules of an element dV; we can represent the total potential energy by the formula
k 1 For the frequency we find | 3 al NE, er eel k Beene 2 7) x 4 V,, @ $= N(22Om) en! je 1 Ny!
or, introducing the function w by means of (e)
3 = NyE,
—n — k Ny Ee ee 2 @ nV) g niee IN (pawerande 2 Ar i
1 Ny!
The formula (ll) is a direct consequence of the last equation.
As we are treating a case in which there are differences in density in the system of maximum frequency, the question arises as to whether these differences have any influence on the value of the function w. If it were so, this function would depend not only on n, but also on the derivatives of this quantity with respect to z.
The difference in question really has an influence on the energy, but in conse- quence of the hypothesis of p.p. 526 and 527 the density changes so little along the length dz, and the value of the exclusions at the limits of dz, is so small in com- parison with the value of those originating from the molecules of dz, itself, that we may consider q, as depending only on ny. This, however, will be true no longer if the sphere of action of the attractive forces is not large in comparison with g; for this case the following theory would have to be modified considerably.
36
Proceedings Royal Acad, Amsterdam; Vol. XI.
( 530 )
is proportional to the absolute temperature of the system). *)
In order to find the condition of equilibrium we have only to determine the values of the numbers n, that make the quantity Sor log & a maximum. Before we proceed to this investigation we have to express the quantity ¢, in the numbers n,.
Let us suppose that ? is a point of the layer dz, We shall try to determine the potential energy for a molecule situated at that point. Consider first the contributions from the molecules situated in two plane layers at a distance vdz from P. We shall indicate these layers by dz, , and dz,4,. We cut from these layers cylindrical rings by cireular cylinders having OPO’ as axis and as basis circles with
OA = ON =r and: OB SOB = r + dr as radii. The number of the molecules in these elements amounts to 2 a rdr dz (n,—y + nx»). Considering as equal the distance of all these molecules from P and representing it by f, we find for their contribution to the potential energy of P — rde dz (Dos BEP). - - 7 a eee Now we have r? + (vdz)* = f? and therefore EN EE EP 5 Taking into account (1) and (3) we can replace (2) by
nde (Drs mep Pf) en ae
1) Cf my dissertation § 4 p. 15.
( 531 )
The ‘total contribution to «, from ali the molecules of the layers de, and deg, is found by integrating (4) with respect to f from vdz to oo. Proceeding in this way we find
—— de (los Dhr) AB (pds) otten B) from which formula the energy per molecule in the layer dz, can be calculated by adding up all the values of this expression which are such that w (rdz) differs from 0.
In this way we find
& = — adz bo (01, + Dep) p(vdz) . . . . (5)
For the potential energy of the system we have the formula k
k pe nee = — ade > Ne Von + nee) p(vdz). . (IIT) 1 1 5
§ 3. We may now proceed to the determination of the condition for the maximum. Consider therefore the change of log § when we give the variation dn, to the numbers n,. These variations are subjected to the equation
k Sdn, = 0 Bnn vk Boy pire ne veen AN 1
In the following investigation we may replace n,! by n, e ~
We find for dlog & k
d log w, flog 8= S| tog n, pe +m, zen +
D 1 x
nde
k Ee | DD ital) (Des + Det) + 1
k J- pak ny, Ae (» d z) (d ny, + d me) os (V)
; … de, a It is easily seen that the two sums, with which @ 8 multiplied
are equal, both consisting of the same terms, and further that each of them is equal to
1 k ae Ex dn. adz pas
36*
(532)
Attending to the condition (IV) in the usual way, we find that the numbers 7, in the system of maximal frequency must fulfil the condition
te + Ny a ed Ap, hse as Aces ee on an; 7) whereas the second variation of log $, 0° log & given by the formula
Ae B dns i d , dlog w, JN
k mdz + dn, w(vdz)(dn,— + dn) , . (VID) Ye
must be essentially negative.
The first conditions are equivalent to those given by van DER WAALS. It is easy to give the equation (VI) the form which is assigned to it by van DER Waars. We have only to introduce the hypothesis that n changes continually with the height and then to calculate the energy &.
We obtain in this way *)
oy, dlogw, Zan, 1 5 l a le ~ = G ed Nn, ou dn, ca 0) 25 } ES
Ì
_d?sn,
EE d 2,78
== 2.7) or VA
1) To calculate e/ we proceed as follows. On account of our hypothesis we can write En ante (vdz)? d° n, ne (vdz)?s d?sn, ey Nn, ty == aD, EPS ea en : oT a! de (2s)! dz?
Introducing this into the formula for €, and putting
0 20 a = [+228 4p (2) de = SU (2s)! w ( ) Qs 3 we find for &, ex d2sn, END En Mals er € (6) 1
I shall write for &, also
5 UD en ae It is only in the capillary layer that the quantity ex differs from zero, 2) We may mention as another advantage in the above deduction of (VI’) that
( 533 }
§ 4. Before Ll proceed to the discussion of the stability I shall consider the equation (VI). Using (6’) we can put for it
oo 2 at d log w, a dans Zeer vr" og — + n, ——— —— — —_=f.. ... 9 Dz dn, 7) 7) ie \
Subtracting the equation (VI) taken for the height z, + dz, from the corresponding one relating to the height z, we obtain
1 d log w, d? log w, 2a dn, 2 de, —— +2 zn 1, ——— SS) BRE 5’ is dn, ane © / de, © dz,
If we introduce the function p, determined by the equation
P — = 1h —" 1"
6) dn 0)
dlogw an’
yh Ott EE 1
— which quantity represents the pressure in every element of a homogeneous system with the molecular density n — we easily see that we can replace (7) by 1 dp, dn, 2 der On, dn, dz, @ dz,” This equation leads to
dn, de, hg AE 5 dz, dz,
The form of this relation recalls the statistical condition of equili- brium namely that the difference of pressure between two planes be equal to the force acting on the mass between these planes.
By integrating (9) from a point of the homogeneous phase (indi- cated by the index A) to a point of the capillary layer (index x) we find
cd ex ao
dt
de, dn Ph — px =2 tn a, dz = Oren — 2 = ec dz, & az mh Zh d2sn we have avoided to prove for each of the integrals fnò 7 zes d2 separately that d 22
d?sn we can put for i on te dz, as is done in the treatise of van pER Waats—
Kounstamm p. 238. In the paper of van per Waats this gives something accidental to the appearing of €, in the condition (VI). This advantage is due to the fact that the hypothesis
of continuous transition and the expansion «, in a series have been introduced after the deduction of the condition (VI).
( 534 )
or €, is zero in the homogeneous layer. Instead of the former formula we may put
2 x
dn Dy == Ph = SBB Of Ede en a an de Zh
Introducing now for «,, the series that follows from (6) and (6’) we find for the pressure
*dn d?sn
d°sn; 1 doe des i Pi WR Oe JW Cop | ——— dz. es ‚a de 2 (5) di dz dzs VEE
00
9
zh
It follows from the above reductions that we obtain for the
C dn, erde Px =P Ny - == Pe BN as Ni Bere, ago ae ) +
S= 0 A=s—l
pressure p‚
In, d2s—*n, 1 dn, \? = Ra Oy ae “na anak me heee (VL S= 2 A=
An approximation for p, may be obtained by breaking eff the series at s=1: we then find a formula, which agrees with one given by VAN DER Waats namely
d De L fan, \* F Pr = Ph + Caf nx de = a ae ts
1) In order to reduce the integrals contained in the sum, we have the formula
Zr Zr
dn ds dn, d*s—! n, In d2s—1y
— == de = SS he ee dz dz?s da, dee! dz* de?!
Zh Zh Where the remaining integral may again be transformed by the same operation. In this way we are finally led to a term in which the integration may be per-
formed namely den ds+1n (—1)s (dsn,\? nf a ; des des! Br Ades
zh It follows from (VIII) together with the above reductions that by integrating from the one homogeneous phase Aj to the other /, we obtain: Ph, = Pho
which is the well kwown condition for thermodynamical equilibrium.
( 535 )
The constant u of the equation (VI) can be determined, if we observe that in the homogeneous phases ¢,,—= 0. Representing the molecular density of these phases by n, and n,, we have
d d log « w, Zan, Wy dlogws Zang
= log — ng ———— at | ere) iad Nn, T Ne d nz qc 0)
Rn n,
which yields one equation between n, and nj. We ean find a second by means of the observation made at the end of the note of p. 534 We have
let DIE pak, ein pS. cen OR We a A (12) where the p’s are known functions of n, and n, (cf. (8)).
After having determined n, and n, by means of the foregoing equations we can use the first to determine yu.
The thickness of the capillary layer depends on the modulus 0, it can be determined by means of (VI); we can also calculate the number of the molecules in this layer. This number being known, the equation (I) enables us to calculate the height of the liquid and gaseous phases.
§ 5. We have now to examine whether the frequency of the system determined by (II) and (VI) is really maximum, in other terms whether the condition of the system is one of stable equilibrium. The quantity 6*logS consists of three parts, the two first of which belong to the elements of the homogeneous phases /, and /,, whereas the third relates to the capillary layer c.
We may put the first parts in the form
dn,” if sl 5 Zar; Glogs dh 5 (34 a i = Fhe 1
h nz dn, where . has to be extended over the elements of the homogeneous
layers Ah, and 4. For the part belonging to the capillary layer we have the formula
2 log 5 dn, : d _ @ log w, nf AS aera am
x dz
2S Wlrde) (dn, Fn EE VEEN)
In order that d? log § be negative, it is necessary that ds, log &, d°h loy & and d*.logS be negative for all possible values of the numbers d7,.
The parts relating to the homogenevus layers may be written in the form
( 536 )
d d log Wy Zan, dn*, dz lo c= —1+4 n*,, ——— F ae ( zl dn, ) + 7) ) De 2n,
ha
where @ is 1 or 2. These contributions are negative, if
d , dlog os 2an, net aken Bree Mare feos re a A
Now, we can transform this condition by means of the function p (c. f. (8)). We then find as a condition for the stability
Re 8 eo
for the homogeneous phases. As for these phases, the function pz represents the pressure, the condition (IX’) is nothing else than the known thermodynamical condition for stability.
Not only must (IX) be fulfilled, it is also necessary that d*,log§ be negative, for there are possible variations in which du, is zero everywhere in the homogeneous layers.
I shall transform the first sum in d° /og§ by means of (VI). I shall write for it
~ dn, 1 1 od d log w, . dn, | — — + = cg og ; 2 ny n, dn, dr,
c
which may be replaced by
dn, d , d log w, Sr: tn (tay 2 ii **)
c
By a transformation of the same kind as that which leads to (7), we can replace the foregoing expression by
dé,
| = dz: J x J ek
0) pS ig dn,
dz,
Introducing the value of ©, by means of (5'), and considering that the differentiation of n,—, with respect to z, gives the same result as that with respect to z,_,, we find for the sum under consideration
a dz dn dn, ane. Gas == Sige as Sw (vdz i) Ke GO ef dn, hae pine) dE ae dze
dz
therefore (VII) may be reduced to
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d To) En. Ze aoe: De w (w de) (dn,_, + dn,4,) — —~ On, dn, QD yts5 dnt, ae oe » dz SATS eran Vir" pa dn, DE 4 G ) (= Az mal ee )
dz,
Now we ¢an easily show that this sum is essentially negative. For this purpose we arrange the terms in the following way. From the first sum we take the term dn, yw (p dz) dn,—,, and also the ter m dn, —,wW(v dz) dn,. These are equal, and their sum is
»y
AS dn, On,—, W (v de). dz
Next from the second sum we take the term dn, dn, dus
— ——_—__— wp (v de),
dv, AZ gy
and also the term
OR 2 5005) 00s
— —- —ap (vp dz). dns dz, dys Adding those four we find ‘ beds doc, dn, Ors EN —— a | rde). dede dee, du, (cine dz, Ha) dn ' This result is essentially negative, for-— has the same sign at all
points. *).
We can arrange all the terms of (VII'") in the same way. Accord- ingly, the whole sum may be written as a sum of essentially negative quantities, and therefore d°‚log& is essentially negative. From this it follows that a system consisting of two coexisting phases with a capillary layer between them is stable, if the homogeneous phases taken by themselves are stable.
$ 6. I shall now determine the entropy aid the free energy of the system considered.
GiBBs®) showed that 4, the constant in the equation (IL) has
1) A similar transformation does not hold for the elements of the homogeneous
dn . phases for there EE = 0.
2) J. W. Gress. Elementary principles in Statistical Mechanics 1902,
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the properties of the thermodynamical.free energy. I shall therefore determine the quantity W, which may properly be called the statis- tical free energy.
Taking the sum of the numbers 5, obtained by giving to the num- bers 2 all possible values, we get the total number N of the systems in the ensemble. I shall represent this sum by >,, so that we have the identity
3 LY Zi 4 Ny &, —n — UL I 9 Rk De (Om nl Noo at _T (w, dzz) OQ N= ) S—(227 0m)" New nl y LT 5 e A e e PN
This equation enables us to determine W. In order to find the value of 2, 5, we may by means of (VID express the frequency
S of an arbitrary system in that ¢, of the system of maximum fre- quency. From (VID it follows that
NS dn: th d , adlogw,
Ss — — —— nN’, = SS eee OTs dn, dn, =F Se
x E en SE Ak Jd n, % W(vde) (On, + dr) de | GO ease 8
Introducing this into the sum ,, we obtain
rens En En: 1 d log w, NS ok So an 1 ieee, ee dd E Ee sm | 27, + dn, dn r ae EN ke 2 T NX +- = , dn, s 43 (dz) (dn, + dn,,) de |
In my dissertation *) I have shown, that this may be replaced in a fair approximation by
ye = sae En nice dl vol En
The quantity §, is given by the equation WW 3 NzEx
Ne (220m) 2 Waan 5 ws Me 2 oO = = Nes REP 1 x
“Ot; (2ar)Flo(n, ...m,...mg)'l2
1) p.p. 111 and 126.
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where the numbers n, and n, have the values following from (VI). We now have
dn,” Ya, a e = (297) (Meese) a n— hs
e
and therefore, using (13) and (14), we find for 4”
U 3 k Ny Ey as 7 n Ne ye gn =(2%Om)*' "|| ( ) e hittin (XJ) n, 9 GaBBs showed that the quantity — defined by the equation ne XII Ne te OS a (XL)
has the proporties of the entropy s. Here the quantity e is the average energy in the canonical ensemble; it is equal to the energy of the system of maximum frequency *). The kinetic energy of this system amounts to 3 —n@. 2 For the potential energy we have written k
Ny &, 5
l and the value of ¢ is therefore
k = ee e=570+) ae ae 2 1
For s we have the equation
k anes pat Tog telog (2 a © m) Fn log np as 3 : i = Const + = nlog O + ee ag
Z 3 w = Const + 5 nlog @ + fn log” de Subst hoot fe a ae ea RE EN n 0
1) GrBBs showed that the average energy in an ensemble is equal to the most common energy in that ensemble. Now not every system with this energy is equivalent to the system of maximum