pt ST ES = : rit jay Mtr BON HSE iat it ih hi i SEEN | ne ae i Ht KI Bn a dn PEW | en talon euch io 4 i a Mis , manta Wi ia Wy ihn Wie aha ait i il eit a Hiatt aut he i H if ana as ae 1 hn " hi i wiek ‘ ht hee tiie} to iM tevoren bel Sas oe eee Tp = x 5 oe Se = Nei 1 Kn in en = SSeS se DEES se i Willi { HAVEL ITS Hf sorters: = RE j ; A ‘ A KAAT Vn! # VANOP ET, TND k NU Navel at Hot 4 1 vA KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN -- TE AMSTERDAM -:- PROCEEDINGS OF THE SECTION OF SCIENCES NG) al ME XI 257 PART — ) JOHANNES MULLER :—: AMSTERDAM JULY 1909 in oes ee et dert Kau 7 a. ke > ; é ty ca En ik ~ "et is A YROTE AES ERS SR a). SR i dn - ae ade en os - 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. ( 446 ) 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 ( 447 ) 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* ( 448 ) 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. ( 449 ) 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, ( 450 ) 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. ( 451 ) 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, ( 452 ) 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 ( 453 ) 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, ( 454 ) 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. (455) 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. ( 456 ) 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, ( 457 ) 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. ( 494 ) 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 ( 495 ) 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* ( 496 ) 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 ( 497 ) 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 ( 500 ) 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. ( 502 ) 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. ( 504 ) 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 ( 505 ) 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 ( 506 ) 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. ( 507 ) 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 ( 508 ) 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. ( 509 ) 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. ( 510) 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, 55 Proceedings Royal Acad. Amsterdam. Vol. XI. (514) 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 (515) 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* ( 516 ) 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 ( 517 ) 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. ( 518 ) 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. ( 519 ) 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. ( 520 ) 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 ( 537 ) 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, ( 538 ) 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. ( 539 ) 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 frequency, but the most common energy is-equal to the energy of the latter system, therefore the same is true for the average energy. This result may also be obtained by determining e directly by means of (VII). ( 540 ) This formula ean be used to determine s, if we know the manner in which n depends on z. We easily see from (XIII), that, just as VAN DER Waars supposed, the entropy in each element of volume depends only on the density n and on the number of collisions in that element'). We could expect this, having found exactly the same condition of equilibrium to which his theory leads. It must, however, not been forgotten that the whole above develop- ment and therefore the hypothesis of var per Waars are only valid, if the assumptions about the attractive forces introduced at p. 526 and 527 are true. The changes that will have to be made in the theory, when these assumptions are relinquished, must be a matter of further examination *). 7. Finally I shall determine the force exerted in a horizontal direction by the system. Consider a system identical with the former ; only let the section be no longer equal to unit of area, but let it be o. It is easily seen that this has no influence at all on the former developments. The density n, and the energy ¢, are determined by analogous equations; the only difference is that 7, (the number of molecules in the layer dz,,) is now given by n, 0dz,. instead of by n, dz,. For 4” we have therefore the formula k yr 4 Wz £, \ == — = Const. En py on, log == = == 0) n, 0) l vA ty o = Const. + fn 1 —— 5) de WE ly eee n () 0 The average component, corresponding to the parameter 0, of the force exerted by the systems of the ensemble is given — as GrBBs showed — by the relation =: dy KT er ea oe er do The force A,, exerted by the systems of maximum frequency, is equal to the average force A,. Therefore equation (XIV) may be used to determine the force in a real system. Before I use (15) to 1) The function w is connected with this number. 2) In this examination the function S (mz, dz-) introduced in my dissertation will have to play a part. ( 541 ) determine ,, I shall put this equation in a new form by means of (VI) namely Z uy d lo an? ne ea re eet: +0 fun — ae 4) ae = 0 Z d lo : A = Const. + (u—1) + fl (n—n' ee ak =) re n 0 with the aid of (10) we can replace this formula by Z Z z EN de eef 2 Dea }d xr ar Ar 15 phaz @ —DE + 7. tes de …/ ) 0 0 Zh For K, we get finally Z z d : K,= p+ { (ne +2 f Feeds) de EN St z 0 Zh ‘ . Cy dn, An approximate value for A, can be found, by putting €,, = — BL This value for K, amounts to Z K=—mZ sf dn dn\? te ‚=p ry re) ELSE Wies 0 Z . d 2 ~ > Pn = rid +C, {x SECs, at ota, aM ene fo oe Oe . dz* 0 When the surface of the capillary layer increases by unit of area the free energy (so far as it depends on capillary action) decreases by ~ Z z dn fre va f Beas) ac We bie avn AEN dz 0 Zh or, if we use the approximate values by Z dn Carne, ee ret Gene) i. e. the free energy increases proportionally to the surface. Only the elements of the capillary layer contribute to the integrals, for dn it is only in these elements that « and = differ from zero. The quantities expressed by (17) and (18), taken with the negative sign, agree with what is commonly called the capillary energy. In this form they also represent the so called surface tension. The quantity Zx dn Ph — DrEcz + > sn e-dz, az Zh or the corresponding approximate quantity (c. f. (16)) A Cy de dn, 2 5 a Eede (PL ee 2 i dz’, de may be called the horizontal pressure in the element dz, at the height z,. I shall represent it by pi. As we can see from (10), the connection between p‚and p„ is given by the formula Wig Yon, Ere LT ee MA The term e‚ being 0 in the homogeneous layer, we have Pix = Pr = Phi = Pho: We can determine the sign of &,, and therefore that of pu —- pr, by means of the equations (VI) and (10). We then come to a discus- sion exactly analogous to that which vaN DER Waats has given on p. 19 of his paper’). If one goes upward from the liquid phase, « is first 0, then positive, then O again, after that negative and finally O in the gaseous phase. By means of the foregoing considerations, we can obtain all the results formerly found by van ber Waats and the above method may also be applied to a spherical mass, whose density is distributed symmetrically around the centre. 1) Cf. van DER Waats-—Kounstamu p. 239, ( 543 ) Mathematics. — “On fourdimensional nets and their sections by spaces.” (Fourth part). By Prof. P. H. Scour. The net (C,,). 1. In the first communication under this title we have transformed the net of the cells C@ into a net C,,,) in two different ways, into 4 J a net of cells C@ and into a net of cells Ge? The difference between these two transformations may be characterized by the remark, that each cell Cf contains as a part the cell Cc? from which it is derived, whilst it is possible to consider each cell OM? to be bodily inscribed in a cell Oke by starting from two nets of Ge each of which fills the space Sp, entirely, related to one another in such a way, that the system of the vertices of the cells of the one is at the same time the system of the centres of the cells of the other and reversely. As we have used the second of these trans- formations in the deduction of the table of relations between the axes inserted in the first paper, we still cling to it here, though it cannot be denied that the advantage of including the cells Cie in boxes C§” is not quite so important as was that of including the cells oo in cells Ce, Cc! all cells (ee of the net corresponding with one another in orientation. We again restrict ourselves to the sections of the net (C,,) by spaces normal to one of the four different kind of axes of one of the cells cy and therefore of all the cells of the net. We remember to that end that the table on page 544 quoted above indicates which diameters of the box CY correspond to the chosen axes OF,,, OK. 24? OF}, ‘Oli, ot CS. We repeat here the part of it relating to the net (C,,) in the form (4) (8) (3) (6.7 (5) (1,2) Peo, Okn(2lT 0)Cs, OF y=OKe=(3,1, | Cs, ORa=OR OFS, indicating by means of. the figures (4), (8),..., (5), (4,2) between brackets the lines of the table, where these results are to be found. By this it is immediately evident, that the series of sections normal to OLF,, and to OK,, involve every time a definite position of the ( 544 ) : (2) intersecting space with respect to the axes of the including cells Cg’, whilst this position can be chosen in two different manners in the cases of the two other series. This gives rise to six different series of parallel sections of a Cer enclosed in a ce, which have to be considered in the following. 2. We adopt here the method followed in the second and third papers and indicate the results of the determination of the section of a cell CS” in two ways. Once more the first plate gives us the projection of the limiting elements of the cell BA on the diameter normal to the intersecting space, and the characteristics of the sections deduced tabularly from these projections; in this only four series of sections present themselves. The second plate shows further the form of the sections in parallel perspective, enclosed in the circum- scribed eightcell; here we have to deal with six different series. Finally a third plate principally contains some diagrams with three- dimensional space-fillings generated by the intersection of the net, whilst the third of these diagrams numbered separately, which domi- nates the deduction of the projections of plate I, has been transferred thither, in order to facilitate comparison. We now proceed to the consideration of the diagrams 1 and 2 of plate III. The manner in which the cell peo is inscribed in the box Ge is characterized by this, that the vertices of are the centres of the faces of Ce ’ We indicate how these points combine themselves by twos to extrimeties of edges, by threes to vertices of faces and by sixes to vertices of limiting octahedra, by indicating these octahedra in the diagrams 1 and 2. It is immediately clear that eight of these 94 octahedra are polarly inscribed in the eight limiting cubes of Ce fig. 1 exhibits two opposite faces ABC, A’B’C’ of one of ty octahedra, whilst fig. 2 shows two opposite faces ABC, A"B'C" o one of the sixteen remaining octahedra. Indeed the vertices of oe divide themselves with respect to the space of a limiting cube of Ce, the central space parallel to it, and the space of the opposite limiting cube into three groups of 6,12,6 points and the central section is evidently the combination (12, 24,14) of cube and octa- hedron in equilibrium; from this can be deduced that the second ( 545 ) limiting octahedron of which ABC is a face has for opposite face one of the two faces of this combination parallel to ABC and then — it goes without saying — the triangular face differing in orientation from ABC. So we get indeed sixteen new octahedra, each of the eight triangular faces A"5"C'" of the combination (12, 24, 14) procuring two of them. The projections given under the headings OL, OK, OF ,,, OR,, on plate I can be easily deduced from plate I of the second paper by means of the projections of the faces and limiting bodies of the including eighteell given there. By tracing the centres of all these faces and the octahedra polarly inscribed in all these limiting cubes we obtain the results tabulated in the four diagrams 3%, 3°, 3¢, 3d of plate I, i.e. in the cases OF,,, OK,, only eight, in the cases OF,,, OR,, only sixteen of the 24 limiting octahedra. As the laws of reci- procity require that the arrangement of the 24 limiting octahedra into groups for the cases OZ, OK,,, OF, OR,, corresponds to that of the 24 vertices for the cases OR,,, OF,,, OK,,, OE,, respectively, the obtained numbers (2, 4, 2), etc. of the octahedra can be completed to the really occurring numbers (6, 12, 6) added between brackets. Then from the projections of vertices and octahedra those of edges and faces are easily deduced. 3. We now proceed to the consideration of the sections represented on plate II in parallel perspective; of these the sections of the CS have been derived from the tables of the plate I, whilst those of the enveloping CS have been taken from the second paper. This plate is divided by three heavy vertical lines into four parts successively concerned with sections normal to OF,,, to OK,,, to OF,,, to OR. Of these parts the third and the fourth are subdi- vided into two parts, in relation to the two possible positions of the circumscribed eightcell. Sections normal to OF,, = OF. If we restrict ourselves here to the sections of transition and the intermediate sections bisecting the distances between these, we have ; 23 4 to deal with five cases corresponding to the fractions 0, B'B'E'E As to the circumscribed Cs” we then find a rectangular parallelopi- pedon, the base of which is a square with side 2, whilst the height 37 Proceedings Royal Acad. Amsterdam. Vol. XI. ( 546 ) 1 3 (V2) is successively 0, ove, 5, ove, 2/2 and as to Cos “ wegeta 1 point, a cube with edge 2 V2, a cube with edge V2, a polyhedron (32, 48, 18) limited by 6 squares and 12 hexagons with two axes of symmetry — which may be characterized as a rhombic dodecahe- dron truncated at the octahedral vertices by the faces of a cube — and this semiregular polyhedron itself (24, 36, 14) with one kind of face, which is also called granatohedron *). Sections normal to OK,, = (2, 1, 1, 0) GC, Here we have to distinguish two series of fractions, one related to the C,, itself, the other related to the box C,. The fractions 0, a = KET placed below on the right hand correspond to the ke. 8 1616’ B placed above at the left hand present themselves in the second case. In our second paper we have explained why the problem of the determination of the section of an eightcell loses one dimension in the case (2,1, 1,0) C, and all the sections are prisms with height 2, the bases of which are the sections of a cube with a series of parallel planes normal to the line connecting the origin with the point (2, 1, 1), seven sections in the first case, whilst the fractions 1 Le. of planes determining segments proportional to, 1,1 on the axis of coordinates. For the seven cases presenting themselves here fig. 4 indicates the form of the bases; so it is not diffieult to draw the prisms represented in the second column of plate IL. As it is not quite so easy to deduce from the characteristics given on plate I the forms of the sections of C,,, the faces of these sections situated in the boundary of the prisms have been determined independently by means of the diagrams 54, 5%, 5e closely connected to fig. 4. If we suppose that ON, OX,, ON, OX, (fig. +) are the four edges of the eightcell concurring in © and that the intersecting space is brought Me YD) os eae oe. _ 1) Here too the vertices of the faces of the section of (24 ‘ visible in the limiting 2) faces of the section of C8 have been brought to the fore; the shaded faces passing into one another by a parallel translation of the intersecting space are shaded in the same way. ( 547 ) parallel to ON, through the seven sections of the cube, it is clear that this space will be cut by the space O(CX, X, _X,) in a plane parallel to the endplanes of the prism, by the two spaces OX, X, X) and OY, X, X,) in planes parallel to the couples of parallel lateral faces of the middle section, by the space O(X, X, X,) in a plane not presenting itself in the middle section that according to its position in our figures — may be called the face behind. We now try to find in each of these four spaces of coordinates a plane normal to the indicated plane of intersection of that space with the intersecting space, on to which moreover the projection, of the limiting cube situated in that space of coordinates and the octa- hedron polarly inscribed in it is as simple as possible. So we get in OX, X, X,) the plane OLK, Y), in OK, X, X,) the plane OLX,_X,’, in OLX, X, X,) the plane OCX, X,), in OCX, NX, X,) the plane OLX, _X,). With omission of the case OCX, XN, Y,) equal to that of OY, X, XN) these projections are represented in the diagrams 5%, 52, 5e where the series of parallel intersecting planes are indicated by their parallel traces. For any position of the intersecting plane the required sections of the octahedra are easily found by means of these diagrams. So the section pgr of the octahedron in the lozenge os of fig. 5% is the hexagon of the endplanes and the section p’g’r’ of the octahe- dron in the rectangle o’r’ of fig. 5° is the deltoid of the lateral faces, of the middle section, whilst the section p'g'r's” of the octahedron in the rectangle of” furnishes to us the hexagon in the face behind of og With the aid of 12 the characteristics of the sections tabulated on plate I we then easily find how the visible faces of the section are to be completed to the total boundary of the polyhedron by means of faces situated within the prism. The form of the polyhedral sections of C,, obtained in this manner is rather complicated and therefore not easily described ; all the forms admit of two common characteristic features: they possess an axis with the period 3, in our figures the horizontal line MN, and four planes of symmetry, three through the axis and one normal to it. This axis MN is no axis for the prismal section of the circumscribed C, the middle section excepted; for this section it is an axis with the period 2 and in connection with this it becomes an axis with the period 6 for the middle section of C,, that admits of seven planes of symmetry. It is easily verified that the length of the axis MN within the seven different sections is successively i the section corresponding to the fractions To And ai ( 548 ) ve : we? S We B Eee LE 7e d 22 ’ 6 ] 6 ay 6 Va ’ Ti 5 6 an . Sections normal to OF,, = OK,. To the five polyhedra presenting themselves here — see the first column of the third part of plate IT — correspond below to the right 1 + 2 the fractions 0, —,...,—, above to the left the fractions —, 8 8 12 3 6 12’ DR) 12 Here too the problem of the determination of the section of the eightcell has lost one dimension, these sections being prisms with a height 2, the bases of which are sections of a cube, this time normal to a diagonal. Here too it is desirable to determine independently the faces of the sections of the inscribed C,, situated in the limiting faces of these prisms. To this end we have to revert to the diagrams 5e, Db, 5° and to replace the series of parallel lines representing the traces of the intersecting planes normal to the planes of the diagrams made up in the supposition of the intersecting space (2,1,1,0) by those which are connected with the simpler supposition (1,1, 1,0). As the new diagrams 5? and 5° become equal to one another, the new series of parallel lines have only been indicated — by dotted lines — in the diagrams 5% and 5’. So we find — entirely in the manner explained above — the section wv of the octahedron in the equi- angular semiregular hexagon fx forming the end planes, the section t’u’v’ w’ in the rectangle s’ 7’ forming the three lateral faces of the prismal section corresponding to the fraction ay ee easily get — once more in the same manner as above — by means of the data of plate Il the total boundary of each of the five sections of C The forms obtained in this way possess the same characteristic properties as those of the preceding group, an axis J/N with the period 3 and four planes of symmetry for the excentric sections, an axis with the period 6 and seven planes of symmetry for the central section. We can only record this difference that here the line MN is an axis for the sections of the C,, and those of the circumscribed C’, together, and that its length within the polyhedra always remains 2. Sections normal to OF,, = (3,1,1,1) C,. Here we find — see the second column of the third part of plate II a derd ™ ( 549 ) — the five sections of C,, already obtained, but enclosed now in parts of rhombohedra, regularly truncated at one of the axial vertices’), and as far as the middle section is concerned — enclosed in an unmutilated rhombohedron. For these sections of the circumscribed eightcell one may compare plate IL of our second paper. Here too the sections of C,, and C, correspond principally with one another with respect to axis and planes of symmetry. Only the plane bisecting the axis MN perpendicularly is not a plane of symmetry for the truncated rhombohedra and the axis maintains its period 5 for the unmutilated one. Sections normal to OR,, = OR,. In this simplest case partially mentioned above we find — see the upper region of the fourth part of plate I! — corresponding successively to the fractions O, DE (24, 36, 14) with one kind of vertex, and the combination (12, 24, 14) of cube and octahedron in equilibrium, always enclosed in an invariable cube. the octahedron, the semiregular polyhedron Sections normal to OR,, = OK, Here the three sections once more must be characterized by pairs | 2 ; of fractions, below to the right O, ener. related to C,,, above to tee the left —, —, — related to C,. Sue Ee The three sections of C,, found above reappear here — see the lower region of the fourth part of plate Il — successively inscribed in a tetrahedron, a semiregular polyhedron (12, 18, 8) with one kind of vertex, and an octahedron, represented in their turn inscribed in cubes for the more simple deduction of the true measures. 1) These rhombohedra bounded by lozenges with acute angles of 84°15'39" are represented in toto, the section with the truncating plane being indicated by heavy lines, In arranging the sections on plate Il I mistook the sections of the Cy occurring here for cubes, though I myself had indicated their true nature in the second paper. After having read the manuscript Mrs. A. Boore Srorr had the goodness to set me right as to the text; but the diagrams could no more be corrected. Happily the difference between these cubes and the slightly elongated cubes that should replace them is hardly perceptible. (550 ) 4. From the two ways mentioned in the preceding paper leading to the knowledge of the threedimensional space-fillings generated by the intersection of the net (C,,) we choose here the more theoretical one, in which is deduced from the section of the intersecting space with a definite cell C,, how this space must affect the other cells’C,,. To this end we indicate in diagram 6 how the boxes C,@) filling twice the space Sp, project themselves on the chosen axes OF,,, OK,,, OF ,,, OR, Of the projections on the axes OF, (2, 1,1, 1) C,, OK,, OR,*) coinciding with the named ones the third case OK, distinguishes itself from the other by this that the centre of the eightcell (2, 6,6, 2) does not project itself in the projection of a ver- tex: in connection with this each space-filling corresponding to OK, deviates from the general rule according to which the difference of the fractions corresponding to sections partaking in a selfsame space- filling is itself a fraction with numerator unity, the denominator of which is equal to the number of equal parts of the projection of an eightcell. So according to this rule this difference is sean the first . le and the fourth case, in the third case of the fig. 6, and it would una : : 1 have been = in the case of OK, but 7s now only = Now we have indicated in general which sections of C,, must generate together a space-filling we can proceed to the treatment of the individual cases; we shall then see that in any of these combi- nations of sections each face occurs twice in the same position, as the juxtaposition of the pieces requires. Space-fillings normal to OW, Here we have to deal successively ma EE a ie ae with the combinations of fractions (5) (5e) Ga so we find three space-fillings, that of granatohedra, that of granatohedro truncated at the six octahedral vertices and of small cubes, that of cubes with edges twice as long. Space-fillings normal to OK. Expressed in the fractions belonging eles Oa Ve | pe), See to C, here the combinations : ) 0, Ek ee ra ava Nr ’ Loe 16> 671 16 16 AC iG 2 6 10°14 ; se | present themselves; as no threedimensional section LEN eb 16 1) In order to make the diagrams fit better on the plate, the order of succession is changed there by interchanging (2,1, 1,0) Os and OK. (551) - 1,2 . of C,, corresponds to 0, TA these symbols, difference in orientation 16 being disregarded EE eN A Ee oO "eoar § ay ; , ETE IOA eN fe | Sea Ie eme isregardea, may je reduced: to 16° 16 16° 16’ 16 16 We now consider each of these cases separately. 4,8 Case (= 75) In this space-filling the middle section (26, 42, 18) 8 with the fraction err oe orientation only, whilst the poly- ? hedron (11,18, 9) with the fraction occurs in two different posi- tions passing into one another by a rotation of 180° about the axis MN with the period 3. In order to make this space-filling perfectly clear we project it successively on a plane normal to the common direction of the axes and on one of the axes. These projections can be found immediately, if we know how the composing polyhedra are to be put in contact with each other. Therefore we indicate first that two polyhedra of different form in facial contact with each other have always a deltoid in common, whilst this contact can be realized for two polyhedra (11, 18,$) by a lozenge only, for two polyhedra (26, 42,18) by a hexagon only; this is clear if we bear in mind that all the axes MN are parallel. The projection of the space-filling on to a plane normal to the axes MN may be regarded as the superposition to one another of two wellknown plane-fillings (fig. 7°), that of regular hexagons and that of equilateral triangles, the vertices of the polygons of the one being the centres of the polygons of the other and vice versa. If this space-filling is cut by a plane bisecting an axis MN normally the result is the plane-filling by triangles or that by hexagons accord- ing as that axis belongs to the form (11, 18, 9) or to the form (26, 42, 18). The projection of the space-filling on to an axis is given in fig. 84 in two layers of which the upper one is related to the axes of the central section, the one below to the axes of the polyhedra (11, 18, 9). The axes MN of the first group with the length 2/2 fill the whole line, whilst the axes J/’N’ of the second group with the length 4 3 V2 leave parts of the line uncovered. On one axis of both groups the projections of the vertices of the polyhedron have been indicated, on MN the points A, B, C, D, on M'N’ the points ( 552 ) A’, B’,C’; of these C, D, N coincide respectively with NW’, A’, BY). ooh, ake ase —,—<:— |}. Hae ids ee forms ( ) : Case i= Te = ach En the three forms (14, 21, 9), (26, 39,15), EL i ; 0 ; : : (32, 48,18) with the fractions 16°16’ 16 oe in two oppositely orien- . . . . 5 tated positions. The shape of the faces proves that two polyhedra & in contact must have an isosceles trapezium in common and two 7 16}? hexagon of the equatorial belt, whilst the required parallelism of all the axes J/N only allows the possibility of two polyhedra 3 E 5 polyhedra & having a hexagon, and two polyhedra & having a hexagon of the equatorial belt in common. Moreover the contact 5 7 of a polyhedron Ga) and a polyhedron (= must find place in 7 an equatorial hexagon, that of the forms (=) and (5) in a deltoid, 3 5 whilst the forms (5) and (= may be in contact by a hexagon. From all this may be easily deduced that the projection of the space-filling on a plane normal to the axes (fig. 7°) consists in the superposition of two plane-fillings, of which the one brought to the 5) fore here contains equatorial sections of the polyhedra € and 4 D ; 3 — |, whilst the more regular one of hexagons, the vertices of 16 which coincide with the centres of the polygons of the former, is built up of equatorial sections of the polyhedron This proves 3 at the same time that no two polyhedra & are in facial contact, : 9 : neither that two polyhedra (5) have a hexagon in common. The projection on an axis is represented in fig. 8’ in three layers, 7 5 3 successively related to the polyhedra (5) (5): (se): In none of the three layers do these axes cover the line of projection entirely. In the manner explained above have been indicated on MN the projections A, B,..., H, on M'N' the projections A’, B’,..., #5 1) These points have been indicated by the same letters on the polyhedra 11 5 and ze of the second part of plate Il; but here the daslies have been omitted. corresponding to >). (S53 ) on M’’N’’ the projections A’’, b’’,..., D’’ of vertices; here the pairs and triples (2, 7’), (FM), (@, A’, A’’), GH; B’, BY), (N,C’ coincide. Ch ; 6 I ne > . } Be : 49. an Te in this case of the unique polyhedron (14, 24, 12) occurring in two opposite orientations, bounded by an equatorial belt of six isosceles trapezia and two polar regions of triples of lozenges, the contact of the polyhedra takes place either by an isosceles trapezium or by a lozenge. The projection on a plane normal to the axes consists in the plane-filling of regular hexagons, each of these hexagons divided in the same way into three lozenges. The projection on an axis (fig 8°) consists of two layers; all axes MN, M'N’ have the same length. By the projections A, B, C, D of the vertiees each axis is divided into five equal parts; the last two segments CD, DN of MN cover the first two segments M'A’, A’ B’ of M'N’. If we consider the portion of the space-filling situated between the two planes normal to the plane of the diagram (fig. 8) according to a and 6 and if we imagine that the halves of the polyhedra (14, 24, 12) lying between these planes are hollow, we have before us a figure in space, imitating very nearly the shape of the honey-comb of the beehive. Indeed a space-filling, the polyhedra of which really are double beecells has been described by A. ANDreErNI'). But the space- filling derived here from fourdimensional space does not characterize itself by the known minimum property of the beecell; it is rather closely connected with the space-filling by granatohedra. If we divide a granatohedron in two equal halves by a plane normal to a diagonal and rotate one of the halves an angle of 180° about that diagonal, we generate a polyhedron also limited by six isosceles trapezia and six lozenges, but the trapezia have another shape. Space-fillings normal to OF. Expressed in the fractions belonging to the sections of C, we have to deal here with the two combinations 0 okt yal 10 ti. 3 i nic! HIS RT)? men which may be reduced to 4 6 3 5 2’ ia) Md (ag? 7) 1) In Anpremi’s memoir “Sulle reti di poliedri regolari e semiregolari e sulle corrispondenti reti correlative’ (Memorie della Società italiana delle Scienze, series 3, volume 14, p. 75—129, 1905) the three dodecahedra filling space are treated in the paragraphs 76, 77, 87. For the rest my study has nothing of any importance in common with his work, which is nicely illustrated by stereoscopic views of threedimensional space-fillings. ( 554 ) : 46 Case (= z) Of the two polyhedra (12, 21, 11), (18, 30, 14) Gd 12 12 orientations. The facial contact of two forms (12, 21,11) takes place by a triangle, that of two forms (18, 30,14) by a hexegon and that of two polyhedra of different shape by a trapezium. The projection on a plane normal to the axes is represented in fig. 9%. In the three shaded hexagons ABCDEF one recognizes 4 6 corresponding to (=) ; (a) only the first one occurs in two opposite ; 6 immediately the upper planes of three central sections =) and in LMN the base, in F,E,B‚A,D,C, the equatorial section of the body + — | lying between, etc. (oa) lying Case (5: =). Here both forms (18, 27,11) and (80, 45, 17) occur in two different orientations. Two polyhedra (18, 27,11) of the same orientation have no face in common, two polyhedra (18, 27,11) of different orientation in contact have a hexagon in common, so as to make their axes one another's production. Contact of two polyhedra (30,45, 17) takes place either by a semiregular hexagon or by a hexagon with two axes of symmetry in a plane inclined with respect to the axes, according as the two polyhedra correspond or differ in orientation. Contact by a hexagon with two axes situated in a plane parallel to the axes MN takes place between two polyhedra of different shape. The projection of the space-filling normal to the axes MN is represented in fig. 9’. In the three shaded semiregular hexagons drawn ; ONE in toto one recoguizes easily the three upper planes of the section 12m the other shaded parts of the figure portions of lower planes of 0 : : : : 5 : the section i in Opposite orientation, in the two larger semiregular hexagons partially covering one another in the centre of the figure 3 ; equatorial sections of the form (5 ‚ for which the thickness, 1. e. the segment on the axis lying within the polyhedron, is half as | 5 large as that of ge , ete. 12 Space-jillings normal to OR,,. Finally we get here two known space-fillings, that of octahedra and combinations of cube and octa- hedron in equilibrium, and that of the body (24, 36, 14) of Lord KELVIN, SS ourth part) L/ | ces.” (F are bs SSS SS En ij Cs HENK HEEEL sen I == u HE XQ idd P. H. SCHOUTE. “On fourdimensional nets and their sections by spaces.” (Fourth part) Eee 3 b Ti b b 2 0) 12 12 3 lee 6 6\* 6 ; 72) 12| Á 6 bs 6 Ó 12 3 3 2 7 TA 6 (6) 6 0) a ee 1 ve 3 : | La 12 2 B Se) RK, 6 Et —{ Cele FF y ee DE ema) 12 eer Wei 3 E ae’ d O| & S| 7 5 jelf EN 12 ; 5 ó 6 Ô Z q é 12 | | 7 J ees 3 EE 3 7 TE LIE ME Ss Ve ANS ej 2 4 BEE: al 3 BBE Sem Hs EE , hs er 7 € Lé 018 / Zn IA | 3 4 AWL #À 1h 4 2 | ó 5 ( 4 uw) b 6 A A ( b | 7 PF of Ó 4 Proceedings Royal Acad. Amsterdam. Vol. XL rth part). \y SIN D AL// j | 4 On N 9 iP 5 z ye \ RN De ESA > SN TN q | EN P. H. SCHOUTE. “On fourdimensional nets and their sections by spaces.” (Fourth part). OE, = Of Proceedings Royal Acad. Amsterdam. Vol. XI. LZ OF = 24 8 GERE 2 i JZ 4 GN AN 3K ol) Ne i Pe ripe 3 0 ‚SC HO UT E Ol fou rdim ensi ion al net s and th eir secti ion s b ys pac es 3” ene urth ea di P. H. SCHOUTE. “On fourdimensional nets and their sections by spaces.” (Fourth part). Plate III, oy WW LZ Á EZS Nd VV A Enk es fe DA ¢ wil SY == ee ; | \P G/ 4 V/ | toceedings Royal Acad. Amsterdam. Vol. Xl. (555) Geophysics. — “On the duration of showers at Batavia.” By Dr. J. P. VAN DER STOK. Since January 1st 1866 hourly observations of different meteoro- logical quantities have been made at the Batavia Observatory and also of the rainfall so that, at present, a series of 40 vears is available comprising some 25000 rainy hours distributed over about 8200 showers. In less favourable climates it is not practicable to have such observations made continuously night and day and, as self-registering instruments are subject to frequent interruptions, this series may be regarded as a unique material for investigation. The purpose of this inquiry is to investigate the distribution of showers of variable duration in different seasons, and to apply to these frequencies suitable frequency-formulae. A rainy hour is defined as every hour during which rain fell, if it were only 0.1 m.m.; the duration of a shower is defined as the number of subsequent hours in which rain was observed: e.g. by a shower of 10 hours’ duration we do not assume incessant rain during this period, but that no hour has passed without some rain having fallen. As during the first decade no observations were made on Sundays, the total amount of hours e.g. for January is not equal to: 40.31.24 — 29760 but to: (40.31—44) 24 — 28704. The results of this inquiry have been summarized in Table I. Average values of quantities so divergent and intermittent as rainfall are hardly sufficient to convey an adequate idea of the way in which this phenomenon affects the climate and it is questionable whether excessive quantities or durations are to be inciuded in the computation, because even the mean values deduced from a long series of observations may be affected to an important degree by one shower and thus the meaning which we attach to averages, loses its value. Consequently the frequencies given in Table II give not only a more complete, but also a more accurate idea of this climatological factor than the average values of Table I. From this summary it appears that in April, almost suddenly, the condition of rainfall shows an alteration such, that the probability of showers of long duration is considerably reduced; in the next months this probability again increases whilst in August the distribution ( 556 j TABLE I. Showers at Batavia, 1866—1905. | Number Percent. Mean dur. Mean numb. RUE of rainy of rainy Nel of showers | of showers hours hours in hours per diem | | | | January. … 28704 LAG 1575 1367 | 3.26 1.14 February... | 26136 4565 17.5 12ste Ii 365 > | a March...... 28680 2026 10.5 939 | 3.92 0.79 20) | VR | 27792 1704 6.1 664 2.57 OE MANNE | 28680 1138 4.0 HOG. ry) O68" POD JUNE veele 27768 1222 4.4 ANT 3.02 | 0.34 Malye. 98798 924 3.9 341 2,4 | 0.98 Augustus ... 28680 | 516 1.8 212 2.43 0.18 September .. 27768 897 ote 324 Paar it 0.28 October ....|| 28704 1219 4.2 480 2.54 0.49 November .. 97768 2039 | 7.3 709 2.88 0.61 December .. 28704 3372 ew 1071 | en heal) 0.90 == neen Ee | En WEAR veen, | 338112 | 25078 7.42 8190 3:06 slo OE shows some resemblance to that of April. The phenomenon of rain- fall, therefore, is subject to an annual variation consisting of a single period as shewn by the percentage of the showers and a semi-annual period with minima in April and August. Further it appears from Table II that showers of 24 consecutive hours rarely occur (14 cases in 8190), but that, although very rarely, showers of 100, 109 and even of 147 hours have been observed in February and March. 2. Besides the numerical representation as given in Table II, also an analytical representation is of some importance as, in the constants calculated, peculiarities of the curve of distribution often oceur which appear neither in the numerical, nor in a graphic representation and, at the same time, these constants can be considered as a quantitative measure of these peculiarities. In treating frequencies of this kind, the number of constants to be used must remain restricted to afew, as every constant necessitates the computation of an average of higher order and constants based on high moments will, in this case, be practically idle. It seems, therefore, desirable not to introduce more than two ( 557 ) TABLE II. Frequencies of duration of showers at Batavia, 1866—1905, En Jan. Eebr. Mrch Apr. May June | July Aug. Sept. Oct. | Nov. Dec. hours | ' | | 1 || 417 | 357 | 277 | 220 | 437 | 442 | 415 | 30 | 89 | 159 | 294 | 316 2 || 304 | 963 | 254 | 485 | 121 | 145 | 89| 55 | 96 | 454 | 193 | 273 3 || 206 | 203 | 438 | 140 | 71 | 55| 62| 36 | 65 | 86 | 12 | 175 4 11430 426 | 89| 59| 38| 46| 26 | 16| 32| 30| 52 | 411 5 || 81] 77| 50| 43| | 33| 13 | 40| 42} 27 | 33) 64 Boll 5) 60) 45) AT 10) 14-43 7| 44 | 13), 34) 38 mileteaaos( dete og | si. | el 10 | 42) 95 ee Ee EN 7) 40 31 A — 48.) 90 Galera ER MT ON EE OR ol. 44 10 16 | 20 5 1 9 2 eae | el lege 3 8 ONE EN ee | a a Bo gl g 12 eee ee ee |} —| —| 4] 2] 4 43 Al 9 2; —| —| 2 ya Mee al tel Fee 4 i} 5| 6| 2) —| —| 4) 2) —| 4) —f 4] - 15 EA AE ed Fea poll EE EH Tol OP EN en ln ene 17 EE de et B a ig | —| 42 —| —| ee —| -| | 1 19 FED ac tio tell a NE EN a 20 Eede UE ay ne EP LO ea OA en me zen en Jet =) er en ie ae en 1 EE ri el en I ed et mee en lend rd anc ik HIK le 24 Be hee | 25 A et men | DER en A ee Á 97 MM ld Wind Mel | len 28 pi ee AT EE EN moe (ere, Pee a) Me SE od" boos |) Me LTS Rag aaa ee eens es el EN er! les 30 Dey ye |b AS Rad) Secor) cee. EE ee VE ile er eee ee ef £ 32 ee eo == 1 Seg DE A ee Been ee 100 Se ae ie ee beets hoe Ph eae Soh | ieee eee | — 409 || - rit ak SEE en ed ei Mes MI | — 1 ed nt ad a end end Mes BEMI Le Total ||1367 host | 939 | 664 | 498 | 405 | 344 | 212 | 324 | 480 | 709 1071 ( 558 ) constants in the formulae, so that no average values higher than those of the second order need to be calculated and, as the phenomenon presents an annual and a semi-annual variation, it is reasonable to expect that this number will be sufficient to arrive at a suitable expression for the law of distribution. Then, in the first place, the frequency-formula known as Type III of Prof. Pkarson’s formulae finds an application. As the quantities under consideration are rainy hours, with the exclusion of hours in which no rain has fallen, the funetion must vanish for «=O if we take the duration zero for origin of coordinates, then for increasing values of v the function will rapidly rise to a maximum-value and decrease in a continuous way without any definite limit. In this case Pearson’s Type III assumes the form: wiee pp Oe hs i eS EN If we put mt =z the expression for the mean of the n‘' order becomes: 0 U n= | er? eet de {in AT { d 0 and KE Mo met! met 0 adi AP(pr1 | ep de = Eden | from which we find for the determination of the two constants m and p the expressions: M= en a al +. 3 en a Hy, u, A being defined so that the area of the curve becomes equal to unity, this quantity must not be regarded as a characteristic of the curve. Applying these formulae to the frequencies of Table I, we find the values given in Table III. ‘ In computing the means it is to be noted that the duration of a shower of say 3 hours is not to be regarded as a duration between 2.5 and 3.5 hours, but as a mean duration of 2.5 hours, because any duration beyond 3 hours would transfer the quantity into the 4 hours group. Further it must be noted that for February and March the excessive durations of 100 and more hours have been excluded from the calculation. ( 559 ) TABLE III. Constants of PEARSON’s formula, Type Ill. | B! | m p | | january {°0:3240 | 0.987 “| 0.408 February 0.8073 0.365 | 0.273 March 0.8483 0.400 | 0.227 April 0.6444 0.854 | 1.194 May | 0.5575 «| 0.734 | 0.94 June 0.3921 0.486 | 0.466 July 0.4863 | 0.600 | 0.695 August 0.6899 0.882 | 41.022 September 0.5044 0.677 | 0.876 October | 0.5443 0.648 0.646 November | 9.4499 | 0.491 | 0.443 December 0:33844- |-- 0.359 | 0.134 3. In the second place we have to consider the formula in series form given by the author in a former publication *) for the case that the function must vanish for «= 0: DE na AS cy a a ea a | and, if the mean of the n‘* order is represented by: b= f ux” da 0 re da = gs Bea Ent Yo PE Unt Vnln—1)! W(nl(n—=2)! nl Al (4) We may now introduce a suitable alteration of the scale value <‚ multiplying form. (3) by the factor A and further writing everywhere zh for x and p,h" for un. bt Pii! uh td nn Ered U i! Eed dl ete. LET A 1) These proceedings Vol X (799—817). ( 560 ) As by the use of these formulae the function and its integral assume rather large and the 721) p= n=? D= n= on EPEN ee BON ee SS SRS en 0.033 0.0506 0.0056 | —0.0266 0.0126 0.0314 0.0482 0.0073" | —0 0257 0.0106 0.0287 0.0459 0.0088 | —0.0247 0.0086 0.0265 0.0437 | 0.0101 | —0.0236 0.0067 0.0244 0.0415 0.0112 | —0.0224 0.0049 0.0225 0.0393 0.0122 —0, 0212 0.0933 0.0207 0.0373" | 0.0130 | —0.0199 0.0018 0.0191 0.0353 | 0.0136 | —0.0187 0.0004 0.0176 0.0334 0 0142 —0.0174 —0.0009 0.0162 0.0315 | 0.0146 | —#.0161 — 0.0020 0.0149 0.0298 | 0.0149 — 0.0149 —().0030 0.0098 0.0220 | 0.0151 | —0.CO9I —0 00%3 0.0064 0.0160 0 0138 —0.0045 —0. 0074 0.0042 0.0144 0.0149 —0.0012 | —0:0070 0.0027 (10081 0.00 8 0.0008 —().0059 0.0017 0.0056 0.0079 0.0022 —0.0045 0.001 | 0.0039 0.0061 0.0026 | —0.0034 0.0007 0 0027 0.0047 (). 0027 —( 0020 0.0005 0.0018 | 0.0035 | 0.0026 —0.0014 0.0008 0.0012 | 0.0026 0.0023 —0. 0004 0.0002 0.0008 | 0.0019 0.0019 0.0000 0.0004 0.0006 | 0.0013 0.0016 0.0003 0.0001 0.0004 | 0.0010 | 0.00413 0.0004 0.0000 0.0002 | 0.0007 0.0010 | 0.0005 ats 0.0002 | 0.0005 0.0008 0.0005 = 0.0001 0.0003 | 0.0006 | 0.0004 = 0.0004 | 0.0002 | 0.0004 | 0.0004 men 0.0001 | 0.0002 | 0.0003 0.0003 = 0.0000 0.0001 0 0002 | 0.0003 = = | 0. 0001 0.0002 | 0.0002 = 0.0001 0.0001 0.0002 = zak 0.0000 0 0001 0.0001 = a =) | oscon" 0.0001 Eat = = | 0.0000 | 0.0001 gt ol En | = 0.0001 == | == | — | = 0.0000 | TABLE VIII. Values of the Integral Brit ( 567 ) je n=0 n=4 a A= 0.0 | 0.0000 0.0000 0.0000 0.0000 0.1 | 0.0047 | —0.0045 0.0044 | —0.0042 0.2 | 0.0175 | —0.0164 0.0153 | —0.0143 0.3 | 0.0369 | —0.0333 0.0300 | —0.0269 0.4 | 0.0616 | —0.0536 0.0465 | —0.0400 0.5 | 0.0902 | —0.0758 0.0632 | —0.0521 0.6 | 0.4219 | —0.0988 0.0790 | —0.0622 0.7 | 0.1558 | —0.1217 0.0933 | —0.0699 0.8 | 0.1912 | —0.1438 0.1054 | —0.0748 0.9 | 0.2975 | —0.1647 0.4153 | —0.0770 1.0 | 0.2642 | —0.1839 0.1226 | —0.0760 1.4 | 0.3010 | —0.2014 0.4276 | —0.0740 1.2 | 0.3374 | —0.2169 0.1301 | —0.0694 1.3 | 0.3732 | —0.2303 0.1305 —0.0631 1.4 | 0.4082 | —0.2M7 0.1289 | —0.0556 1.5 | 0.4492 | —0.2510 0.1955 | —0.0474 4.6 | 0.4751 | —0.2584 0.1206 | —0.0379 1.7 | 0.5068 | —0.2640 0.1144 | —0 0284 1.8 | 0.5372 | —0.2678 0 1074 | —0.0188 4.9 | 0.56683 | —0.2700 0.0990 | —0.0092 9.0 | 0.5940 | —0.2707 0.0902 0.0000 2.4 | 0.6204 | —0.2700 0.0810 0.0088 2.2 0 6454 | —0 2681 0.0715 09-0170 2.3 | 0.6692 | —0.2652 0.0619 0.0245 9.4 0.6915 | —0.2613 0.0523 0.0314 2,5 | 0.7497 | —0.2565 0.0428 0.0374 9.6 | 0.7326 | —0.2514 0.0335 0.0427 2.7 | 07513 | —0.2450 0.0245 0.0472 2.5 | 0.7689 | —0.2384 | 0.0159 | 0.0509 9.9 | 0.7854 | —0.2314 | 0.0077 0.0558 3.0 | 0.8009 | —0.2240 | 0.0000 | 0.0560 3.4 | 0.8153 | —0.2165 | —0.0072 0.0575 3.2 | 0.8288 | —0.2087 | —0.0139 0.0584 3.3 | 0.844 | —0.2008 | —0.0201 0.0587 3.4 | 0.8532 | —0.1929 | —0.0257 0.0585 35 | 0.864 | —0.1850 | —0.C308 0.0578 3.6 | 0.8743 | —0.1771 | —0.0354 | 0.0567 3.7 | 0.8838 | —0.1692 | —0.035 | 0.0554 3.8 | 0.8996 | —0.1615 | —0.0431 | 0.0533 3.9 | 0.9008 | —0.1539 | —0.0462 | 0.0512 4.0 | 0.9084 | —0.1465 | —0.0488 | 0.0488 4.4 | 0.9155 | —0.1393 | —0.0511 | 0.0463 42 | 0.9220 | —0.1323 | —0.0529 | 0.0437 4.3 | 0.9281 | —0.1255 | —0.0544 | 0.0409 4.4 | 0.9337 | —0.1188 | —0.0555 0.0380 4.5 | 0.9389 | —0.1125 | —0.0563 0.0352 4.6 | 0.9437 | —0.1064 | —0.0567 0.0323 4.7 | 0.9482 | —0.1005 | —9.0569 0.0294 4.8 | 0.9523 | —0.0948 | —0,0569 0 0266 4.9 | 0.9561 | —0.0894 | —0.0566 0.0238 0.0000 0.004 0.0133 0.0241 0.0343 0 0425 0. 0481 0.0507 0.0505 0.0478 0.0429 0.0363 0.0285 0.0198 0.0107 0.0016 —0).0073 —0.0157 —0.0254 —0.0302 —0.0361 —0.0410 —0. 0449 —().0478 —0.0498 —0.0508 —0.0509 —0.0504 —0.0491 —0.0472 —0.0448 —0.0420 — 0.0389 —0.0354 —0.0318 —0.0281 —0.0244 —0.0206 —0.0169 —0.0133 —0.0098 —0 .0064 — 0.0033 —0. 0003 0.0024 0.0049 0.0072 0.0093 0.04114 0.0127 (568). TABLE VIII (continued). Th n=1 oe N= | n= | | } | | 5.0 0.9596 | —0).08i2 —0 0562 0.0211 0.0140 od 0.9578 —0.0793 —0.0555 0.0184 0.0151 ee 0.9658 —0 0746 —0.0547 0.0159 | 0.0162 5.3 0.9686 —0.0701 —0.0538 0.0135 0.0169 5.4 0.9711 | —0 0659 — (150527 0.0112 0.0175 DD 0.9734 | —0.0618 00515 | 0.0090 0.0179 5.6 0.9756 \ —0.0580 —0.0503 0.0070 0.0182 5.7 | OO7%6. "| —0.0544 —). 0489 0 .Q050 0.0188 wo, OO 1% —0.0509 —0 0475 0.0032 0.0182 gd | OSL — 0. 0477 — 0.0461 0 0016 0.0181 6.0 0.9827 —0.0446 —0.0446 0.0000 | 0.0178 6.5 | 0.9887 | —0.0318 —0.0371 —0.0060 | 0 0154 7.0 | 0.9927 | —0.0223 —0 0298 —0.0095 0.0149 1.5 | 0.9953 —0 0156 —0.0233 —0.0107 0.083 8.0 | 0.9970 —0 0107 —0.0179 —0.0107 0.0050 8.5 | 0.9984 —0 0074 —0.0135 —().0100 0.0024 90 | 0.9988 | —0.0050 | —0.0100 | —0.0088 0.0005 975 | 0.9992 | —0.0024 | —0.0073 | “—0.007% | —0.0008 10.0 0.9995 —0.0023 | —0.0053 —0.0061 —). 0015 10.5 | 0.9997 | —0.0015 | —0.0038 | —0.0048 | —0.0019 AAO 0.9998 —0.0010 —0.0027 —0.0038 — 0.0020 4 A) 0.9999 —0.0007 | —0.0019 —0 0029 —0.0019 42.0 | 0.9999 | —0.0004 | —0.0013 | —0.0022 | —0.0017 42.5 | 41.0600 | —0.0003 | —0.0009 | —0.0017 | --0.0015 13.0 | — —0. 0002 —().0006 —0.0012 —0.0012 13.5 ad —0.0001 | —0.0004 | —0.0009 | —0.0010 14.0 = —0.0001 | —0.0003 | —0.0007 | —0.(008 14.5 -- —0.0001 —) 0002 | —0.0005 | —0.0007 15.0 — 0.0000 —0.0001 —0.0003 —0.0005 1555 — -- —0.0001 —0.0002 —0.0004 16.0 -- -- |} —0.00M —) .0002 —0.0003 16.5 — — 0.0000 —0.0001 | —0.0002 17.0 as = =) | SO EON 1170 — | — | — —0.0001 | —0.0001 18.0 — — — 0.0000 = —0.0001 485 — | — | — — —0. 0001 19.0 — | — —- — 0.0000 Mathematics. — “On curves of order four with two fleenodal pomts or with two biflecnodal points.” By Prof. JAN DE VRIES. 1. The points of a binodal curve of order four, C,, are projected out of the two double points O, and U, by two pencils in corre- spondence. (2 , 2). So such a C, is determined by the relation ag, Ayu? ta, Aa, Ha, AW Ha A+ a, 94+ a, +a,,uta,, = 9, where Vee enh ey 2, \ According to a well-known property the eight singular rays (4) ( 569 ) are in four Ways projective to the eight singular rays (u); conse- quently through QO, and Q, pass four conies bearing each four points of intersection of two tangents out of ©, and O, and at the same ume four points of intersection of rays out of O, and O, to the points of contact of those tangents (double-rays of the (2, 2)). If O,O, is a branchray for both pencils, one of the four conics degenerates, in which case C, has cusps in OU, and O, (see my paper “On bicuspidal curves of order four’, Proceedings of the meeting of Dec. 24th 1908, Vol. IX, p. 499). We suppose that 0,0, is conjugate as double-ray to the branch- rays O,O, and O,O,. The equation of correspondence must then furnish for 20 and for «= 0 the equations u? =o and2?—=o: Heneeaaes dr Oa =O a = 0. The equation of C, can now be written in the form Bt, + 20,0,a, (0,2, + 5,2, + bewo) + 2,4 = 0. In each of the two double points one of the branches has an inflectional point; the corresponding tangents are wv, — 0 and x, = 0. Out of each of the two jlecnodal points three more tangents can be drawn to C,. They are represented by bite,’ + 26,b,4,?a, + (6,7—1) w,a,? — 2b, 2,' beta,’ + 26,5, w, + (b,7—1) a,a,? — 2b,2,' By eliminating z,* we find beta? — B,*«,*) + 2e, (0,20, — b,2a,") + be, (b,2,— b,0,) = 0. So on the right line 6,7, — 6,x, lie three points of intersection of the tangents out of V, with the tangents out of O,. We shall indi- ll || oc cate it by A. It is evident that these three points and the point OU, are the branchpoints for the two collocal series of points in correspondence (2,2), determined by the pencils (V,) and (Q,) on the line h. So according to a well-known property this (2,2) is involutory. Indeed, we find out of wt 26,Vuwt+2b,4u74+ 2b,A4n+1=—0 and BiA =De; that the (2,2) is indicated on 4 by the symmetric relation buu + 2 6, 6, (wu e+ wu?) + 20,5, due’ Hb, = 0 between the rays projecting it out of O,. 2. If Q, Q’ is a pair of the involutory (2,2) on h, then the points FO 0.0) cand Li (0,Q’, 0,Q) lie-on ¢,,.The line P,P, (570 ) intersects 0,0, in a point H, separated harmonically by the line / from O, and O,. So the pairs of points P,, P, form on C,a fundamental involution F., of which each ray through M/ contains two pairs. The coincidences of /’, are the points of contact of the tangents out of H(y, = by, = — b,, y; = 0). The polar curve of H has as equation (b, ©, — 6, 4) (@, 2, + 2, by) = 0, so it consists of the line 4 and the conic et, + #,6,—0. The points of intersection of this conic with C,, DE 4-28, @, 2, De HL, = 0, he ont? = O-and on +27 == be: By combining 0, == EP, with the equation of C, we find @, z, + z,’)? =0. So H is the point of intersection of two double tangents. The points of contact of these double tangents forming two pairs of PF, and being generated by the conics 2,4 position is at hand that #, can also be determined by means of the pencil of conics se a == 0; the su ’ 2 EZ. Indeed, the movable points of intersection of these conies with C, lie on the rays (1 + 0) 2, + 20% = 0, passing through 7, whilst the line A, Ot = bi is the polar of H with respect to each conic D= BE Resuming we can say: Of a C, with two fleenodal points O, and O, two double tangents meet on the connecting line O,O, of the double points. The points of contact of the four tangents which it is possible still to draw out of their point of intersection to CU, lie on a right line, which contains moreover three points of intersection of the tangents r,,s5,,t, out of O, with the tangents r,,5,,t, out of O, and the point of intersection of the inflectional tangents f, and f, in O, and O,. 3. From (/,7, 5, 4) =(272 5 é) follows (i r, Sy t,) == Gs. ie ty 8,) == (s, ts LF r.) = (t, Sa Ts Ay (571 ) By this three conies 9,,6,,7, through O, and QO, are determined containing in succession the quadruplets of points Mahe, bikes Uta. Gant ems Fei: Pity ijs Er; Fate 1) Maple ey: NE On these too the pencils (O,) and (V,) arranged in (2, 2) determine involutory (2,2), which then again are connected with fundamentat involutions on C,. The pairs of such an involution lie on rays through the pole ZS, 7 of O,O, with respect to the corresponding conic 9, OG, T,. This pole is the point of intersection of two double tangents ; this follows amongst others from the fact, that the point of contact of each tangent of the C, drawn from F must lie on the conic 9, and must be a coincidence of the involutory (2,2); the number of these tangents amounts thus to four, so that the remaining tangents out of A must coincide two by two in two double tangents. For further particulars about the properties which can be deduced from these observations | refer to my paper mentioned above and to the paper named in it published in “N. Archief voor Wiskunde, XIV.” 4. We shall now suppose that 0, and 0, are bijlecnodal points. Let us choose the point VY, in such a way, that the tangents in ©, and in O, are separated harmonically by O,0,,0,0, resp. by O,0,,0,0,, then the equation of C, has the form Ba, — GET —a,?a,72,7 + b,0,2,0," + 5,2,2,° + b,2,2,° + C°, = 0. If O, and QO, are to become biflecnodal points, then we shall every time have to find‘when substituting 7, + ar, and z,= + 4,2, that z,* —0. For this is necessary 6, = 4,5, = 0 and 6, + a,b, = 0, ns), Os aud ba) *). So we have to deal with the equation DE, — Aw U — wt He, =0. If we write for this ° (w,* ae ats’) (w,” TER ao.) sie (c* a= a,*a,") a," — 0, and if we put moreover C= ie ed it is evident that C, can be generated by the projective involutions of rays bh The six points 7189, 8,72, Sila, 482, 472, Mil lie on a conic; for, through 111g) S1S2, tylg passes the line h. 2) We find moreover that C, cannot have at the same time a flecnodal point and a bifleenodal point. td e= AVEN ji, = —_ as. — A 2,7), In this C, thus oo! quadrangles are described having all O, and O, as diagonal points. The vertices of these quadrangles evidently form a fundamental involution HF, Out of #,7=( at Hide”, Ao CN nd KP we find for the diagonals of the quadrangle (2) the equation (Aa,* —j) 2,7 = (da, 2 HAf) z°. So all quadrangles have in OV, their third diagonal point. At the same time it is evident from this that we can build up the © above mentioned #, out of pairs of the fundamental #, of which each ray through QO, contains two pairs. If the two pairs coincide then the ray which bears them is a double tangent. The pairs on the ray z, = or, we find out of fH + (a,*—a,°0°) 4+ fo? = 0 Thus for a double tangent we have (2,7 —a,707) = Ao", or ao’ + 2fo — a,? — 0. So O, ts the point of intersection of four double tangents corre- sponding to de + rr, — az,’ = 0, or, what comes to the same, to 2 2 a Bie a, U, == Zer, x, el i a, Ls a Q, The eight points of contact he on a conic. For, the polar curve of OU, degenerates into 2, =O and the conic 2 2 2p 2 nn aa, + a,*2,7 — 2c?z,? = 0. 5. We shall show now that the remaining four double tangents are connected with two fundamental mvolutions of pairs which can be generated by conics. r The curve C, can be generated by the projective pencils (e, — 4,7) (4, — 4,”,) = ofz,’, Q (z, zin a,x) (z, =e a.) = Edge Evidently the two variable points of intersection of conjugate conics lie on the line 2) , : 3 } ag (ar, ay at) + (9 Di 1) fw, 0, passing through the point /7, having as coordinates (a,, — a Each line „ 0). a,v, + a,v, + ofa, = 0 bears two pairs of the fundamental involution which can be generated by each of the two pencils of conics; for we have 9? — 2690 + 1—= 0. For @ = +1 these pairs coincide and we find the double tangents ats das, = fr, = 0. In a similar way the pencils (w, — a,#,) (2, + 4,00) = oft,” o (ze, + a,#,) (w, — a,v,) = — fa,? determine a fundamental involution which is also generated by the rays out of the point 7, (a,,a4,,0), through which at the same time the double tangents pass. The four double tangents form a quadrilateral having O,O0,0, as diagonal triangle. 6. The polar line of (a,, + a,,0) with respect to the conic (7, — at) (w, = az) = ofa,’ is represented by ttr a=, 0. From this ensues that the pencils (H,) and (//,) determine two involutory (2,2) on these two lines A, and A, Their branchpoints are generated by the nodal tangents and the tangents which can still be drawn out of O, and O,. If we write the equation of C, in the form (z,?7 — ar) wv, — (ast, — Cw) «,? = 0, it is evident that the lines ar =a ca,” touch it on rv, = 0. In an analogous way the lines at, — a Pie have their points of contact on #, = 0. And now we see directly that these two pairs of rays intersect each other on the lines /, and /,, at, =a,z,= 0, (574 ) which bear at the same time the points of intersection of the nodal tangents Se. and ws Bee The remaining points of intersection of the two fourrays lie on the conic Gt, = & — (a,%a,? + ¢’) 2,7 = 0 This is immediately ae if we eliminate out of the equations (aren Ne —a;72,°) = 9, (a?o, — Pa) (c,* —4,7z,7) = 0 the quantity «,* The coincidences on /, and 4, here also originate from the tangents out of H, and H,. Indeed we find for the polar curves of H, and H, awe,’ — aw, t°) + a,(2,72, — a,°4,2,") = 0, or (are, Ears =e 22,2, Nd. From this is evident at the same time that the conics PE ddr == generate the points of contact of ihe double oe meeting in A. and. A. By combining the equation 7a. a, 6, 2,° == 0 with the equation of C, we find that the eight points of contact of the four double tangents are situated on the conic 21 ar = (a,7a,* + &) a,’ 7. The curve of Hess is represented by (a,?2,7+ a,?x,”) eer, ie ee Za De. ve — (a,72,7+4,7u,7)} + (a,2a,?— 2c?) (a,7#,?+-a,°2,7) x," (SUR ea If we eliminate w,*v,° out of en equation and the equation of C,, “a7 — (a,72,7 + 0e) ez, Hea, = 0, it is evident that the points which the two curves have in common besides 0, and Q, are situated on the conic (da daj dee. The eight points of injleaion of a C, with two bifleenodal points are points of intersection with a conic. They lie two by two on four right lines through the point of intersection OV, of the four double tangents of the first group. The polarcurve 7, of the Ee (y) is represented by (yey EY, )eyety—Y ay 2ey? Harte stad ag) + ery, =O. ( 575 ). As it is touched in O, and Q; by the lines Yots 4 y;t, =O and voer, — a,7y,2¢, = 9; we find that Vita — U YY TD — A YY Tg + c*y, wv,” = 0 represents a conic 13 touching the polar curve in O, and Qs. If (y) lies on C,, then HY — GY Ys) TAN Ya Heys = 0, i. e. (y) also belongs to #,. The tangent (y) to 1, has as equation Vai eo Y2%,) x (az Q Has Yi YsUs—Ys (0 Yot Has yow) +207 y,*a, = 0. As when (z) and (4) are exchanged it determines the polar curve 1, it represents at the same time the tangent in (4) to C,. In each of its points C, ts touched by a conic which touches the polar curve of that point in the biflecnodal points. The curves C, and 4, have two more points in common. If / is their connecting line, then the pencil determined by C, and 4, + / contains a curve composed of 4, and a second conic. From this ensues: Zhe points of contact of the sin tangents out of a point of C, can be connected by a conic. 8. The projective involutions of rays (O,) and (O,) have as double rays N= Ds, Be OF Ho. Ten Pean 0. B= 0; a = 0, and eae! Mt Oe eet Oe When the double rays 0,0, and 0,0, are conjugated to each other, their point of intersection becomes a third double point of C,. This takes place when we have Seca seats — + —=0, or = 0. a, £ The C, is then represented by Gt. BE dd = 0. So it has three bifleenodal points. As is evident from the above we can describe in this C, oo quadrangles having the three double points as diagonal points. The double tangents of the first group are now replaced by the tangents in VO, (§ 4). In each of the bifleenodal points the tangents are harmonically separated by the lines to the remaining two double points. The C, with three bifleenodal points have been extensively treated by Lagurrrn (Nour. Ann. 2° série XVII, 1878) and by Scuoure (Archiv der Math. und Phys. 2° Reihe, H, Il, IV, VI, 1885— 87). ( 576 ) Mathematics. — “On curves which can be generated by projective involutions of rays.’ By Prof. JAN pe Vries. 1. By the symbol (a,@, + dgag)™ we shall indicate a homogeneous form of order 7. By the projective involutions of rays (agen + as) 4 A (age, Has) = 0, (be, = b,x) +4 (Biz, + Bv) — 0 a curve Co, is generated in which o' 2n-sides are described pos- sessing in OV, and Q, n-fold vertices. For brevity I call such a 2n-side bisingular. QO, and 0, are n-fold points of the curve. The tangents in 0, form a group of the first involution which is conjugated to the group of the second containing the ray O,0,. These two groups determine a singular 2n-side, where (, replaces }7(n-+-1), and Oy replaces 4 n(n—1) vertices. If we can describe in a Cx, with two n-fold points one bisingular 2n-side it bears an infinite number of those figures. For, if the indicated 2n-side is represented by the two groups of rays (az, + agt) el | age (ie -|- bz) ml | ~ and if «,—= ma, is one of the rays of the first group, then the sub- stitution must furnish w‚” (bie, + b,ma.)”) = 0, wv, = me,, O, being an n-fold point. Hence the equation of (>, must have the form (a,2, == at) (3e, se Bw) = (bie, ie be) (a2, + ae) : (1) But then the equation can be formed by elimination out of (asv, + a,a,)™ + A(a,e, + a,x,)™ = 0, : : 2 (bie, ln boe) ar (3,2, = Boe) == G and the curve contains the oo! bisingular 27-sides which can be indicated by these two equations, 4 varying. 2. We shall now investigate under which condition two projective involutions of rays will generate a curve Co with three n-fold points O;,, so that n? points of intersection of two conjugate groups of rays are vertices of three different bisingular 2n-sides having each two of the points O7 as n-fold vertices. In that case we must be able to bring through the points of inter- section of (577) (ayx, -{- ast) =") and (be, L been) = i a group of rays (en, + eg) = 0 It is now at once evident that this is only possible when the first two equations have the following form agrar — agter = 0, a" — ara,” = 0, so that we have ate,” — Agg = 0. Out of ara” — a,*x,” + 2 (agra — A7”) = 0 and art aa,” Ala — 00) = 0 follows (aan ad) CEE —,"@,") pee (arta a. Par") (a@y".vy"—a,"a,") —0 or in transparent notation (aa), #7," 2" + (aa), v,"2," + (aa),7,"7,"— 90... . . (8) The tangents in 0, are represented by (aa), vt + (aa)ger ” = 0. If wy = ke, is the equation of one of these tangents, then the sub- stitution in the equation of the Cy, evidently furnishes ‚2 = 0. In each of the n-fold points each tangent has thus (2 + 1) points in common with the corresponding branch. For each value of 4 we find a figure consisting of 3n lines (of which however only 3 or 6 are real, according to » being even or odd) and (n? + 3) points (of which 4+ or 7 are real). *) 1) We have in particular for 73 a configuration (125, 9,). From this ensues, by the way, that of the configuration (9,, 125) corresponding dually to it only 3 points and 4 lines can be real. From the above itis evident that the 12 lines of the (94, 123) can be represented by Ei —0, == 0 Ea ane 0, En O | and Ei =— gk, == ils, where 3 — i! is. The three lines & = Eg = &, & = Fo = PE, Fo =e, = ef, contain together the 9 points. They are also indicated by Lj + La + de == 0, Li a EXy + 23 = 0, Li |- "Xo =} eLs = 0. The 9 points lying also on 2,27; =0, they are the base-points of the pencil (a + ay + Hg) (+ La 7g) (@ + 27g + ed) + M My HqXy = 0. And so here we have found back the canonical equation of Cs. 39 Proceedings Royal Acad. Amsterdam. Vol. XI. (578 ) 3. The projective involutions of rays (a7, + a) + Avg’ (a,x, + Ge) (biz, + be) + Ark (8,2, + Bee) A @ generate evidently a Cor, which has O, and O, as (n—k)-fold points and as equation (asc, + a,2,)™ (Biz, + Bor) D= (be, boz) (a,z, Ha, . (5) The two multiple points are for £>>1 of a particular kind. For the tangents in QO, are represented by (ar, + 0) = 0, Zand each of them has as is evident from substitution (4 + 1) points in common with the corresponding branch of the curve. For 2, = 0 we find atk an—k (a,8,7,*—b,a,0,") = 0. Therefore the curve is intersected by O,Q, in a group of the invo- lution J; which has O, and QO, as Z-fold points. If we can deseribe in a Co 7 with two (n—k)-fold points a bisin- gular 2n-side having those multiple points as n-fold points it has an equation of form (5). But then it can be generated by two involu- tions of form (4) and it bears therefore oo bisingular 2n-sides. 4. For k=n we find a C, which will in general not possess any singular points. Yet it is in general not possible to generate a C, by two involutions of rays of order 7. For, the centres O, and 0, of the involutions must be n-fold points of an involution T,, of which the points of intersection of C, with 0,0, form a group. But then the polar curve of 0, would have to have (x—1) points in Q, in common with the right line OO, and this is not possible for a general Cn. But each cubic curve can be generated by two projective cubie involutions of rays. Their centres O, and Y, are conjugate points of the curve of Hessr, for the two double rays which Q, possesses (besides the threefold ray 0,0,), bearing each of them the points of contact of three tangents out of O,, form the polar conic of O,, whilst the rays which complete the two double rays to groups of the involution form the satellite conic of 0, Let us now take inversely O, and Q, as two conjugate points of the curve of Hesse. We regard 0), as centre of a cubic involution which has 0,0, as threefold element, whilst a second group is formed by three tangents the points of contact of which lie in a line 7, so that their points of intersection with C, are situated on a line s. The line counted double and the line s we unite to a group of a cubic ( 579 ) involution (O,) having O,O, as threefold ray. We now make the two involutions projective in such a way that the threefold rays correspond, that the group (77s) is conjugated to the group of the three tangents and that finally the groups are assigned to each other which are determined by the rays to an arbitrary point of C,. The two involutions then generate a C, having with the given C, ten points in common, thus coinciding with it. In each general cubic curve we can thus describe op* bisingular heaagons. Their threefold points lie on the curve of Hrssx. 5. If the ray O,O, counted double belongs to corresponding groups of the cubic involutions (O,) and (0), these involutions generate a C, which has O, and 0, as points of inflection the tangents of which meet each other on the curve. For, out of a,ty° + Sa ag a, + daga,a,7 + agt, + Avg? = 0, bet, + 36,2,72, + 3bo¢,¢,7 + bz,’ + Awr, = 0 we find (avg? +a, rg, HBagrgers Hasan), = (ber, 43, 2,7@,43b97,2,'+b,2,')ao, and this is satisfied by Pee ande BO, aa =), According to the rule found in § 3 0,0, is harmonically divided by (,. Inversely, when two stationary tangents of a C, intersect each other on the curve whilst their points of contact are harmonically separated by C,, then those points are threefold vertices of op* bisingular hexagons described in C;. For, in that case the equation of C, has the form (em, + 225) EL + (fe, + fate + favs) 7,027, + (9,7, + goa,)a,* = 0. If we replace it by : 2 id iy Ghd id id 3 id {c,v,° + AF Ze + (Bs + 02,4," + 9420 es + bew + frr, + (47, — v)a,2,? + Git, ta, = 0} it is evident, that the curve can be generated by the pencils en + f,a,7@, + (4f, + @) 2;2,7 + 9,2,° + Az,z,7 = 0, Ct, + fits %s BE Dit ER Q) Ty,” he RCN ma Ax yx,” = 0. Here we can still replace (4+ 09) by u. 39* (580 ) Physics. — ‘On the law of the partition of energy in electrical systems.” By Prof. J. D. van peR Waats Jr., (Communicated by Prof. J. D. vaN DER WAALS). It is well known that MaxwrLr was the first to pronounce the thesis that in the case of statistical equilibrium every degree of freedom will possess on an average the same amount of kinetic energy. In my opinion BOLTZMANN's and GaBBs’s investigations have raised the validity of this thesis for the systems to which their theory applies, above doubt. And yet this result is difficult to recon- cile with the experimental data. It has already long been known that the gas molecules must undoubtedly possess more degrees of freedom than would follow from the specific heat at constant volume in connection with this law. BOLTZMANN *), GiBBs ®) and others have expressly stated that systems, as they occur in nature, always exhibit important points of difference with the systems for which the said law was proved. For BOLTZMANN and Gipss have drawn up their statistical theory exclusively for systems which have a finite number of degrees of freedom, and for which the equations of classical mechanics hold. The systems occurring in nature, on the other hand, always contain electrical charges; so we have always to deal with the ether with its infinite number of degrees of freedom. Moreover for the changes of the coordinates no longer the mechanical laws are exclusively to be taken into considera- tion, but also the fundamental equations of the theory of electricity. In view of this an extension of the statistical method, in such a way that it applied also to electrical phenomena, was urgently required. An attempt at such an extension was made by JHANs *) and by Lorentz *). They arrived both at the conclusion that also in this case the law of equal partition of energy holds. Their considerations, however, pointed out a new difficulty. When every degree of freedom possesses the same amount of kinetic energy, the ether with its infinite number of degrees of freedom would finally acquire all the energy. A consequence of this would be that in case of equilibrium between a material system and the ether, as is found in all heat phenomena in consequence of the radiation, the velocities of the molecules and the electrons would become zero. Moreover the ether would then have to contain the energy in such a way that all the 1) L. Botrzmann, Wiener Sitzungsb. LXIIL p. 418. a. 1871. 2) J W. Gress, Statistical Mechanics. p. 167. 3) J. H. Jeans, Phil. Mag. Series VI. Vol. X, p. 91, 1905. 4) H A. Lorentz, Nuovo Cimento, Series V. Vol. XVI. energy had aceummlated on the side of the infinitely small wave-lengths. It is clear that here too theory is in direct contradiction with experience. Still it is easy to see that we must come to this conclusion if we assume the three following suppositions : Oe Oh . A. The relation %~ “* =O is satisfied. Here n represents the dome cl total number of independent variables, p, an arbitrary of these variables. In mechanies both the generalized coordinates and their time varia- tions or the corresponding momenta are to be considered as independent, so that in a mechanic system m represents twice the number of degrees of freedom. B. There is statistical equilibrium. C. All the phases (in the sense of GrBBs) representing the same energy, lie on the same path. By Jpans') and by W. Rrrz’) attempts have been made to explain this contradiction between theory and observation. In a certain sense also by M. Pranck, though the latter does not start from Grpps’s statistical method in his theory of radiation. As none of these theories satisfies me entirely, I shall state here another direction in which a solution of the contradiction might be sought. Before proceeding to this, however, I consider it my duty-to indicate why the three theories mentioned do not satisfy me, as else 1 should not be justified in adding another view to the number of existing ones. On a superficial consideration Rrrz’s theory makes the impression to look for the solution of the difficulty in this that it rejects suppo- sition C. For Ritz wants to reject a great number of states of the electromagnetic field which are compatible with the field equations, because in his opinion they cannot occur. Thus he assumes only waves originating from electrons, not such ones as converge on them, because, if the latter existed, the electron would be a perpetuum mobile. From these words it appears already that it is really suppo- sition B that is rejected by Ritz. The system he considers, constantly loses energy, and so it is not, in equilibrium. If we think the material universe to oceupy a finite space surrounded by an ether which reaches to infinity, it must of course lose energy, and cannot be in equilibrium. But then we knew already that on account of the prevailing differences of temperature the material universe is not in equilibrium. We have always only to deal with limited systems which are surrounded by other systems, and which are only for a 1) Comp. i.a. Le. and Phil. mag. Series VI. Vol. II, p. 638. 2) Rirz. Phys. Zeitschr. Vol. IX. No. 25 p. 907. Anno 1908, ( 582 ) certain limited time excluded from external influence, and may be considered to be in equilibrium. During the time, however, that the equilibrium exists, every particle must on an average absorb as much energy as it emits. So an inward radiant vector must exist during that time. But, says Ritz, when we consider a finite system, we must always think it enclosed within walls which contain a finite number of electrons, and which therefore can reflect the radiation in fewer ways than Lorentz’s totally reflecting walls can. So Lorenrz’s theory is not applicable to the natural systems. This really offers a difficulty. Even the y-rays of radium pass through metal sereens in a consi- derable degree, and it is possible that vibrations of still smaller wave-length possess so great a penetrating power that they are never in equilibrium of radiation in our experiments. Yet it seems to me that there are facts which indicate that in Rrrz's observation, which is quite correct in itself, the clue to the explanation of the normal spectrum is not to be found. For it seems to me that if Rivz’s explanation was the true one, the spectral formula of RAYLEIGH would have to hold for all wave-lengths which are still regularly reflected by the walls, which is by no means the case. Moreover, we should then have to expect that this formula would be fulfilled with the greater degree of approximation as the walls were thicker, and so more wave-lengths were approximately in equilibrium of radiation; then we should not find a definite spectral formula in- dependent of the thickness of the walls. We may finally imagine the walls to be infinitely thick, so that they would contain, an infinite number of electrons, and Ritz has not shown that also in this case his restriction of the number of possibilities in the inward radiation originating from the walls, is justified. While Rrrz’s theory deals only with the normal spectrum, JEANS tries at the same time to find a solution of the difficulties attached to it, and of the difficulties attached to the specific heats. He, too, thinks that he has to find the solution in the fact that supposition B is not satisfied. He thinks, namely, that the coordinates of a system may be divided into two kinds: 1 those which we may call conservative coordinates, which possess an appreciable kinetic energy ; 2 those which we may call dissipative coordinates, which can receive energy from the conservative ones only exceedingly slowly, and which lose the energy they have received so rapidly by radiation, that they never possess an appreciable quantity of energy. Then the kinetic energy which must be ascribed to systems agrees with their number of conservative coordinates and so is less than would cor- ( 583 ) respond with the total number of degrees of freedom. So this agrees with what we observe with regard to specifie heat. An ovjection to this theory is, that it is inexplicable how such a system could be. heated by radiation. For then the dissipative coor- dinates would first absorb heat, and then transfer it to the conser- vative coordinates. But they can only transfer it, when on an average they have more kinetic energy than the conservative coor- dinates. So if such a body was exposed to the radiation of a hotter body, it would absorb very much heat before the temperature began to rise, and on further heating by radiation the dissipative coordinates, too, would have to receive energy, and the specific heat would therefore be greater than when the heating was done by conduction. On account of these and similar conclusions which might be made from JEANS’s theory, this theory did not seem satisfactory. Finally in Pranck’s theory it is not possible to ascertain which of the three suppositions is to be rejected. Still it is clear that we can never obtain PLANck’s spectral formula, if we accept the three suppositions. His suppositions must, therefore, be incompatible with the three suppositions given here, and probably the hypothesis that the energy can only be absorbed in fixed energy-quanta, will be irreconcilable with our suppositions. Of course, this is no objection to PrANCK's theory; the assumption of the three suppositions bringing us into conflict with experience, one of them must in reality not be fulfilled. © Of more importance seems to me the objection that the supposition of these fixed quanta of energy which can only be absorbed or radiated as a whole, and which, moreover, have a different amount for radiation of different wave-length, clash altogether with all our ideas on the behaviour of vibrators, and that it is difficult to see how it could be reconciled to the ordinary laws of radiation by vibrators, of which PranckK also made use in his theory. Nor would this objection perhaps suffice to make us reject the theory, when there were urgent reasons why we should have to assume .the existence of these energy-quanta. In my opinion, however, these urgent reasons are wanting. PLanck used the supposition of the existence of these energy-quanta to bring two equations into harmony, which have been derived in an entirely different way. One’) has been derived from the laws of BOLTZMANN (STEPHAN) and Wien, which are again derived from the 2°¢ law of thermodynamics in connection with the fundamental equations of the theory of electricity, and which, therefore, hold for all the systems for which these laws hold. 1) M. Pranck. Vorlesungen über die Theorie der Warmestrahlung p. 149 equation 223, ( 584 ) Dae ' ; This equation runs S=F{ } in which S represents the entropy, u 5 the energy, » the frequency of the free vibrations of the vibrator and F'a é in which ¢ represents a small quantum of energy, into which the total energy of all vibrators with frequency pv is divided, after which these atoms of energy are distributed over the different vibrators accord- ing to the laws of probability. If these equations are to be in harmony, « must really be = Ar, as PLANcK assumes. The question, however, is, whether there is sufficient cause to assume that these still unknown function. The other equation!) may be written S=F' be equations do harmonise. And in my opinion this is not the case. For the first equation holds for real bodies, the second for the fictitious systems of vibrators, which most likely do not occur in the real bodies. In the first place so many different kinds of vibrators with so many different free periods are hardly to be assumed. And moreover every vibrator will, no doubt contain a moving electron whose motion strictly speaking is controlled not by a differential, but by an integral equation, so that the vibrator has not one, but a whole series of periods for its free vibrations. Now PrarckK asserts that it is of no importance whether his radiating systems agree with those really occurring in nature. For, he says’), Kircuyorr’s law teaches that we always get the same normal spectrum independent of the nature of the walls. This, however, seems to me an inaccurate interpretation of the law of KircHHorr. For this law states only something about walls occurring in nature, but it does not decide anything about the spectrum that would be formed in a space inclosed by walls with fictitious properties which deviate from what really occurs in nature. Hence the interpretation of Krecunorr’s law that the spectrum would be independent of the nature of the walls, is to be rejected. The real gist of the law is much better rendered by saying that all walls occurring in nature have such properties that they give rise to the same spectrum ; what these properties are which all real walls have in common, is not yet quite known. Only on special suppositions did Lorentz *) succeed in examining this. That this acceptation of KircuHorr’s law is really the correct one appears from this that in his cited paper Lorentz succeeded in imagining walls of such a nature that the thermodynamic laws of radiation are not fulfilled in their mutual radiation. He imagined, 1) loc. cit. p. 153 in the middle. 2) loc. cit. p. 100 and 101. 35) H. A. Lorentz, These Proc. IX p. 436, 1900. ( 585 ) viz., two uniform systems, of which the linear measures of the second are «-times larger, all masses g-times larger, and all charges of corresponding parts ~/a-times larger than of the first. it now appeared that when the velocities of the parts of the systems were the same, that then the ratio of the electrical forces in corresponding points was 13/a', so the ratio of the densities of energy 3/a*. Then, however, the ratio of the temperatures is as that of the kinetic energy of corresponding particles, i. e. 3:1, so that the law of BourzMANN is fulfilled only when 9‘ = #/a° or B= 1/a. As, however, it is always possible to imagine these systems in such a way that « is not 1/2, it appears that the thermodynamic laws of radiation are not fulfilled for arbitrarily chosen systems. So when we make arbitrary suppositions concerning the nature of the walls, we run a great risk of choosing them in such a way that the spectral distribution with which they would be in equilibrium, does not agree with the real normal spectrum, this could only be incidentally the case. And so there is no ground to assume that the two formulae of PLANCK mentioned represent the same spectrum, which removes the ground for the assumption of the energy-quanta. JEANS’) considers it a difficulty to assume that walls could be imagined for which the thermodynamic laws of radiation are not fulfilled. I do not see the difficulty. The thermodynamic laws are only empiric laws. And when we come to the conclusion that the radiation of arbitrarily imagined walls does not satisfy the second law of thermodynamics, whereas experience teaches that the real radiation does satisfy it, we have simply to conclude from this that such walls do not occur in nature. We should, indeed, meet with a difficulty, if we could show that the laws of thermodynamics had to be applicable to all conceivable systems. The statistical derivation of these laws seems really to imply this. This, however, is only seemingly the case. For any fictitious system, whatever properties we ascribe to it, a state of statistical equilibrium will, no doubt, exist which is characterized by the fact that a certain quantity, which we may call the probability, is maximum. If we call the logarithm of the probability entropy, then for every system the theorem will hold that with given energy and volume this entropy is maximum. But it has not been proved a priori that the entropy defined . rt: dQ ’ in this way, is always represented by es OE at least it has only been proved for mechanical systems, and not for electrical ones, 1) J. H. Jeans, Phil. Mag. Series VI Vol. XII p. 57, 1906. ( 586 ) which we are now discussing. It is not even self-evident by any means that a temperature could be defined for every conceivable system. And we even know that two walls of the nature Lorentz has loc. cit. imagined, when their particles had the same kinetic energy, so that they were in equilibrium of temperature as regards their conduction of heat, would transfer heat to each other by radiation. And reversely, they would not be in equilibrium of con- duction of heat when they were in equilibrium for the radiation. As now in the derivation of the law of SrepHan with the aid of the cycle of operations described by Barrorr and BOLTZMANN use is made NÀ dQ of the supposition that 9 = ae this law need not hold for an arbitrarily imagined body, and ean only do so incidentally. In virtue of the above I think that other explanations are called for, which might be able to reconcile theory and observation. And it seems to me that such an explanation might be found in this direction that we assume that the supposition A is not fulfilled, and this we may do without introducing new hypotheses, without coming in conflict with the current theory of electricity. It is true that Jrans and Lorentz have come to the conclusion that supposition A is satisfied, but it seems to me that their considerations only refer to electrons which possess a mechanical mass; they do not seem to apply to electrons without mechanical mass. To show this we shall consider systems agreeing with those examined by Loruntz'). A number of electrons are enclosed in a parallelopiped space with totally reflecting walls. It is true that the objection advanced by Ritz to such walls, is not to be entirely refuted, but without this supposition we can never imagine a system in statistical equilibrium, and when we think the space so large that the radiation reflected by the walls, has long been absorbed before it has reached the central parts, the condition in the central parts will most likely not be influenced by the walls. Besides, the reflected radiation will behave in almost the same way as if it was emitted by a medium of the same nature outside the enclosed space, so that also the parts lying nearer the walls will probably be in the same condition as if the walls did not exist, and the medium extended also outside the walls. 1 shall assume the electrons to be spherical, so that their position is entirely determined by the cartesian coordinates \ YZ of the centre. As further data I shall choose the electrical and the magnetical forces in the different elements of space. In their stead we might 1) H. A. Lorentz, Nuovo Cimento I. c. ( 587 ) also have chosen the coordinates used by Loruntz; this would have been more symmetrical and more elegant. The coordinates chosen by me however, may allow us to determine the quantities, which we wanted to determine, in a somewhat simpler manner. For the rest the result of the investigation is the same in the two cases. It is, however, clear that we may not choose the three components of the electrical force © and the magnetic force 49 quite arbitrarily in every point. For must satisfy the equation Div. = 0, and Div. € is also determined in every point if Y, }, and Z are given for every electron. So if we take &,, E, , and 1D, as independent variables, €z and 9. are determined, with the exception of the constant of integration. This constant, however, is not arbitrary, but determined by the conditions that the normal component of ) and the tangential one of © must be zero at the walls. These conditions yield more equations than the number of constants at our disposal. Hence we have still to diminish the number of independent variables by considering still fewer components as independent variables in the elements adjoining the wall. However, I do not think this will affect our further reasoning. Of course the conditions Div»€—=eo and Div § =O cannot be rigorously satisfied, when we really think € and § constant within the elements. We can then take it e.g. in such a way that we understand the mean values in the elements by the given components of € and §, while inside these elements © and # are linear functions of xz, y and z, and they do not show any discontinuities on the boundaries of the elements. The changes of siate in our systems are now determined by the following equations: "1 di wd c Rot € = — — ce Rot 3) = — + ov dt dt fe (€ Sf [v D1) des mo, fe (€ + [v £])] de = Mw. In these latter two vector formulae, which yield six scalar equations when written down for the different components, r represents the radius vector from the centre of the electron to an arbitrary point, m the mass, JM the moment of inertia, and v the velocity of an arbitrary point, so that » =», —{rw], when rv, denotes the velocity. If this value of v is substituted, six equations are obtained in which the six components of », and w occur linearly. We have here at once to distinguish two cases: 1. m and M are not zero, i.e. we assign a real mass to the ( 588 ) electrons. In this case the equations serve to determine the accelerations, the velocities are to be considered in this case as independent variables. 2. For some electrons m and M are zero. In this case the accele- rations disappear from the equations. Now we can solve the com- ponents of », and w in determinant form, as they form a set of six linear equations. The elements of this determinant are integrals which are to be extended over the electron, and are known if € and ® have been given in every point. So », and w appear not to be independent variables in this case. The number of variables is, therefore, smaller than in the case that the masses are not zero. And this is not strange: in the expression of the kinetic energy the velocities of the electron do not occur when we ascribe the energy to the medium. Only if by the supposition of a quasi-stationary motion a connection is established between the motion of the electron and the forces of the field, the equation of the energy can be reduced to such a form that », and w oceur in it. The values of v, and w thus found must now be substituted in d dE the equation c Rot ) = 5 + ov to find the value of Sa dt z If we now examine in how far supposition A has been fulfilled, it is at once evident that this is certainly the fact in the first case. itn cee | Then — =O even for every variable separately. Op . . ET Över . In the second case, however, it is different, then ox 38 not zero. For the integrations occurring in the elements of the determinant which determines v,, must be extended over another region when the electron is displaced. When differentiating we should bear in mind that ©. is determined from o, whereas g changes in the points of the space if the electron is displaced, so that Eis to be considered as: function. of X,Y oz R dS) cf AS é ‘ ms Ie follows from SS hots that EE is independent of .9,so dt dt d5 dx dt dt that the terms ——— are zero. The same holds for the terms —— 0. dE, for elements which fall outside the electrons; but not for terms inside : dE. dE : them. For there is determined by cat = Rot, ) — EV: and vx at at : dE, being dependent on &,, this is also the case for Sie It is true that t ( 589 ) d&,. dt ; t= hn DE becomes very small for every element, and zero in the limiting A 19 case that we take infinitesimal volume-elements, but the sum of these quantities for all the elements falling within a certain electron does not become zero, but when the volume-elements become infinitely small it verges to a certain limit, which is not to be neglected by : . Oye L the side of DX Here too, we must remember that we must not only differentiate with respect to €, where this quantity occurs explicitly, but also that ©. depends on &,. For by varying €, we vary o, unless we take care to bring about a suitable modification in €. in the surrounding elements. : } 1D r—=n O7 7 : Equation ——- = — = — now indicates that we have a stationary D dt see Op, 7 state if on every path covered by a system the density of phase loa. a —— fl Ed fa? 5 Op is made to satisfy De f , Op form which 2 assumes it seems hardly possible to draw further . P conclusions, unless we succeed by a felicitous reduction’ or by making a better choice of the independent variables from the begin- ning in rendering this sum in a much simpler form. For the present [ must confine myself to pointing vut that it seems that in this direction a solution is to be found of the contradiction which has existed up to now between theory and observation. And if this should prove to be the only way in which this can be done, it is a qualitative proof of the existence of electrons without mecha- On account of the intricate nical mass. We must, however, point out that it can only prove that some electrons do not possess any mechanical mass, not that this would have to be the case for all electrons. For as soon as electrons without mechanical mass occur, the relation >? <0 is not fulfilled. The supposition that some, e. g. the positive Sieciions will possess mecha- nical mass, and others e. g. the negative electrons not, is by no means excluded by these considerations. If we should take Kircnnorr’s law as holding for arbitrarily imaginable walls, the partition of the energy in the normal spectrum would of course never enable us to conclude anything concerning the nature of the bodies by which the radiation is emitted. If, however, ( 590 ) this view is the correct one that the bodies give the same spectrum only because they have certain properties in common, then it must be possible to learn something about these properties from the partition of energy in the spectrum. And thus we may hope that a further development of the application of the statistical method to electrical systems may tend — not to find an accurate formula for the spectrum, which would only be possible if we a priori perfectly knew the nature of the electrons — but rather to test whether a hypothesis concerning the nature of the electrons gives rise to the correct spectral formula, and so whether it is to be accepted or rejected. Botany. — “Some remarks on Sciaphila nana Bl” By Prof. F: A. F. C. Went. While working at the 7’riwridaceae, collected by Mr. G. M. VeRrsTEEG during the expedition to Southern New-Guinea in 1907, I have also examined the plants of the same order, which are found in the Botanical Museum of the University of Utrecht. In so doing I came across alcoholic material of a Sciaphila brought from Buitenzorg by Mr. Purrr and collected at Tjiomas. When an attempt was made to name this plant, it at once became evident, that it was not Sciaphila tenella Br. and it was therefore surmised that the other species described for Java, namely, S- nana Br, had been met with. Now the diagnosis of BruMme is of such a nature, that it is impossible with its aid to recognize the species"); nor are the figures of his plate XLVIII conspicuously clear. I soon found, by comparison with Bxccari’s monograph of Malay Triuridaceae, that the specimen in question evidently agreed completely with his S. corniculata’). I will shortly give detailed proof of this identity, but first remark that Brccarr himself had noticed the agreement between BLuME’s S. nana and his own S. corniculata, for he speaks of S. nana*) as follows: . “Non ho visto questa Specie, ma dalla figura lasciata da BLUME mi sembra poterla includere nel gruppo della S. corniculata e della S. Arfakiana”. In order to obtain greater certainty I have examined the original specimen of Brume’s in ’s Rijks Herbarium at Leiden. Under the name of Sciaphila nana Bu. there are here found, 1) C. L. Buume. Museum Botanicum Lugduno-Batavum I. p. 322. 1849 —1851. 2) Q Beccart. Malesia III. p. 336. Tav. XXXIX. Fig. 5—13, 1886 —1890, 8) O. Beccari. |. c. p. 338. ( 591 ) pasted on one sheet of paper, three plants, numbered 1, 2 and 3. REICHENBACH had written there: Mihi specimina 2—3 sint tenella, specimen 1 nana Br tantum H.G. Re. fil.’ I completely share this opinion; there can be no doubt, that 2 and 3 are specimens of S. tenella Br, while the other specimen is the one, which served for Biumr’s diagnosis, as it is exactly the same as that used for the illustration of the habit. Only in this illustration two flowers are still present, whereas the specimen now possesses but a single one. This renders the Leiden herbarium specimen of little use for deter- mination, as one would at most be justified in sacrificing a part of it, when preparing a monograph of the order, supposing also that one had sufficient reason for assuming, that no new species of Sciaphila will be discovered, a by all means remote contingency. As was mentioned, the specimen of the Utrecht museum is cer- tainly identical with Brccari’s S. corniculata. | will now mention the reasons for this conclusion. Since staminodes are wanting in the female flowers, and the rudiments of pistils in the male flowers (which have three stamens), and since the style is found on the top of the ovary, it is clear that our plant belongs to the subgenus Hyalisma. Here several species are further excluded, because in the centre between the stamens there are no sterile organs, which, according to Brccart, are appendages of the staminal connectives. There then remain S. nana, which for the above-mentioned reasons we will leave out of account for the present, S. Arfakiana, in which the segments of the male perianth terminate in appendages, which are here wanting, while the style in also fixed on the ovary in another way than in the specimen, with which we are here concerned, and S. corniculata. Of the characters, given by Brccari as typical of this latter species, all are found in the specimens from Java. | mention them here in suecession. Small low plants, with somewhat strongly branched shoots and thick fleshy roots. Only the extreme tips of the shoot-branches bear flowers; of these the two or three lowest flowers are female, the upper ones male. The latter are present in larger numbers, but the uppermost generally remain buds. The perianth of the male flowers has six lobes and the latter are provided at their top with a few long fine hairs, resembling cilia; the filaments of the three stamens have more or less grown together. While the male flowers have definite, albeit short peduncles, the female flowers may well be deseribed as sessile in the axil of a bract. Most characteristic are the pistils, which, as Brccart indicated, are sigma-shaped, while the upper part of the ovary and the style are more or less papillar; the description might perhaps still leave some doubt as to the identity, ( 592 ) but the figures of ovary and fruit: 10, 11 and 12 are quite similar to the specimens in the Utrecht museum, as indeed all the other figures. Only the cilia at the top of the perianth leaves are figured somewhat shorter; this is, however, intelligible, as Beccarr had dried plants to work with and I had excellently preserved alcoholic material at my disposal. My conclusion is therefore that the plants found in Tjiomas belong to Sciaphila corniculata Byccart and that the distribution of this species is consequently not limited to New-Guinea, as Brccart had imagined. If we may now assume that the figures of S. nana given by BLUME are not very accurate — an assumption which does not seem to me to be very hazardous —, and if we further eliminate from BLUME’s deseription the unbranched shoot, which was probably due to an acci- dental property of the specimen described, then it seems to me, that we may well assume, that S. nana of Brumr and S. corniculata of Breccarr are names for one and the same species, especially as so far no other species of this genus have become known from Java except the so widely different S. tenella Br. There is however no complete certainty on this point, and as long as this is not the case, it will be best to affix the name of the accurately described Sciaphila corniculata Brccart to the specimen in question, and for the present to regard the name of Sciaphila nana Br. as not sufficiently well characterized. Possibly a future monographer, having many more data at his disposal, will be able to restore this name, but at present it is better to reject il. Utrecht, December 1908. Astronomy. — “Zhe Solar Vortices of Hare”. By Mr. A. Brestsr Jz. Communicated by Prof. W. H. Junius. On the more or less cyclonic configuration of the hydrogen flocculi around the spots on the spectroheliographs of the solar atmosphere and on the shifting and the becoming invisible of one of these flocculi at a short distance from a spot, Hare recently founded the hypothesis that the spots are vortices, which from the solar atmos- phere continually absorb the hydrogen, which there comes back every time as new protuberances or flocculi outside the spots. *) ) Hare: Astroph. Journ. Sept. 1908 — Contrib. from the Mt. Wilson Sol. Obs. No. 26. ( 593 ) At the outset I think I ought to observe that this hypothesis, which is considered by Hate himself, with the laudable caution characteristic of him, as still very uncertain *), 1s in a hardly explicable contradiction with the equality of the angular velocity of the hydrogen- floeculi in every latitude, which Hare has made probable in an earlier investigation *). For if, between these flocculi and. the spots there is the connection that HaLr supposes, we should not expect the same angular velocity, at each latitude, but rather very different angular velocities, which would have to answer to the great aequatorial acceleration of the spots. But the hypothesis that the spots are absorbing vortices, has often been proposed, but has always turned out very improbable. For a vortex leads us to expect first of all that it rotates. But generally nothing is seen of this rotation in the spots. CARRINGTON, SrccH1 and Youne have more than once intentionally set this forth. According to these observers some indication of a cyclonie configuration is shown in only 2 or 3 per cent of the spots, and this configuration is most times such that it would prove a rotation in opposite direction in different parts of a same spot and consequently an impossible rotation for the whole spot.*) Moreover Mircuent *) as well as Hate and Apams,’*) in their investigations of the spot-spec- trum, have found the gaseous substance of the spots generally in almost perfect rest. Besides the spots, as a rule, do not seem to be concave, but convex. °) Although these clear facts, which have been known a long time already, make it very improbable that the spots are to be considered as absorbing vortices, in Hare’s paper on “Solar Vortices” this improbability is demonstrated also in other ways. If there were in reality absorbing vortices above the spots, it would be impossible 1) Hare: Contrib. 26 p. 14. 2) Hae: Astroph. Journ. April 1908. 3) Youre: The Sun 1895 p. 126 — Seccur: Le Soleil I. p. 89. 4) MircHELL: Astroph. Journ. 22 p. 38. 5) HALE und ApAms: Astroph. Journ. 25 p. 87. 6) Already at the first discovery of the spots CHRISTOPHORUS SCHEINER drew the attention to their often occurring convexity and to their origin as through the bursting of bubbles. (Rosa Ursina 1626—1630 p. 461, 493, 513 etc.). See further: How.ett: M. N. Dec. 94 -- Sidgreaves M. N. March. 95 — Wirson: M. N. 55 p. 458 — Frost: Astr. a. Astroph. Il. p. 734 — MAUNDER: Journ. Br. Astr. Ass. 17 p. 128 — Corrie: Astroph. Journ. 7 p. 248 — Moreux: Bull. Soc. Astr. de France Janv. 1907. 40) Proceedings Royal Acad. Amsterdam. Vol. XI, ( 594 ) of course that their absorbing action would only be shown by a single one among the many ‘floceuli which HALE saw floating above the spot studied by him. Yet we should have to believe in that impossibility, for among all the other floceuli above the spot Hare could not observe one, not even among the smallest and nearest to the spot, that showed the slightest advance towards the spot ’). In my opinion it is not at all certain even that this single dark floceulus, which Hate thought sufficient to prove the absorbing action of the Solar Vortices, actually disappeared in a spot. It is also quite possible that this floceulus, amidst other incessantly renewing and shifting floeculi, has been covered up by them in consequence of which it has become invisible. On very close and unprejudiced inspection, to be sure, we see that much of the quasi-absorbed flocculus is left on the clichés obtained after the supposed absorption. But even more clearly than by the hydrogen flocculi the non- existence of material vortices is proved by the imperturbable rest of the calciumfloceuli, which never show the least trace of a cyclonic configuration *), although according to Harre and also as appears from their angular velocity *), which Fox found to be somewhat smaller, they are probably even a little nearer to the spots than the hydrogen floeculi. So it is on account of all these old and new direct evidences that I have come to the conviction that the spots are no material vortices., Neither the spectroheliographs of Hare, nor his discovery that there are lines in the spectrum of the spots, which most probably show the Zeeman effect ‘), „have been able to indirectly weaken my con- victign. So, if, according to me the spots are no material vortices, but when the cyclonic configuration of the hydrogen-flocculi still reminds us in some degree of such vortices (vortices to be sure, according to Hate “so complex’, I should prefer to say “so impossible” that, not unfrequently they show opposed motions in neighbouring places) *) we have now to explain how, also without material vortices, such a quasi-eyelonie configuration can originate. A few years ago already I showed a way to come to that expla- 1) Hate: Contrib. 26 p. 15. 2) Hate: Astroph. Journ. April 1908. — Fox: Astroph. Journ. Sept. 1908. 3) HALE : Contrib. 26 p, 1, 6; Plate XXXVI. 4) Nature, Aug. 20 1908. 5) Hare: Contrib. 26 p. 6 “Although most of the points in a given region appear to move together, there are a sufficient number of apparently oppose motions to weaken seriously the value of the evidence”, ( 595 ) nation, when I pointed out as follows how the Polar Auroras originate on the earth through the spots on the sun. *) If we want to follow this explanation, we must suppose that there are radio- active substances on the sun. This supposition is surely not in the least extravagant, since we know how generally such substances are found on the earth and also take in consideration, how as a rule the same substances which we know on the earth, are also found on the sun. Moreover it is probable, also according to Ruruer- FORD, that on the sun the radioactivity of matter will show itself even more energetically than on the earth, which is so much cooler. *) If now there are radioactive substances on the sun — and also the presence of Helium is in favour of this — such substances will remain hidden under the photospherical shell, owing to their great weight, just as all other elements of great weight. So, under this shell their a, @ and y rays will originate. But for the greater part these rays will be prevented by this comparatively thick shell from escaping from the sun. Only where there are holes in this photospherical shell and so especially where we see spots, this impediment will not be so great. And so, out of each spot just as out of the leaden vessel in the investigations of Mrs. Curm a bundle of more or less parallel 8 and y rays will come forth, vertically going out into the wide world. If now such a bundle, which is often many times thicker than the earth, comes in contact with our atmos- phere, it will bring here about all these electric and luminescence phenomena which have already been considered by BIRKELAND, PaurseN and Arruentus 1 as caused by kathode rays of the sun and 2 as the cause of our Polar Auroras and of our magnetic disturbances. *) If, by means of these supposed strong bundles of rays there is such a simple connection between these earthly phenomena and the cavities of the solar spots, we understand at once: 1. why these earthly phenomena have the same period of 11 years as the spots; 2. why also every year these earthly phenomena show maxima 1) De Nieuwe Courant 19 Febr. 1907 — Bull. Soc. Astr. de France Juin. 1907 p. 283 — Essai d'une Explication du Mécanisme de la Périodicité dans le Soleil et les Etoiles rouges variables. Verh. Kon. Akademie van Wetenschappen te Amsterdam IX. 6 p. 19—21 (1908). 2) RUTHERFORD : Radioactivity 1904 p. 344 — MAUNDER: Knowledge Nov. 1903 p. 255. 8) ARRHENIUS: Lehrb. d. Kosm. Physik. p. 152 — PaursenN: Bull. Soc. Belge d’Astr. Oct. 1906 p. 381. See also my Essai of 1908 p. 20—23 referred to above, 40% ( 596 ) in March and September and minima in June and December. (For the axis of the sun is in such a direction that in March and September those bundles of rays which are surest to reach us, are most numerous because then they are emitted from the parallels of 7*/,° respectively southern and northern latitude, which are compara- tively rich in spots, while such bundles occur most rarely in June and December, because then they must proceed from the equator which has very few spots). 3. why these earthly phenomena also have a period of 27 days © which agree with the synodical rotation of the spots, and 4. why also these earthly phenomena often become more powerful suddenly, when a great spot appears on the sun. My hypothesis that Polar auroras will originate here when bundles of 8 and y rays thrown out by the solar spots reach our atmosphere, is considerably strengthened by the important fact discovered by Sir and Lady Hueerns, that when also here in our laboratories the rays of Radium come in contact with our atmosphere, they cause in if a luminescence, which spectroscopically show the same four nitrogen lines, which have also been found among the most important of the Polar Aurora by PAULSEN ’). Though the Polar Aurora shows many distinct phenomena, which agree very well with my explanation of its origin, it also shows many other phenomena, which, although very mysterious still, are also of the highest importance for the theory of the sun. Such mysterious phenomena are the rapid motions which the light confi- gurations of the Polar Auroras so often show. What it is that in the Polar Auroras causes their bows to wave, their curtains to fly, their brilliant sea of flames to trill, their bundles of rays to flash out suddenly, we do not know. But we do know (and that is the thing really of the greatest importance for the theory of the sun) that all these rapid motions cannot be ascribed to material changes of place. In the time of von Humsonpt, who tells it to us’), the inhabitants of the Shetland Islands may have considered such motions as caused by a “merry dance in Heaven”; the astronomers may still go on taking rapidly appearing rays on the sun for “terrible eruptions’, here on our calm earth such fantastic speculations are 1) Sir W. Huaeerns a. Lady Huveerns.: Astroph Journ. Sept. 1903. On the spectrum of the spontaneous luminous Radiation of Radium at ordinary tempera- tures. — The four nitrogen lines photographed in this investigation and found among the most important of the Polar Aurora by PAULSEN are the lines 3372, 3575, 3918 and 4285. ARRHENIUS: loc. cit. p. 910. 2) vy. Humpotpt: Kosmos 1st vol. 2nd part p. 200. ( 597 ) too naive. For us it is impossible to see material eruptions in the bundles of the auroral rays which often shoot up as quick as light- ning. In all such sudden shiftings of light the molecules of our atmosphere remain comparatively at rest and probably it is only electrons or ions that move. But if our Polar Auroras are such movable electric configurations, which originate when bundles of 8 and y rays, sent out by the solar spots, come in contact with our atmosphere, then it is quite conceivable that analogous movable configurations will originate also in the solar atmosphere itself around the spots, if there these same bundles, just escaped from the spots and consequently much more powerful even than here, pass through the solar atmosphere. Thus the Protuberances of the sun and the rays of its Corona would have the same canse as the Polar Auroras of the earth and the “Solar Vortices” of Hate would be “Solar Aurorae”. In all these phenomena only ions would move, and, as I have already maintained these 20 years *), the matter would remain at rest. For that identical origin of on the one side the Protuberances and the Corona of the sun and on the other side the Polar Auroras on the Earth, which identical origin | have already discussed in my last Essai *), pleads also the remarkable agreement, which Srassano 1) As the fundamental principle of my theory of the sun I have aiways demon- strated the impossibility of the dogma of the solar eruptions. That demonstration will be found and will be seen to become more and more powerful in the following papers: Verklaring van de veranderlijkheid der roode sterren p. 9—11. (Mei 1888) — Essai d'une Théorie du Soleil et des étoiles rouges variables p. 20 (Dec. 1888) — Théorie du Soleil. Verhandelingen Kon. Akad. v. Wetensch. te Amsterdam [. No. 3. p. 1—80 (1892) — Astron. a. Astrophysics Dec. 1903, March !894 p. 218, Dec. 1894 p. 849. — My last Essai of 1908 referred to above p. 1—31. Het 17de Jaar- verslag van het Technologisch Gezelschap te Delft, p. 87—124, Een theorie van de zon. 2) Essai d'une Explication du Mécanisme de la périodicité etc. 1998 p. 20—23, 84, 125. In this my last Essai I have shown on p. 21, that, if the solar spots throw out the bundles of ‘rays which I suppose, it is very clear why the same period of 11 years of the solar spots is also observed in the 3 following lumines- cence-pheromena; 1. in the protuberances and the corona of the sun, 2. in the Polar Auroras on the earth and 3. in the Comets. Thus it appeared in the inves- tigations of Bersericn that during the maximum period of the spots (so, when the Comets have the greatest chance of being brought to greater luminescence by the bundles of rays meant by me) the radiance of the Comet of Encke is greatest and that then also the discovery of very small Comets is most successful. (Astron. Nachr. n°’, 2836 and 2837). The sudden variations of light, which the Comets sometimes show and which now have been seen again so distinctly in the Comet of Morenouse, may also be explained perhaps by their temporary contact with the bundles of rays thrown out by the solar spots. On the same page 21! of my ( 598) discovered in the spectra of these three sources of light. Stassano 9 has found, that while B of the lines in the spectrum of the Polar Aurora must be aseribed to Neon, Argon, Krypton and Xenon, also the light of the Protuberances and especially of the Corona greatly emanates from the same newly discovered elements of the Zero-group. Among the spectral lines, which have been found in the Protuberances by DesraNDrrs and Hare, there are, according to STASSANO, 44 which belong to this Zero-group and nearly all the 339 corona lines, photo- graphed by Humpareys during the eclipse of 1901, are also lines of this group.*) And so it is the same elements which (according to me also for the same reason) cause the same light to shine on the outside of the sun and the earth. If there is therefore great reason to take Hare's “Solar Vortices” for Solar Aurorae, the configurations and the motions of the hydrogen- floeculi in these Vortices do not at all clash with the improbability of the existence of material vortices. For these flocculi then agree with the electric light configurations of our Polar Auroras and like these they will move without any change in the place of the molecular matter. Hatr’s Solar Vortices instead of weakening my idea about the rest of the sun, give on the contrary unexpectedly a splendid support to this idea. For they help to remove the principal objection, which has always wrongly been raised against this idea and has been derived from the shifting of the spectral lines. For if, for the many reasons developed above, we consider these Vortices as Aurorae, they lead us to the conclusion that, although a gas is at rest, yet it will show shifted spectral lines, if only it contains enough ions rapidly moving in the line of sight. The correctness of this conclusion, at which, on other grounds also ScuusteR has lately arrived,*) was a few years ago experimentally proved by Srark, when, in examining the light of hydrogen in the Essai is also illustrated the characteristic change of shape of the corona with the period, and the rays of the corona are not taken for real eruptions (as SCHAEBERLE does), but for luminescences, analogous to the rays of the Polar Aurora. In my Essai (p. 84—88) has also been treated the repartition of the Protuberances, little agreeing with the repartition of visible spots. Openings, too small to be seen as spots throw out nevertheless their bundles of rays which form their Protuberances. If now these openings, as my theory tries. to demonstrate, are smallest at a latitude from 60 to 65°, then with that the constant minimum of the Protuberances at this latitude is explained too. 1) ARRHENIUS: Lehrb. d. Kosm. Physik. p. 911. 3) ScHUSTER: Nature 29 Oct. 1908. ( 599 ) direction of channel rays which he led through it, he photographed at the same time 1. the normal lines of the hydrogen at rest and 2. the strongly shifted lines of the hydrogen-ions in motion. *) And so finally it appears that the relative tranquillity of the sun, never disturbed by terrible eruptions, as has been proved so clearly by numerous important solar phenomena and has been demonstrated especially also in the last year by the rotation-investigations of ADAMs, Hare and Fox, *) is not even in contradiction with a Dorrie shifting of the spectral lines of the Protuberances. Delft, the 18* of January, 1909. Physics. — “On the course of the isobars of binary mixtures.” By Prof. Pu. Konnstamm. Communicated by Prof. J. D. van DER WAALS. 1. In these Proceedings of June 27th 1908 van DeR WAALS ais dp dp showed that only if a’,,< a, a, the curves ae = 0 and ae Q can touch for volumes larger than 30, the critical volume of the mixture taken as homogeneous. On the supposition a,a, —=a’,, the point of contact lies at a value v = 6. Now at higher temperature the well-known diagram of isobars (These Proc. IX p. 630) leads to the intersection d D of the two branches of Ree on the line Dadian = 0, which takes dv dv du place at the minimum critical temperature of the system under dis- ; ee dl ses cussion. Then the line aa = 0 divides into two branches, which we av can now denote as the lefthand branch and the righthand branch. 5 dp The lefthand branch necessarily intersects the line —-—=0 in two At ; 4 dp points, and as it contracts more and more, while the line — = 0 wv moves towards the right with increase of temperature — tbe asymp- p 8 : k da ‚dd tote of this locus being given by 7 MEI — contact must av Ak take place, and that for a volume larger than that for which the line 1) StarRK: Astroph. Journ. Dec. 1906, p. 362. 2) Apams: Astroph. J. November 1907, April 1908. — Hare: ibid. April 1908. — Fox: ibid. Sept. 1908. : ( 600 ) d de “P —0 has its tangent parallel to the v-axis, and which is therefore VU larger than 34. So it would seem to follow from this diagram of isobars, in connection with the just-mentioned theorem of VAN DER Waats that the possibility of a minimum critical temperature is excluded on the supposition a,a,=a’,,. However, already in his Théorie Moléculaire van per Waars derived the condition for the existence of a minimum critical temperature, viz. : nes a a Din NEER Se Oe b, ce b, It is clear that it is easy to satisfy this condition also in the case of a?,, =a,d,, e.g. — if we assume’) 2b,, = (6, + 6,) — by the values bib, anda; = 3a,’ trom-“whieh B, 20,5; dj, = 4, VS; 80 LEEK the two conditions (1) become: 1 ZVL Now it is true that the case will not easily occur that of two substances which have the same critical temperature, the one has molecules three times as large as those of the other, and a physical theory which does not intend to investigate all mathematically possible combinations of a’s and #’s, but only those which really occur, need perhaps hardly consider this point. It would indeed be very desirable for us to bave an insight into the way in which the a’s and 8’s of simple substances are connected, and for mixtures into the way in which a,, is connected with the a’s of the components, so that the theory of the mixtures need only reckon with realisable combinations. Now, however, we do not possess this knowledge, and it seems hardly possible as yet to indicate in what direction such an insight might be gained. Under these circumstances it seems to me most advisable to develop as completely as possible the conclusions which proceed from the different possible suppositions for the dependence of a,, on a, and a,, and to compare these results with the results of observation, in order to try and get an indication in this way of the last-mentioned dependence. No doubt we shall treat a great many suppositions and combinations in this way which will appear to be of no physical signification, but it seems to me that under the given circumstances this difficulty is unavoidable. In this sense the following investigations concerning diagrams of isobars, deviating from those examined up to now and cited above, are to be considered. 1) In fact we must do so, because the theorem of van per Waars mentioned only holds for this supposition. ( 601 ) 2. It appears from the fact mentioned in 1 viz. that the diagram of isobars of fig. 1 loc. cit. in connection with the theorem of VAN DER Waals mentioned excludes the possibility of a minimum critical temperature for the case a,, =4,4,, whereas after all also on this supposition a minimum critical temperature is not impossible, that the diagram of isobars mentioned is- not the only one possible. Now the shape of this diagram is in the first place controlled by d ep the line and the question suggests itself if in general another x shape of this line is also conceivable. In the determination of its course it was derived from the equation: da eter hee SB eet Leb PEN MRT — da that an asymptote must exist for the value of xv determined by: da db Ss ERT de da and that to the right of this point everywhere a positive value of v greater than b is to be found satisfying this equation. In this it has db da been tacitly assumed that for the value of x, for which ae MRT a v & b is still positive; for if 6 were negative at this place, only a high negative value of v could satisfy for the values of 2 somewhat larger : f da db dp than that for which — — MRT —, and hence the course of — =0 dx dx dx would become an altogether different one. So though naturally that value of & for which 6 becomes =O, can never lie within the realisable part of the diagram of isobars, it yet appears that the situation : ; dp. + Se of this point can determine the course of ae and with it of the at isobars in the realisable region. 3. In the complete (extended) diagram of isobars such a point must probably always occur. This is self-evident if we should be justified in considering the dependence of 6 on « as linear, and it is also easy to show it if we assume Lorentz’s well-known formula for b,,. For then: Pe (SV ON 12 2 ( 602 ) and we have to prove that this value is larger than W5,b,. If we now put 6, = 7°b,, the condition which is to be satisfied, is: n'+3n*+3n?+ 1 : 8 or n° + 3n* — Bn? + 3n7 + 1>0 or n= Vi (nt A Bn? Gn? An JD) 0: It is clear that for positive values of m this condition is always fulfilled, so that 6,,2 > 6,6,, and the equation: b, (1 — a)? + 2b,,2(1 — 2) + 5,27 = 0 : has always real roots. / / 4. It has now been assumed in the general diagram of isobars (loe eit.) that these roots always lie on the leftside of that value of da x for which 7 —0. To what change will this diagram be subjected v in the opposite case? We begin with determining the course of dp : ‘ da , EN, ; ea in this case. 5O as 7 is positive, according to our suppo- + DH HH sition, for that value of w for which 6=0O, we can always think the temperature so low that for this value of z, which we shall call 2°: dp Then we get for the course of 7 = 0 in the neighbourhood of «, Ak v—b Ss Senor tl = ny =DE v Now the value of 5 is positive for somewhat higher value of z than z,, whereas ) becomes negative for somewhat smaller value of d x. So we see that two branches of == pass through the point Lv a= 2,, v=. The two branches lie on either side of the line v = b, and have both positive v fore >>, negative v for «1. (See fig. 2). For comparison we reproduce the complete diagram for the ordinary case in fig. 3, which will not require any elucidation. Only the transition temperature d db between fig. 1 and 2, for which URI just at e= #, calls Xv v for discussion. To simplify the calculation we introduce as origin ot the coordinates the point «=—w2,, vO; in its neighbourhood we may put: da db db — = MRT — + MRT — Ce de da da and We Cia where da _ de’ db 1 MRT de and db C,=— daz d. so that the equation for el becomes : wv = 3 1 a ae == Ce v 14Cz ( 606 ) if we neglect the second powers, from which follows for the two roots : Gie OF: Cia Cm i {eo oe eee v 2 7) En 2 db \? : 2MRT| — 2C, dx Co gfe va =e Ir =—b C, da 2 dx’ So we have one finite root and one root equal to zero from which : : 1 follows fig. 4 for the course of = ==); a“ Fig. 4. 5. In the second place we have to examine the course of d, : 5 = in the ease now under consideration. We may write the 2] equation of this curve in the form: MRT v? — 2a(v—b)? = 0. It is very easy to separate the roots of this equation. For, when a is positive, the first member is negative for v — 0, positive for vb, and positive for y=. So there is a root between O and bh, and either two or none for v > 6, as is known, according as the critical temperature is below or above the critical temperature for the mixture under consideration. When 6 becomes equal to 0, both the product of the three roots and the sum of the products taken by twos becomes equal to zero. So there are two roots v =O in this 2a case. And the third root assuming the value UR’ the two branches ( 607 ) passing through the point «= .w2,, v =O, appear to be the liquid branch and the branch v < 6, which has no physical signification. These two branches touch the line vy = 4% in the point mentioned as appears from the fact, that the product of these two roots is in the neighbourhood of this point 4’, and the sum of these roots 20d. Besides we can also prove this directly from the direction of the tangent. For: 9 da dp 2MRT dd da dv dode ais (v—b)? de v3 DP OM RT Ga. 2 dv* (v—b)? oy! If we substitute in this equation the value for (v—d) from the dp equation for aa = (0, we get: ) MRT\?/: eS vile db 2a 2MRT — 2 dv x v? BEAP bs HORTA En ba vile i 2a 2MRT — —— vt When we approach v == 0 the second members disappear in numerator and denominator, so that we keep: dv = db | dx a =“ da: b=—=0 ae did) d, So for 2 little greater than «,, - = 0 will have greater volume av dy than st for the same wv. If, however, there should occur a Vv minimum critical temperature in the system, and we shall see later on that this is very possible, there will be a point of intersection of d d “P —0 and = = 0, which will, of course, constitute a fundamental Vv point for the diagram of isobars. Before proceeding to a discussion of the shape of the isobars them- selves, we shall have to indicate for a complete elucidation of the problem discussed in the beginning, which gave rise to this inves- ( 608 ) d; d, tigation, in what way the lines - = Oand = = 0 get quite detached Hij U in this case. For this purpose we must ascertain what the relative position of these two curves will be at the temperature, at which da db... i . a i ee just for «,, and for which, therefore, fig. 4 holds. & U Now slightly on the right of z,, where 5 has very small values without a approaching to zero, the critical temperature is very higb, d so the two branches of 5 = 0 well certainly still exist on the right wD of z,. But this curve will be closed towards the righthand side, ie. passing from x, to the right we shall first have mixtures which are below their critical temperature at the temperature considered, then mixtures which are already above it, and still further to the right we may sometimes meet with mixtures which are again below their critical temperature, sometimes not. 6. It is very easy to prove this on the supposition b,, = 4 (b, + 6,). In this supposition we can give a very simple construction for the mixture with minimum critical temperature. Let the curve on which A lies (fig. 5) represent the values of a, the right line BD the Fig. 5. values of 6, then: a — bt tg ABC — —tg DBC b in the point A. EN a . . . . a . C . As tg DBC is constant, ty ABC is minimum if ; is minimum ; hence we find the mixture with minimum critical temperature by tracing a tangent to the curve from B. For this point of contact: 27 db ga ln MRT», Tre de JC 8 da ( 609 ) According to a well known property of the parabola the point B da lies halfway between MZ and C, (fig. 6), and a being equal to zero av in 4,” and: increasing ae with a: da (B db ee ee = a MRT —- B de) dx db la So for the asymptote of oe — Oto be found in B, so (=) = MRT — oF, de ) 7 B dx ae Carole : we must raise the temperature to te FT. A fortiori the thesis holds, of course, if, instead of «,,* = a,a,, as was put here, Ay? > a,a,. For instead of the combination of the curve with the right line HBC we get then the combination of the first-mentioned with the right line through B’, and B’ lying to the right of B, the temperature will have to be raised still higher than just now, for ad to, bees MRT in the point A’. f Also in the general case for 5 we can demonstrate the property mentioned, and it will appear afterwards that for these general considera- tions it is desirable not to replace the quadratic form of 5 unnecessarily by the linear one. We treat the case a,,? > a,a, at once, so that a==0 has two real roots. We choose the point 5’ as origin; we db da call the abseissae of the points where re 0, aa ==) anda = Om absolute value resp. 2, #,, #,, then we can write the equations for a and b (see fig. 7): a= a, (z Da ze) an (w, AD és)” a 1 (2° “tr 2a ®, “i ded, te #,”) | b=), (@ 4-4) — b, #,? = 6, a + 26, &, 2 dp The temperature at which the asymptote of a =O reaches the Hi point B’ is determined by: 41 Proceedings Royal Acad. Amsterdam. Vol, XI. da ward ef oe es db bm, dit 4 ho Now we must investigate if there exist mixtures for which this temperature is the critical one on the right of B’. And so: Utr MRT — 8 a oe a, ie ened ned Vi : be, Bib 27 b, (a? + Zea.) 1 So for the determination of « we find the equation: : 8 38 8 ; ode as nate a + 27 an Pe a (27,—a,) — 0. If for the sake of brevity we call the coefficient of 2? A, the roots are: or POs ch nee i c= — 27 A = A DE za + 57 A z‚e, (2%,—2,) If A is positive, the roots are real, as according to the supposition uv, IS De, and the expression under the radical sign being larger 19 : : than — «,4,, We get a positive and a negative root. So this means 7 that one mixture on the right of 5’ has its critical temperature at ‚dp eee the said temperature. Hence the line ign Q has a direction // v-axis Ke) d ; , at this 2, and —0 does not exist any longer on the right of this av mixture. If A is negative, both roots remain real, for then we get under the radical sign: LOPE ae ga GAL 97° A2 1E 27 at bed a (2e, —e,) En 972 oS oak a 27: a As 2, >a, the second term is positive, and the third is smaller than the first. So the expression under the radical sign is positive, 19 but smaller than By 2 The first term of the expression for the roots now being positive, we have now two positive values of wv, 1 e. on the left of B’ we have first a region of mixtures which are below their critical temperature, then a region of mixtures which are already above if, and on this follows again a region of mixtures below their critical temperature. So the line ( 611 ) d : = = 0 has split up into two branches. We need not concern ourselves v ; i } ‚dp about the righthand branch at least now, for the detaching of = == (0) & d Re and aes For the point with minimum volume of this branch — v = . which has the well-known shape — lies on the line = 0, and so avat d di at greater volume than == 0. The line 2 = 0 can, therefore, inter- v LX sect this rightside part only in the branch of the two curves where dv is oe Seas and this intersection does not offer anything noteworthy for v Et 4 dp our present investigation. So we have only to examine how ER & d; intersects the leftside half of al and detaches itself from this av left side half or the only one that is left at this temperature in the case just discussed that A is positive. : d, 7. Now we saw before that in fig. 4 the point where =C U intersects the line «= v,, lies at a value of wv: db\? 2 MRT (a) der dx v, ZE Er a e dx? dp At a temperature somewhat but very little lower, == 0 will av have nearly the same course on the right of «—., as in fig. 4, then, very near the valuexv=w,, v=v, it will abruptly turn upwards and pass through the point = #,,v —0. This follows also from the coefficient of direction, which approaches oo according to formula (3). At somewhat higher temperature the first part will remain almost unchanged, but the curve, having got very near wv =,v=v, will now turn abruptly downward to an asymptote lying somewhat to the right of w,. So the question whether for this temperature a double intersection of a= 6 and aen Q will exist, which will dx dv necessarily lead to a contact afterwards, before the curves get quite ( 612 ) detached, is entirely dependent on the fact whether the point : Ne = 2,,v =, lies outside or inside — = 0 for the temperature con- av sidered, as appears clearly from the figs. 8 and 9. Now as we saw BPN ata before the point where EE O intersects the line «=a, is deter- v mined by : 2a MRT Fig. 8. ( 613 ) Accordingly the question whether for a temperature somewhat ; F é ; da ate higher than that for which for «, aa MRT —, there will be double Av AL intersection and then contact, or no intersection of the two curves, will depend on the fact whether the expression : ; db\? 2MRT ( =) da aa Vz 2a i MRT ; da db or, since we have here — = MR] ae will be smaller or larger than 1. So for the case a,a, =d’, there is no longer any question of intersection above the temperature da dx db dx Tr MRT =S because then v, is twice as large as v,, and a,, must have conside- rably descended below this value, before there can be question of this. Just as VAN DER Waats derived (These Proc. June 1908) we get contact if v, = v,, and so (=) de zl. da a eN b It appears that the value of —, which belongs to the point of (bi contact (loc. cit. fig. 32) becomes equal to zero in this case, not because the denominator becomes infinite, but because the numerator becomes zero, ( 614 ) In a subsequent communication I hope to indicate the course of the isobars in the case given here, and to examine by the aid of a general survey of the possible combinations of a’s and 6’s,, whether besides the diagram of isobars given by VAN DER Waats and the one treated here there are other diagrams of isobars possible for mixtures of normal substances with «’s and 6’s which are quadratic functions of 2. ERRATU M. p. 294 line 20 read 6 ba de, for 6beda. 22 read 8idcha, for 8dabce. ? (February 25, 1909). KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN TE AMSTERDAM. PROCEEDINGS OF THE MEETING of Saturday February 27, 1909. De (Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige Afdeeling van Zaterdag 27 Februari 1909, Dl. XVII). OMR EE B ARES. F. A. H. SCHREINFMAKERS: “The system: Copper sulphate, Copper chloride, Ammonium sulphate, Ammonium chloride and Water at 309”, p. 615. J. G. SrerswigK: “Contributions to the study of Serumanaphylaxis”. Ist Communication. (Communicated by Prof. C. H. H. Spronck), p. 621. . alia ; Lucien Gopeaux: “The types of bilinear »’ -complexes of M9 in Sp”. (Communicated by Prof. P. H. Scmoure), p. 625. Hexrp. DE Vries: “The plane curve of order 4 with 2 or 3 cusps and 0 or 1 nodes as a projection of the twisted curve of order 4 and of the Ist species”, p. 627. Miss WinirreD E. Cowarp: “On Ptilocodium repens, a new gymnoblastic Hydroid epizoic on a Pennatulid”. (Communicated by Prof. Max WeBER), p. 635. (With one plate). G. C. J. Vosmaer: “On the spinispirae of Spirastrella bistellata (O. 8.) Ldfd.”, p. 642. (With one plate). J. W. Morr: “Carbon dioxide transport in leaves”, p. 649. C. Eyxman: “Investigations on the subject of disinfection”, p. 668. Eve. Dusors: “On a long-period variation in the height of the ground-water in the Dunes of Holland”. (Communicated by Dr. J. P. van DER STOK), p. 674. W. pe Sitrer: “On the periodic solutions of a special case of the problem of four bodies”. (Communicated by Dr. E. F. van pe SANDE BAKHUYZEN), p. 682. : J. D. van per Waats: “Contribution to the theory of binary mixtures”. XIII, p. 698. Chemistry. — “The system: Copper sulphate, Copper chloride, Ammonium sulphate, Ammonium chloride and Water at 30°”. By Prof. F. A. H. SCHREINEMAKERS. (Communicated in the Meeting of January 30, 1909). Although the above mentioned five substances take part in the construction: of this system only four need be considered as components, as between the four salts exists the relation ; CuSO, + 2NH,Cl = (NH,),SO, + CuCl,. The system constructed from the four salts only ought, therefore, to be. looked upon as a ternary one; we will now see first of all how we may represent such a system. 42 Proceedings Royal Acad. Amsterdam. Vol. XI. ( 616 ) Let us take the four substances: AB, AC, BD and CD between which may take place the double decomposition : AB+ CD= ACH BD If we represent the substances AB, AC and BD by the angles of a rectangular isoxeles triangle the rectangular side of which has the length 1 [fig. 1] substance CD is represented by the point CD so situated that the four points representing the four substances form the angles of a tetragon. What will now be the composition of a Pig. 1; phase represented by point p? This may be represented in various manners according to the nature of the three substances in which it has to be expressed. 1. We take as components AB, AC and BD. The point p then represents z Mol. BD, y Mol. AC and, therefore, 1—a2—y Mol. AB. 2. We take as components Ab, AC and CD; p then represents: 1 —y Mol. AB, « Mol. CD and, therefore y — 2 Mol. AC. 3. We take as components AC, CD and BD; the point p then lies outside the triangle having these substances as angles; it then represents 1 — y Mol. BD, 1 — x Mol. AC and, therefore, a + y—1 Mol. CD; «+ y—1 is negative in this case. 4. If we take as components AB, BD CD p again lies outside the observed triangle; it then represents y Mol. CD, 1—e& Mol. AB and, therefore, c—y Mol. BD which latter quantity is negative. We may, therefore, assume in four ways that the phase p is constructed of three of the four snbstances; it is obvious that each ( 617 ) of the four assumptions leads to the same composition. This may be explained also in the following manner. If, for instance, we calculate the quantity A of this phase in the four ways described we always get the same result: According to 1 the phase contains y Mol. AC and 1—«—y Mol. AB, therefore 1—zx Mol. A; the same is found according to 2, 3 and 4. We also find in each of the four cases that that phase contains 1—y Mol. B, y Mol. C and « Mol. D. If we draw from p the perpendicular lines z and w these are _ also significant. For we find: 1 1 ay (l—#—y) V2 and w= = (y—a#) V2. For the composition of the phase p we may write according to 1. «2 Mol. BD, y Mol. AC and 1—a«—y Mol. AB. As, however, between the four substances exists the relation AB + CD= AC+ BD we may express the composition also in the four substances, for instance : a—n Mol. BD, y—n Mol. AC, 1—«—y + n Mol. AB and n Mol. CD. From this it follows that 1—«#—y represents the number of Mols. AB minus the number of Mols. CD, while y—«a represents the number of Mols. AC minus the number of Mol. BD, Therefore: 1 1 En (Mol. AB—Mol. CD)V2 , u= 7 (Mol. AC— Mol. BD) 2. 1 The half diagonal of the square is now 5 V2; if, however, the half diagonal is taken as 1 we have: | z= Mol. AB — Mol. CD and u= Mol. AC — Mol. BD. The composition of the phase represented by point p may, therefore, be deduced in two ways. 1. From the situation of p in regard to one of the four triangles whose angles represent the solid substances. The length of the sides of the square is then called 1. We then obtain the composition expressed in those three substances which form the angles of the observed triangle. 2. From the situation of p in regard to the two diagonals of the square. The length of the half diagonals is then taken as 1. If we now add a fourth component, this may be placed on an axis in the point O perpendicularly to the plane of the square; if 42* ( 618 ) on this we take a piece OW =1, point W then represents the fourth component JV. The different phases occurring in the system will then be represented by the points within the prism JV. AB. AC. CD. BD. As in the system now investigated the relation Cu SO, + 2 NH, Cl = (NH,), SO, + Cu Cl, takes place between the four salts we will, in the case of ammo- nium chloride, take the double molecule (NH, CD, and not the single one. If we proceed along the circumference of the square in a constant direction the angles represent successively the components: CuSO,, CuCl,, (NH,Cl), and (NH,),SO, the water will then be indicated by point W in space. We may now project the different points in space on an arbitrary plane; for this we choose a projection on the square and then obtain something like what is represented in Fig. 2. As the drawing would become too large the diagonals have only been drawn as far as necessary; the sides and angles of the square have been omitted. We will now consider first the different ternary systems. ‘1. The system: water, CuSO,, (NH,),SO,. In this system investigated by Miss W. C. pr Baar three solid substances occur at 30° in equilibrium with the liquid, namely, (NH,)80,, Ca'50;.. 5 HO and CuSO, (NH), SO, „GHO: The isotherm theretore consists of three saturation lines, namely : =O cle EBD Ashe i An ( 619 j de the saturation line of the (NH,), SO, CO at 5 Beba oe OR SOL ANE BREE EE OB i; é Paki CU DO EE 0 If we unite the point O with the point representing the double salt CuSO, . (NH,),5O,.6H,O we intersect the saturation line of double salt the latter being, therefore, soluble in water without decomposition. 2. The system: water, CuSO,, Cu Cl,. Only two solid substances namely CuSO,.5H,O and CuCl, .2H,O occur in this system as solid phases by the side of liquid; point % is the solution saturated with both salts, 4z is the saturation line of the CuCl, .2 H,O, ka that of the CuSO, .5 H‚O. 3. The system: water, CuCl,, NH,Cl. The isotherm of 30° has already been determined by Dr. P. MeerBure. As solid substances occur, by the side of liquid, CuCl, . 2 H,O, NH,Cl and CuCl, ,2NH,Cl.2H,O. The saturation line of the CuCl,.2H,O is represented by zh, that of the CuCl, .2NH,Cl.2H,O by hy, and that of the NH,Cl by fg. The line uniting point O with the point representing the double salt intersects the saturation line of the double salt the latter being, therefore, soluble in water without decomposition. 4. The system: water, (NH,),SO,, NH,Cl. In this system only (NH,,5O, and NH,Cl occur as solid substances; the saturation line of the first salt is represented by de, that of the second by ef; e represents the solution s&turated with both salts. After this short review of the four ternary systems we can now discuss the quaternary system. At 30° the following substances can occur as solid phases in coexistence with liquid: CuSO, .5 H,O, CuCl, . 2 H,0, (NH,),SO, NH,Cl, CuSO, . (NH,),SO,.6H,O and CuCl, . 2 NH,CI. 2 H,O. As the solutions, which in a quaternary system are saturated with solid matter are represented by a surface, there must be six saturation surfaces; their projections are indicated in the figure, namely : abmlk is the saturation plane of the CuSO, . 5H,O Dn ve ne ets ONO. AHO GONE 55 ss 9 Bits pels, HCl BEDE na nn 03 » 9» 9» (NH,),SO, hlmog ,, » is Paes, UC ANH Ol) DEED bmope ,, 5; re on wh, CuSO, ONE 0H-O ( 620 ) In order to facilitate the survey, I have indicated on each of the saturation surfaces the solid substances with which the solutions are saturated; for the sake of brevity Cu, stands for CuCl, . 2H,O. The lines in which the surfaces intersect each other two by two represent the solutions saturated with two solid substances. They are the following: ep, the saturation line of (NH,),SO, + NH,Cl Cp: mee ‘3 poy (NEY) SO, 4 CuSO,” NH SO Gag bies En » » CuSO,.5H,O + CuSO,.(NH),S0,.6H,0 PAAL 4 » » ©usO,.5H,O + CuCl, . 2H,0 ia » CuCl, .2H,O + CuCl, .2NH,Cl. 20,0 OG, os és. » » NH,Cl+ CuCl, . 2NH,Cl.2H,O mo, 5; 5 2; CusO,,(NH)S80, 16HO + + CuCl, . 2NH,Cl . 2H,O Dld J >» CusO, .(NH,),SO, .6H,O + NH,CI Mis, * 3° ay, OO, SH, OS Cull. INHSCI ZRD The points in which the saturation lines meet three by three are the saturation points; they represent the solutions saturated with three solid substances. We have; p, saturated with CuSO, . (NH,),SO, . 6H,O + (NH),50, + NH,Cl 0, ys » CusO,.(NH,),SO,.6H,0 + CuCl, .2NH,Cl.2H,0 + B + NH,Cl m, 44 , CuSO,.(NH,),SO,.6H,O + CuCl, .2NH,Cl.2H,0 + + CuSO, .5H,0 l, zh » CuCl, .2NH,Cl.2H,0 + CuSO, .5H,0 + + CuCl, . 2H,0 We can now see plainly from the figure the solid substances by the side of which a defined solid substance can exist in saturated solution. We notice that CuSO,.(NH,),5O,.6H,O can exist in coexistence with (NH, ,SO,, NH,Cl, CuSO, .5H,O and CuCl, . 2NH,CI. 2H,0 but not together with CuCl, .2H,O; CusO, . 5H,O can exist by the side of CuCl, .2H,O, CuCl,.2NH,Cl.2H,0 and CuSO, .(NH,),SO,. 6850 but not together with (NH,),SO, or NH,CI. It further appears that both double salts behave in regard to each other and to water as single substances; at 30° we may have a series of solutions saturated with CuSO, .(NH,),SO,.6H,O, a series saturated with CuCl, .2NH,Cl.2H,O and one solution saturated with both at the same time; the latter is represented by point n. Many other conclusions may be drawn from the figure, but this I must leave to the reader. ( 621 ) Physiology. — “Contributions to the Study of serumanaphylaais” (Ast communication). By J. G. Srueswisk, Foreign Member of the Pasteur Institute at Brussels. (Communicated by Prof. SPRONCK). (Communicated in the meeting of January 30, 1909). Of late the problem of anaphylaxis has attracted the particular attention of more than one investigator of immunity. On the one side the purpose is to answer the purely scientific question, how hypersensibility has to be explained, which in an organism may appear with respect to very different albuminous substances after such a material in some way or other has formerly been assimilated by the organism in question. But on the other hand the practical serumtherapy wishes to be delivered from the difficulties of the serumdisease, and tries to find means of preventing the dangers which, already with the first injection, but still oftener with an injection that is repeated not too soon, threaten the patient. In the meantime the sphere of investigation has been examined in many a direction, the literature is increasing, but theory has still too frequently to complete what is wanting in a useful supply of facts. Therefore an extension of the latter is very desirable, if new points of view offer themselves there. This communication is to furnish a contribution to this. It contains in a few words some results of the first part of an investigation which was made in the Pasteur Institute at Brussels and which had the phenomenon of the serumanaphylaxis for its subject. The literature will only be referred to, as far as this is strictly necessary to elucidate my explanation *). It was THEOBALD SmitH who had observed that guinea-pigs which had served for the titration of diphtheria-serum, and which accordingly had been treated previously with small quantities of diphtheriatoxine and antitoxic horse-serum, after a certain period of incubation had become extremely sensitive to a second injection of horse-serum, that they reacted thereupon as upon the administration of a strong poison and — in proportion to the dose — very often perished. Orto proved that with nothing but horse-serum (without toxine) this hypersensitiveness was also obtained, whilst RoseNav and ANDERSON proved that also with the aid of other sera such an anaphylactic state could be called into life, and that for each serum in a specific 1) For an ampler discussion about the present state of the problem I beg to refer to a critical study from my own hand, which is shortly to appear in the “Zeitschrift fiir lmmunitätsforschung und experimentelle Therapie’. ( 622 ) sense. Since that time the guinea-pig has become the fit test-animal for such investigations. I have provisionally confined myself to the study of horse-serum, also because the knowledge of this in connection with the origin of our therapeutic sera has the most practical importance. While a normal guinea-pig bears an intraperitoneal or sub- cutaneous injection of 5cm’ horse-serum without any perceptible symptom of disease, such an animal (of 250 to 300 grammes) mostly perishes however under typical symptoms of intoxication, when about 12 days before it has been treated with a small dose of the same serum fe.g +7/,,, Cm’). Instead of immunity (prophylaxis), which usually follows on the administration of a larger dose, here a state of hyper- susceptibility or anaphylaxis (Richet) has arisen. The borse-serum completely harmless in itself, plays in this case for the sensitized guinea-pig the part of a heavy poison. The first sensitizing injection must therefore have caused such changes in the organism as to change the second serum-injection into a toxic one. This process of reaction no doubt belongs to the symptoms of immunity, and consequently it ought to be studied with the aid of the methods that the doctrine of immunity has procured. It was therefore a matter of course that the question was asked: Is in the process in question alexine fixed? Orro') answers this question in the negative, in my opinion wrongly. For repeated observation taught me that a sensitized guinea- pig, which reacts upon the second serum-administration with symp- toms of intoxication some time after that injection produces a serum that is exceedingly poor in haemolytic alexine (sensitized red corpuscles serving as test-object). A short time (5—10 min.) after the toxic injection the alexie power of the pig-serum is still the same; after this it decreases gradually and rather rapidly, so that after '/,—1 hour it has become minimum. In this period the animal mostly dies. If it recovers, however, the alexine is also seen to increase again, so that 1’/,—2 hours after the injection it has returned again to the normal level or even higher. This course of things might be graphi- cally represented by means of a curve. In a normal, not anaphylactic guinea-pig the alexine-quantity of the serum remains constant under the same experimental circumstances. Now, if the blood is not examined at the right moment, or not at several moments during the stage of intoxication, the chances are that one is too early (when the alexine has not yet disappeared) or 1) Münch. med. Woch. 1907, no. 34. ( 623 ) also too late (when it has recovered itself again). I presume that Orro has thus been led astray. I have still to add here that, if the second toxic injection is applied not in the abdomen or subcutaneously, but in the circulation — in consequence of which the symptoms of intoxication show themselves very soon and pass very quickly — these symptoms may already be present even before the alexine has disappeared from the serum of the animal. From this may be concluded that the symptoms of poisoning are not the consequence of the loss of alexine, but that these two are processes running parallel, independent of each other, but both having a common cause. And this can be no other but the reciprocal influence of the horse-serum administered (the antigen) and the reaction-products, specific for this, of the sensitized organism, arisen after and in consequence of the first injection of the alien serum. This being settled, we continue asking ourselves: where are these reaction-products to be found — probably a particular kind of anti- bodies? Where do we meet with such materials as show a particular and specific affinity to horse-serum ? In order to answer this question, of course the first thing done was to examine the serum of sensitized guinea-pigs, but without any special result. For in not a single combination such serum gives a precipitate with horse-serum. Another possibility for the disappearance of the alexine from the serum of the intoxicated ani- mals might still be found in the presence in their circulation of antialbuminoid sensibilisators of Guncou. But also these seem to be wanting; I have repeatedly been able to convince myself that ana- phylactic serum, again in not a single combination with horse serum, is able to fix alexine. On the other hand | have been able to prove that the serum of the sensitized pigs reacts antialexically with respect to fresh horse-serum, and especially during the stage in which after the toxic injection the original alexine has disappeared from the circulation. Although I now reserve to myself the duty to revert to the meaning of this fact on a future occasion, yet it seems to me that this formation of antialexines (which we also meet with at the usual serum-immunity) does not bring us much nearer to the explanation of just the anaphylactic complex of symptoms. But if not the fluids, it is perhaps the cells that can bring us a step onward? — | have applied to the erythrocytes of the guinea- pig, and it has appeared to me that washed normal pig-blood, brought in contact with horse-serum, whilst a sufficient quantity of physiological solution of sodiumchloride is present, is able to fix ( G24 Ye from the serum the substance that is toxic for sensitive animals. *) This procedure was already for another reason known in the immunity-literature, because in this way also the alexine from the horse-serum is fixed upon the blood of the guinea-pig. *) Serum treated thus has for anaphylactic animals lost its poisonousness, and this fact seems to me to open a new point of view. For it proves that there exists affinity between the toxic principle of horse- serum and cellular elements already of the normal pig-organism. The supposition does not seem to be too bold that also other elements of tissue or organs of the guinea-pig are subjectable to such a fixation, and that this affinity is still enhanced in the anaphylactized animal. The reaction between the horse-serum and the sensitive elements — especially those of the central nervous system — would then give rise to the action of the anaphylactic shock, whilst by the side of this the secondary fixation of the alexine would be the consequence of this reaction to be observed in the serum. Starting from these facts and considerations I continue my investigation in this direction. In the meantime it is worth while to point out here that already some time ago v. BEHRING drew the attention to the paradoxical fact that a horse containing abundant diphtheria-antitoxines in its blood, can yet react upon a relatively small dose of toxine with symptoms of poisoning and even with death. Therefore v. Benrinc presumed the existence of an Aistogenetic hypersensibility, which hypothesis, in connection with what precedes, grows more probable. To the many attemps made by different investigators with varying results, to deprive horse-serum of its toxicity by the help of physical or chemical means, I have tried to add another, which had a satis- factory result. | have namely submitted to dialysis horse-serum in so-called “Fischblasencondome”. From this it appeared that the arising precipitation, dissolved in a physiological salt-solution, shows no trace of toxicity with respect to sensitized guinea-pigs, whilst the serum floating on the surface and free from salt (before the animal- - experiment reduced to isotonical proportions), gradually loses its poisonousness during the process of dialysis. Now the proof for the non-poisonousness just of the filtrate is not devoid of importance, because former investigations have shown that in dialysing antitoxical horse-serum the diphtheria-antitoxines (which 1) Take for 1 vol. serum: 11/3 vol. blood and 2 vol. salt solution. — On simple dilution with salt-solution in the same proportions, the serum retains its toxicity. 2) See about the meaning of this phenomenon: Ehrlich and Sachs, Berl. Klin. Woch. 1902, no. 21 and Bordet and Gay, Annales Pasteur 1906. ( 625 ) are fixed to the soluble globulines) are to be found back quantitatively in the filtrate. Thus the way has been paved to obtain an antitoxical solution, at the same time free from anaphylactic by-actions, —- which might be of great use to the serum-therapy. Erelong I hope to be able to give further information about this subject. Mathematics. — “The types of bilinear w'-complexes of Ms in Spr.” By Mr. Lucten Gopraux, at Liege. (Communicated by Prof. SCHOUTE). (Communicated in the meeting of January 30, 1909). I have been recently *) investigating which were the essential charac- teristics of the most general type of the bilinear complex of conics in Sp,; it is now my purpose to extend my work to the linear space Sp, with » dimensions. Let there be oo” varieties Ms with r—2 dimensions and of order 7. Any one of these varieties is entirely situated in a linear space Spr 0E the fundamental space Sp,. Let us say that these oo” varieties form a oo’-complex. The characteristics of such a complex are: 1. The number u of the M/-s situated in a general Sp,— of Spr 2. The number v of the M"_» passing through a fixed point and the Sp: of which passes through a Sp,—s containing the chosen fixed point. The aim we have here in view is the determination of the essential properties of the most general o,-complex ZL having the charac- teristics b= be pk Let us notice that all the varieties Al,» of Sp, are the sections } by the Sp, of the varieties V,; with r— 1 dimensions and of J / n+r—l order n of a linear system ( ) = times infinite K. n The M,". of L are evidently situated on the Vs; of an o”-system K' contained in K. 1) Determination des variétés de complexes bilineaires de coniques. Bull. de Acad. Roy. de Belgique 1908. ( 626 ) Turorem I. — The M;-20f L situated in the Sp, passing through a fixed Spy generate a variety yet of r—1 dimensions and of order n + 1. Let d be a linear space Sp,—s, Each Sp,—; passing through d contains a J/;~». Space d belongs to the variety generated by these M;"», for » =1. We deduce from it the above theorem. f n r . Tueorem I]. — A Vo of the system K’ contains generally but n ? r a os oye one My» of L. Let us suppose a V;_, of K’ containing two My» of Z and let us denote by «, 8 the Sp, containing these two Ms. The M5 > of which the Sp,—; pass through the Sp,—2 common to « , yn-+l ° 3 and 2 generate a Vet! on which the points common to a, 8 and to the two Mf-2 are multiple of order two. From this ensues that through a point of the Sp, > common to a, B generally no M'_» of ZL will pass of which the Sp, would pass through this Sp‚—s, which is contrary to the hypothesis y= 1; hence the theorem. ConcLusion : We see that 1. An Sp, contains a single M;» of L, thus to an Sp corresponds a single V;. of A’. 9. A Vi. of K' contains a single M‚ > of L, thus to a Vs of K' corresponds a single Sp. Hence: A oe”-complex of M ;-_ with characteristics u=1, »v—=1 ss the intersection of the elements of two varieties in birational correspond- ence; one of these varieties is composed of the Sp, of the space, the other is a homaloid system of Vi, r=times infinite. Liege, Oct. 1908. ( 627 ) Mathematics. — “The plane curve of order 4 with 2 or 3 cusps and O or 1 nodes as a projection of the twisted curve of order 4 and of the 1% species.” By Prof. H. pe Vrizs. (Communicated in the meeting of January 30 1909.) 1. If two quadratic cones are situated arbitrarily with respect to each other, they intersect each other in a twisted curve 7* of order 4 and of the 1“ species. If we suppose the plane t to be brought through a point O of the nodal curve of the developable belonging to rf in which plane lie the two tangents of r* passing through QO, then this plane must intersect the two cones according to conics k?,, k*,, touching each other in the points of contact O,, O, of the two indicated tangents with rf. We shall now suppose the first cone to be deterinined by the base-curve 4’, and the vertex A, the second by 4°, and the vertex S. The plane r is a double tangential plane of vr‘, so it must be a tangential plane of one of the four quadratic cones passing through rf; i.e. in t, and on the line O,0,, lies the vertex H of a third double projecting cone of7*; and finally the vertex 7’ of the fourth cone must then lie in the common polar plane of H with respect to the cones [R] and [S], and this plane must pass through O, because the double curve of the developable of r* consists of four plane curves of order four situated in the faces of the tetrahedron RS7 H, and O, as a point of this double curve, must thus lie in one of those faces, namely in the polar plane RS 7’ of H, because the points 0, and O,, whose tangents intersect each other in QO, lie on a straight line through H. The cone | 7’] intersects t in a conic A, likewise touching in O, and OQ, the lines OO,, OO,; the cone [H] on the contrary has with t only the line 0,0, counting double in common. 2. If we project 7‘ out of O on an arbitrary plane 2, then the projection is a plane curve £* with two cusps in the points of inter- section of this plane with OO,, OO,; it is convenient to take for this plane of projection the polar plane of O with respect to the cone [H], because then O,, O,, together with two other important points — of which we shall soon hear more — coincide with their projections; the cuspidal tangents are nothing but the traces of the osculating planes of 7* in O,, O, with 2. The plane 2 intersects the cone [H] in two generatrices ; one is O,0,, the other intersects r‘ in two points D,,D, coinciding with their central projections on zr, and in which &‘ touches the ( 628 ) line D,D,, because the plane through this line and ( isa tangential plane of [H]; so D,D, is the double tangent of £*, and H is the point of intersection of this double tangent with the connecting line O,O, of the cusps’). Each generatrix of [H] contains two points of r*, lying harmoni- cally with respect to the point H and the point of intersection with the polar plane RST of H; so if we call / the line of intersection of this plane with a, it ensues immediately that each line of x through H contains four points of k*, lying harmonically in two pairs with respect to H and h; each pair originates from two points on a gene- ratrix of [H]. If we consider 0,, O,, D,, D, as base-points of a pencil of conics, then for each curve of this pencil H is the pole of A; each curve containing the cusps and the points of contact of the double tangent of k*, it cuts this curve in two more points P,, P,, whose connecting line passes through MH. These pairs of points determine on £* a fundamental involution in such a way that on each ray through M lie two pairs, originating from the two pairs of points of 7* on two generatrices of {| situated with O in one plane; the conics of the pencil are thus arranged by the rays out of H in pairs of a quadratic involution, whose double elements correspond to 0,0, and the double tangent d; the former consists of the conic of the pencil touching in V, and O, the cuspidal tangents, a curve which together with A forms the first polar curve of H with respect to /*; the second must break up into the lines 0,0, and d, because this conic must touch the line d in D, and D,. By the pencil (H) and the pencil of conics (O,,0,, D,,D,) paired involutorily conjugated to it £* is generated as the locus of the points of intersection of corresponding elements, where besides £* also the line d appears. *). However, k* can be generated in still another way. Let us imagine through O,O, instead of 2 another plane; this will intersect r* besides in O,,0, in two more points P,, P,, whose connecting line passes through # and is divided harmonically by these three points and the plane RS 7’; so the central projections P’,, P’, are situated likewise on a line through M, and lie harmonically with respect to H and h. Let us now consider the pencil of conics (0,,0,, P, P’,). The different conics of this pencil are likewise involutorily paired by the pencil (H); the branchrays are again 0,0, and d, the double conics conjugated to them are the conic of the pencil touching in 1) See the paper of Prof. Jan pe Vries (Proceedings of Amsterdam of Dec. 1908 p. 499): “On bicuspidal curves of order four”. 2) J. pe Vries, |. c. p. 500. ( 629 ) O,,0, the cuspidal tangents and the one passing through D,, D,. The locus of the points of intersection of corresponding elements consists of k* and the line PP We can finally allow P,, P', to coincide with the cusps, we can thus consider all the conics touching the cuspidal tangents in O,,0, ; these too are paired involutorily by the pencil (/7) whilst the branch- rays are again represented by d and O,0,; to d is again conju- gated the polar conic of H, to 0,0, a conic having in each of the two cusps four coinciding points with &* in common; so it can be nothing but the line O,O, counted double. As the locus of the points of intersection of corresponding elements of both pencils appears this time besides £° still the line O,0,. 3. Among the planes of the pencil (O,0,) are of special impor- tance those containing the cone-vertices R, S, or 7’; the former e.g. is the polar plane of O with respect to the cone [ 2] and contains therefore the two generatrices RO,, RO,. Each of these cuts r‘ in one point more, eg. Rk, and A; the tangential planes along these generatrices to [A] pass through O however; and from this ensues that the central projections of RO, and RO,, cutting each other in the projection R’ of A lying on A, must touch £* at the points R',, R',. We can, however, also bring through O two tangential planes to the cones [S|] and [7’], so: out of each of the two cusps three tangents can be drawn to k*, the traces of the tangential planes through O to the cones [R], |S], [7]; these tangents intersect each other in pairs in three points Rk’, S’, T’ of h*) (this also follows from the harmonic position of the whole figure with respect to Hand h), the projections of the three cone-vertices R, S, T. Remark. If we take for 4%, £,? (see $ 1) two concentric circles, and if we then project the figure in space on a plane zr (§ 2) which is parallel to t, the oval of Descartes is generated and R’, S’, 7” pass into the foci. The twisted curve r* can be generated in six different ways as intersection of two of the four cones; let us take in particular [R and | /7]. If we make to pass through the line RH two planes u, u, harmonically separated by the planes RHS and RHT, the points of the two quadruples of points lying in these planes are situated two by two on four straight lines through each of the four cone-vertices ; we now take for u, the plane passing through the points O,, O,, R,, R,, and we shall call the four points in the other plane S,, S,, 7,, 7,. The 1) J. pe Vries, l.c. p. 501. ( 630 ) latter four points lie thus with the former on four straight lines through S and on four others through 7’; let us now suppose e.g. that O,S, passes through S. The tangential plane along the line SS,O, to the cone [S| has then as trace with r the tangent in O, to &,’, i.e. the line OO,, and from this ensues that the four points S,, S,, 7,, 7, are nothing but the points of contact of the tangential planes through O of the cones [S] and [7 with r*; however, we know now that these four points lie on two lines through B, so in projection the points of contact S’,,S’,,7’,, 7’, of the tangents out of the cusps not passing through R’ lie on two straight lines through R’, namely S’, with T’,, and S’, with T’,.') That they lie also on two lines through H follows moreover again from the harmonic position with respect to H and A. The polar plane of R with respect to the remaining three cones is simply the plane STH; it intersects the plane AST in the line ST (whose central projection falls on A), and the two lines of intersection with u, and u, lying in this plane in two points R*, R**, lying on ST, and situated harmonically with respect to S and T: R* is then the point of intersection of O,R,, O,R,, R** that of S,T, and S,7,. If we transfer these results to the projection, we find: of the complete quadrangle O,O, R', R', two of the diagonal points are R'; R*' (the third is H), and of the quadrangle S',T',T'S, likewise R', R**'; the points R* and R** lie harmonically with respect to S' and T'. A similar property of the same points holds if we combine S’,,.S', with the cusps and then regard the other four points; or finally if we couple 7’, 7’, to O,, O, and join the other four to a complete quadrangle. And finally all six points #’,... 7", lie on a conic in consequence of the harmonie position with respect to h and H, whilst the points of intersection not lying on / of the six tangents out of the cusps lie likewise on a conic and at the same time in pairs on three straight lines through /. 4. Besides the group of 8 points just considered consisting of the two cusps and the points of contact of the six tangents passing through these points, there lie on 4* still an infinite number of other such- like groups which have with respect to &', S', 7',H the same properties. Let us for instance suppose that the plane u, ($ 3). is still passing through RA but for the rest arbitrary, and let us suppose uw, again as harmonically conjugate to u, with respect to HRS, HRT, then on 7* a new group of 8 points is generated 1) J. pe Vries, |. c. p. 498. dE ( 631 ) Jying in pairs on four lines through each of the 4 vertices, and whose central projections thus possess the same property with respect to the points R’,.S’, 7’, H. If we divide the 8 projections into two quadruples, in such a way that one belongs to the four points of u,, the other to those of wu,, then the two quadruples form two complete quadrangles with the common diagonal points R’ and M, whilst the others, R*’ and R**’, lie harmonically with respect to S’ and 7”; the pairs of points R*’, R**’ on h form therefore a quadratic involution with the double points S’, T'. Similair properties hold for the two other possible divisions of the group of 8 points into two quadruples, namely with respect to the points S and 7’. A special group of 8 points is found by choosing for the two planes w the tangential planes through the line RH to the cone [H], for these are likewise harmonically separated by HRS, HRT, but they furnish instead of 8 points 4 pairs of coinciding points of r*, namely the points of contact of the 4 tangents out of A to 7“. These points of contact lie in the polar plane o($3) of A and on two generatrices of the cone [$S], and likewise on two of the cone [7']; the tangents themselves pass in projection into the four tangents of k* through R’ not passing through the cusps, so: the pomts of contact of the four tangents of k* through R’ not passing through the cusps are the vertices of a complete quadrangle whose diagonal- points are the points S’ 7’, H; the corresponding points A’, R**’ are the points of intersection of the two sides of that quadrangle passing through H with h. Another special group of 8 points is generated if we choose for the planes u the tangential planes through RH to the cone [ZR]; we then find the four points of 7* in the plane RST, and therefore in projection the points of intersection of r* with A, whose tangents indeed pass through H, in consequence of the harmonic position of k* with respect to h and H. 5. A group of 8 points of &* must be determined by one of these points; for the connecting line of this point with O intersects 7* in one point, which determines with the line AA the plane u,; and by u, at the same time u, is determined. Planimetrically we can deduce out of one point of a group the other ones with the aid of the following property. The cone [RR] intersects the plane ge = STH in a conic 7’, and we find that 7* lies harmonically with respect to this and the point &, in that sense that the two points of r‘ on a generatrix of |R] are always harmonically separated by R and the point of intersection with 7’; in particular 7’ contains 43 Proceedings Royal Acad. Amsterdam. Vol. XI. ( 632 ) the four points of intersection of 7* with 9, whose tangents pass through FR, and also the points harmonically conjugate to A with respect to the pairs 0,,R, and Q,, R, lying in the polar plane O,0,R of O with respect to the cone | F]. By passing to the projection we find out of this the following property of k*: through the points of contact of the four tangents out of R' not passing through the cusps, a conic r* can be brought touching the two tangents which do pass through the cusps in the points harmonically conjugate to R' with respect to the cusps and the corresponding points of contact; k* now hes harmonically with respect to R' and r® in such a sense that the four points of intersection of k* with an arbitrary ray through R' arrange themselves into two pairs, each lying harmonically with R' and one of the points of intersection of that ray with r*. Of course also the points S’ and 7” possess such a conic, but the point H likewise, namely the line 4 counted double; for, the polar plane of H with respect to r* is the plane RST, and the section with the cone [MZ] lying in this plane is projected out of O into the line A to be counted double. With the aid of the conic 7” it is easy to deduce out of one point of a group the seven other ones. If point 1 is taken arbitrarily, we find four others by the harmonic position of 4* with respect to the points &',S', 7’, H and the corresponding conics; on the line R1 e.g. are lying besides 1 still 3 points of £*, but among these is only one forming with 1, R', and one of the points of intersection of R'1 with 7? a harmonic group. The three then still missing points we find by simply combining in proper fashion the already found one Withee. is oor i, Just as in §2 by rays out of H now, too, a fundamental involution is generated on &* by those out of R' (or S, or 7") in such a manner that on each ray lie two pairs, originating from the two pairs of points of 7* on two generatrices of the cone [Zi] lying with O in one plane; each pair is harmonically separated by A’ and one of the two points of intersection of the indicated ray with #°. If now in two points of 7* lying on a straight line through A we draw tangents, then these intersect each other in a point of the plane STH, and the locus of this point of intersection is a plane curve of order four, double curve of the developable of r*, with double points in S, 7, H and having with r* the four points of intersection of r* with the plane S7'H in common. 1). J. pe Vries, |. c; p. 498. END ( 633 ) Let us consider in particular the points O,, R, (§ 3), lying on a straight line through A, and let us remember that the tangent in O, to r* passes through OQ; we shall then find on OO, a point of the double curve of order four, whilst the tangent in that point is the line of intersection of the osculating planes of 7* in O, and R,; that one of O, passes through QO and furnishes in projection the cus- pidal tangent in O, to &*; so the central projection of the double curve will contain the cusps of 4*, and will touch the cuspidal tangents here. So, summing up we find: the point of intersection of the tangents to k* in two conjugate points of the involution generated by the pencil (B) generates a curve of order four with double points in S', T', H, situated harmonically with respect to h and H and having with kt m common the tangential points of the four tangents out of R' not passing through the cusps and likewise the cusps and the tangents in these. The new curve of order four cuts £* besides in the cusps (count- ing together for six points of intersection, and the tangential points of the four tangents out of 2’ not passing through the cusps, in six points more, of course again situated two by two harmonically with respect to h and H; now these six points lie twice with two of the four above mentioned points of contact (those namely whose connecting line passes through //) on a conic. For, the conic through two of those points of contact touching in QO, and QO, the euspidal tangents of £*, contains 2 XX 3 + 2 = 8 points of intersection of the two curves of order four; so the other eight must also lie on a conic. Also to the points S’ and 7” belongs such a locus of order four ; as to the point H we can observe that in space the locus belong- ing to H lies in the plane ARS 7’ and passes through O, so it passes by central projection into the line A counted three times, because each line passing through O and lying in the plane RST intersects the curve besides in O in three more points ; indeed, through each point of / pass three pairs of tangents io /*, whose chords of contact pass through #7. The central projection of O itself however is undetermined, and so / is covered with points for the fourth time; the locus belonging to AH consists thus of the line 4 counted four times. Remark. The properties found in this § hold in a some- what more general form also for £* with two nodes, because they have been simply arrived at by centrally projecting the complete figure of the tetrahedron of Poncerer; the points #’, S', 7", H have then however not such a simple position as for the bicuspidal 4. 43* a ( 634 ) 6. If we bring the vertex S of the cone [S| on the surface of [FR], then 7 coincides with S, whilst the point S= 7’ becomes a node of r*; so 4* possesses besides the two cusps a node and in this point S' and 7” lie united. Out of each cusp only one tangent more can be drawn to 4*, and these two tangents intersect each other in the point R’ lying on h. Through this point pass two more tangents to 4*, the projections of the two generatrices of the cone (R] touching r*. The points of contact of these two tangents with 7 lie in the plane e (§ 4), the common polar plane of A with respect to the cones {S| and [MZ], and so on the conie r? which has this plane in common with the cone {[#]. This conic contains the vertex S; so now k* is harmonically situated with respect to the point R' and the conic r* passing through the double point and the points of contact of the two tangents out of R' not passing through the cusps, and touching the tangents out of R' which do pass through the cusps in the harmonically conjugated points of R' with respect to the cusps and the points of contact. If we bring through the line HA an arbitrary plane u, (see $ 4), then the harmonically conjugate u, always coincides with the tangential plane HRS to [H|, which plane contains no other point of r* than the node; of each group considered in $4 of 8 points there are four coinciding in the node, whilst the four remaining ones form a complete quadrangle with the diagonal points #, H, R*. The tangents to r* in two points on a straight line through /# intersect each other in 9g and the locus of this point of intersection is a plane &* containing the node of r*, having in this point a cusp (with cuspidal tangent in the plane RST), passing through H and having with #* two points in common, whose tangents pass through &. So if we pass to the central projection of r*, we find that by the rays of the pencil (R') on kt again a quadratic involution is generated in such a way that the two points of each pair form with R’ and one of the two points of inter- section of the ray under discussion with r’? a harmonic group; the point of intersection of the tangents in the points of a pair moves along a cubic curve lying harmonically with respect to h and H, containing the cusps of k* and touching here the cuspidal tangents, passing through the points of contact of the two tangents out of R’ not passing through the cusps, passing through H and having in the node of k* a cusp with cuspidal tangent h. The cusps, the point of contact of the two tangents out of A’ and the node represent all the twelve points of intersection of the cubic curve with 4f. ( 635 ) 7. If, still imagining that the cone-vertices S and 7’ are united in a point of the surface of {H], we suppose the cone [S| to be such that it touches the tangential plane in S to [|M], then 7* gets a cusp in S and R coincides with S and 7’, so that through #* pass but two quadratic cones; the plane RS 7’ remains determined, as the polar plane of H with respect to the cone [Rk] = [S]= [7], and in this plane lies the tangent in the cusp 2. The central projection now becomes a £* with three cusps, and the cuspidal tangent in R' is the line A. The double tangent through H remains determined as a trace of the plane of projection with the second tangential plane through O to [MH]; however, as is easy to prove from the stereometric diagram, the points of contact must necessarily be imaginary. As k* is situated harmonically with respect to h and H, the tangents in the cusps O,, O, must intersect each other on 4; however, / is the tangent in the cusp &': so the three cuspidal tangents pass through one point. The point H forms with the cusps O,,9,, and the point of intersection of the line O,O, with 4 a harmonic group; but now the same must hold for the two other sides of the AO,O,R’ of the cusps; the double tangent d is therefore the so-called harmonical line of the points uf intersection of the three cuspidal tangents with respect to the triangle of the three cusps; and the points of contact of the double tangent are the nodes of the elliptic involution on d, of which the points of intersection of d with the sides of that triangle and with the cuspidal tangents in the opposite vertices are three pairs. With respect to each cuspidal tangent and the point of intersection of the opposite side of the triangle with it the curve lies harmonically with itself. Zoology. — “On Ptilocodium repens a new Gymnoblastic Hydroid epizoic on a Pennatulid.” By Miss Winierep E. Cowarp B.Sc. Victoria University of Manchester. (Communicated by Prof, Max WEBER). (Communicated in the meeting of January 30, 1909.) Order. Gymnoblastea — Anthomedusae. Family. Ptilocodiidae. fam. nov. Ptilocodium repens: gen. nov., sp. nov. “Siboga” Expedition Stat. 289. 9° 0,3 5. 126° 24,5 E. 112 metres. Among the Pennatulids sent to Professor Hickson from the Siboga ( 636 ) Expedition were two specimens of Ptilosareus sinuosus (Gray)., and growing over the tips of the leaves of them a small epizoic hydroid was discovered. To a description of this new hydroid, the present paper is devoted. | The occurrence of an epizoite on a Pennatulid is in itself an interesting fact as the Pennatulids have usually been regarded as being peculiarly free from any such growths. Only two specimens of Ptilosarcus were received from the Expedition and the hydroid occurs on both of them. The other Pennatulids of the collection have heen carefully iooked over, but on none of them has an extra- neous growth of any kind been found. Ptilosarcus belongs to the Pennatuleae, the section of Pennatulids which are distinctly bilaterally symmetrical and have the autozooids in rows, with their body walls fused to form leaves. Along the free edges of the leaves of the given specimens of Ptilosarcus, the hydroid Ptilocodium grows (fig. 1). It is quite visible to the naked eye, though in a cursory glance over the leaves of the Pennatulid, it probably would not be noticed. The hydroid affects the free edges of all the leaves of the Pennatulid, even those at the free extremity. It is suggestive that it does not spread over the rachis of the Pennatulid nor even over the main surfaces of the leaves, but grows only over the oral ends of the autozooids. The hydroid is devoid of any kind of skeleton and spreads over the distal parts of the antozooids composing the leaves of the Ptilo- sarcus (figs. 2, 3). The colony grows by means of spreading stolons. These stolons run singly over the spicular projections of the autozooids and along their tentacles ; (figs. 2, 3, 4) but over the part immedia- tely below this, they branch and closely anastomose forming a more or less continuous sheet of basal coenosare. The hydroid exhibits the phenomenon of dimorphism, the gastero- zooids and dactylozooids being quite distinct. The zooids are sessile, arising directly from the stolon or basal coenosare as the case may be. The dactylozooids arise at very short intervals along the stolon and are far more numerous than the gasterozooids. Gonozooids occur at frequent intervals. They are much fewer. in number than the Note by Professor Hickson. The hydroid described in this paper was found on the only two specimens of Ptilosarcus in the Siboga collection. As they appeared to be of very great interest, and I could not part with the Pennatulids which are themselves under investigation, I considered it to be advisable, in the interests of science, that a description of them should be prepared in my laboratory without undue delay. [| wish to express my hearty thanks to M. Bittarp, to whom the description of the Hydroidea of the Expedition has been entrusted for kindly giving his sanction to the publication of this paper independently of his memoir. —_— ( 637 ) gasterozooids but in every case the gonozooid arises, not from the stolon but from the base of a gasterozooid (fig. 7) in close proximity to the stolon or basal coenosare. Stolon. The stolons are tubular in structure (fig. 3). Their walls consist of superficial ectoderm and a lining of endoderm separated by a structureless lamella, the mesogloea. The ectoderm exhibits no traces of a perisare. It possesses a few scattered nematocysts of the smaller kind .008 mm. > .005 mm. (vide infra). Basal Coenosarc. The basal coenosare is formed by the anastomosing of stolons running over the parts of the autozooids immediately behind the tentacles. When two stolons run together or cross, the ectoderm of the dividing walls disappears so that the upper ectoderm of one stolon becomes continuous with the upper ectoderm of the other and the lower ectoderm of the one becomes continuous with the lower ecto- derm of the other. Thus the basal coenosare of the hydroid consists of superficial ectoderm and lower ectoderm separated by endodermal tubes (fig. 5). The structure arrived at is thus the same as in the coenosare of Hydractinia except that Ptilocodium has no chitinous skeleton. Nematocysts. The hydroid possesses two sets of nematocysts. The larger kind is found in the dactylozooids. Here the nematoeysts are oval in shape and measure .017 mm. & .008 mm. The smaller kind occurs in the ectoderm of the basal coenosare and of the gonozooid, the size of these nematocysts being .008 mm. > .005 mm. Gasterozoovds. Gasterozooids occur at frequent intervals on the basal coenosare and are sessile (fig. 7). They vary in size from .213 mm. high and 106 mm. broad, to .373 mm. high and .026 mm. broad. They are much reduced in structure. There are no traces of tentacles. The zooid is simple and sac-like; the mouth is a simple pore leading from the exterior into the cavity of the zooid. The gasterozooids show no nematocysts. The endoderm cells near the mouth of the gasterozooids are comparatively short whilst those lining the remainder of the gastral cavity are long and narrow. The material is not sufli- ciently well preserved to make out clearly the histological structure of the cells but it seems probable that the digestive functions are preformed by the long, narrow cells of the basal half of the gaste- rozooids. ( 638 ) Dactylozooids. The dactylozooids are very numerous compared with the gastero- zooids. They occur at irregular intervals; there seems to be no defi- nite relation, as regards arrangement on the basal coenosarc, between the dactylozooids and gasterozooids such as we find in Millepora or Stylaster. The dactylozooids are short and broad and each bears four capi- tate tentacles crowded with large nematocysts (fig. 6). The zooids do not vary much in size, the average size being .186 mm. >< .106 mm. The smallest zooids measure .106 mm. X .053 mm. The capitate tentacles are .038 mm. in length and .033 mm. broad. The nema- tucysts of the tentacles are of the larger of the two kinds possessed by the hydroid and measure .017 mm. X .008 mm. The ectoderm of the remainder of the zooids shews no nematocysts. The endoderm of the dactylozooids and tentacles is solid and scalariform, there being no trace of a cavity or oral opening. (fig. 6). Judging from the preserved specimens the dactylozooids seem little, if at all, contractile. Gonozooids. The gonophores which are adelocodonic, arise in each case, as before described, from the base of a gasterozooid. Thus the base of a gasterozooid functions as a blastostyle (fig. 7). The gonozooids vary little in size, the average being .3873 mm. X 186 mm. They are considerably reduced in structure, having the form of closed sporosacs. All the gonophores on the two specimens received are female. The ova are borne between the ectoderm and endoderm of the manubrium and have a diameter of .017 mm. They are practically all of the same size but it cannot be said whether or not they are ripe. The superficial ectoderm of the gonozooid and the ectoderm lining the cavity corresponding to the sub-umbrella cavity of a medusa, are separated by an endoderm lamella, which shows traces of radial canals (figs. 7 and 8). There is no velum and there are no sense organs and only traces of four rudimentary tentacles. Nematocysts of the smaller kind occur in the superficial ectoderm of the gonozooids. The Relation between the Hydroid and Ptilosarcus. The specimens of Ptilosareus sinuosus on which the hydroid is growing seem practically unaffected by it. The autozooids are well developed, showing no signs of degeneration. | The hydroid spreads only over the oral ends of the autozooids, that is, it keeps near the tentacles of the same, and does not run far over the leaves of the Pennatulid. Correlated with this is the er ( 639 ) fact that the gasterozooids of the hydroid are devoid of organs for catching food. These facts at once suggest that the Ptilosarcus bene- fits the hydroid by helping it to secure food. On the other hand, on looking at a preparation of the hydroid, one is struck by the great number of dactylozooids which are so well provided with large nematocysts. Such a protection as these are capable of affording is probably more than is required by the small hydroid. The Jarge projecting spicules of the autozooids of the Penna- tulid also, would protect the hydroid. Therefore I would suggest that the Ptilocodium is of use to the Pennatulid in the warding off of enemies and in stinging prey by means of its batteries of large nematocysts and that the Ptilosarens by means of its projecting spicules, protects the Hydroid. Thus the Ptilosareus and Ptilocodium are mutually benefited. : Systematic position. Ptilocodium seems to have some affinities with Hydractinia, Podo- coryne and Millepora, as shewn by the sheet-like, encrusting basal coenosare from which the zooids arise independently ; but even in this character Ptiloeodium stands alone in having no chitinous or caleareous skeleton to protect it. In other respects Ptiloeodium is unique. The dimorphism of Ptilo- codium is quite distinct from that of Hydractinia and Millepora, and probably originated independently. In the first place, the gasterozooids of Ptilocodium are extremely unlike those of Hydractinia and Mille- pora. They are short, sessile, sac-like structures, without tentacles. Those of Hydractinia are long, filiform structures, provided with a crown of tentacles. The gasterozooids of Millepora are also much longer than broad, not at all sac-like in form, and are provided with knob-like tentacles. The dactylozooids, also, are very different in structure, in the three genera. Those of Ptilocodium are short and broad, and are furnished with four characteristic capitate tentacles. The endoderm of the body of the zooid and of the tentacles is solid and scalariform, giving the dactylo- zooids a marked appearance. The dactylozooids of Hydractinia are long and slender and very muscular so that they are capable of coiling and uncoiling themselves. They have also a central cavity and according to Miss Corcurr *) are provided with a terminal mouth. The dactylo- zooids of Millepora are very similar to those of Hydractinia. In possessing a solid scalariform endoderm, Ptilocodium resembles the Stylasterina. All the genera,except one, of this group, have the endoderm of the dactylozooids solid. 1) Quart. Journ. Micro. Sci. No. 157, 1897. ( 640 ) In all other points except the dimorphism, however, Ptilocodium differs from the Stylasterina, and on the importance of dimorphism as indicating close relationship too much stress should not be laid. The old group Hydrocorallinae affords an illustration of the result of laying too much stress on this factor. In this group, the Mille- porina and the Stylasterina were formerly united, but it has since been pointed out that differences of more importance necessitate their being classed as separate orders. The next point to consider in discussing tbe relationship between Ptilocodium, Hydractinia and Millepora, is that of the gonophores. Ptilocodium resembles Hydractinia in having adelocodonic gonophores but differs from it in having the gonophores arising from the base of the ordinary gasterozooid and not from a specialised individual or blastostyle. In Millepora there is no blastostyle and the medusae arise independently of the gasterozooids, from the surface of the colony. In Perigonimus, which, in the important respect of the structure of the basal coenosare does not closely resemble Ptilocodium or Hy- dractinia, the gonophores arise from the hydrocaulus bearing the gasterozooid or from the hydrorhiza. This case may therefore be taken as an indication that too much stress must not be placed on the position of origin of the gonophore in the colony. Ptilocodium stands quite apart from other genera of Hydroids with epizoic habits. The epizoic Hydroid Stylactis minoi described by Arcock *) was found on the fish Minous inermis, but differs considerably from Ptilocodium. It has bydranths crowned with numerous tentacles rising from the hydrorhiza. It is not provided with dactylozooids of any kind and its gonophores are in the form of sporosacs arising from specialised individuals which bear tentacles. . Ptiloeodium also differs considerably from Hydrichthys mirus, a Hydroid described by Fewkes?) as epizoic on the fish Seriola zonata. This hydroid exhibits structures of two kinds arising from the basal plate. In the first place there are long, filiform hydranths, which are looked upon as degenerate gasterozooids and secondly clusters of botryoidal gonosomes. Dactylozooids do not occur. Ptilocodium has obviously no affinities with the hydroid Nudiclava described by Lroyp®), nor with Moerisia lyonsi, a Hydromedusan from Lake Qurun, described by C. L BOULENGER *). 1) Arcock A. Ann. & Mag. Nat. Hist. 1892 vol. X p. 207. 2) Fewkes. Bull. Mus. Comp. Zoöl. vol. XIII p. 224. 3) Lroyp R. E. Records of Indian Museum. Vol. I. Part IV. 4) Boutencer C. L. Quart. Journ. Micros. Sci. Vol. lij Jan. 1908. Fig. 5. er. © WINIFRED E. COWARD. ‘On Ptilocodium repens, a new gymnoblastic Hydroid on a Pennatulid.” W. FE. Coward del, Proceedings Royal Acad. Amsterdam, Vol. XI, Fig. 3. OU =p mes, ect, End can %420 A liz nem ect, Tad can. upper: ect, tower ect 7 end can Supect . sub ect ff 210 —.-” * 4 ( 641 ) In consequence of these considerations the only course to adopt is to regard Ptilocodium as the representative of a new family. The above piece of work was undertaken at the suggestion of Professor Hickson, to whom I wish to express my sincere thanks for his most generous advice and assistance. mA) EXPLANATION OF PLATE. aut — autozooid of Ptilosarcus. bas coen. — basal coenosare of hydroid. dact. — dactylozooid of hydroid. ect. — ectoderm. ect. man — ectoderm of manubrium. end. — endoderm. end. can. — endoderm canal. end. lam. — endoderm lamella. end. man — endoderm of manubrium. gast. — gasterozooid of hydroid. gon. — gonozooid of hydroid. mes. — mesentery of autozooid of Ptilosarcus. nem. — nematocyst. ov. — ovum of Ptilosarcus. rad. can. — radial canal. spic. — spicule. stol. — growing tip of stolon. sub. ect. — ectoderm lining the cavity corresponding to the sub- umbrella cavity of a medusa. sup. ect. — superficial ectoderm of gonophore. fent. — tentacle. t. p. — tentacles of Pennatulid. External view of parts of two successive leaves of the Pennatulid, shewing the extent of the surface of the leaves, affected by the hydroid. Drawing of a microscope preparation of part of the free edge of a leaf of Ptilosarcus, shewing the hydroid running out over the spicular projections of the autozooids X 20. Drawing of a preparation similar to >< 68. Longitudinal section of a Pennatulid leaf shewing the hydroid growing over the free edges of the leaf. Vertical section of basal coenosarc of the hydroid shewing superficial and lower ectoderms, and endoderm canals. Optical section of dactylozooid of hydroid shewing solid endoderm, and the characteristic four tentacles. Longitudinal section of gonozooid arising from base of gasterozooid, Transverse section of gonozooid. ( 642 ) Zoologie. — “On the spinispirae of Spirastrella bistellata (O. 8S.) Ldfd.” By Dr. G. C. J. Vosmarr, Professor at the University of Leiden. (Communicated in the meeting of January 30, 1909). Some years ago (1902) I drew attention to the fact that there is confusion with regard to the terminology of certain sponge-spicules, and tried to clear this up. I arrived then at the conclusion that the spicules, which are generally called “spirasters”, far from being a sort of ‘asters’, i.e. polyaxon spicula, ought really. to be considered as monaxons, the axis of which is a helix screw. I proposed for that kind of monaxons the term spirazon (Le. p. 105 and 112). In order to avoid further confusion I called the spined forms: spinispirae. I had some doubts about the supposition of some authors, that transitions between true asters and spinispirae really existed, because I never found them and failed to find any proof in literature (le. p. 105). On the one hand authors make a certain distinction between true asters (euasters) and “spirasters”, but on the other hand consider both forms as belonging to the same group. Thus Torserr (1900 p. 21) distinguishes the genera Hymedesmia and Spirastrella on account of the fact, that the microscleres of the former genus are “euasters’, of the latter “spirasters”. It is generally — accepted that the microscleres of Hymedesmia stellata are euasters ; but with regard to H. bistellata there is diversity of opinion and confusion. I believe this to be due to an erroneous conception of the spicules under consideration. Although I was convinced for myself, that these spicules were by no means (polyaxon) asters, but (monaxon) spinispirae, I have tried nevertheless to produce proofs for my statement by carefully studying the spicules treated in various ways. More especially I was led to do this in order to settle the question between LENDENFELD and Topsent about the sponge, which Oscar Scumipt first described under the name of Tethya bistellata. Is it, as LENDENFELD suggests, a species of Spirastrella, or, as TOPSENT believes, one of Hymedesmia? Of course it is no Tethya; so far everybody agrees. Scumipt (1862 p. 45) described a sponge, which he called Tethya bistellata, a name which he altered himself into Suberites bistellatus (1864 p. 36). Now LenpenreLp believed to have traced the sponge in his collection from Lesina and called it Spirastrella bistellata (1897 p. 55). From this Torserr dissented in 1898, alleging that ( 643 ) Tethya bistellata O. S. must be transferred to Hymedesmia, and consequently he called it Hymedesmia bistellata (4900 p. 125) 5. Now I possess in my collection from Naples a sponge, which is beyond reasonable doubt Scumipt’s Tethya bistellata. 1 can affirm this especially because the spicules absolutely agree with those of a preparation I made at the time in Graz and which is labelled “Suberites bistellatus O. S. Origin. Schmidt”. We may suppose, therefore, that LENDENFELD, Topsent and myself really examined the same sort of spicules, albeit that I must acknowledge that there is ‘no absolute proof. Topsent says, that the microsclera under consideration are euasters ; he writes (1900 p. 123) that they are “spherasters de forme parti- culiere ... Chacune d’elles résulte de la congrescence latérale de deux spherasters a actines nombreuses, coniques, pointues et lisses.’”’ And later (p. 127): “les sphèrasters sont doubles. O. Scumipt a insisté sur ce caractère important, auquel l’espece doit son nom.” The question arises whether Scumipt’s statement is of great value. In 1862 he said (p. 45) that some are “ganz eigenthümliche Zwillingsgestalten”; and further: “es sind also Doppelfiguren, welche einige Aehnlichkeit mit den Euastern haben.” It must, however, not be forgotten that ScnMipr at that time was unconcious of the sort of spicula which he called later (1868 p. 17) “Spiralsterne” or “Walzensterne’ of which he mentions as characteristic ‘dass ihre Strahlen nicht Radien eines Centrum sind, sondern in Spiralstellung sich folgen.” According to LENDENFELD (1897) are the spicules under consider- ation “spirasters’” and he gives some illustrations (1. c. Pl. VI fig. 59) which clearly show his conception of the thing. Both from his illustrations and from his description it follows that the axis is sometimes longer, sometimes shorter. LENDENFELD does not believe that ‘“‘euasters” occur and suggests that ScuMipT was perhaps misled by an optical illusion. If spicules are examined “deren Axen im Praeparat aufrecht stehen und daher verkürzt gesehen werden” they simulate euasters. In spite of the fact that Torsenr himself remarks, that in minute microsclera it is much more obvious that the centre is a line and not a point, this author does not consider them as spirasters but as double euasters. “Plus elles grandissent, plus la tige d’union se rac- courcit. Sur les plus grosses, les deux centrums sont directement 1) Actually the course of events was this: Topsent (1900 p. 113) writes with regard to Tethya bistellata O.S.: “ie l'ai mise à sa place naturelle en 1892.” However, in that article nothing is mentioned but the name Hymedesmia bistellata without reference to Tethya bistellata. ( 644 ) aceolés...” Of course Topsent has not overlooked that spicula have a different appearance whether seen “de profil” or ‘‘de face”; but evidently he did not pay attention to intermediate positions such as can be seen if one allows them to turn over. I drew attention (1902 p. 170) to the fact that in almost all cases the twisted character becomes plain enough by applying the above device. However, there are some more methods to make out the shape and the structure of spicules (Cf. Vosmaer & WiJsMaN, 1905 p. 745). One of these methods is heating. In using this method it is, however, not indifferent in what way it is applied. If isolated spicules (e. g. styli of Zethya) simply dried in the air, are heated on a platina spatula immediately above the flame, a brownish colour soon becomes visible. If they are further heated the brownish tinge turns into white. It then frequently occurs that a crackling noise is heard and that spicules or portions of spicules jump off from the spatula. Such spicules, seen under the microscope, generally appear to be cracked or broken; they are brown or black, some were quite misshapen as if the spicopal had been partly melted. How can these phenoma be explained? BowERBANK ascribed the brown or black colour to carbonised organic matter, but KOLLIKER proved that the colour can certainly not wholly be explained in this way. Indeed, in some parts the colour is brown only in transmitted light, whereas it is white in reflected light; consequently KOLLIKER declared those parts to contain microscopical air-bubbles. Quite correctly „Börscnrr (1901 p. 240) remarks that KöLLiKER where he speaks of “Luft”, in fact means “Gas”. Wijsman and myself have demonsirated (1905 p. 28), that spicopal is a form of hydrated siliceous acid, which can give off water in an atmosphere dried by P,O,. It is, therefore, very likely that when the spicules are treated as described above, a portion of the water becomes water-vapour. The tension of the heated globules of steam of course can be great enough to make the spicule explode. This explains at the same time the crackling noise and the jumping off from the spatula. Still, it need not come to this; hence we see some spicules only slightly cracked, not broken or deformed. If, however, spicules are not simply dried in the air, but, by slowly warming on asbestus for several days or by P,O,, water is taken from them, and they are afterwards very carefully heated, then it is possible to prevent any cracking. Spicules treated in this way show quite other details. First of all the carbonised central thread is clearly visible. In some spicules the rest of the spiculum remained quite transparent; in others a brownish colour is to be seen : ( 645 ) on special places. Seen with reflected light these places are white; on the whole those places have a granular or frothy appearance. *) As a rule the lamellar structure is very conspicuous. In certain cases the carbonised spiculum sheath is likewise visible. It also happens that the carbonised and shrunken central thread is seen as a flexuous, continuous or broken, black string lying within the central canal (fig. 20—21). Of course the best microscopical figures are obtained if the spicules are examined in a medium the index of re- fraction of which is equal to or comes very near that of the spicopal. Controlling experiments sufficiently prove that no artefacts or any- thing of that sort come into play through which no conclusion can be drawn about the structure of the spicule. The spicopal being in slowly by dissolved the method published by Wissman and myself, the mieroscope reveals facts wnich are in perfect accordance with those obtained in the way described above. One may also combine the two methods — heating and dissolving; again the results are the same if one follows the process under the microscope. Suppose one observes in heated spicules a black central thread with a brownish surrounding; suppose the object is mounted in glycerine of about the same index of refraction as the spicopal, the external limit is clearly visible as a delicate dark line (fig. 22). Some time after the action of the hydrofluoric acid the spicule appears as drawn in fig. 23. The silica begins to be dissolved as soon as the hydrofluoric acid has penetrated the spiculum sheath; the external delicate line remains visible but at some distance the limit of the spicopal, now thinner, becomes visible. The distance between the sheath and the limit of spicopal becomes gradually larger, the brownish surrounding of the central thread disappears and finally nothing is left but the carbonised central thread and the likewise carbonised sheath (fig. 24). This proves, that the brownish colour arround the axial thread does not originate from carbonised organic matter. 1) Biirscutt admits as is well known, that in spicules which are not heated jikewise little holes occur and that these holes simply become larger by the process of heating and consequently better visible. He says (le p. 248): “Das Auftreten der feinwabigen Struktur beruht darauf, dass eine solche auch schon in der nicht gegliihten Nadel besteht, jedoch zu fein, um mikroskopisch sichtbar zu sein. Beim Gliihen tritt eine Verdampfung des in den Wabenhohlräumchen eingeschlossenen Wassers ein und damit eine Erweiterung derselben bis zur Sichtbarkeit. Für diese Ansicht spricht vor Allem die Beobachtung, dass wenigstens in einem Fall auch eine nicht geglühte Nadel.... den wabigen Bau der Schichten deutlich zeigte.”’ Apart from the question whether in unheated spicules a frothy structure really occurs or not, it is certain that the dark colour of heated spicules is due to little holes, void of air or filled with some gas, say water-vapour. ( 646 ) The experiment can be modified in the following way. Isolated spicula are brought into acid fuchsine; if the hydrofluoric acid is now allowed to act on the spicules the spicopal will be dissolved, whereas the sheath and the central thread will be stained red. In both eases the silica is dissolved; in the former case the thread and the sheath are visible because they are black (carbonised), in the latter case because they are red. In the original experiment the spicopal is only optically dissolved. What has been said for the styli of Tethya holds true m.m. for other spicules of Demoterellida. The structure of several spicules — monaxons, tetraxons or polyaxons — is fundamentally the same; in details there are important differences. However, I do not wish to speak about them in this paper. I have only mentioned as much as seemed to be necessary to show that by the described methods we are able to demonstrate most plainly the central thread. This can be done also in those cases in which the thread is not visible under ordinary circuinstances, e.g. if the spicules are very minute or irregularities of the surface prevent it. Thus, for instance, in Tethya no central threads are visible in the oxyasters or at any rate they are not present beyond doubt’). If these spicules are heated with great precaution they look under the microscope like fig. 19. It depends, as in other spicules, on the grade on heating whether the thread will be blackened only or with it its surroundings. Independently of this it is evident that the axes originate from one point. Applying the heating method to the spicules in question of Spir- astrella bistellata (O. S.) Ldfd. the microscope reveals pictures as drawn in fig. 1—8. It is most evident that we have here an axis exactly like that which unquestionable spinispirae possess. Such images are entirely unexplainable if the spicules are considered as congrescences of two euasters. They fully exhibit their true nature of spicules belonging to my group of «-spiraxons (1902 p. 112). Although I suppose this to be convincing, I applied moreover the dissolving method. It seems rather a paradox that the shape and the structure of a siliceous spicule can be cleared up by dissolving the silica. Still it is a fact, as I have frequently learned. Wijsman and myself (1905 p. 18) confirmed Bürscuarrs observations of 1901, that the dissolution of spicopal may proceed in more than one way. Only we have given another explanation of the fact. According to our 1) On the whole spongiologists speak about the central thread as a constant feature of spicules. As a matter of fact stands that the presence of a thread is proved only in some cases and that in numerous microscleres nobody saw it, As far as | know only KOLLIKER found it in oxyasters of Tethya (1864, Pl. IX, fig. 2). ( 647 ) conception the spicopal which limits the central canal is more easily dissolved than that of subsequent layers. It seems that this is likewise the case for the radii of oxyasters of Tethya. It is probable that this depends on a difference in the quantity of water the “gel” contains. Now we observed that in pointed undamaged spicules, where conse- quently the central canal is shut, the funnelshaped dissolution is not seen, at any rate not at the very beginning of the process. The apex simply becomes thinner and thinner till the dissolving agent reaches the neighbourhood of the central canal, in which case the “funnel” often appears. Consequently we have herein another method to prove the existence of a central canal. On the other hand we may conclude from this, that, if a funnel never appears there is no central canal resp. no special layer of spicopal in the centre. Thus, for example, in spicules with spines, the latter disappear gradually and the spicule becomes gradually thinner. [ have observed this phenomenon very distinctly in acan- thostyli of an Zctyon from Naples. If the latter method is now applied on Spirastrella bistellata (O.S.) Ldfd. we see, that the pointed processes of the spinispirae become thinner and shorter, and finally disappear whereas the rest of the spiculum later becomes thinner (fig. 4—18). The microscopical images one sees during this process leave no doubt with regard to their structure. The more the spines dissolve, the more it becomes evident that we have to do with spinispirae. Moreover, it follows from the above experiments that the spines of these spinispirae are of quite another nature than the actines of the Tethya-asters. In the former case (Spirastrella) we have to do with local extuberances of spicopal destitute of any central thread or canal. In the latter case (Tethya) we have organic axes. Indeed, the former spicules are monaxons, the latter are polyaxons. Consequently the microsclera of Spirastrella bistellata (O.S.) Ldfd. are indeed spinispirae. Since LENDENFELD, TorseNr and myself believe to have found sponges, which are identical with Tethya bistellata of Oscar Scumipt, the species belongs as little to Hymedesmia as to Tethya. For the moment there is not sufficient evidence not to bring it to Spirastrella. The name for Tethya bistellata O.S. has to be, therefore, Spirastrella bistellata (O.S.) Ldfd. I believe with Torsexr that it is identical with Spirastrella cunctatriz O.S. 44 Proceedings Royal Acad. Amsterdam. Vol. XI. ( 648 ) LITERATURE. 1841 (a) BoweERBANK in Ann. & Mag. N. H. VII p. 72—74. 1862 Scumipt, Die Spongien des adriatischen Meeres. 1864 K6LLIKER, Icones histiologicae. I. 1864 Scumipt, Supplement der Spongien des adriatischen Meeres. 1892 Topsent.in Résultats Campagnes Prince Monaco. 1897 LENDENFELD, Clavulina Adria. 1898 TopsENT in Bull. Soc. Scient. Ouest. 1900 TopsEnt in Arch. Zool. expérim. (3) VIII. 1901 BürscuHur in Z. W. Z. LXIX. 1902 VosMAER in Kon. Akad. Wetensch. Amsterdam. Proc. meeting June 1902. 1905 VosmMAER & WijsMAN in id. Proc. meeting May 1905. EXPLANATION OF THE PLATE. (Fig. 1—21 are drawn 500 times magnified; fig. 22—24 still more magnified), 1—3 Spirastrella bistellata; spinispirae carefully heated. In 3a only the central thread (carbonised) is drawn, lying in the central canal. . 4—18 Id. Influence of hydrofluoric acid. In figs. 4, 5 and 6 the acid has acted for a short time; only the spines begin to be dissolved. In fig. 7—12 the process is advanced; the ““axis” becomes more and more obvious. In fig. 13—16 this is still more the case. „19 Tethya tincurium; oxyaster after carefully heating; distinct, (carbonised) central thread. . 20 1d. Middle piece of a stylus, carefully heated. Black (carbonised) central thread, entirely filling up the central canal. 21. Id. Id. Shrunken, bent and broken central thread in the somewhat brownish central canal. ig. 22—24 Id. Id. Slightly more heated and brought into the hydrofluoric camera of Vosmaer & WIJSMAN; d. central thread, b. brownish layer around the central canal, s. brownish spiculum sheath. In fig. 22 it is seen at the begin- ning of the experiment; the sheath lies immediately on the external spicopal; the little granula are adhering particles, not belonging to the spiculum. In fig. 23 the hydrofluoric acid has penetrated the sheath and dissolved the peripheral layers of spicopal, the limits of which are marked c; the sheath remained in its place. In fig. 24 all spicopal is dissolved; only. the carbonised sheath s lies as a very delicate cylinder around the central thread. Leiden, 2 Jan, 1909, dr ad. C.J. VOSMAER. “On the spinispirae of Spirastrella bistellata (0.S.) Ldfd.” as \ 7 ae / ; /o / B KN / 33 Jb je > a B A4 Ee 17 jg EN 7 ny ne 9 1 20 2/ Proceedings Royal Acad. Amsterdam. Vol. XL. “et ( 649 ) Botany. — Prof. J. W. Morr presents the dissertation of Mr. K. ZIJLSTRA, Assistant at the Botanical Laboratory, Groningen, entitled: “Kohlenstiuretransport in Bittern’, Groningen, 1909, and with reference to this he communicates the following. *) In 1877 the speaker published the results*) of investigations, which proved, that the carbon dioxide, which is found in considerable quantity in soils containing much humus, and which is at the disposal of the roots, cannot lead to starch-formation in the leaves, when the latter are in a space, free from carbon dioxide; nor can this carbon dioxide appreciably accelerate starch-formation in the open air. From the experiments, which led the speaker to this result, he further concluded, that a leaf or a portion of a leaf cannot form starch in a space devoid of carbon dioxide, even when parts, organ- ically connected with, and bordering immediately on the portion in question, are placed in an atmosphere which is many times as rich in carbon dioxide as ordinary air. The experiment, which appeared to prove that starch cannot be formed even from carbon dioxide, which is offered to immediately adjoining parts, was the following. A starch-free leaf was placed between the greased edges of two similar crystallizing dishes in such a way, that the apex was within the space enclosed by the crystallizing dishes, and the base was outside. The lower crystallizing dish contained potassium hydroxide solution; over the apparatus there was placed a bell-jar, containing air to which about 5 percent of carbon dioxide had been added. After the leaf had been exposed to the light for some hours, the base was found to contain much starch, but in the upper portion, even right up to the edge of the erystallizing dish, which was 3 mm. thick, no starch whatever had been formed. From this the speaker concluded that when carbon dioxide is abundantly present in any given part of a leaf, this can nevertheless not lead to starch-formation in an immediately adjoining portion, when the latter is in a space free from carbon dioxide. This result was remarkable, for the stareh-formation in the basal portion proved, that the carbon dioxide, offered to the leaf had indeed been taken up, and the presence of many intercellular spaces in the des Travaux Botaniques Néerlandais. 2) J. W. Mow. Ueber den Ursprung des Kohlenstoffs der Pflanzen. Land- wirthsch. Jahrb, VI. 1877, p. 327—363. 44* ( 650 ) mesophyll led to the supposition, that the transport of the carbon dioxide so taken up, need not be impossible. A want of further experimental data made this question remain unsolved up to the present. The necessary data have now, however, been collected in the investigation of Mr. Zijrsrra, who has shown that the facts, previously observed by the speaker, had indeed been correctly described, but that other results may also be obtained, provided one works with different plants from those which the speaker happened to have used, or arranges the experiments in a different way. Mr. Zirsrra was able to show that in the experiment described, starch-formation may sometimes indeed occur in the space free from carbon dioxide. The above conclusion as to the impossibility of starch-formation at the expense of carbon dioxide derived from the immediate vicinity, has therefore been found to be incorrect. The other results formerly obtained by the speaker, and especially the chief deduction, regarding the impossibility of starch-formation in the leaves at the expense of carbon dioxide, taken up by the roots from a soil rich in humus, were, however, completely confirmed and further elucidated by. Mr. Zuusrra’s investigation. An explanation of the experiment described above, was also suggested and generally speaking Mr. Zuusrra succeeded in solving pretty completely the question of the possibility and occurrence of carbon dioxide transport in leaves. How this was done the speaker wishes to communicate below. Not unnaturally it seemed desirable to begin the investigation with a repetition of the above described experiments. This was first done with the leaves of Polygonum Bistorta and of Cucurbita Pepo (experiment LUI and LIV}'), which were also employed by the speaker in his above-mentioned investigation. In these and in all later experiments Mr. Zirsrra demonstrated the formation of starch by the so-called Sacus-ScuimmpER metbod, according to which the entire leaves, after decolorisation, are examined for starch content with the help of an iodine-chloralhydrate solution. This method was unknown in 1877, so that the speaker used microscopic sections, which is very cumbrous and gives less complete, albeit equally certain results. The speaker limited himself to applying the starch reaction to sections of the apex of the leaf, which was in the space free from carbon dioxide, and to the base, which was in the air rich in carbon 1) The numbers of the experiments are here and in what follows the same as those in Mr. Ziustra's paper. ~ ae ( 651 ) dioxide. He did not, however, examine the strip, 3 millimeters broad, which was in the grease between the edges of the crystallizing dishes. Now when Mr. ZijrsrrA examined the entire leaves, it became apparent that starch-formation extended continuously from the leaf base upwards over part of this strip, in an area bounded by a sharply defined line, somewhat like a zig-zag, but that it nowhere extended so far, that starch had also been formed in the space, free from carbon dioxide. The speakers observations were therefore confirmed, but were found to have been incomplete. When, however, Mr. Zamsrra repeated the experiment with a Dahlia leaf (exp. LV) the result was different. In this case also the starch-formation was found to extend into the greased strip, but in addition it extended here and there for some millimeters into the space freed from carbon dioxide. The border line of the starch reaction was here also somewhat zig-zag, but not everywhere equally sharp. Finally, when the experiment was performed with a leaf of Pontederia cordata (exp. LVI) starch-formation extended uniformly for 0.5 centimeters into the carbon dioxide free space. The limit of the starch reaction was in this case not zig-zag, but bent regularly and was not sharp, since the dark colour of the reaction disappeared towards the leaf apex by a gradual transition. Sranr’s cobalt test showed, that the stomata of the apex were closed at the end of the experiment. The possibility now existed that in these experiments the greased join had not been absolutely tight, although it was not very probable that this fault should have revealed itself in the experiments with Dahlia and Pontederia and not in others. In order to exclude this possibility two pieces of appa- ratus were constructed, which, with the aid of mercury, per- mitted of a completely airtight separation being effected between the space free from, and that rich in carbon dioxide. Both pieces of Figure 1. apparatus were used in the inves- tigation, but here the speaker will only describe the better of the two. (See fig. 1). In a Petri dish a, of 15.5 cm. diameter a smaller one 5 of ( 652 ) 9 em. diameter was fastened down with resin and wax. Mercury was poured into the annular space g and concentrated caustic potash into the small Petri dish /. A glass bell-jar 4 of 3.2 litres’ capacity was placed in the mercury, so that the air inside the jar was kept free from carbon dioxide. It was now possible to introduce the apex of a leaf, which had been freed from starch, into the carbon dioxide- free space through the mercury. The leaf base and the petiole thus remained outside the bell-jar 4, and the petiole was always immersed in a dish of water in order to keep the leaf fresh throughout the experiment. A piece of metallic gauze was always placed over the small Petri dish 4 in order to prevent the leaf from coming into contact with the caustic potash. The base of the leaf could now be left at will in the open air with its unlimited supply of carbon dioxide at great dilution, or the base could be surrounded with an atmosphere ricber in carbon dioxide. For the latter purpose the whole apparatus was placed on a tripod in a large porcelain dish, partly filled with. water. A large glass bell-jar of 38 litres’ capacity was placed over the apparatus containing the leaf, so that the space in the large bell-jar was cut off by the water in the porcelain dish. Into this space any desired quantity of carbon dioxide could be introduced. Finally attention may be drawn to the tubes 7 and &, which connected the interior of the small jar with the free atmos- phere. This prevented differences of pressure between the interior and the exterior from raising the jar and establishing a communication between the outside and the inside. Moreover a stream of air, free from carbon dioxide, could be passed by these tubes through the inner jar. For this purpose the tube £ was connected to an aspirator, and the tube 7 to absorption tubes for the carbon dioxide of the atmosphere. With this apparatus a number of experiments were carried out, in some of which the base of the leaf was in the ordinary atmosphere, and in others in an atmosphere with much carbon dioxide. The portion of the leaves which was underneath the mercury had in all the experiments a length of 3 centimeters. These experiments led to the result, that not only in Dahlia and Pontederia leaves but also in all the other leaves investigated, a narrow strip of starch was formed in the space free from carbon dioxide at the border of the mercury. This strip was generally coloured jet black by iodine. The following table summarizes a number of experiments performed in this manner. In all these experiments the small starch strip was of course bordered on the side of the mercury by a straight line, but towards the apex heh, har ( 653 ) TABLE I. Bint | Number | | vaal Din | ‘Width of tie a | of exp. The leaf base in: of exp. starch strip. Se ee eS eer eee eee eee Dahlia Yuareszii | I 5 °/, CO? | 5 hours 3 mm. ” pe II open air adt 2-3 „ „ (Cactus) Thuringia XVI a DO To 34 „ Aster macrophyllus V open air En 0,5 Sisymbrium Alliaria VI Kd : Di fig hi tg je 2 X pe pe Be A Polygonum Bistorta VII : 2 as er Aesculus Hippocastanum VIII ” > ba OFS: ting pn Pavia IX si 7 SiGe 6.57: Wy Acer campestre X fe 5 40% Wee fe Sambucus nigra XII 21/,°/, CO? | Ik ns Zie " st XIII : s SS ty Yi Juglans regia XIV AE Fi Sy Liv Acorus Calamus XV 5 » tee (di, se Heliopsis laevis | XVII | ZU, Gs | a of the leaf the border was a zig-zag line, which often coincided with the presence of veins in the leaf. In those places the border-line was sharply defined. In Acorus alone this line was also straight, without clearly depending on the veins; in this case moreover, the border line was not sharp, but towards the apex the reaction became weaker by imperceptible degrees and soon disappeared completely. Since in these experiments the apex of the leaf was in a space free from carbon dioxide we must assume, that the starch strip had been formed at the expense of the carbon dioxide, which had been transported from the portions of the leaf nearer the base. The most plausible assumption now seemed to be the following: that the carbon dioxide which had been absorbed by the leaf base, had been conducted through the parts of the leaf under the mercury, and having arrived in the part of the leaf exposed to the light, had there given rise to the formation of starch. In order to test the validity of this assumption, comparative experiments were undertaken, in which the one leaf was placed with its base in air containing ( 654 ) 2—3°/, of carbon dioxide, and the other in ordinary air, containing therefore very little carbon dioxide. If the hypothesis just brought forward were correct, one ‘might expect that owing to a more copious supply of carbon dioxide, a wider strip of starch would be formed in the former case than in the latter. It did not seem probable that the increased supply of carbon dioxide would manifest itself by a stronger iodine reaction, for, as we have seen, in most of the experiments described above, the reaction, if it appeared at all, was as strong as possible. These comparative experiments were carried out with two sets of apparatus of the kind deseribed, both of which were provided with a large bell-jar, including therefore the one in which the leaf-base was in ordinary air. This was done in order to maintain as far as possible the same temperature in the two smaller jars, for it was found that in these experiments the temperature had a great intluence on starch-formation. The result of these experiments varied considerably in different plants. In some cases it could be definitely shown, that carbon dioxide, supplied to the base, had influenced the manufacture of starch in the upper part of the leaf, but this could not be demon- strated for other plants. The speaker first considers the experiments with a positive result, summarized in the following table: TABLE II. == Nl — ——E nn ee — EE = = Width of the starch strip Number | Duration | in the CO, free space. Name. of ex- of ex- | periment. | periment. Base | Base in 2 COR ordinary air. nn Pontederia montevidensis XLVIII | 7 hours, G/)) sam. 2 mm. Eichhornia speciosa XLIX 7 5 laet iem: Zen ” ”» L 51/, ” | 1,5 ” 1 ” Eucomis punctata LI 9 " | 1 5 3% The first two experiments were carried out with leaves which were as nearly as possible equal, and the last two each with the halves of one and the same leaf. All these experiments very clearly demonstrate the influence of the increased supply of carbon dioxide by the formation of wider strips of starch. (655 ) In the above experiments the leaf base in the space with much carbon dioxide was exposed to the light and accordingly this part of the leaf was full of starch at the end of the experiment. We might now expect that on darkening the leaf-base, more carbon dioxide would remain available for transport to the apex, and that a wider strip of starch would be formed than in the former case. If this expectation were realized, it would be an additional proof, that carbon dioxide is transported from the base to the apex. An experiment, carried out with the two longitudinal halves of the same leaf of Lichhornia (exp. LIL) indeed gave proof of this. Here both bases were kept in air with 2°/, carbon dioxide, but one was darkened, the other not. In the latter case less carbon dioxide remained available for transport on account of the consumption of carbon dioxide for starch-formation in the base, and in the course of 4 hours a strip of starch , 2—5 mm. wide, was formed in the upper part of the control half, as compared with one 5—8 mm. wide in the other half of the leaf, of which the base had been darkened. In these experiments the strip of starch never had anything like a sharp edge on its upper side, nor was there any connexion between the limit of the reaction and the veins. The reaction simply became weaker and weaker at the edge of the strip and soon stopped completely. A series of experiments with other leaves yielded, however, a totally different result. Here the strip of starch in the two leaves experimented on was always equally wide, no matter whether the leaf base had been placed in ordinary air or in air with a high carbon dioxide content. In some of these experiments leaves of the same plant individual were taken, in each case as nearly as possible equal. The subjoined table summarizes them. TABLE III. —- — << qh: fe TT —— — = —— Plant | Number ‚ Duration, | Width of the starch à of exp. | of exp. strip in both leaves. Sambucus nigra | XIX | 5 hours 3 mm, Juglans regia XX | er An | l £ | Acorus Calamus XXI | ale 2 se Zea Mays XXII | Oi | VD orn Hordeum vulgare XXIII Ps | | ™ ( 656 ) The negative result thus obtained, could now, after the experience with the influence of darkening on the leaf base of Michhornia in experiment LII, be attributed to the fact, that the carbon dioxide absorbed by the base, had been wholly used up for starch-formation on the spot. Experiments were therefore also made, in which the leaf ‘bases were darkened with black paper right up to the edge of the fmereury, and therefore remained free from starch. But these experiments also gave exactly the same result; the strips of starch were equally wide in the leaves with bases in air containing much carbon dioxide, as in those with their bases in ordinary air. A survey of these experiments is given in the subjoined table: FABLE IV. Plant. | Number Duration | Width of starch strip of exp. of exp. | in both leaves. | | Triticum vulgare | XXIV | 6 hours | 1.5 mm. Zea Mays _ XXV ae Ae ae Dahlia Yuarezit | XXVI 5 » 4 n Aesculus Pavia | XXVII | Je, Sad ee Tradescantia virginiana XXVIII aya 1 h In order to eliminate more thoroughly possible inequalities between the leaves compared than could be done by careful choice, a few experiments were also performed, in which the longitudinal halves of the same leaf were used for comparison, and were darkened at the base. These experiments ‘yielded the same result, as given below : TABLE V. nee Ti EP ae ge - Plant , Number | Duration Width of starch strip in a | of exp. } of exp. both halves of leaves. eS Dahlia (Cactus) Thuringia XXIX 5 hours not noted Heliopsis laevis XXX | Den | 155. Mit. Finally two experiments may be mentioned, which were performed under conditions, similar to those of the last named, but in which the caustic potash was omitted in the smaller bell-jar. In all the previous experiments the carbon dioxide of the leaf apex would, ( 657 ) if it were at all possible diffuse into the surrounding space, which remained permanently free from carbon dioxide. This, however, did evidently not take place so fast, but that in all experiments starch was formed near the mercury. Nevertheless the possibility was not excluded, that a wider strip of starch might appear, if the carbon dioxide absorbing potash were absent. Such experiments were accordingly undertaken, and in order to make it possible that there should even be some accumulation of transported carbon dioxide, the inner bell-jar of the apparatus, which hitherto had been of a capacity of 3.2 litres, was replaced by one of only 0.8 litres’ capacity. In both experiments two halves of the same leaf were taken in each case. The two bases were placed in 3°/, carbon dioxide, one being exposed to the light and the other darkened. The two halves gave exactly the same result and the starch strips were not wider than in all the previous experiments, as is shown by the following summary : d TABLE: VI. Piant Number Duration | Width of starch strip in ; of exp. of exp. both halves of leaves. | Dahlia (Cactus) Thuringia | XXXI 4 hours | 3—4 mm. Heliopsis laevis SEKI 4a. ol | 05-1 , It is obvious from all these experiments that with a number of leaves from widely different Monocotyledons and Dicotyledons a totally different result is obtained from that of the experiments described first. Carbon dioxide, even when abundantly supplied to the leaf base of these plants, cannot bring about the formation, at about 3 cm. distance in the apex of the leaf, of a wider strip of starch, than would have been formed without the addition of this carbon dioxide. Attention may further be drawn to a difference, which again clearly showed itself in these experiments, between the reticulate Dicotyledons and the parallel-veined Monocotyledons as regards the edge of the starch strips on the side nearest the apex. In all leaves with reticulate venation these edges were zig-zag and in most places sharply defined, owing to the presence of small veins. In the leaves of Grasses, of Acorus and of 7radescantia on the other hand there was a gradual transition to the starch-free apical portion, without any relation to veins, and there was no zig-zag border. ( 658 j It would certainly be going too far to deduce directly from these experiments, that the carbon dioxide, which is supplied to the base of these leaves, cannot at all contribute to starch-formation in the apex. Nevertheless the results obtained gave a clear indication as to further experiment, intended to show, beyond dispute, if possible, what was the state of affairs in these leaves. The way in which such experiments should be performed was now clear. The base of the leaf must be deprived of any supply of carbon dioxide, and the question was whether, as might almost have been expected from the above, a strip of starch would be formed above the mercury. If this were to take place, and in the same way as before, the proof would have been given, that the part of the leaf, placed under the mercury, itself produced the carbon dioxide, required for starch-formation in the adjoining part, which was exposed to light. The experiments which supplied an answer to this question were made by Mr. Zijrsrra in various ways. In the first place an experiment was made with the apparatus described above, but without the large bell-jar. A small leaf ot Dahlia Yuarezii (exp. XXXIIT) was placed in the apparatus in the ordinary way, but the base and the petiole were immersed in water which had been poured on the mercury outside the small jar. In the space free from carbon dioxide the apex now produced a strip of starch, similar in all respects to that in the experiments previously described, and in addition, a starch strip was formed under water in that part of the base which adjoined the mercury. Similar experiments were also made with a simpler apparatus, which moreover permitted of the water, in which the leaf base was placed, being kept quite free from carbon dioxide. Fig 2 gives a representation of this little apparatus. It consisted of a rectangular glass box, measuring 9 by 4.5 cm., and 5 em. high, which is represented in the figure in section, the wall being indicated by 7. With resin and wax a vertical plate of glass was fixed longitudinally but it did not reach to the bottom. The box was filled with mercury to slightly above the lower edge of the glass plate G. The leaf Figure 2. B was introduced underneath the vertical plate, so that its base was on the right side of the figure. There a layer of boiled water W was poured on the mercury, so that the ( 659 ) base was completely submerged. The apparatus was now placed under a bell-jar with air, free from carbon dioxide, and the whole was exposed to the light. After some hours starch had been formed in this apparatus in Dahlia Yuarezit (exp. XXXIV) and in Populus pyramidalis (exp. XXXV); there was a strip of starch in the apical portion along the mercury, quite like that of the previous experiments; there was a similar strip in the basal portion under water, also along the edge of the mercury; and finally a narrow strip of starch had appeared in the mercury itself, at the place where the leaf had been in contact with the lower edge of the vertical glass plate and had therefore received light. The carbon dioxide, which in this case had been used up in starch- formation, could only have been derived from the portions of the leaf under the mereury and could scarcely be anything but carbon dioxide of respiration from these parts. When this had once become evident, a still simpler arrangement naturally suggested itself. For this purpose pieces of leaves, free from starch, were placed on a layer of mercury, were partially covered with black paper, and finally pressed under the mereury by means of a glass crystallising dish with flat bottom. The light could now reach those parts of the leaves which were not covered by black paper, through the bottom of the glass dish. Such experiments were made with Dahlia Yuarezi (exp. XXXVI, XXXVII and XLI), Sambucus nigra (exp. XXXVI), Syringa vul- garis (exp. XXXIX) and Tila platyphyllos (exp. XL). In all these experiments, which lasted about 5 hours, strips of starch were formed in the ordinary way at the border of the black paper in the lighted portions of the leaves; the border of the starch on the side opposite the paper was again sharply defined in many places by veins. We may hence assume, that in all the experiments of the second group, in which the result was negative, starch had been formed exclusively at the expense of the carbon dioxide, resulting from the respiration of the parts of the leaf under the mercury, which carbon dioxide had been transported a certain distance to those parts, which were exposed to light. In this case the supply of carbon dioxide from outside was prevented by the mercury, which also kept off the light. The re- spiratory carbon dioxide could therefore not be immediately reduced on the spot, but could spread and thus reach the lighted portions of the leaf in fairly large quantity. It should further be noted, that the epidermis of the darkened leaf-fragment was closed hermetically by ( 660 ) the mercury, so that the carbon dioxide formed could not escape from the leaf, but had to move sideways. The question now arose whether this closing of the epidermis was a necessary condition for the success of the experiments. From the nature of things this might be considered probable, for otherwise the respiratory carbon dioxide would follow the line of least resistance through the stomata and epidermis. Experiments made with this object in view have completely confirmed this opinion. A leaf of Dahlia Yuarezii was introduced into the apparatus first described, with its apex in the small bell-jar without caustic potash. One longitudinal half of the apex was uncovered, but to the other half a strip of black paper was fastened, by which a transverse strip of this half was darkened; this strip measured 17 mm. from the mer- cury in the direction of the tip (exp. XLII). In the same leaf it was therefore possible to compare: a lighted portion adjoining a darkened one, the epidermis of which was shut off by the mercury; and a lighted portion adjoining one darkened by paper, the epidermis of which was therefore not shut off. A starch zone was indeed formed along the edge of the mercury, but not along that of the black paper. When the experiment was repeated (exp. XLIII), with the portion of the leaf under the paper smeared with a mixture of cocoa-butter and wax (according to Sranr), the result was quite different. This mixture is known to close the epidermis almost completely to carbon dioxide, and the well-known starch zone now also appeared at the edge of the paper. \ The use of cocoa-wax finally led to some experiments which may be called extremely simple, and which once more confirmed the result obtained. Leaves of Aesculus Pavia (exp. XLIV) and of Juglans regia (exp. XLV), free from starch, were completely covered with cocoa-wax, in which condition, according to STAHI.’s experiments, extremely little or no starch is formed during exposure to light in the open air. The leaves were, however, exposed to the light by Mr. Zijrsrra after they had been partially covered with black paper or tin-foil, and now, as was to be expected, black borders of starch were formed along the edges of the paper and of the tin-foil. A rather pretty modification of these experiments was finally obtained by using variegated leaves, of which the colourless portions were quite white, and did not therefore contain any carotin, by means of which carbon dioxide might be decomposed, even in the absence of chlorophyll. It is further necessary that such leaves, in order to be suitable for the experiments in question, should possess ( 661 ) in their colourless parts a well-developed parenchyma, capable of producing by respiration a proper amount of carbon dioxide. If such leaves, after having been freed from starch, are smeared with cocoa- wax to prevent the escape of the carbon dioxide formed by respiration and are exposed to the light, borders of starch are developed in the green portions, at the edge of the colourless patches, in the same way as in the earlier experiments, but without any portion of the leaf having been darkened. Such experiments were performed with leaves of Cornus tartarica and Elaeagnus Frederici (exp. XLVI) and the best result was obtained with Pelargonium zonale (Mad. Salleroi) (exp. XLVII). It appears from the above, that in all the leaves investigated a transport of carbon dioxide was possible to a greater or lesser extent, and that this might lead to starch-formation in the portions of the leaf exposed to the light. The experiments were arranged in such a manner, that this starch-formation generally showed itself in more or less broad strips of the leaf. But the carbon dioxide which led to the formation of these starch strips was found to be of dual origin. In the majority of the leaves investigated, and in all the experiments of the various tables, except those of table 2, we had to assume, that the starch strips were exclusively formed at the expense of the respiratory carbon dioxide, which had been formed in the neighbouring darkened portions, hermetically shut off by mercury or by cocoa-wax. In the experiments with water plants, mentioned in table 2, the respiratory carbon dioxide must no doubt have also contributed to the formation of starch borders. These experiments did show, however, that another source of carbon dioxide also cooperated, namely the supply of carbon dioxide, which had been added to the air surrounding the leaf-base, which was absorbed by this base, and which was trans- ported through the 3 cm. long portion of the leaf under the mercury into the space free from carbon dioxide, and was assimilated there. In other words: in most leaves, of J/onocotyledons as well as of Dicotyledons, only a very limited transportation of carbon dioxide is possible. But in these leaves one has an excellent method for the study of this transportation, by utilizing the respiratory carbon dioxide of the adjoining parts. In a few parallel-veined leaves of water plants on the other hand, a much wider transportation is possible, which can be demonstrated with the relatively rough apparatus employed. The question now arose, how these two varieties of carbon dioxide transportation must be imagined and on what the difference of the two categories of leaves referred to, depended, ( 662 ) Mr. Zijrsrra succeeded in giving a complete account of these phenomena by the study of the anatomical structures of the various leaves used in the experiments. The speaker begins with those cases, constituting the majority, in which starch-formation only took place at the expense of the respiratory carbon dioxide. The carbon dioxide of respiration, produced by the living cells, will of course diffuse into the intercellular spaces, and as these are connected up for longer or shorter distances, we may indeed assume that a transportation of carbon dioxide will in the first place take place by diffusion along this route. Further we must consider, that the veins generally have far fewer intercellular spaces than the parenchyma; indeed, they may have none at all. In this connexion a fact deserves notice, to which repeated attention has already been drawn in the above, namely, that the starch strips were generally sharply defined by veins on the side opposite to the carbon dioxide supply, so that it gave the impression, as if these formed a barrier across which the starch-formation could not extend. The edge of the starch strip was therefore frequently toothed in an irregular manner. In the Dahlia leaf, which is coarsely reticulate, the strips of starch came out largest in all experiments; especially in those places where the veins happened to be a little more remote from the border-line between carbon dioxide production and carbon dioxide consumption, the starch had spread furthest, and then there was often no sharp delimitation. In leaves with very fine meshes between the veins, such as those of Aesculus and Acer, the starch zones were correspondingly narrow. In parallel-veined Monocotyledonous leaves on the other hand, as for instance in the experiments with Acorus, Zea, Hordeum, Triticum and Tradescantia, there was no relation between the edge of the starch strip and the small transverse veins. On the contrary, in these leaves the starch strip was generally seen to be straight on the side facing the apex of the leaf and not sharply defined; it faded away gradually, albeit fairly rapidly. 8 These observations led to an anatomical investigation of the various leaves used in the experiments, especially with a view to answering the question, to what extent their veins, in the absence of intercel- lular spaces, formed barriers, across which the carbon dioxide could not move at all, or only very slowly. Should this really prove to be the case witb the leaves employed, then the simplest interpretation of the observed facts would be, that the carbon dioxide can indeed ( 663 ) be readily distributed through the intercellular spaces by diffusion, but that this distribution can be limited by the wholly or partially closed tissue of the veins. In that case one would come to the con- clusion, that such leaves are divided up by smaller and larger veins into areas, within which carbon dioxide transportation can readily take place. The passage from any one such area to another is difficult however, or quite impossible. The distance across which carbon dioxide can be transported in such a leaf will therefore depend very largely on the average size of the transport areas in the leaf. The anatomical investigation showed, in the first place, that in the case of net-veined Dicotyledonous leaves, the conception which has been worked out above, completely explains the phenomena observed. Mr. Zijrsrra indeed found, that in these leaves veins, which take up the whole thickness of the leaf, are devoid of intercellular spaces. A leaf such as that of Dahlia, in which similar transverse veins occur only at comparatively long intervals, will have large transport areas, and will be able to form relatively wide strips of starch. On the other hand in leaves like those of Acer and Aesculus, in which numerous vein branches occur close together, and take up the whole thickness of the leaf, we can only expect to find narrow starch strips such as indeed occur. The transport areas, even in the Dahlia leaf, which isin this respect in the most favourable condition among reticulate leaves, are neverthe- less very small, certainly much smaller than 3 cm. in diameter, as simple inspection of the leaf shows. It is therefore evident, that in the first apparatus, in which the part of the leaf under the mercury measured 3 em, these leaves were bound to give negative results, as regards the conduction of carbon dioxide supplied to the leaf base. In all experiments with these leaves, an idea of the carbon dioxide transportation could, however, be obtained from the starch-formation which took place at the expense of respiratory carbon dioxide, derived from other parts of the same transport area. It is also evident that in these leaves the edges of the starch strips must often follow the irregular course of the veins and must suddenly cease at the veins. Only in those cases, in which only a small portion of the ieaf area was in the dark and so could produce but little carbon dioxide, it might be expected that the large lighted portion could not fill itself completely with starch, and that the edge of the starch strip would not be sharp. Places in which this could be observed, were indeed pretty frequent in Dahlia leaves. An important question now arose, as to the condition of the various parallel-veined leaves, which, in the experiments described above, aS = Proceedings Royal Acad. Amsterdam, Vol. XI. Ld ( 664 ) also formed rather narrow and not very sharply defined starch strips. As regards the grass leaves of Hordeum, Triticum and Zea Mays, the transverse veins, which connect the longitudinal veins, are here indeed insignificant; the vascular bundles by no means fill up the whole thickness of the leaf and much parenchyma remains above and below. Investigation showed, however, that in transverse section the intercellular spaces of this latter parenchyma are very narrow, although they extend pretty far longitudinally. When passing through this parenchyma above and below the veins, the carbon dioxide is therefore checked very much more than in the general parenchyma between the veins, and its course must be much less rapid, although it is not stopped completely. In accordance with this, only narrow starch strips were formed in these leaves in the limited duration of the experiments, which lasted generally for 6, or at most for 7 hours. In other words, the carbon dioxide transportation was very limited, so that a transference of carbon dioxide across a greater interval than 3 em, in the apparatus first described, was an impossibility. The transverse anastomoses of the veins did not however, sharply define the starch strips. The leaves of Acorus and Tradescantia behaved similarly in the experiments performed. In the green parenchyma of Acorus only small intercellular spaces occur and here some veins moreover take up the whole thickness of the leaf, the colourless central parenchyma of the leaf contains it is true many large spaces which extend longi- tudinally, but at frequent intervals they are shut off by transverse cell-layers, diaphragms without intercellular spaces. Lastly Tradescantia has a very spongy assimilating tissue, but in it many vein-anasto- moses occur, which only have minute intercellular spaces. In these cases too therefore the agreement between the anatomical structure and the experimental result was sufficient to warrant the acceptance of the above view. Finally the question arose how the intercellular spaces are distri- buted in the leaves of Pontederia, EHucomis and Eichhornia, which, as is evident from the experiments of table 2, are much better adapted for carbon dioxide transportation, so that in the apparatus employed this gas could he carried from the leaf base to the apex. The anatomical investigation of these leaves yielded the following result. The leaf of Mucomis is parallel-veined, the whole of the leaf-parenchyma is very spongy, and the longitudinal as well as the transverse veins are very insignificant, so that carbon dioxide can everywhere pass freely. There are still better gas passages in the curved-veined leaves of Eichhornia and Pontederia. In both these ( 665 ) species the parenchyma contains air-channels, running continuously from the base to the apex and taking up from one third to one half of the area of a transverse section. In these channels there are, it is true, diaphragms of one cell thick, but these are themselves also provided with many wide intercellular spaces. | It is therefore merely a matter of course, that in these leaves carbon dioxide can be carried over much larger distances; indeed, it is clear that this carriage could extend from the lowest part of the base to the very tip, given a sufficient duration and suitable arrangement of the experiments. It is likewise quite natural, that no veins form a sharp boundary to the starch strips on the side towards the leaf apex. These observations therefore also completely confirmed the view, that the above conception of carbon dioxide transport in leaves is the correct one. At the same time it will be clear that fundamentally this repre- sentation is the same for all the leaves examined. In net-veined leaves the transport areas are small and very sharply defined; in the parallel-veined leaves of Grasses, Acorus and Tradescantia they are small but less sharply defined; in the leaves of Hichhornia, Eucomis and Pontederia the whole leaf is one transport area. If it were possible to make experiments with the two first-named categories of leaves in an apparatus of the first type described, but of much smaller size, positive results as regards carbon dioxide transportation would then be obtained as readily as bas now been the case with leaves of the third category only. Lastly there is the question over what distance the carbon dioxide transport can extend in various leaves. We have seen that in Hucomis, Pontederia and Hichhornia carbon dioxide can be transported through a piece of leaf 3 cm. long under mercury and then even 1.5 em. farther through the apical portion, which was placed in air, free from carbon dioxide and of which the stomata were closed. As has already been said, we may assume that this distance is by no means the maximum one, but that it might be increased at will, on condition that the duration of the experiment were also increased as much as possible. In all the other leaves, however, it was found that the carbon dioxide could not reach the apex through the 3 cm. long portion. With some of these leaves experiments were now made in order to determine the maximum distance, through which carbon dioxide could be transported during the course of the experiment. These experi- ments were arranged as follows, The leaves, freed from starch, were 45* ( 666 ) completely covered with cocoa-wax and then, here and there, with strips of paper of various widths; between these strips portions of the leaf remained exposed to the light. ‘Borders of starch were then formed along both sides of the black strips of paper, since the respiratory carbon dioxide, formed under the paper, escaped on both sides. The width of these starch borders was slight in the case of the narrowest strips of paper, because beneath these but little carbon dioxide was formed; it increased with the width of the strips and of course reached its maximum as soon as the half-width of the black strip of paper corresponded more or less to the maximum distance through which carbon dioxide could pass during the time of the experiment. The half-width of that strip of paper, at which the starch borders just reached their maximum width, was therefore a measure of the distance through which the carbon dioxide in the leaf could pass under the given conditions. Such experiments were first made with parallel-veined leaves, in which the carbon dioxide transport was only limited by the small dimensions of the intercellular spaces. The result was, that in 7’riticum (exp. LVII) the carbon dioxide could be transported in 6 hours over at least 2.5 cm., in Acorus (exp. LVIII) in 6 hours over rather more than 1 em, in Tradescantia (exp. LIX) in 5 hours over less than 1.5 em. There is every reason for the assumption, that in these leaves, in experiments of longer duration, somewhat higher values might have been obtained. In net-veined Dicotyledonous leaves the case is somewhat different, for, as we have seen, in consequence of the absence of inter- cellular spaces from veins of a certain order, the carbon dioxide transportation is strictly limited to definite areas, the size of which may be very different in different leaves. If these areas are minute we observe with the narrowest strips of darkening paper also the maximum width of the starch borders, the absolute dimensions of which are likewise minute. This was the case in Juglans, Aesculus and Tilia, in which the distance of the transport could not be accurately determined, but certainly did not exceed 2—3 mm. In Dahlia and in Sambucus the transport areas referred to are relatively very large, the starch borders along the narrowest strips of paper are narrowest, and they increase to a certain maximum width -along the wider strips. Here, however, this method cannot give very accurate results. For if a transport area is darkened over its greater part, ther much carbon dioxide is indeed formed in it, ( 667 ) but this can only cause starch-formation in a small part. Conversely, if only a small part of the area is darkened, the starch-formation can be observed at a relatively large distance, but then too little carbon dioxide is formed in the small darkened portion to give rise to starch-formation in the more distant parts. With this reservation an experiment may be mentioned, which was made with the leaf of Dahlia (Cactus) Thuringia (exp. LX), with the result, that the carbon dioxide can here be carried over through at least 0.5 cm. In summarizing his communication, the speaker points out that Mr. Zisustra has shown that transport of carbon dioxide is possible in all the leaves examined, and that it takes place through the inter- cellular spaces. The transport is completely dependent on the size and extent of these spaces in the leaf. In some parallel-veined leaves, such as those of Hichhornia, Pon- tederia and Eucomis, the intercellular spaces are very wide and extend in an uninterrupted series throughout the whole length of the leaf. By the use of suitable apparatus the leaf base can be made to absorb carbon dioxide, which moving on through the intercellular spaces by diffusion can give rise to starch-formation a comparatively long way off in the leaf apex, when the latter is exposed to light. In the great majority of leaves however, carbon dioxide transport cannot be shown with the relatively crude apparatus employed. In such leaves the carbon dioxide transport can be studied by another method, which utilizes the fact that respiratory carbon dioxide, which is formed in a darkened and shut off portion of the leaf, can diffuse from there through the intercellular spaces to neighbouring lighted portions of the leaf and can there cause starch-formation. Such leaves possess limited transport areas, which are formed by spacious and connected intercellular spaces, and wliich are either connected at their margins through much narrower intercellular spaces, greatly retarding carbon dioxide transport (Grasses, Acorus, Tradescantia) or the areas are completely cut off by veins, which have no intercellular spaces (net-veined Dicotyledonous leaves). In these leaves with limited transport areas a carriage of carbon dioxide over a distance of from 2—3 mm. to at most 2.5 cm. is possible. Finally the speaker draws special attention to the impossibility of carbon dioxide transport being of any advantage to the plant in nature, in the first place, because this transport is so extremely limited in the majority of cases, and in the second place especially because for transport it is necessary that the conducting part should not itself assimilate, and also, that the epidermis should be impervious ( 668 ) to carbon dioxide. These conditions will doubtless never be fulfilled in land plants, in water plants perhaps very exceptionally. It has therefore been established by Mr. Zisistra’s investigations, that the speaker was wrong when, in his above cited paper, he came to the conclusion, that a leaf or leaf fragment cannot form starch in a space free from carbon dioxide, when parts organically connected with it, or even immediately adjoining it, are placed in an atmosphere very rich in carbon dioxide. Mr. Zijrsrra’s results are however, in complete agreement with the main result, formerly obtained by the speaker, according to which the carbon dioxide of the’ soil, even if it should be absorbed by the roots, cannot appre- ciably contribute to the synthesis of organic matter in the leaf. Groningen, January 29%, 1909. Microbiology. — “/nwestigations on the subject of disinfection”. By Prof. C. ErkMan. Last year I communicated results of experiments ') from which it appeared that the resistance against high temperature of bacteria of the same pure culture is individually very different. While for example the majority die off in a few minutes, some may remain alive after ‘/,, */, hour, etc. If the times are noted on the absciss and the corresponding numbers of survivors are drawn to it as ordinates, we get as “curve of survivors” a line which in general has the form of a \. In a slow process, as it occurs when the mortal temperature is taken relatively low, the first part of the curve shows itself clearly as an horizontal line and therefore represents a latent stage of incubation. Notwithstanding this the period within which the first half dies off, is much shorter than the following, in which the second half passes away. In a quick process, as is observed when the temperature is far above the physiological limit, the duration of the incubation will become so brief that it easily escapes notice. In connection with the inevitable circumstance that the number of observations in this kind of experiments cannot be increased arbitrarily, but is confined within a rather definite period, the curve may, instead of the \ form, assume the shape of a \. The latter has also come to light in investigations published the other day by MADsEN & Nyman, *) which differed from mine in so 1) Biochem. | Zeitschrift, Bnd. XI, Hft. 1—3, Festband Dr. H. J. HAMBURGER gewidmet. ® Z.f. Hyg. u. Inf. Kr. Bnd. LVI, (669 3 far as they were not made in vegetative forms of bacteria, but in (anthrax) spores and that the dying off for the greater part was not brougbt about by heat, but by a chemical means of disinfection, viz. sublimate. . The said investigators think that they are able to give a mathe- matical formula for their curve. They assert namely that the same formula is applicable here, which holds good for the so-called monomolecular reactions, e.g. for the inversion of cane-sugar by acid: dx K ae (a—x). In this formula a represents the number of living anthrax spores that was originally present, w the number that has died off after a space of time ¢, and K a constant, expressing the velocity of reaction, i.e. the velocity of disinfection. In other words this formula means that during the entire process the number dying off at any moment, is in a constant ratio to the number of living individuals present at that moment. Therefore this A’ would yield a very suitable measure to judge about the action of a disinficiens under certain circumstances (of temperature, concentration, ete.). A much better measure than the one customary up till now, viz. the space of time necessary to destroy all germs. For it follows from what precedes that this space of time is to a high degree dependent on the number of germs which in the experiment has been started from. With this number the chance increases that there are some among them which ‘offer resistance extremely long. On the other hand & is not to that degree dependent on the number of germs used in the experiment *) and in order to calculate it, the experiment need not even be continued till all germs have died off, but two determinations of v at arbitrary points of time would suffice. For 1 A—e MS - In —— +) t,—t, A—aQ, 2 These experiences of Mapsen and Nyman have been not only corroborated by an English investigator, Miss Harrinrr Crick *), for anthraxspores and sublimate, but she has stated a similar course of the curve also for the action of three disinfectants on vegetative 1) Our experiences render it probable that a great number of germs per unit of volume somewhat retards the process of dying off. 2) See the text-books about physical chemistry. 3) Journal of Hygiene, 1908. ( 670 ) forms of bacteria, only with this deviation that towards the end the velocity of reaction was decreasing, instead of remaining constant. On the score of her experiments she is inclined to attribute this deviation to a difference in resistance between the individuals of various ages in the same culture. There would exist, as it were, an old and a young generation by the side of each other, the latter of which dies off slowest. Such a difference in connection with the age does not exist in spores to the same degree. Of the curves published by me, it is, however, not only the tail, but also the head that shows a deviation. The course is here, still apart from the incubation, much slower than according to the formula. At the most the middle part is, stating roughly, in accord- ance with it. Meanwhile it seems to me that this kind of investigations is hardly fit for a mathematical treatment. ; MADsEN & Nyman, for example, avail themselves of means, resulting from numbers of three values found, which deviate 25°/, and more from these means. An example from many *): found: 193, percentage of the average: 74.5 330, 5 ee jn 127.4 ka. WO, iy eae a 98.1 average: 259 100 And if the numerical results of Miss Crick are looked at somewhat more closely, they, too, do not appear to be more exact. Sometimes the errors in the observations are so great that, instead of the expected gradual decrease, here and ihere an increase of the number of survivors was in course of time to be noted’). It may be called objectionable, as Mapsen & Nyman do, to rid oneself of the deviations between the numbers determined experiment- ally and those calculated according to the formula, by remarking: ‘Wenn man die grossen Versuchsfehler, die an dieser Art von Unter- suchungen kleben, in Betracht nimmt, ist die Uebereinstimmung eine recht gute’. It is true, a line may be drawn between a number of points determined experimentally, leaving one point to the left, another again to the right, but when, as is not very seldom the case here, the deviations from the regularity are considerable, imagination and arbitrariness will get too large a scope to inspire confidence in the correctness of a curve construed in this way. 2 Haler Table XI: 1. e. Table III and X, 9 x Seeing that my results did not well agree with those of the above investigators, and this difference might possibly be based upon the fact that the dying off of the microbes was brought about by heat and not by chemical means, I have extended the investigation in this direction. Again bacillus coli communis was made use of, a bacterium forming no spores, while as disinfectant was used phenol in a concentration of at most 1 °/,, generally only */,°/,. The use of higher concentrations would make the process of dying off pass so quickly, that the time for a sufficient number of determinations would be too short, and among others the stage of incubation, if at all existing, would easily escape observation. In order to have no great differences between the individuals and accordingly to render the conditions as little complicated as possible, as a rule a fresh (broth) culture, only a few hours old, incubated at 37°, was taken for the experiments, which culture, in its turn, had also been obtained by inoculation from a fresh culture. For the same reason the broth culture was slowly moved to and fro in a tube specialiy made for this purpose, which in an apparatus moved by a time-piece had been placed in the thermostate. Consequently all individuals were in well nigh equal conditions of development, so that the results of the experiments were more likely to be equivalent. Before we used the culture for the experiment, it was centrifugalized in order to remove the clots of bacteria, which were probably to be found in it and for obvious reasons would have a disturbing effect. Besides it was strongly diluted (+ 1000 times) with physiological common salt-solution. It would be necessary, in order to prevent sowing too many bacteria, to take of the non-diluted broth culture such small samples that, in measuring these, inevitably relatively too great mistakes would be made. The vessel with the diluting fluid, provided with the necessary quantity of the disinfectant, had already beforehand got the required temperature in a waterbath with a toluolregulator and an automatic stirring-apparatus. After the inoculation with the broth culture the mixture was constantly kept in motion by a glass stirrer, in order to make the disinfectant work as equally as possible upon all germs. As it is of great importance, to take the samples in rapid succession and just in time, I availed myself for this purpose of a peculiar kind of pipettes, which in case of immersion fill themselves automati- cally to the required height, so that the measuring, which takes up so much time, was avoided. The samples were put in Petri-scales and sown in melted, lightly alkaline reacting meat-agar. The develop- ment took place at 37°. The phenol put in the culture-plate together ( 672 ) with the samples was bound by the alkali and, also because the the quantity was relatively small, it did not disturb the development of the colonies, as appeared from control experiments. In the graphical representation of the results, to render a mutual comparison easier, a number of 1000 living germs has been started from and the values found experimentally have been reduced accordingly. As proceeds from the figures, the type of the “curves of survivors’ is in our disinfection-experiments with phenol quite similar to the one which was found in dying off by heat; very clearly the \ form is again to be recognized in it. 200 700 Ar (am Bac. coli 1000 PES £ 900 2 800 ee AD So 600 =. S00 < El 400 beki as BS ae 0 6 oO Ler} 5: lal LV S N 2% bo B | oO en! Co Nay as S Nv 172.73 14.13 Minutes Fig. 1. As in our experiments nothing has been left undone to put all the individualls, both in the process of incubation and that of disinfection, under quite the same conditions, a very marked accumu- lation of deaths might have been expected on either side of an average. In reality, however, this was not the case (most, though, with a in fig. 1) and again considerable differences of resistance between individuals of the same culture came to light. For this I see no other explanation, though it remains for the present only a mere supposition, than that the power of resistance during the development between two successive divisions undergoes changes. It may for example be imagined that under for the rest equal circumstances a daughter-cell just formed is, on account of her relatively ‘larger surface, more vulnerable than a full-grown cell. And because the length of generation is relatively short, amounting, in strong multiplication, to less than half an hour, all stages of development will occur by the side of each other. ( 673 ) Bac. coli 70009 00 a ea : g 800 Sui eS ee. Ge 3 =" 600 _ 8 2 Be ad Bd x se ge 5 00 5 aal 3 Bit ee wd z 5 0 Cs ® Minutes Fig. 2. As in our former experiments, when the bacteria were killed by heat, we now also experienced, while using phenol as a disinfectant, that, though cultures of one stock are used at every time, yet we did not succeed in coming to somewhat equal results. This is taught by a look at fig. 1, in which a number of eight curves have been drawn, referring to experiments, made at different times each with fresh cultures of the same colistock. The concentration of the phenol and the temperature at which the disinfection took place, were in all cases the same and yet for the greater part quite different curves were obtained. Bac. coli 1000 Wee Rn A A La Neha desta rolde eee en : Boole EE FE El Sa a ° Ps ede fe eae a ak [bn & Ss 4500 3 A — a TS SE Ss ve = 400 } ESR | O0 rea 0 En 5 Eed ae EEN ea zE el 0 Pore ei 8 x 0 2 4 6 8 10 1214 16 18 20 Minutes Fig. 3. Owing to this, it will not do merely to compare the results of one experiment with those of another, unless both are made with the same culture and about the same time. Therefore strictly speaking parallel experiments are necessary, if one wishes to study the influ- ( 674 ) ence of some factor or other, as the concentration, the temperature, on the course of the process. MADSEN & Nyman and also Miss CrrcK derive from their obser- vations that the influence of the above factors may likewise be expressed in formulas. Thus the well-known formula of ARRHENIUS in which the relation between temperature and velocity of reaction is expressed, would also hold good in this connection. It seems to me, for reasons already mentioned, to be prudent not to follow them on this path. Therefore we refer, with regard to the points meant, to the figures 2 and 3, without commenting on the subject. Geology. — “On a long-period Variation in the Height of the Ground-water in the Dunes of Holland.” By Prof. Eve. Dusots. (Communicated by Dr. J. P. VAN DER STOK). Unmistakable and obvious is the lowering that the height of the ground-water in the dunes of the provinces of North- and South- Holland has undergone in consequence of the lowering of the level of the water at their east border (the making dry of the Lake of Harlem and of a large part of the IJ) and of deep cuttings in the dunes themselves (North-Sea Canal), furtheron, not less, by the collecting of large quantities of water supplies for some cities and towns. From these causes there resulted a lowering which may be called a permanent one, inasmuch that soon they have brought about a new state of equilibrium with the supply by the part of rainfall which soaks in, and the flowing off. This really did take place in each case in which certain limits were not transgressed and as long as the collecting of water did not increase. Side by side with these artificial changes of the height of the ground-water in the dunes, there exist also changes by natural, viz. climatal causes. These, in this as in other cases, are not continuous, but they do occur in periods. Indeed, in the latest historical past, as far as data are available, very clearly dry and wet epochs alternate with one another. The Commission which, in 1891, inquired in the supplying of water from the dunes to Amsterdam pointed out, in their report, that from 1849 till 1856 there was a period of much rain, from 1856 till 1868 a dry period, again followed by the rainy years of 1869 till 1882. They showed also (for Utrecht) that under the combined in- fluence of rainfall and evaporation such wet and dry epochs are ( 675 ) found, with maxima about 1855—56 and 1882—1883 and a mini- mum about 1869—70. In his “Hyetography of the Netherlands” Mr. ENGELENBURG *) inquires if there exists any relation between sunspots and rainfall. He finds that we can admit that such a relation, if it exists at all, does not appear very distinctly. It is the merit of Dr. LAaurENs Vuyck, in his elaborate treatise on the vegetation of the dunes,’*) to have submitted the problem of the change of height of the ground-water in the dunes, as it also appears from older writings, to a close investigation. He however thinks of a progressive drying up of the low places in the dunes, having slowly taken place from as long ago as the end of the eighteenth century. From a careful consideration of the problem he arrives at the conclusion that the cause can only be found in a continuous and imperceptibly ‘slow filling up of those low places with eolian sand. The earliest intimation of the dunes drying out, which ought once more to be quoted, is from the end of the 18 century. In the report of a committee from that time to the Government, the reporter, JAN Kops*) makes mention of the fact, that at the time of his inquiry, in 1797, the obstacle to the culture of the dunes arising from an excess of ground-water has been removed for a great deal. “In all our inspections, in the North as well as in the South, the most experienced people told us unanimously that during the last ten years, from year to year, less and less water than before is found in the plains amidst the dunes. They showed us places which for- merly stood two or three feet under water and became extensive icefields for winter-sport, but now were only somewhat muddy in winter. In other plains, only four years ago, there stood still water in spring, from which nothing now is to be seen, and so it is with almost all the dunes. This particularity has raised our highest atten- tion and surprise, as on account of the well known and alarming rising of the level of our rivers and inland water, we should expect the sinking down of the water of the dunes to be checked and prevented by it. Nobody was able to explicate to what cause this decrease of the dune-water should be imputed and we too could not trace out the true cause of it. But where it is to be sought for, this circumstance is most favourable for all following undertakings in the dunes.” (p. 114 and 115). 1) Physical Transactions of the Kon. Akad. v. Wet. Amsterdam 1891. 2) Laurens Vuycx, De plantengroei der duinen, Leiden, 1898. 3) Rapport van de Commissie van Superintendentie over het onderzoek der Duinen van het voormaalig Hollandsch Gewest. Leiden 1798/99. ( 676 ) So it cannot astonish us that in 1805 A. P. Twenr’) makes mention of great dryness, making the birches in the plains amidst the dunes die at the tops, and that for three years there had been no water in places where in earlier times it always had been found, even in summer. “Considering that the sea does not be lower now than before, as shown by all circumstances, of which not the least certain is that the outlets of the inland waters are not improved in this part, this matter deserves double consideration by the naturalists”. In 1816 and still in 1828 a quite different state of things prevailed, as appears from the prize-essay on the making accessible of the valleys in the dunes along the coast of Holland by D. T. Gevers, ?) an answer to the question: where and how to drain the water from the plains in the dunes and at the same time facilitate the access to them, in order that they may no longer lie useless and uncultivated. This, namely, was imputed for the most part to the want of the necessary evacuation of the water. This inconvenience then was very great, and its removing indeed was the chief purpose of the large treatise. Concerning a following dry period in the dunes no direct infor- mation has come to my knowledge. It appears however that from 1831 till 1840 the rainfall at Zwanenburg (Halfweg), that is very near the dunes, has been considerably below the average. *) Certainly the level of the ground-water in the dunes of the com- munes of Zandvoort, Bloemendaal and Velzen, as well as in the dunes of the province of South-Holland (cf. Vuycx, ].¢. p. 184) was very high about 1845, so that for instance people skated in the dune-plains near Zandvoort. On the contrary, about 1860, the ground-water in the dunes stood only little higher than in the present period which is get very dry ; the water holes which contain now but little water were sufficiently but not abundantly provided. Though after 1858 the water supplies to Amsterdam, from the dunes, became less than in the former years and remained so till 1864, yet it was necessary to make new water collecting canals. Then again follows a wet period, during which many plains and lower places in the dunes became marshy or were drowned in winter, frequented by a number of waterfowl (ducks, pool-snipes) and in some spots remained occupied by water even in summer, so that water- 1) Wandeling naar de Zeeduinen van Wassenaar tot digt aan Scheveningen, p. 5. 2) D. T. Gevers. Verhandeling over het toegangbaar maken van de duinvalleien langs de kust van Holland, uitgegeven door de Maatschappij ter bevordering van den Landbouw te Amsterdam opgerigt. Deel 18, Amsterdam 1826. 5) Nederlandsch Meteorologisch Jaarboek voor 1878, p. 288. ( 677 ) plants could flourish there. And such a state of things obtained in all the dunes, outside the influence of the large water works, arriving at a maximum about 1880. A few years later the present dry period commenced, by which, also independently of each artificial cause for a lowering of the ground- water in the dunes, its height decreased so much that in the present winter water is only found at 2 M. below the level surface of plains, which about 1880 were flooded in winter. Certainly no less than 2 to 2'/, M., in some places probably more, the ground-water is now lower, by natural causes alone, than it was in those wet years. The periodical changes of the level of the ground-water in the dunes, thus appearing during more than a century, agree in striking conformity with the thirty-five-year period discovered by Prof. Ed. Brickner, according to which, in almost all the countries of the earth, the rainfall and the height of the water of lakes changes. Really between 1786 and 1805 a dry period occurs (for the Netherlands too this appears from the rainfall as determined at Zwanenburg). So in the beginning of the 19 century a minimum of rain and the lowest watermarks were reached. Then follows an epoch of much rain between 1806 and 1825, again a dry period from 1826 till 1840, a new period of much rain from 1841 till 1855, a dry period from 1856 till 1870, with a minimum about 1860, the latest period of much rain from 1871 till 1885, with a maximum about (880, finally again a dry period, with a minimum about the end of the nineteenth century. A few years ago we still were in this dry half of the cycle. Since, some years ago Dr. WirLiam J. S. Lockysr*) proved, that in the amount of spotted area of the sun also, a thirty-five-year period could be traced, from 1833 till 1900, the discovery of Brickner surely has ‘still gained in importance. On the other hand we now understand better what may be the cause of the few temporary or lasting deviations of some countries. If the better insight we now have got in the cause of the phenomenon discovered by briickNer is well adapted to increase our confidence that we have to count with it in future, we also need not suffer ourselves to be prevented from this by the deviations in question. Lately, from the extensive study of the rainfall in Germany by Dr. G. Hettmann’) and its discussion by Brickner *), it again appeared how in our vicinity wet and dry periods, and pretty weli simultane- ously with those in the dunes of Holland, alternate with one another. 1) Proceedings of the Royal Society. Vol. 68. (1901). p. 285—300. *) Die Niederschläge in den norddeutschen Stromgebieten. 3 Bande, Berlin, 1906, °) Meteorologische Zeitschrift. Wien 1906, p. 565. RAINFALL. | Progressive decadal sum above or below mean | Millimeters decadal sum over 50 years, percents. = regs, | Soma = a SIS JES EE CLA |) eee = az SSE = = 1849 735.3 1850 812.3 1851 681.2 1852 1048.(./876.0 1853 |489.7| 737.5/662.9 | 53/54 3 1854 |687.9) 821.1/715.2 | 54/55 2 | 1855 |543.6| 640 9/478.2 55/56 | —1 2 2 1856 |517.7| 760.7/651.2 56/57 | —2 1 NT 1857 |385.8) 449.4|459.1 57/58 | SSD IS 1858 |417.1| 634.9/637 1 5850) cl trio |e 8 1859 |508.7| 677.9|569.4 59/60} > Tops —b ABR eta | 12 4860 |628 5| 810.8/730.7| 874.8|| 60/61 | —9 | —6 | —18 | --12 | —9 1861 |510.0] 663.4/670.3] 710 7|| 61/62} —8 | —5 | —15 | | —4 1862 |566.2| 590.8/6 0.5} 659.0/| 62/63} —3 ie es RE 1 1263 |434.9| 524.7/622.0] 588.8/| 63/64 | —A Ni re, 0 164 |363.3] 459.7/448 0} 551.9/| 64/65 | A hal bi) 2b 2 1 1865 |549.5| 710.2|696.4| 822.u/| 65/66| 2 | A | —6 | —7 3 2 1866 |722.3| 812.6/961.4/1065.5|| 66/67 | —3 | —A | —6 | —7 2 2 1867 |717.4| 686 1/780.8|1047.7|| 67/68 | —2 On hr ees 4 7 4868 |513.3| 674.4/602.4| 7:5.6|| 68/69; —3 | — | —5 | 2 4 | 40 1869 |625.5) 786.9/695.7/1013.0)) 69/70 | —2 0} 3 2 7 1 ae 1870 |583.7| 744.1|791.4| 904.7|| 70/7! 0 3 1D 3 KE 1874 |504.3| 640 1/603.2| 764 0|| 74/72} —4 ON RE 2 2 | 44 4872 |607.4| 878.7|785.7|1075.5|| 72/73 | —A 1 0 4 3 | 10 4873 |505.8| 576.3/628.1| 803.9|| 73/74 0 Sie eae 5 6 | 12 1874 |481 6! 784.41644.3! 817.9|| 74/75 4 AED 3 6 | 40 4875 |724.4| 787.4|565.4] 208.2|| 75/76 3 6 2 4 2 8 1876 |631.8| 703.7|754.0| 845.8] 76/77 3 Ghee 6 lone 10 4877 |746.6| 826.4/845.4| 986.8|| 77/78 6 B 80 7 5 | 40 1878 |711.4| 733.2|827.7| 923.8]| 78/79 vj 9 | 44 7 4 | 10 4879 |561.3| 698.0/664.5| 803.4|| 79/80 8 9{ 42 5 4 8 Millimeters RAINFALL Progressive decadal sum above or below mean decadal sum over 50 years, percents. = = - = | ERMEER = ~ rs = Year £ 3 3 2 | Year £52 Sm5 5 3 3 3 EE chor fie Eat | SAS oh Da Bi he | ee 1880 |602 8| 770.7/536 0| 790.9) 80/81 7 10 3 5 8 1881 |730.2| 773.2/627.4) 913.9) 81/82 7 7 7 3 5 8 1882 |799.0/ 952 6/953.3/1029.5\, 82/83 4 4 Fal gee 1 3 1883 (624 6| 618.0/601.9/ 841 0, 83/84 4 3 Bee 0 1884 |568.0| 607.6|585.3] 682 2!| 84/85 3 2 Pi. Al ee 0 1885 |560.1| 633.3/656.4 805.4) 85/86 4 0 Ota oe en 0 1886 |495.9] 702 3|746.4| 801.1), 86/87 9 1 6 AT Sted 1887 |6l4 9| 473.3|601.4| 6243/8788 | —2 | —4 ey ees ee et 1888 |660.0| 668.2/597.6| 667 0/| 88/89 | —2| —5 | A | —2] —3| A 1889 |742.8| 874.1|639.4| 834 4| 89/90 | —1 | —4 0 0 Oi ve 4890 |701.7| 777.7/606.0| 743.7|| 90/91 | —2 | —4 2 1 4) 8 1891 (522.3| 788.3/587.1| 854.2) 91/92 | A | —4 2 A Ee | ee 1892 |506.7| 750.8/771.7| 916 .4l| 92/93 Oils 1 5 oy =3 1893 |542 7| 711.9|702.3| 710.4/| 93/4 | A | —6| —2 6 a at 1894 |633.8] 747.1|847.7| 848.3|/ 94/25 | —1 | —6 | —6 3 {ate L8 1895 |668 2! 743.1/678.2] 772.7, 95/96 | —1! —6 | —7 2 Dn Oe 1896 |495.9| 681 4|632.1| 628 4l| 93/97 —8 3 9) —5 1897 [553 9| 733.0)658.2| 794.5/| 97/98 =F 1 Di Sn 1898 |468.2| 724.1/651.8] 791.2/ 98/99 5 4 EN en 1899 |527.5l 691 01637.0| 633.8 99/1900 A) Blk auth eo 1900 |630.6| 724.2|704.7| 767.6||1900/01 5 Fae bag 4901 |467.0| 817.6/608.8| 730.4|| 01/02 as A eee A 1902 |527.8| 631.3/604.9| 659.0/| 02/03 0 a ee lore 1903 |700.3| 925.9/887.8| 893 6) 03/04 3 4). de 40 1904 |481.0| 596.1/580.3) 660.4 1905 |782.4| 776.0/579.5| 777.6 1906 (7725) 723.9 654.1] 808.4 1907 |896.7| 639.7/607.3| 632.3 1908 |630.6/ 643.8/539.2) 629.9 | | Av. |596.1| 713.2|672.3| 797.0 46 Proceedings Royal Acad. Amsterdam. Vol, XI, ( 680 ) In the accompanying table I tried to make clear the alternations of dry and wet periods of rainfall for some stations in the Netherlands, by the decadal sums, progressing from year to year, according to the method made use of by Brickner. In the first place we take Maestricht having the most continental situation of all the meteoro- logical stations in the Netherlands with many years’ determinations of rainfall, and being nearest to the basin area of the lower Rhine in Germany (Cleves, Bonn, Triers, Nancy), then Utrecht as intermediate between that station and Helder and Leiduin, which most of all the old stations may approach to the conditions obtaining in the dunes. The fifty-years’ average rainfall of the three first-named Dutch stations is computed over 1859 till 1908, the average of Leiduin over 49 years only, beginning with 1860, the averages for Germany are over the years 1851 till 1900. The double years indicate the middle of each decade. Now, what we observe is a close agreement and conformity to the rule of Brtckner, generally, till 1882. In the last (dry) period, however, Helder and, in a still higher degree, Utrecht show important deviations. Maestricht, on the contrary, agrees very well and Leiduin tolerably with Germany and at the same time with by far the majority of all the countries of the world. Also after 1896, till 1900, also in the basin of the lower Rhine in Germany, just as at Maestricht, the rainfall remains considerably below the fifty- years average. About the beginning of the twentieth century the expected change took place every where. In the lower basin of the Rhine in Germany the rainfall in the years 1896 till 1900 exceeds the fifty- years’ average by 4, 8, 10, 6 and 4°/,, and in the latest lustrum the rainfall at Maestricht was nearly 10°/, above the average, but not so at the three remaining stations, where deficits of 5, 11 and 12 °/, were observed, the wet period evidently not yet having com- menced. On account however of the agreement with regular regions, during so long a time, as well as of the circumstance, that anticipation or retardation of an epoch is accustomed to be regained in the next period, the probability must not be called small, that for the whole of the Netherlands an epoch of increased rainfall and of higher ground-water levels is at hand, and that especially in the dunes too the ground-water will soon be rising. | Undoubtedly not only the annual rainfall, but also the evaporation is of consequence for that rising. But we know that evaporation is relatively small in periods of much rain and that, generally, the ground-water rises and falls with the amount of rainfall. Variations in the rainfall are very strongly indicated by the height ( 681 ) of the ground-water in the dunes. This not only is a consequence of this, that water enclosed in sand must rise three times as much as in an open basin, the supply being equal; but also of that circumstance, that the sand in question is particularly loose and composed of grains with equal dimensions, thus readily absorbing and giving way to the downfalling water. Further favourable conditions are that the ground is very uneven, not admitting of superficial flowing off, and only thinly covered with mosses, grasses, shrubs of Hippophaé rhamnoides, Salie repens and Ligustrum vulgare and with loose-crowned trees, especially birches. Really Mr. pr Bruyn found that, even during the dry years 1895 till 1902, at least half the rainfall served as a supply to the ground-water 5. Moreover the geological constitution of the dune region, where eolian sand reposes upon the very little permeable fine clayey marine sands (Old Sea-clay and Sea-sand of Staring), favours very much the accu- mulation of the excess of rainfall. The circumstance that so many low plains amidst the dunes, having subsisted during centuries, have undergone simultaneously quite the same up and down variations in the height of the ground- water, proves, as it seems to me, that we should not, generally, impute the becoming dry of the dunes to a successively filling up of those low places with eolian sand. Another proof of no less strength I find in the phenomenon, already observed and rightly explained by Gevers, that, on the whole, the surface line of the plains in the dunes runs parallel to the line of the ground-water, descending toward the sea and toward the polderland. It is indeed unconceivable that those remarkably flat and pretty well horizontal, often very extensive low grounds amidst the dunes, commonly called in North-Holland “vlakken” and “velden”, have had another origin than the sand being blown off — before the time that such blowing off was prevented by the planting of sand-binding grasses — till the level was reached where it was moistened by the ground-water, raised by capillarity to about thirty centimeters above its free level. Really we observe, as far as natural influences prepon- derate, that generally only where the character of the underground changes, making the water sink down accumulate in such places, in other places, these modifications in the geological structure modify the line of the ground-water, but at the same time, in consequence thereof, that of the surface of the dune plains. 1) Handelingen van het 9de Natuur- en Geneeskundig Congres, ‘s-Gravenhage, 1903, p. 148. 46* (632) Astronomy. — “On the periodic solutions of a special case of the problem of four bodies’. By Prof. W. pr Sirrer. (Communi- cated by Prof. E. F. vAN DE SANDE BAKHUYZEN). The special case considered in this paper is that of a central body and three planets, or satellites, whose masses are small compared with the mass of the central body, and whose orbits are all situated in one and the same plane, the mean motions (in longitude) being roughly proportional to the numbers +, 2 and 1. This special case is realised in nature by the three inner Galilean satellites of Jupiter, if the inclinations, the influence of the sun and of the fourth satellite, and the compression of the planet are neglected. This latter restriction is not essential, since the compression does not disturb the periodicity, provided only the motions take place exclusively in the plane of the planet’s equator. Neglecting at first the relation between the mean motions, we will consider the periodic solutions of the problem thus generalized for the case that the masses of the satellites are zero, i.e. for the unper- turbed problem. These may be divided into two kinds, analogous to PorrcarÉ’s well known classification of the periodic solution of the problem of three bodies. In the solutions of the first kind (sorte première of Poincaré) the (unperturbed) orbits of the satellites are circles, in those of the second kind they are Keplerian ellipses with arbitrary excentricities. The solutions of the first kind exist, if the differences of the mean motions are commensurable, thus: vy, — TY; = pr, Yv, — Ps = qv, p and g being integers, mutually prime. This condition can also be expressed by saying that the mean motions must satisfy a linear equation of the form av, + Br, + yr, — 9, where a, 2 and y are mutually prime whole numbers, satisfying the relation atp+y=—0. The mean motions can then be expressed thus: hy So, De t= ¢, Ps VY, = CV — %, 1 where ¢,, ¢,, C,; are again whole numbers. We have then: Led OC p = Cs — ¢, y= CT; De ts gt Then, if we put ( 683 ) and if we count the time from the instant-of a conjunction of II and III, and the longitudes from the common longitude of these satellites for that instant, we have A= Ct —v A, == CT —V A, =cr—vtK aA, BA, + yA, =X. After the lapse of the period 2% T= — v the relative positions of the four bodies are the same as for the instant ¢—= 0, the whole system being rotated in a retrograde direction through the angle x7’. By a reasoning entirely similar to that used by Porncars') for the solutions of the first kind of the problem of three bodies, we find easily that the condition, that these solutions shall remain periodic if the masses have small finite values, is KS 00° ar "4805. In other words, there must be a symmetrical conjunction or opposition of the three satellites at the beginning of the period. *) The reasoning by which the existence of these solutions for small values of the masses is proved, fails in only one case, viz. when = = 0 or a whole number. This exceptional case is analogous to the well known exceptional case for the periodic solutions of the first kind of the problem of three bodies. For the special case of Jupiter’s satellites we have tka 3,7 — 2, K = 180° A, = Art — v + 180° A, = 2t — v Ast 0 Td, — 4, == A, aA, In the system of Jupiter we find that v is small compared with r. We have roughly (in degrees per day): po dl .0571 Pe 0.1900. 1) Les méthodes nouvelles de la mécanique céleste, tome I, § 40. 3) See also Les methodes nouvelles, t. 1. 8 50. ( 684 ) It is owing to this particular circumstance, that the motion of the satellites can also be considered as a periodic solution of the second kind, as will now be shown. In the periodic solutions of the second kind of the unperturbed problem the excentricities are arbitrary, and the mean motions (not only their differences) are mutually commensurable. In other words we have here x = 0. If the masses are not zero, these solutions may also remain periodic. In the perturbed motion we must then distinguish the mean motions in longitude and those in anomaly. Let nit + lio be the mean anomaly ASP t= As if then a; be the longitude of the pericentre,-we have nml + aij Stare , longitude, Inquiring into the conditions that these solutions shall remain periodic for small finite values of the masses, we find again that there must be a symmetrical conjunction at the beginning of the period, i.e. for t= 0. The angles im I I must all be 0° or 180°. One of the angles /;, (e.g. /;,) can always be made identically zero (or 180°) by a convenient choice of the zero epoch. There thus remain 4 angles, each of which can have one of two values. We have thus 16 combinations which may a priori be expected to give rise to periodic solutions. 20 80 CL uke ; A 2x Now if ie were zero, then at the end of the period 7 = — Yv the configuration for tO would be exactly restored, as it ts in the unperturbed problem. It is, however, sufficient to insure the periodicity de of the solution, that the value of en integrated over a complete period shall be the same for the three satellites. In addition to the conditions of symmetry we have therefore the conditions ge 1 7 as dt = le dt = Ts dt = a 1 Eo ee 0 0 0 After the completion of the period the whole system is then rotated through the angle — «7, as in the solutions of the first kind. ( 685 ) The mean motions in longitude are the same as in the solutions of the first kind, viz. : Dy == CV — x, The mean motions in anomaly remain rigorously commensurable.') I will now restrict the discussion to the special case represented in the system of Jupiter, viz. : ty a i a ek For the general case similar results will be found, which I do not however at present propose to investigate. Moreover I will limit myself to the consideration of small excen- tricities, which is the only case that is of immediate practical value. Whether the conditions (1) do also admit solutions with large excen- tricities, is a question which can only be answered by a special investigation. Under these restrictions we find that out of the 16 combinations satisfying the conditions of symmetry, there are only 4 which also satisfy the conditions (1. For two of these x is positive, and for the two others it is negative. Further, if the quantity 4,-—84,+24,=—K is formed for each of these solutions, it will be found that one of the solutions with a positive x has K = 0° and the other has K = 180°, and similarly for the solutions with a negative x. Of these four solutions that with A —= 180° and x positive (the case of nature) is the only stable one. These solutions of the second kind thus appear, on both sides of the exceptional point x = 0, as the natural continuations of the two possible solutions of the first kind (A = 0° and A= 180°). In the solutions of the first kind the unperturbed orbit is circular, the perturbed orbit is affected by a ‘great inequality”, with the argument ct. In the solutions of the second kind this inequality appears as an equation of the centre’). In the solutions of the first kind we have the condition that the unperturbed excentricity must be zero; corresponding to this the excentricities in the solutions of the second 1)’ These solutions are based on the same principle as those investigated by Scuwarzscuitp (Astr. Nachr. 3506). Scuwarzscuitp, however, only considers the case of two planets, one of which has an excentricity, and at the same time an infinitely small mass. Consequently the orbit of the other planet, which is acircle, is not perturbed. 2) In the integration by the usual method, this inequality presents itself as a perturbation of the excentricities and pericentres. Besides this “great” inequality there are, of course, a number of others, whose arguments are multiples of >, which are the same in the two solutions. ( 686 ) kind are not arbitrary, but must be determined from the equations (1). When the value of x is the same for both cases, the two solu- tions are entirely equivalent. In order now to investigate these solutions according to the theory of PormcarÉ, we must write down the conditions of periodicity T dE; — a = 0 a (5 0 where for E; we must take successively each of the elements of the system. If further 8; be the small correction to be applied to the value of B; (for f=0) in the unperturbed orbit, in order to retain the periodicity in the perturbed orbit, then the stability of the solution depends on the roots of the equation 0 0 Ou, ie + 28 bil a) Paes Co wel eli re v = dp, DB, 2, Ow Ow, Ow \ AE 5 aen an Per ef = =S ” a ee OB, ©) dum i) Be Ou, 2 eae 1 — 08, 03, 03, a . | If we put s—e*! (or, approximately, 1—s——aT), then the condition that the orbit shall be stable, is that all the values of a are real and negative (with the exception of one or more, which may be identically zero). I will introduce the elements (Ord 1 EA a te of which the meaning is L7=mpy a OH; = L;V1 — e? 1; = mean anomaly zt; — longitude of pericentre. Supposing the units to be so chosen that the constant of Gauss and the mass of the central body are unity, the equations of motion are : dh; dF CUT; OF KE Ee Te dl; OF da; OF dt ae OL; dt En Ò FSE h Pane m,* m,° m,° , replies. aes BS ame 0 ( 687 ) In the unperturbed motion we have et Te en and the constants a; must be such, that n, = 4p N= EP he The integral of areas is ® — I, + IH, + H‚ = const. By means of this integral we can eliminate /7,, and diminish the number of degrees of freedom by one. For this purpose we introduce Ga TT, GE Oni Ee Ip dg Tg The equations then preserve the canonic form. *) In forming the equations w; =O we need only those terms of A, whose integral over a complete period does not vanish, i.e. those in whose arguments the mean anomalies do either not occur, or occur only in the combinations ! t=1,—21, f=l,— 21, of which the mean motions are zero. The constant term will not be required in what follows. Of the others, we only require the terms of the lowest degree in the excentricities. Thus, introducing the further notations | 5 Bn dg De Rs HS. Gs | 8 we find that A can be replaced by __ mm, \ { — Ag, cos (l + 2m) + Be, cos (l + w)} + mn, d, “ft { — Ae, cos (! + 2w') + Be, cos(l! + w')} . . . (3) as where A=a' (4 A@) + A,@) 2 B (3 Al) + A0) — ==) Waa The symbols AY? have the usual meaning (LeVERRIER, Annales de Paris, tome J], p. 260, 262), and must be computed for the value h The integral of areas still exists, if the compression of the planet is taken into account, provided only the motion takes place in the plane of the equator. Also those terms of the perturbing function which are here used, remain the same. The conclusions reached below, thus can be applied unaltered to the case of a compressed planet. ( 688 ) ha (::)"= (:)"= 5) — 0.630. n 2 a’ ny : The coefficients 4 and ZB then are pure numbers. Their values are A= + 2.381 B= + 0.964. The meaning of the symbols ¢; occurring in A’ is hinde == Va 5 ES mm! 5 Pis or approximately 1 See The expressions of the various differential coefficients of A’ are: OR 2a; OR’ cosg; 1 OR da; air ese Va; &; de; OL; mi OR' 1 1 oF’ OT; rad mi Wai e; Òe; oR OR AR ÀR _ÒR DR ET UL 0m 0G) a 0 ne òR OR OR OR OR or OR pa Os as or ee dR OR OR OR OR dg, dw dg, ml dar ae Jl The quantities 8; and W; will be supposed to be correlated to the different elements as follows: To: DORI Dy Sr ae! ONS Peake Pars Fae yen ariel eet ote correspond : a ’ B, ’ B, ’ B, ’ B, ’ Be ’ B, ’ Be B, ’ Den and: Wi; wW, ’ w, ’ w, UW, If in R’ and its differential coefficients the elements are replaced by their unperturbed values, these funtions become constant. Con- sequently the first terms of the developments of the functions y; in ’ We 1 w, ’ Ws ’ Ws ’ Wo powers of the masses are of the form jh | adik 0 where D represents a differential coefficient of A’ in which, after the differentiation, the unperturbed values of the elements have been substituted. Now we have ( 689 ) dL; _ OR’ dG; _ OR; dy DR a = Raga The functions ,,Y,,W,.w, and yy, therefore contain only sines of linear combinations of the angles /, /', w, w'. Further we have Or’ OF, 2 OZD and therefore the equation wy, = 0 is a necessary consequence of w, = 0 and w, == 9. The four remaining mutually independent equations | eae a, 0 oe Le correspond to what has in the beginning of this paper been called the “conditions of symmetry”. If we put L=(, —2l,),=ea v= ( — 21), =e Je (x, ry Xs), EE 8 0, = (x, ik Xs) — B, (the subscript O indicating the unperturbed value, or the average value over a complete period in the periodic solution), then the equations (4) are satisfied, if each of the angles eae aes ed is either 0° or 180°. These conditions being satisfied, we can, in the differential coeffi- cients of R’ (after the differentation) replace the angles J, !, w, w' by a;'a', B, B'. The developments of the functions p,, ... W‚ in powers of 3; are ò’R' On 07 R’ terms of », =F |B dl, +8, 01,01, de cls has 01,09, AG higher orders and similar formulae for w,,w,,w, and w,. Then we find easily or Pr B ST: + terms of higher order, OL, and similarly w, and w,. These equations give 8, = B, = 8, = 0, in other words the mean motions in anomaly (n;) are not changed. Finally we have T ME dt OR OR IE tam om WED OR Y,, = — Trae OI, The equations W,=w,,—0 are thus found to represent the conditions (1), since ( 690 ) it dx; dR’ er dt OTK 0 From the value (3) of A’ we find easily (remembering that Qh =n; == tte or nd rl a) yi 1 dx, a, A Ve dt = — m, —— cos a, 2 dt a, €, 0 FT Bd el Re as Aran mn OE OSE m Mr COST ee Dare) “dt Te °) "as & . T Sioa tr al iets ‘ SS nl I COS CL, Dd). teh hy vat ie eee 0 The conditions (1) can thus be written xT x S= = 2S = =—- 27 p == | a | p a B QO | Oo > wey wit > | 2. (1) Os OLS Oe? OF | — | 0) +] aapessible, (2) 0 0 | 0 180 |— ,O | — | conditional. (3) 0 0 180 0 — + —| impossible. (4) 0 0 180 180 |— + + | impossible. (5) 0 | 180 | 0 0 os + | impossible. (6) 1-0.) 480° | | 4g0 |l =| paseible: (7) Oel 480.4 480/170 — | O | — | conditional. (8) 0 | 180 } 480 | 480 | — | O + | impossible. (9) [190 0 | 0 | 0 — | + | impossible. do) [aso | o | o | 480 jess _— | impossible. (14) | 180. | 0 | 480 | © impossible. (12) $480 | oO | 480 | 480 | + | conditional. + | conditional. (13) | 180.) 4802 pO — 40 > Go a) 14) | 180 | 480 | 0 (15) | 420 | 180 | 480 | 0 | — | impossible. dd tk + O O O O | — | impossible. as Be (16) | 480 | 480 | 180 | 4180 ‚+ | possible. ( 691 ) of a,a’,3 and 2, we find the following summary. Only those com- binations can give rise to periodic solutions, in which 2, , 2, and &, are of the same sign. The letter OU stands for undetermined. Out of these 16 combinations there are only two, (6) and (16), for which the perturbed orbit can remain periodic for all values of the masses. There are four: (2), (7), (12) and (13), for which the periodicity is only possible if a certain condition involving the masses is satisfied. For all solutions we find from the equation y, = 0 el pe en 6.225. It needs hardly be pointed out, that this is only a rough approxi- mation, the higher orders of ¢; having been neglected. In the system of Jupiter’s satellites we find actually (see these Proceedings, March 1909 ele, — 0,11. Further if we put EN Ze then we find, for the solutions (6) and (16), from w,, == 0 ay B &, B €, 2nA If the longitudes are counted from the apocentre of III, and the time from a passage of [IL through this apocentre, we have, for t= 0, 7, = 180°, /, = 180’, therefore 2, = 0°. For the corresponding values for II and I we find, for t=O, for solution (6): RIN ne, == 1807 l= 0 bl d=0 A == 160 K =a, — 34, + 2A, = 180° and x is positive: the mean motion in anomaly exceeds the mean motion in longitude. This is the case of nature. For the solution (16) we find: mr 0° t, = 150 b = 180 i = 180 4, == 180 A= 0 == 180° x negative. 1) The expressions there given are based on Sovuittarr’s theory. The quantities s;‚ Which here appear as excentricities, are thus there considered as perturbations, and are called 2), 2, 2g. ( 692 ) The possibility of the solutions (2), (7), (12) and (13) depends on the sign of 2,. In all these cases cos (a + 8) and cosa’ are of the same sign. Thus if we put Q — U; Es Zin u, B, a, we find that for positive values of Q the solutions (2) and (12) can exist, (7) and (13) being impossible; for negative values of Q (2) and (12) become impossible, but (7) and (13) are possible. We find for these solutions: Solution (2) Solution (12) ge | Hi Ee Ev l= 0 i= 0 {== 180 f= 180 A= i= Ae == 160 4; = 160 x positive. % negative. Solution (7) Solution (13) = 180% + == 0" £180" oe de L, == 180 Er 0 is 180 ds An A, == Lee dts) x positive. x negative. All four solutions have K = O°. For the solutions (2) and (12) we find Ss Jog se E, B &, 2a,A and for (7) and (13) Beg Ge E, B é, 2a, A For Q=0 (or, if higher orders of ¢; are taken into account, for a value of © in the neighbourhood of the value for which Q = 0) [Hs we have e,=0. The solutions (2) and (7) then become identical, and similarly (12) and (13). We thus find that the two cases (2) and (7) form together one continuous family, which exists for all my, values of . The same thing is true of (12) and (13). Ms, Thus all that has been said above regarding the existence of the periodic solutions has now been proved. It remains to investigate their stability. For that purpose we must form the equation (2). We introduce the notations : l—s be Orde, eR Tien, ) Oo eo ( 693.) where x and y represent two of the variables /;, gi and p and gq two of the variables ZL, G;. The quantities (wy) are of the order zero in the masses, the quantities [pq] are of the first order. With the aid of the values of y;, which have been derived above, it is now easy to write down the determinant A(s). The differential {ie ie Or coefficients such as ———-—— will have m, in the denominator. To 03, Vie : remove this, and to make all terms of the same type also of the saine order in the masses, the five lower rows have been multiplied by m,. Then the five upper rows, have been divided by /m,, and the last five columns by m,Vm,. Finally every term has been divided by 7. The equation then becomes | —e 0 0 0 0 COMA mer RN pee -¢ Bed 0 U (a4) Cal) Caos) Cans | 0 0 =P 0 0 (hls) (lala) lala) Clam) (laga) 0 6 0 —F 0 (hm, Can) (lag) gam) (9,92) | ” 0 0 0 =F Chg2) laga) laga) (raa) (gga) | AAT AANED AT EE VRT oe sa —[ZyLe) Ko-[Lala) —[Lale] —-[LoG] —[LeGa] 0 —p Û 0 0 —[L,L£3) —[Lel,) K3-(Lsl3] —[/sGi] —[ZeG.] 0 a 0 0 | [AG] (4G) HA LAAT LAG] 0 0 0 —e 0 CRD AE TEN EE Ee For brevity we have put 3 f 3 d 3 k, = — —-~ et , tae 9? 2 2 we, a, ENGE To simplify the determinant (5), we may use the relation, which has already been mentioned above, (Le) HAU @) + (U, 2) =0, where x represents an arbitrary element. We perform the following operations, which are here, in order to save space, only indicated (the ordinary figures refer to the columns, the roman figures to the rows) : To (8) add 4.(6) + 2.(7,, From (VZ) subtract 4.(VJZ/) EE) wos ELEN ” II) ” 4 (1) at (ITZ), ” (1) ” 4.(3), 9 (2) ee ee) The determinant then becomes divisible by 9, and the columns (3) and (8) and the rows (III) and (VIII) drop out. For the sake of ( 694 ) clearness I will, however, continue to indicate the remaining columns by their original rotation-numbers. Now we have’) vt) = (4), (9:91) = (ww), Vl) = — AW), (9192) = — (wa), Ul) = AU) + Ul, (19) = (wo) + (ww) (Lg) = (lo), (/,9,) = — (lo), (l.9,) = — WUlw), (Lg) = 2(lw) + Uw’). By means of these relations the determinant can be still further simplified. We perform the following operations To (7) add 2.(6), From (VJ) subtract 2.(VZ/) » UI)» 2.(L), ” (1) ” 2.(2) > (10) ,, ° (9), mA) (X) » (V) » (ZV), coh) ee (5). If now the remaining rows and columns are rearranged in the following order dE DEE GE c A) reen A0 EAA VA MUI I VE ee then the equation becomes =o 0 A 0's 20s - 07) Copan 0 SOE De OR (BE 0 O (lw) 09 —e 90 (ww) 0 Oe, 0. Cay 0 or een u ln A EO 7 AA where the meaning of the coefficients is as follows (I mention only those coefficients that will be used below, those omitted all contain m, as a factor): A. = iK + terms of higher orders Ke ee hee eee A. = = IGG IG [6,61 TT LGG] mie [G,G,] A= GG 1) These formulas suppose (Ul!) = («w') = (/'@) = (/w') = 0. This is only true if the third and higher orders of e are neglected, ( 695 ) The expressions [pq] all contain m, as a factor. Thus, in order to derive the term independent of m, in the development of @, we take all those expressions = 0. The determinant then becomes divisible by 0‘, and is reduced to its first four columns and rows. Four of the eight roots of our equation thus appear to be divisible by Ym,. The first terms of the other four are the roots of the equation: —o 0 s 0 0 -—e 0 s' = 0, An A. cage 0 B As 0 —oe or o* Tik, (ar 8 zi Aln s) o Sin (A,, Ay. a AG ve) ss = 0 ic 2 2 (7) where we have put, for brevity: C35 MCR = st. The solution can only be stable, if the equation (7) has two real and negative roots. Now A,, and A,, are negative, and A,, 4,,—A’,, is positive. The necessary and sufficient condition that the equation (7) shall have two real and negative roots is therefore, that both s and s’ are positwe. Now we have sat l Acme Beale +a | a, : : ER al) pier Ye, ' + ( s' = —{ de, cos a — Be, cos (a +2) | a, l For the six possible combinations we find the signs of s and s’ as given below | : a a | 8 | bet 8! | s | 8 | | | sr (6) | 0° 180° 0°) 1809 ie ae (16) | 480 | 180 | 480 | 180 | — | —] unstable. IR A EN (130 | oO | + mo |19 | 18: | 0 |+/0 (12) [180 | o | 480 | 480 | O | — | unstable. (13) | 180 | 480 | O 0 — | O | unstable. +7 Proceedings Royal Acad, Amsterdam, Vol. XI. ( 696 ) The solutions (16), (12) and (13), ie. those with a negative value of x, are thus certainly unstable. For (2) s will be positive if Ae,— Be, > 0. é . By using the value of — found above, this leads to the condition é, eae Q< 2— B = 7.41. a, Similarly we find for (7) that s’ will be positive if Q EP Uh oan ET Foe For the family consisting of the solutions (2) and (7) we thus find that s and s’ are both positive for all values of Q between the limits — 0.46 and + 7.41. For the Jovian system we find Q=- 4.14. For the solution (6) both s and s’ are always positive. This is, however, not sufficient to prove the stability of these solutions. We must also consider the four remaining roots of our equation (6). To determine these I divide the last two rows and the 5th and 6 columns of that determinant by Wm, Introducing then oe Wins Aij= m, Bij, the equation becomes | —p'V Mm, 0 (Ul) 0 0 0 (lo) 0 Oe sans 0 (iv) 0 0 0 (et) A); Ajo —p'V mg, 0 ByV img ByuV me 0 0 Ay Aas 0 Vn, Bagh mg BosV me 0 0 A @)= = 0 0 0 (lo) 0 —p’ 0 (aw) 0 0 0 0 (Ua) 0 —f-' 0 (wo) Buma Bam 0 0 Bas Bas il 0 Buma Bama 0 0 Ba Bu 0 =e If now again we neglect all terms which appear multiplied by Vm,, and if we perform the operations From (7) subtract .(3), from (8) subtract (il) . (4), we find —o 0 on ih) Ao en WA On Aan BB Sd Bar 0 —@ where we have put lw)? lo’): Ee) tn eee (il) (er) We thus find that 9’ is determined by an equation very similar to (7). For the coefficients B;; we find easily 6 = (ww) — B,, = i, ae H,, bi =H B,, = H,+ H where oR ie ieee ne, O77’, 16 u‚a,a, €? òR 1 u, B Us ! i ea Tea 16 RR ete : = E es B Een For the cases (6), (2) and la ), which are the only ones that we need investigate, all these expressions are negative. For H, and H, this is at once evident. For H, we find: Sol. (6) Sol. (2) Sol. (7) i Lie zr Oe ee en ie a : 16a’,€*, 16a',e°, 16at.e, which is also negative in all three cases. The equation determining the first term of o/ now becomes oe — (4+ 4,)6+(4,+4,)o}y" + [4,.4,+4,.H,+ H,.H,}00'=0, . (9) The condition that this equation shall have two real and negative roots is again that o and o’ are both positive. Now we have 6 = (ww) — ee) ; 6 = (ww!) — ey s s It is only necessary to investigate those cases where s and s’ are both positive. The conditions of stability thus become s.(ww) > (lw)?’, s'. (w'w') > (Vo)? The values of s and s’ have already been given above (8). For the other quantities we find 47 * ( 698 ) (ww) = 4 Ae, cosa’ — Be, cos (a + a). 4 Aeg, cosa — Bestel. 2 A £, cos a — Be, cov (a+ py], - loy=" | wo) = — | | | 2 A &, cos a — Be, cos (a) + 8) ' from which s. (oo 4 Ate’ + B'e,’ —5 ABe, eg) (Lw) a A | 4 A e* + Be, — 4ABe, e, cosB ' a, Therefore u, 3.0 = SAB, &, cos B, 2 and similarly so = — aA Be, &, cos @' 3 The only stable solutions are thus those in which 8 and 9’ are both 180°, and the only solution which satisfies this condition is (6). This solution, i.e. the case actually occurring in nature, is thus found to be the on/y stable periodie solution. It needs hardly be mentioned that all the proofs given above suppose, that the developments in powers of ¢ and m; converge so rapidly, that the sign of the various quantities used is determined by their first term. What the upper limits of « and im; are satis- fying this condition, cannot be stated without a special investigation, but nature teaches us, that for the values occurring in the system of Jupiter the solution (6) still exists as a stable solution. Physics. — “Contribution to the theory of binary mixtures, XIII.” By Prof. J. D. vAN DER WAALS. We have considered the closed curve, discussed in the preceding Contributions, as the projection of the section of two surfaces, viz. Py aw ==) ene dx? dv? axis. Let the z-axis be directed to the right, the v-axis to the front and the 7-axis vertically. The projection of these sections on the other projection planes will now also be a closed curve, in general =— 0, constructed on an z-axis, a v-axis and a T. en” il ( 699 ) with a continuous course. We shall here chiefly consider the pro- jection on the 7’, «-plane. This projection will possess a lowest and a highest point, and be enclosed on the right and on the left between a minimum and a maximum value of 2, which two values of « are the same as those between which the v,2-projection is enclosed. But the highest and the lowest point of the 7’-projection is no special point in the v,v-projection. Only in this v,2-projection the points 2 d mentioned have the property that a line er =0 and also a line Vv aw : ENE eis ; i touches this v,v-projection at the minimum or the maximum x temperature. At all temperatures between this minimum and this maximum temperature the v,v-projection is intersected by a line dp B dp f —— = 0 in two points, and also by a line —- == 0. But this contact dv? dx? can take place e.g. for the minimum temperature in a point that lies either on the left or on the right of the point in which v has the minimum value, and even, but in special cases, exactly in that ae? | point. So the quantity a, en be both positive and negative for the & point in which 7’ is maximum. This holds also for the point where 7’ is maximum, but generally the first mentioned point is of greater importance. dv If for this first-mentioned point a is positive, this is also the case Hij dee rde with 7 for the point in which —=0 touches the closed curve, & v dp d*w dv d? ; ———_ + —— = 0, th anti and as EEP UE ST ‚the quantity en will be negative in the 3 point in which 7’ is minimum. In the same way the quantity ER & d is positive, and is positive for that point because also the line U a? d? uae 0 touches, and so men da” dax? dw dv be bata k Ido © and the contact takes d? place in such a way that the whole closed curve lies inside Te =0, av If the minimum temperature should just happen to be in the point dv of the closed curve where ao eu we have at the same time ax ( 700 ) dr a’ dv : d? = 6 an a iat If, on the other hand — is negative, a dadv dx* dx dx dv ns dw is negative and also — . de? If the whole curve has contracted to a single point, this applies also to the two other projections and for this case it is easy to express these projections in the value of £, and e, and 7. Then, as was Ve nV &, found before, «== ——-— , and 1 —#= WV UE Then the value of 1 v (n =| is Ve, We, . C 1 ee es seg or equal to 1+-B, o1 E 1 + n (MEt WVEN Both for «,—O and for e,=û is Vv — is equal to poe 1 v-b (nl? Ve Ve bn Wat Ve v—b=0, and as we have to do with a point lying on the line or d : i T=0O. A maximum value of v does not occur, but a i v . . . . . maximum value of 5 does. The easiest way to find this is by retaining the form: v (n —1)? « (1—e@) 2 eee = bh = : b (lr + ne)? If v could be maximum, then: db dB da da als b <5 fa OS or Ld? nr? nn n—1 [lees oe 0 1+(n—l)z 1 (n--1)?2(1 - 2) 2e [Lt De} After reduction we should find „== 0. But the maximum value of v ‚dB b il =. or of —— = 0 requires nr =d Or ¢ = — =. b aL nl If for « and 1—az we put the value We, and n/é,, we find as Uv condition #€,=e,, and so pr, — Pts. Then the value of ; is equal fn 1)? : to 1+ eine == a ie When n is small, 5 is only little greater An An b than 1, and so 7’ much smaller than 7. But for very high values of n, e.g. about 10, the critical volume can be reached, and so 7’ ( 701 ) : ea 7 De 5 ad “ e se =i . v can be = 7. With constantly rising value of n, the quantity 5 can, indeed, increase indefinitely, in which, however, 7’ becomes an ever smaller fraction of 7). The value, however, which &, and # 7 VV Java r U €, will have, and consequently the value of 2, and 7, cannot be chosen arbitrarily. Besides that «, and e,‚ must have such a value that the point denoted by them, lies on the parabola OPQ, the condition must also be fulfilled that a, = /a,a,. For the case that ? =1, the values of ¢, and e, are easy to calculate. Then the point (€,,&,) must also lie on a second parabola, congruent with PO, and shifted by an amount 1 along the ¢, and ¢,-axis in negative direction. These parabolae having their axes parallel, there will only be a single point of intersection. The equations which are to be satisfied, are then: (e, — ne) = 4n'(n — 1) (e, — ne) and Erne, tn — 1)? = 4ufn — 1) (Ee, — ne, FR — 1). Then we find: n+3 ee Feeye and on+1 n he ay Tren ales 1), n+3 A da+1 . ea ——— £ tE 7 i or w m+) and x Tmt) The value of 7 obtained in this case is smaller than the one calculated above if we take e‚ =e,. If <1, e, increases, of course, and «, decreases and reversely. A value of / might be chosen so that 7’ assumes a maximum value, but to this we come back later on. But in any case the values of | dy dw &, and ¢, may be such that the two surfaces —- ==0 and — da” dv? touch only at one single temperature, without intersecting further. And if ” is not very large, this temperature lies very low. Thus from v—b\? (=) ans Moe , and the supposition /=1 we cal- the formula MRT = 2 - b ‘i 1 culate for n = 2 the value of De Td about, and for other values k ( 702 ) T of ? this value of = becomes but little higher. But e‚ and e‚ might k be such for higher value of m, that v approaches to 3b and 7’ to T;.; this might be the case for n — 10. So we see here the following possibilities for the phenomena of non-miscibility, dependent on the value of ». For low value of n, contact of the said two surfaces may take place at so low a temperature that observation is impossible on account of the occurrence of the solid state. For increasing value of 7 this tempera- ture rises, and for a certain value of n, it may have risen to */, or '/, Tk and so the observation will no longer be prevented by the appearance of the solid state. As, if contact takes place of the two surfaces at certain temperature, two plaitpoints make their appearance already at lower temperature, which vanish again at higher temperature than that of the contact, three-phase-pressure will exist between two tempera- tures. A precise determination of the value of m at which this is the case, is not possible, if it were only on account of the fact that we have not been able to determine what relation exists between the tempe- rature of contact and that at which the double plaitpoint begins to appear or disappears, and moreover because we have not been able to determine how long the double plaitpoint must have been present before the plaitpoint appears or disappears on the binodal line. But for small value of n the lowest temperature at which non-miscibility sets in, can certainly not be observed, at least not if the cause of non-miscibility is to be ascribed to the circumstance discussed here. So in the 7',z-projection there is only a single point for which the value of z will be found in the left half, in the case discussed here. But if we besides draw the 7e-projection of the plaitpoints which are the consequence of the existence of the point of contact dy dy . of == 0 and Pept we obtain again a closed curve. Probably the projection of the point of contact lies, especially as regards the value of 2, very eccentrically with regard to this curve — possibly even to the right outside it. The lefthand branch of this curve is the projection of the irrealisable plaitpoints, and these will always have considerably moved to smaller values of 2. But if the projection is a closed curve, they must rapidly approach the points of the righthand branch at higher temperature. However, another case may be expected. In the case that the projection of the plaitpoints remains below the curve indicating the course of 7, the closed curve is to be expected — but if the value of 7 should rise so high that the curve 7% = f(a) would be cut, the lefthand branch of the projection would meet the ordinary plaitpoint, which approaches from the side of the component ( 703 ) with the lowest value of 7). The result is then that the projection of the plaitpoints forms a curve which begins at «= O and 7’= 7%,,, rises from there to the highest double plaitpoint temperature, then falls to the lowest plaitpoint temperature, and ascends again- from there to 7;,. This last case has been treated more extensively These Proc. Vol. VII. The transformation of a branch plait ete. Figure 39 gives a schematic representation of the 7'r-projection for the first case. The point P represents the point of contact of . aw dp the two surfaces ne and ET The full line represents the locus of the plaitpoints, the point P., is the lowest double plaitpoint, and Pz is the highest. In the points Q, and G, the realisable plaitpoint appears or disappears on the binodal line — and then there is three-phase-pressure between the temperatures of Q, and Q,. The dotted curve, which has its lowest and its highest : Fig. 39 ( 704 ) point in Q, and Q,, denotes the concentrations of the coexisting liquid phases at every 7, and the curve Q',, Q', indicates the concentra- tion of the third coexisting phase (vapour phase). The curve 7), = /(z) has been drawn higher in the figure. It follows, however, from the remark, Contribution XI, p. 440, Vol. XI, that the point P? need not even be present, and that yet the remaining part of the figure, but then between narrower tem- perature limits, may continue to exist. We might even imagine the circumstances to be such that the points P,, and P.q coincide, but then .Q, and Q, and Q’, and Q’, would have coincided already before. The second case is represented in fig. 40. Again Pis the projection d 2 on the 7‚v-plane of the point of contact of the two surfaces ou ==) ‘ip : and ils a The: fill ‘curve AQ: Pea Pan Q, Bete, ae: that of dv? G X=0 Nat Fig. 40. plaitpoints. The points P,, and P.¢ are the double plaitpoints. So 4 a; ( 705 ) 4 Al there are three plaitpoints between the temperature of A, viz. 7, and that of P4, unless /, should lie higher than A, in which case Pq would take the place of 7. The curve 7, = f(z) is also inserted in the figure. It will have to intersect the plaitpoint curve at «<1, and that twice. The first point has not been indicated by a special mark, but the second point of intersection is supposed to be in the neighbouchood of B. If we draw a p‚r-section of the surface of saturation, and add to it a line indicating the pressure at which there would be coexistence if the mixture behaved as an homogeneous substance, the extreme point of this line would lie at the same value of w as that of the plaitpoint, at the value of 7’ of the second point of intersection just mentioned. For higher value of 7’ we have then again the rule that for a given mixture 7’, > 7), which is generally considered as the normal case. This being really the case for v very small, and « nearly 1, when there is intersection of the curves 7). = f (2) and 7.= gp (2), this will have to take place twice. For the points Q, and Q, the plaitpoint lies on the binodal line, and between the temperatures Q, and Q, there is three-phase-pressure. The concentration of the three coexisting phases is indicated by the dotted line Q',Q,Q,Q.. We might call the part QQ, of this line the vapour branch. The vapour branch presents a particularity in the drawing which has escaped attention so far, viz. that it can contain a point in which w has a minimum value. I have not drawn this particularity in the vapour branch of fig. 39, because it is less probable there. This applying to a circumstance which has not been noticed as yet, and which is yet not devoid of importance, a digression to show the possibility of the existence of such a point with minimum value of x, may be useful. The more so, because in the discussion properties will be mentioned the knowledge of which is necessary if we want to understand the full meaning of different particularities occurring for the three-phase-pressure. Let us call the concentration of the point representing the vapour phase, wz,, and let w, and w, denote the concentrations of the liquid phase — and let us put 2,< #,<2,. Now the following equations hold: 1 aS Va, dp — (z, Tac ae) B 3 Yar dT 5 P and * vn dp = (2, — 2,) i ts +, dT. ay ‘eg ( 706 ) For the three-phase-pressure both equations hold, and we get the dy value of = for this three-phase-pressure which we shall indicate by - a d hie ae ae by eliminating dz, from these two equations. If we divide the 123 first equations by #,—.w,, and the second by a2, —.,, and if we subtract the quotients from each other, we get the well-known formula: % — th | Kk? dp So dU, AE U, Ls eae | ay al ii 5 dT nn Pati Jet Ven. Ei vs U, Us — vy : ap If we substitute this value of ——— in the two equations given ( 123 we get: dp as dx, Yai a ee + Ney and dp fs ae da Bant ae = («,—2,) ( 4 ) oa + Ns: de, de. Jor dl After division by v,, and v,, we may also write: d°5 de, dp derd ( Op ) < al Nia Den oT 2e, and Os de, dp En Op AE FF Ys1 3 Gre Be, or dp __(dp\ da, Op aT 5, ar a are if we represent by & and =) the ratio of the increase of rye BD p and the increase of 7’ on the vapour sheet of the surface of saturation for a section with constant value of «= z,, respectively for the coexisting equilibrium between the phases 1 with 2, and 1 with 3. So the diff dp a Itiplied b zn el MN with o the difference = fe ar), ultiplied by SE 1 (707 ) indicates the sign of dz,. In the same way if we change 2 into 3. Now it is true that the surface of saturation has been greatly modi- fied by the existence of the three-phase-pressure. But this modifica- tion is restricted to values of 7’, between those at which this pres- sure begins and ends, and also within these limits of temperature, the surface of saturation consists only of a lower sheet and an upper sheet, if we leave the metastable and unstable coexisting phases out of account. So every section for given value of 2, is again, except for the modifications inside the said limits of tempe- rature, the well-known figure in which the lower branch passes continuously into the upper branch. Let us now think the line p,,, as function of 7’ traced in every section. Only for so far this line lies between the upper and the lower branch of the section of the modified surface of saturation, the mixture of the chosen value of « can split up into three phases. If this line intersects either the upper branch, or the lower branch, and if therefore part of the line p,., lies outside the surface of saturation, this must be considered as a parasitic branch, at least for the mixture chosen. So the dotted lines of fig. 39 and fig. 40 represent the values of 7’ for which the line p,,, intersects a chosen section of the surface of saturation. And so the question whether in fig. 40 the situation of point Q’, is such that another point occurs in the dotted curve for this value of zr, coincides with the question whether there exist sections for 21 which the line p,,, intersects the saturation curve twice. As U, = x, V and —— are negative on the vapour branch according to the formula L 3 alr 1 da for the calculation of — a negative value of this quantity is attended a uf dp dp with a positive value of —— or with the line p,,, entering dE, Bd, the heterogeneous region with increasing 7. Reversely a positive value of shows that the line p,,, enters the homogeneous region TT ad with increasing temperature, and therefore appears further only as va ‚(dp parisitie branch. Now in the point Q’, the value of (35) is : 123 dT saturation for the z of the point Q,, as follows if in the formula dp for = — we put xv, + de, for 7,, V.,tdV, for V,, and 7, + dy, G+ 138 dp equal to the value of (4 ye as it is on the section of the surface of ( 708 ) for 7,;. Then we find namely 1 Shy Sn. v — dt a dp h ia 7 ef de, dn . . . r, . ° 7 coincide, the two values of 7’ coincide in the 7’, z-projection. The values of 7’ assume a simple form for these limiting values r ryy B , . of «x, because then VX =O, i MRT = 2 - 5 or TB of course this value must also hold for the case that these limiting values of w coincide, which we treated above. We can even simplify this form ( 710 ) of MRT for the calculation and obtain the form: 2e 1 Et b‚(n —1)? l d n* | bag (n—1)? « 1 (n—1)? l—a« 1 az (1—z) oe b? bts Wile ate ; by writing ea Cia TX? A for 5 pier a a) If we seek the maximum value of 7’, we find for the determination of « an equation of the 3'4 degree, viz.: 3n—1 3—n EE — n?z* = 0 (1 —2)* + z{l—e) ; nr and putting Rn k: —z£ a di 1 + k——— kt? 2n sk = 0. op For n=1 we should have k=—=1, for n= 2 k=—= 1,22; but for k 1 very high value of » — approaches to ze This implies that forn =1 n iy ate. 1 ; 1 the maximum value of MRT lies at a and for n= ata 1 1 For all other values of 7 zr lies between 5 and = By the aid of this value of « we can then calculate the highest value of MRT for the points where X = 0. But the conclusion is not different from that at which we arrived above: viz. that only with 2 appreciably larger than 3 the value of the temperature can rise to 7), or even to ae The value which we found in general in equation (1) for the tem- perature of the points of the closed curve is too intricate to be fully discussed. We can, however, foresee what in general the shape of the T,«-projection will be. For a curve of small dimension the point P of fig. 89 and fig. 40 is to be replaced by a smaller chosed curve which extends according as the former curve itself assumes greater dimensions. Of course the other lines experience the influence of this. Thus in fig. 39 the point P,, will descend and P ascend. For every value of 2, so of a, 6, B and X, the first part of MRT in a B'4X(1—2B) b (14+ By indicates the value of the arith- equation (1), viz.: 2 ( Til ) metical mean, so half the sum of the lowest and the highest tem- perature; and the second part, viz. 2 7 (26B— B + X)V-X denotes the amount that the really occurring temperatures lie above or below this middle value. This second part is imaginary outside the limiting values of x. For between these limiting values of 2, X is positive, and beyond them negative — but the first part exists over the full width. The course of this first may be given in the main points. Beginning with 7'=0 and «=O it ends also with this value at 2—=1. But for very small value of z or 1—~., provided it be outside the limiting values of «, this first part is negative. For the limiting values of 2, where Y —0, it has the above treated a 2 positive value MRT = OIB values of a a value equal to 0 must occur; we conclude to this by noticing that if # or 1—, is very small, 5? and XB may be neglected by the side of B, while X is negative beyond the limiting value of x. The curve which represents the first part begins with an ordinate equal to zero, then descends below the axis, but intersects the axis again before the smallest value of x is reached for which X is equal to zero, then rises to a maximum value, after which it descends below the axis, and finally terminates with a value zero. So if we draw the curve 7}, as in fig. 39, this curve is of course the limit above which 7’ cannot rise for any point of the closed curve. The closed curve being the section of two surfaces which have each a “contour apparent” on the 7'z-plane, the projection of the sections cannot fall outside this outline. So the 7,2-projection can have either one or two points in common with the curve 7%, in which But just beyond these limiting ; v points it must touch this curve. In these points of contact <= 3. os If there are two points of contact De > 3 between these points. v The observation that eS 3 in the points of contact enables us to show that this circumstance cannot occur for low value of n. First of all not for n <2, because, as we saw before, v must there be smaller than 6, = 2,. If we introduce into the equation: 6 fe 2 de RO b b the condition In 3, we get: | 4=-9A-—B. 48 Proceedings Royal Acad. Amsterdam. Vol. XI. Now 1 Oe ar aie n 1 GD | DIe and A 1 ae 1+e, n(1+e,) 1 ite (n=l) toa Let us take two extreme cases: 1. the case that ¢, and e, —0; Crate n'e, l De CAS INL ee ee KU: (n—1)?x (n—l}1 1 In the first case B == A, and so B= 5? or it must be possible to ee | n? 1 find real values for « from the equation : ln en (n—1)?ae (n—1)?1—a which values must, moreover, lie between the limiting values of 2, in this ease «—0 and «—1. For the roots to be real yas 1 must be > or ntl a 0 ve Ol V3+1. hee = Vak if the sign > is changed into =, there is only one root at a, Die 3 n 1 v P p= — and 1—z= Ws So for n about 3.75 x will be n—l n—l if =S 3) ioe BRE if ¢, =e, = 0. Then the closed curve touches the curve 7). in the Zw-projection. But then the Jower branch of the Tv-line will have descended to 7’—0O. Then we have to expect fig. 40, however with this modification that P,, lies at a height = 0 and the three-phase-pressure is already found at all low temperatures. en Sn (n-1)? 2 (n-1)? le ~ In the second case, in which the supposition Vv however, involves the assumption that the point for which re (713 ) lies at a value of « which is just one of the limiting values of x, B (on A and the equation 4—=9 A—B yields the value 2 for B. Pie ee s (n—1)?a (n—1)?l—a 2 Sf 3 n-1 SO ) must be > jor ni. > A0; Le Now, however, if we assign values to «, and «, the condi- tion of the second case will in general not be fulfilled, and sk n'e, T + ea eae (n—-1)?n = (n—1)? 1—a excl (n—1)?n a3 Geil a will have a value between 1 (that of the first case), and O (that of the second case). And the result will then be that the condition must then yield real values for z, and will not have risen to 1 — but we shall 2 ne, have to put <1, or equal to 1—a, in which Hij =< will require a value of m which is greater than 3.75, but which need not rise to 10. But I shall not continue the calculations required for this. If we review what precedes, it appears sufficiently: 1 that the case that three-phase-pressure exists between temperatures that differ little, may occur for all values of n — but that if is small, these two temperatures lie too low to be observed. It is not possible to give the exact value of 7 for which these temperatures if they exist, can be observed, before the ratio is known between the temperature at 2 2 which the two surfaces = 0 and mt =0 touch, and the tempe- rature at which the double plaitpoint has appeared or disappears. 2. That for the case of fig. 40 the required value of m may be estimated as at least 4. 3. That as e,‚ and e descend below the parabola OPQ, the two temperatures between which three-phase- pressure can exist, diverge further, and that only if ¢, and ¢, (we only deal with positive «, and «, here) have become equal to 0, the lowest temperature has descended to the absolute zero point. If we further take into consideration that the point ¢€,, €, lies on the curve a’,,=/?a,a,, which represents in ¢, and &, an ellipse, a parabola or a hyperbola according as # <1 or > 1, and that of this curve only those points which lie in the triangle OPQ (below the parabola) yield a closed curve which we have treated, we see that the phenomena discussed do not only depend on n, but that besides special relations must exist for a, and a, and a,,, which are ( 714 ) a 1e, ae B D 1 represented by ¢, and e,‚ positive in the equations — = = — and c (n—1)? a, n*( l He.) oe ney? the curve a’,, = /a,a, lies outside this region, and the occurrence of the discussed phenomena will, therefore, have to be considered as comparatively rare. If we descend in the region OPQ, so that either «, or €, or both become negative, then (but the consequence of a negative value of ¢, and «, has not yet been discussed) three- phase-pressure is to be expectel already at 7’= 0. If we go upwards along the curve a’,,=fa,a,, and if we get above the paraboia with ¢, and «,, there is perfect miscibility. (For the values of ¢, and é, required for perfect miscibility consult Contribution XI p. 448). As the upward movement along the curve a*,, = (a,a, is attended and n—1 >Ve«,-+nVe,. By far the greater part of A Pleo e e Fy . with decrease of zor it follows from this that if in analogous cases Ph, > Pka _ N ° af en erris pl the ratio ~~ decreases, we pass trom non-miscibility to perfect Pk, miscibility. The conclusions in the derivation of which we have supposed to treat only normal cases, viz. such for which no chemical action takes place between the two components, or for which each of the com- ponents behaves normally, are quite corroberated by the observations. I know only of one exception, namely that the case of fig. 40 occurs also in the observations of KueNeN for mixtures of ethane and aethy1- aleohol, ete. In this case we have to put nm either below or just above 2. How it is that the abnormal behaviour of alcohol has here an influence as if 2 were increased, cannot be accounted for as yet. But in the cases of Bicuner for mixtures of carbonic acid and organic liquids, tor which also fig. 40 gives the course schematically, 7 will, no doubt, have the value found by calculation. (Böcunrer, Thesis for the doctorate 1905). In conclusion a few remarks. 1. In fig. 30 of Contribution VI I have already given the course of the plaitpoint line for the case of fig. 39, and also of the three- phase-pressure of 7. 2. The upper and the lower sheet of the surface of saturation are subjected to some modification in the case of fig. 39 only between the two temperatures between which there is three-phase-pressure. The modification for the upper sheet consists in what follows. Between the limiting values of w of the dotted closed curve of fig. 39 the _ ( 715 ) upper sheet is raised. At the limiting values of « this rise is still equal to zero. But for values of w, which differ from these limiting values, the rise assumes certain values, at first, however, only between tempe- ratures which differ little. But this is accurately rendered by fig. 39. The consequence of all this is that if a certain increase of pressure is applied, e.g. if we observe above the maximum pressure of the modified liquid sheet, the total non-miscibility has disappeared. If we lower the pressure, the non-miscibility may reappear but at a pressure which is only slightly less than the maximum pressure it exists only over a very small range of temperature. In other words there the dotted curve of fig. 39 has greatly contracted. In this two cases will no doubt occur, either real minimum pressure occurs, or the pressure in the point Q is the highest. At higher temperatures, however, splitting up into vapour and liquid is still possible. 3. If in fig. 40 the circumstance occurs of minimum value of v on the vapour branch, there exists for some mixtures, if we take care to follow. the three-phase-pressure, retrogression of the condensation. For the mixtures which show the above discussed non-miscibility between two temperatures, both a’,, may be >a, az, and a’,, may be a, a,, the chance to non-miscibility is smaller. In this case the points ¢,, ¢, lie on a hyperbola which intersects the space OPQ below the parabola close to the point Q; and as the intersection takes place nearer to Q, the distance between the parabola and the ¢,-axis is smaller. And as soon as the value of 9° would become so large that the intersection of the hyperbola a,a, with the e,-axis takes place past Q, non-miscibility will be quite Arges 1 (n?+1)? excluded. So if a Bite For the full discussion of the d? =O and chee, it now remains to 2 dv? d? intersection of the surfaces 7 examine the cases with negative values of ¢, and 6,. (March 25, 1909). KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN TE AMSTERDAM, PROCEEDINGS OF THE MEETING of Saturday March 27, 1909. DOC (Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige Afdeeling van Zaterdag 27 Maart 1909, DI. XVID. CO AES AT SS H. J. Hampurcer: “The permeability of blood-corpuscles to calcium”, p. 718. A. K. M. Noyons: “About observations on the electro-myogram and form-myogram under the influence of fatigue”. (Communicated by Prof. H. ZwaAaRDEMAKER), p. 723. (With one plate). A. P. H. Triveru: “A contribution to the photo-chemistry of silver (sub-)haloids”. (Commu- nicated by Prof. S. HooGewerrFr), p. 730. J. C. Krurver: “An integral-theorem of GEGENBAUER”, p. 749. Jan DE Vries: “A family of differential equations of the first order”, p. 756. S. H. Koorpers: “Polyporandra Junghuhnii, a hitherto undescribed species of the order of Icacinaceae, found in ’s Rijks Herbarium at Leiden (Plantae Junghuhnianac ineditae II)”, p. 763. J. J. van Laar: “On the solid state”. (Communicated by Prof. H. A. Lorentz), p. 765. (With one plate). O. Posrma: “On the calculation of the pressure of a gas by the aid of the assumption of a canonical ensemble”. (Communicated by Prof. H. A. Lorentz), p. 781. J. G. Steeswisk: “Contributions to the study of Serumanaphylaxis”. 2nd Communication. (Communicated by Prof. C. H, H. SPRONCK), p. 784. L. E. J. Brouwer: “Continuous one-one transformations of surfaces in themselves”. (Com- municated by Prof. D J. Korrewxe), p. 788. Pu. Konnstamm: “On the course of the isobars for binary systems”. IT (Communicated by Prof. J. D. vaN DER WAALS), p. 799. W. Karreyn: “On a class of differential equations of the first order and the first degree”, p. 813. J. D. van DER Waats: “Contribution to the theory of binary mixtures”. XIV. Double retro- grade condensation. p. 816. P. H. Eykman: “New methods of stereoscopy”. (Communicated by Prof. K. F. WeENCKEBACH), p. 832. (With 3 plates). 49 Proceedings Royal Acad. Amsterdam. Vol. XI. ( 718 ) Phy iology. — “The permeability of blood-corpuscles to calcium.” By Prof. H. J. HAMBURGER. (Communicated in the meeting of October 31, 1908). It is beyond doubt at present that red blood corpuscles are permeable to anions. This was first established for chlorine by quantitative chemical determinations‘), and also with respect to other anions no reasonable doubts exist as to their power of permeating the blood corpuscles’), This is not the case with metal ions or kations. It is generally believed that blood corpuscles are absolutely impenetrable to these. This opinion seems to be founded on an ex- periment by Gtrper*). This investigator led carbonic acid through a suspension of red blood corpuscies in a NaCl solution, and found that chlorine penetrated into the blood corpuscles, but that the amount of potassium and sodium in blood corpuscles and liquid remained the same. The blood corpuscles thus would seem to be impenetrable to both these kations, and this supposition has the sooner found belief as in normal conditions the potassium is found chiefly in the blood corpuscles, the sodium in the serum. Tacitly the impenetrability of erythrocytes to Na and K seems to have been extended to the other kations and various authors have even quite recently expressed themselves to this effect *). Investigations, however, carried on jointly with HekMA, concerning the influence of various substances, and especially of small quaatities of Ca on phagocytosis have shaken my belief in the truth of this view. The fact that phagocytosis and, as I afterwards discovered, chemotaxis as well are greatly increased by traces of Ca suggests the question whether Ca does not enter into the wnite blood corpuscles.’) The striet proof of this could only be given by quantitative chemical determinations. But the difficulty is that white blood corpuscles are hard to obtain in great quantities. This does not apply to red blood 1) HAmpurcer. De permeabiliteit der roade bloedlichaampjes in verband met de isotonische coefficiénten. Proceedings of the Royal Academy of Science. Series II, Vol. VII, 1889. — Over den mvloed der ademhaling op de permeabiliteit der bloedlichaampjes. Ibid. Vol. IX, 1891. — Zeitschr. f. Biologie 26 1889 S. 414; 1892 8. 405. 2) Hampurerr. Report of the meeting of the Royal Academy of Science. Oct. 27, 1900. Hampurcer und van Lier. Archiv f. (Anat. u.) Physiol. 1902 S. 492. 3) Gürser. Sitzungsber. d. med. physik. Gesellsch. zu Würzburg 25 Febr. 1595. 4) Cf e.g. Héper in the Handbook of von Koranyr and Ricarer, 1907. p.p. 287 and 288. >) Hamburger and Hexma, Proceedings of the Royal Ac. of Science. June 29, 1907, ( 719 ) corpuscles, and as these have never shown themselves different as to permeability from the white ones it seemed advisable to make our investigations on the former. Various methods have been chosen for this purpose. In the first place red blood corpuscles after being mixed with an isotonic cane sugar solution were washed with NaCl solutions, and it was then found that Ca had entered into these salt solutions. It might however be objected to these experiments that the red blood corpuscles, after being washed in a sugar solution, are no longer in a physiological condition and on account of that had become permeable to Ca. Therefore the experiments were repeated in another way viz. by adding slight quantities of substances to the serum oceurring in it also under normal conditions. First we added to the blood 0.024°/, and 0.012°/, CaCl,. Then serum and blood corpuscles were examined as to the amount a Ca they contained, and this was compared with the amount of Ca found in serum and blood corpuscles of the blood used in our first experiment. If no Ca entered into the blood cor- puseles, all the Ca added would be found in the serum. If part of the Ca passed into the blood corpuscles, this would be discovered as well, and as a test it might be investigated to what extent the joint increase of Ca in blood corpuscles and serum corresponded with the Ca added. We subjoin a table which will need no further explanation. This table also contains the result of an experiment answering the question whether erythrocytes, having absorbed Ca from a serum containing Ca, lose their Ca again when brought back into their normal serum, in other words whether blood corpuscles also are ertrameadble to Ca. In explanation to this table must be added that the Ca determina- tions have been made by burning the dried masses and extracting the ashes with dil. HCl and adding alcohol and H, SO, to the clear filtrate, Ca being thus precipitated as Ca SO,. Finally we must add that in these experiments 80 cem. of cow’s blood were used consisting of 32.4 eem. blood corpuscles and 47.6 eem. serum. (1). 32.4 ce. blood corpuscles of blood mixed with 0.024°/, CaCl, supply after being washed with a canesugar solution 0.0458 gr. CaSO,,. (2). 32.4 ee. blood corpuscles of blood mixed with 0.024°/, Co ‚Cl, supply after being washed with serum and afterwards with a canesugar solution … rt 4 2 OOSEL sro Caso... 3). 32.4 ce. blood eos | oy normal blood supply after Haine washed in a canesugar solution . . . 0.0854 gr. CaSO,. (4). 32.4 ce. blood corpuscles of blood mixed with 0.012°/, CaCl, | supply after being washed in a canesugar solution 0.0898 CasO 49% ( 720) (5). 47.6 cc. Serum of the blood mixed with 0.024°/, CaCl, supplied 1) Ont tl or 00499 arr CASOE (6). 47.6 ce. Serum of De “blded mixed with 0.024°/, CaCl, sup- pleads ak dl Pisce oe 0.0492 or. CasQ,. (7). 47.6 ce. of the KR nd serum supplied 0.0369 gr. CaSQ,. (8). 47.6 ce. Serum of the blood mixed with 0.012°/, CaCl, supplied. . .. » «oh eee = a. 0.0487. cn CE 9). 47.6 ce. Serum to which the same amount of CaCl, had been added as to the corresponding 80 cc. blood as sub (J) and sub (5) and (6) supply}. . =... . . 00596 er (ADE Comparison of the values in this table shows: 1. that Ca has entered into the blood corpuscles (Cf. (B) with (1) and (4); 2. that the blood corpuscles give up this Ca when brought back into normal serum. (Cf. (3) with (1) and (2). 3. that the entire amount of Ca added to the blood is found back in blood corpuscles and serum. Moreover this table demonstrates that in the blood corpuscles of normal blood Ca is found (Cf. (3)). This last result clashes with the general opinion that in blood Ca is exclusively found in the serum. In the well known tables of ABDERHALDEN on the quantitative analyses of various kinds of blood, for instance, we find that everywhere Ca is being stated as absent from the blood corpuscles, and in FRANKEL: “Deseriptive Biochemie” p. 557 we read: “Das Calcium ist lediglich im Serum enthalten.” What may be the cause of this contradiction? We think that it is to be found in the method used for the quantitative determination of Ca. We know the metal has been determined as a sulphate or an oxalate, and it was tacitly taken for granted that these compounds are quite insoluble in the fluids in which they are found or very nearly so. This is by no means the case, especially not when the volume of the fluid is considerable. Close determinations of the solubility of CaSO, in acid alcohol have shown me that, besides a slight precipitate always visible after 24 hours, a great part of the CaSO, remains in solution. When we take this solubility into consideration it is found that the blood corpuseles contain a by no means negligible quantity of Ca, as is plainly shown by the figures in the preceding table. Still another method was applied to investigate the permeability to Ca, ( 721 ) If the view expressed by J. LorB and others is a correct one, viz. that when a NaCl solution is added to cells an interchange takes place between Na-ions of NaCl and other kations of the cells, it may be expected that an addition of NaCl to the bloodserum will bring about a transition of Ca-ions from the blood cells to the serum. To investigate this, 0.1°/, and 0.16°/, NaCl were added to the blood serum of a known amount of blood, increases also occurring in normal and a fortiori in pathological life and leaving intact the life of phagocytes (HAMBURGER and HekKMA, 1. ce). Next the amount of Ca in the serum was determined and compared with that of the original blood. The Ca amount was determined by adding ammonium oxalate and measuring the volume of the Ca oxalate in funnelshaped capil- lary tubes. The result of these Ca-determinations however by no means answered our expectations. Instead of increasing a decrease was found, as shown by the last column of the following table. Na Cl added to Dilution of Decrease of Ca amount the serum. serum. of serum. 0.1 Op 9.6 0/, 21.8 %/p 0.16 % 122% 29.5 0/0 To what could this decrease be attributed ? To the dilution of the serum perhaps, which being made hyper- isotonic, had extracted water from the blood corpuscles? In the second column the percentage of the dilution is stated. If this were entirely responsible for the decrease in percentage of the Ca amount, the values in the two columns would entirely agree. We see that this is not the case and that the Ca amount in the serum has decreased more than is consistent with the dilution. We must infer from this that under the influence of NaCl, Ca has passed into the blood corpuscles. It seemed to me that this could only be explained by assuming that in consequence of the loss of water a modification in the dis- sociation had taken place in the blood-corpuscles and that this modi- fication was to be held accountable for the fact. If this were the case the addition of an isosmotic quantity of a non-electrolyte to the serum, of canesugar for instance, would equally result in a transition of Ca to the bloodeorpuscles, and that the decrease of Ca would be even greater than where NaCl was added. Thus the following mixtures were made : (722) ] 100 serum + O.1 er. NaCl. 2 TOG?" t: Be OO ns 3 100: ED ae - 4 100 is OSE canesugar 5 LOU, ce See ieee HA 6 100: ces Me DN © fe Of these 6 solutions and for the sake of comparison also of normal serum, 20 cc. were added to 20 ec. of the same bloodcorpuscles. After being mixed, the suspensions were left to themselves for an hour, after which the amount of Ca in the serum was determined. A summary of the results is found in the foltowing table: Volume of / Decrease of Ca in serum Ca-oxalate | in 0, Unmixed serum. | 39 1. serum +0.1 0) Na Cl 33 | 15.3%, pea ho) tae on SP Bb | 25.60, DRE OB een i 2 | 36. 9, 4. » » 0.884 ,, canesugar | 30 | 2e 5 eo sd, ze 28 | 28.3 0/, SEN ae ee | 22 | 43.6 0/0 Examination of this table will show plainly that indeed the addition of canesugar has caused a considerable transition of Ca from the bloodeorpuseles to the serum, and further that in accordance with our supposition the isosmotic NaCl solutions caused a somewhat slighter transition. The principal cause for the transition of Ca into the blood corpuscles by the addition of some substance to the serum must be sought in the fact that the blood corpuscles lose water on account of this addition; in other words: an increased osmotic pressure of the serum causes a transition of Ca to the blood corpuscles. If the increased osmotic pressure is caused by a salt (electrolyte) a movement of the calcium in the opposite direction takes place under the influence of the kation of the salt; but this movement is much less important. In this connection it must be pointed out that the phagocytary - ( 723 ) power is considerably. impaired already hy a slight increase, of osmotic pressure of the medium, and that the nature of the substance occa- sioning the increase of osmotic pressure is not without importance indeed, but plays a subordinate part. ') More explicit communications on the investigations discussed in this paper will appear elsewhere. Physiology. — ‘About Observations on the electro-inyogram and Jorm-myogram under the influence of fatigue.’ By Dr. A. K.M. Noyons, assistant in the Physiological Laboratory at Utrecht. (Communicated by Prof. H. ZWAARDEMAKER.) (Communicated in the meeting of December 24, 1908). As a consequence of my investigations in which the independence of the electro-cardiogram with regard to the form-cardiogram *) came to light, the question rose whether also for a skeleton-muscle a similar independence exists between electric changes and changes of form under certain definite circumstances. In order to get data with respect to this I started from the principie that in the course of a process of fatigue, arisen by regularly repeated stimulation, different forms of contraction with changes in erescente and different forms of contraction-state as tetanus and tonus can be revealed. It is the mutually deviating physiological conditions in the course of a process of fatigue which, as might be expected, with the existing independence of mechanical and electric reaction upon the same stimulus, would sooner or later break the congruence of these reactions prevailing at a given moment. Not long ago Bricker *), testing the data of Marrius,, WALLER, GARTEN and Duric pointed out cases of troubled parallelism. These investigators, in order to make the action-current visible, have made use of the capillar-electrometer, whereas I regularly availed myself in my investigations of the string- galvanometer of EINTHOVEN (EDELMANN’s small model). In order to cause contraction of the muscle both mechanical and 1) Hampuraer en Hexma. These Proceedings June 1907, *) Noyons., A. K. M. About the Independence of the electro-cardiogram with regard to the form-cardiogram. These Proceedings October 31, 1908. 5) Brücke, E. Th. v. Ueber die Beziehungen zwischen Aktionstrom und Zuckung des Muskels im Verlaufe der Ermiidung. Archiv f. d. ges. Physiologie Bd. 124, 1908. (724) electric stimuli have to be considered. The stimulation takes place indirectly with the M. gastrocnemius of Rana. The electric and form-changes are registered by means of photography. The muscle is with the knee fastened to a substratam; the M. gastrocnemius is prepared free from the other muscles, whilst the Achilles-tendon by means of a thread is connected with a small lever bearing a counterpoise which is moved up and down at every muscular motion along a pulley. The muscle acts at a distance of 2,6 cm. and the weight to be lifted at a distance of 5 cm. from the ful- crum of the lever. The long lever-arm moves past the slit of the registration-box and thus throws a shadow on the sensitive paper of the registration-box, while at the same time the string-movements and the time-writer are photographically registered. The M. gas- trocnemius is carefully kept free from the other muscles to avoid the influence of the contractions and electric phenomena of the other unencumbered muscles. The M. gastrocnemius is led to the string-galvanometer by means of the unpolarizable magazine-electrodes with moveable cotton-seeds. One electrode usually finds itself on the thickest part of the muscle, while the other is placed more proximally. In the last series of experiments the electrodes-seeds with an alteration in SAMOIJLOFFs manner, were, by means of a thin thread, drawn through the muscle-fascia, fixed to the surface of the muscle, in order to be sure that always the same points were led to the string-galvanometer. § 1. Mechanical stimuli. The mechanical stimuli have been obtained in one series of experiments by falling drops of mereury from the mercury dripping apparatus of ScHAFER*), whilst in another series of experiments the tapping hammer of the tetanomotor of HeIDENHAIN produces the mechanical stimulus. a. Mwpernnents with falling drop of mercury. SCHAFER’S apparatus is placed in such a way that at first, the drops are let fall on a sheet of glass, inserted between nerve and opening of the drip-tube. At a given moment when a some- what regular dripping has been obtained, the sheet of glass is 1) Scudrer., E. A. A. Simple apparatus for the mechanical stimulation. Pro- ceedings of the physiol. Society. Jan. 1901. (URS ) drawn av ray and the nerve itself is hit by the drops, whilst then the registration takes place. Along a somewhat sloping substratum, on which the nerve rests in a groove, the drops of mercury rapidly flow away. By changing the frequency of the drops and altering the height of their falling, often also spontaneously without any alteration in the external conditions for the experiment, all kinds of discongruencies may be seen to show themselves between the mechanical and electric reactions of the muscle. Beautifully regular curves of fatigue, are in this way difficult to obtain, as it is not always possible to bring about an always equal fall of the drops of mercury, as to frequency and direction. The curve in fig. 1, got in the above way, proves that the electric phenomenon of the succeeding contractions in the ‘ase in question every time has the same course and the same amplitude. In contradistinetion to this the mechanical effect is repeatedly unequal. b. Kuperiments with the tapping hammer. The tetanomotor of HerIDENHAIN is thus linked in a chain with a chronoscope as interrupter that only twice a second the hammer hits the nerve in the isolated ivory groove. To preserve the nerve as much as possible from too rapid lesion, a bit of muscle tissue, to break the thrust of the hammer, is laid across under the nerve. The nerve itself is, by means of a little windlass, slightly strained. On the whole it is very difficult to regulate the fall of the tapping hammer and the tension of the nerve so as to make the stimuli every time follow by a regular series of mechanical and electric reactions. Now and then, however, it is possible to make a good series, as fig. 2 subjoined shews. It is seen how the series in fig. 2 begins with a large, initial, mechanical and electric reaction, followed by the others which are directly much smaller. At the same time there takes place after the first reaction a removal of the zero-position, first in one direction and then in the opposite direction. On the whole the form- and electromyograms are well nigh congruent. Now and again we see how the muscle with a compound contraction reacts upon a single stroke of the hammer. To dose exactly and administer a better, always the same, local stimulus, the electric stimulus was in the course of the further inves- tigations made use of. (426) § 2. Electrical stimuli. As electric stimuli were used the opening- and closing-currents either of a faradaic or of a galvanic current, which was interrupted either by means of an interrupter fixed to KaGrnaar’s chronoscope of */, second with swinging platina-contacts, or by means of the rheotome-apparatus of ENGELMANN connected with a kymographion. With this rheotome-apparatus the stimuli were on the whole less frequent and the contactstoppers were placed so as to make the closing- and opening-stroke follow each other with well nigh equal intervals of 2 seconds. The following objections, however, can be raised to these methods of stimulation. In the course of the process of fatigue the muscle sooner or later, owing to the weight to be raised, the size of the stimulus, its frequency and the previous history of the muscle itself, comes into such a condition that the muscle is perhaps still potentially able to execute contractions, but is accidentally prevented from doing so by tetanus respectively tonus. Further we are, especially with the rheotome-apparatus, at which the stimuli as a rule cannot nearly be deemed equivalent, not entitled to make comparisons between the respective mechanical and electrical reactions at different periods of the process of fatigue. Therefore another series of experiments was made, in which the muscle, it is true, in the manner described just now was tired by means of opening- and closing-stimuli of a constant current, but in which during the whole course of this process of fatigue the muscle was at regular intervals examined in its mechanical and electric reactions by means of a single closing-induction-stroke. Every time, however, before this closing-induction-stimulus was administered, the muscle was first allowed to become pertectly lax. In making the experiment a Ponr-swing without a connecting cross was made use of, by which the nerve could be stimulated at will, either by the periodical interruptions of the constant current, or by the single elosing- stroke, obtained by the falling-apparatus of BERNSTEIN, in the primary chain of an inductorium. The usual provisions were made, see GARTEN '), to prove that the electric phenomena do not originate in artificial current-loops. As a demonstration I give here a short review of one of the experiments from the last series. Brains and spinal marrow of a Rana fusca have been destroyed. The M. gastrocnemius, arranged for the experiment in the above manner, is stimulated indirectly, alternately with a breaking galvanic 1) Garren S. Elektrophysiologie. Handbuch der Physiologische Methodik p. 470. ( 727 ) current of 2 volts, or with a faradaic current of 2 volts at a distance of secondary bobbin on 57 75 grammes. The string-galvanometer with permanent magnet is used at moderate strain of the string (15 from 60 degrees). The experiment is begun by stimulating the muscle with a single closing-inductionstroke, after which the periodical galvanie stimulating follows. The weight to be raised amounts to In the beginning of this galvanie stimulating the muscle answers with separate contractions, but is soon in tetanus. After some time the muscle is again stimulated by the closing-inductionstroke. The mechanical and electrical effect hereof has increased. The photo- graphic registration shows how the electric phenomena as an answer to the stimulus by means of the interruptions of the galvanic current in general have increased, also how the reaction on the single inductionstroke begins to decrease again in size. The tetanus is attended with a total removal of the rest-position of the string. In fig. 3 it is seen, how the musele under the influence of the fatigue now shows a slight electric phenomenon as a reaction upon the periodical opening and breaking of the constant current, besides how the electric phenomenon as an answer to tbe closing induction- stroke is pretty considerable, but is not accompanied by a mechanical reaction which is not even to be observed as tetanus at the galvanic stimulus. The muscle namely has entirely become slack. The electric phenomenon of the induction-stimulus has become complicated and stretched. 1°/, hours after the commencement of the experiment the muscle shows neither mechanical nor electric reaction either on galvanic and on faradaie stimulus. Only when the faradaie stimulus is strengthened the electric reactions become visible again. Summa summarum, the following, with respect to our subject, most important facts may be gathered from the series of experiments mentioned before and partly described there. In the first place it appears that under definite circumstances, in this case fatigue, also for a skeleton-muscle like the M. gastrocnemius, an independence comes to the front of electric and form-changes of the muscle. It further appears that the more or less congruency of the above reactions is dependent on different factors, among others the strength of the animal. In general I got, even with the slack M. gastrocnemius of Rana, the same results, as to change of length and mechanical reaction, as Bricke found for the M. sartorius, as to change of thickness and ( 728 ) electric reaction expressing himself as follows: “Während der Ermüdung nehmen Zuckung und Aktionstrom ab, und zwar ging die Abnahme der Aktionströme bei wenig kräftigen Muskeln der der Zuckungen annähernd parallel, wenn man die Zuekungshöhe und die electro- motorische Kraft der Aktionströme dem Vergleiche zu Grunde legt. Bei besonders kräftigen Tieren war dagegen deutlich zu erkennen, dass diese Parallelität keine strenge Gesetzmässigkeit darstellt, denn in diesen Fällen hielten sich die Aktionströme auch dann noch auf ihrer ursprüuglichen Stärke, wenn an den Zuckungen schon deutliche Ermüdungszeichen zu erkennen waren”. The grade-formations of the electric and mechanical phenomenon are not identical. It further appears that the effect of an induction stimulus working now and then upon the slack muscle is changeable. In the beginning of the process of fatigue namely the effect increases, to decrease later on again. This decrease in size is attended by a stretching both of crescent and decrescent. The electric phenomenon now and then yields a complicated image. Fig. 3. Already Durie *) knew such a complicated reaction on a single stimulus; also GARTEN, HOFMANN, BRÜCKE *), and Samonorr’*) have observed this in a normal muscle. It appears that in the course of a process of fatigue the proportion of KS > AO as to the electric reaction ofter undergoes a change, and that in such a way, that as a rule the difference between KS and AO gradually becomes smaller so that KS and AO are equal, which generally does not happen before every separate mechanical reaction of the muscle has entirely left off. It sometimes happens that the KS under the influence of the fatigue is in the end < AO. If we make experiments with rather weak currents, it may happen that in this way, both the electric and mechanical reaction on the opening-stimu- lus do not show themselves for some time, to appear again towards the end of the process of fatigue only with an electrical reaction. Some of the above mentioned facts have in passing been indicated by SamoLorr. These facts might be considered as the manifestation of an “Entartungsreaction”. In the same way the slackened crescent and decrescent (of the mechanical as well as of the electric reaction) may be looked upon. Further we often see that in the course of the process of fatigue ') Durie A. Ueber die elektromotorischen Wirkungen des wasserarmen Muskels, Prrücer’s Archiv, Bd. 97, 1903. 2) Bxtcxe E. Th. v. 1. c. 3) Samoutorr A., Einige Elektrophysiologische Versuche. Le Physiologiste Russe 1908. Vol. ¥, N°. 86—96. A. K. M. NOYONS. “About Observations at the electromyogram and form-myogram under the influence of fatigue.” Fig. 1 0.5 sec. Raff vn NEN MADAASS AO An electr mech. Fatigue-curves of mechanical and electric reactions as answer to stimulation by means of falling drops of mercury. The time is given in '/, seconds. The middle line gives the electro-myogram, the third curve the form-myogram. Fig. bo elect. mech. Fatigue-curve of mechanical and electric reactions as answer to the mechanical stimulus of the tetamotor-hammer. The upper line gives the electro-myogram, the next curve the form-myogram and the lowest line the time in '/» seconds. Fig. 3. 0.5 sec. electr. a Fatigue-curve of mechanical and electrical reactions on the faradaic stimulus (a) and on the galvanic stimulus (0). The second curve gives the electro-myogram, and the third the form-myogram. 15'/) minutes after commencement of the experimenent. Proceedings Royal Acad. Amsterdam. Vol. XL. (729 ) the original diphasic character of the action-current gets lost and is replaced by a monophasic image. We see as it were one top of the electromyogram wear off, whereas the other top increases, as already Lee *) observed for the M. sartorius. Also Brernstern and GARTEN *) saw this. The peculiar fluctuations that are often seen at the end of a mechanical fatigue-curve, also show themselves in the electric fatigue- myogram, even in those moments when every mechanical reaction has already disappeared. These fluctuations appear sooner and are much more intense, when the whole: animal is narcotized with chloroform or ether. In nareotization the action-currents on the whole remain longer in existence than the mechanical effects. Warming the whole animal. with due regards to provisions to prevent even the slightest desiccation, make the mechanical as well as the electric reactions on the periodical stimulus increase strongly in size, notwith- standing the fatigue ought already to manifest its influence on the size of both the phenomena. Lastly it may still be mentioned here that, after a short fatigue of muscles at weak electric reflex-stimuli and a small weight to be raised, I saw, when stopping the artificial stimulus, spontaneous contractions appear, which continued for some time’). What was striking here was that these spontaneous mechanical phenomena were as large as those at electric reflex-stimulation, but that at the same time the accessory electric phenomena were much smaller compared with those obtained by artificial stimulation. 1) Lee F. S. Archiv. fiir Physiologie 1887. 2) Berstein, Garten e. a. Handbuch der Physiologie des Menschen von W. Nacet, 2e Hälfte, Erster Teil, 4e Band, 1907. 3) It may be provisionally mentioned here that to complete Samoyrorr’s work I succeeded, also in a reflectorical way, in obtaining considerable mechanical and electric reactions from the M. gastroenemius, both in consequence of chemical and mechanical reflex- stimulation. ( 730 ) Chemistry. — “A contribution to the photo-chemistry of silver (sub-) haloids’. By Mr. A. P. H. Trivets. (Communicated by Prof. 5. HooGewerFr). (Communicated in the meeting of January 30, 1909). Introduction. Previous investigations into the photo-chemical decomposition of silver haloids led to the following formula: 2 AgHal — Ag,Hal — 2 Ag. It is true the great variability in the chemical composition of the photo-chemically formed subhaloid has given rise to doubts as to the correctness of the formula Ag,Hal, so that J. M. Epmr') even thinks it possible that the subhaloid has the formula Ag,Hal,, Ag,Hal, or perhaps Ag,,Hal,,, but a satisfactory explanation of these deviations was found by Gunrz*) in the partial and only superficial photo- chemical decomposition, so that when quantitative determinations were made, the subhaloid dealt with was never pure, but always contaminated with AgHal and Ag. It is a faet that with continued light action the photo-chemical decomposition of the silver chloride never reaches the formula Ag,Cl, but approaches it *). Thus Ricur *) found, after silver chloride had been exposed to the action of light for 1'/, years, a composition corresponding to the formula Ag,Cl,. In 1895 ©. Wierer *) discovered the colour formation of the Seebeck-Poitevin photo-chromics by mechanical colour accommodation, which can only set in, if there are a number of mutually different subhaloids. E. Bauer ®) took these to be four “modifications” of one and the same subhaloid. A little more than a year afterwards J. M. Eper?) in his excellent investigations as to the substance of the latent image came to the conclusion that there exist a number of subhaloids, which consecutively originate photo-chemically one from another, react differentiy on developers, sodium thiosulphate, ammonia, ete, and are formed anew by oxidation (nitrie acid) in the opposite order (as compared with the photo-chemical deeompositon). These results, however, require a slight correction, because J. M. Eper started from the idea then still generally held, but now anti- quated, that the process of development was nothing but a simple reduction of the unstable subhaloid-bearing silver haloid grain, whereas according to the more recent view, enunciated by W. Osrwarp 5), K. Scuaum and W, Braun’), and confirmed microscopically by ( 731 ) W. ScHerrER '°), every development consists in the precipitation of the reduction products from a dissolved silver compound upon the germ-subhaloid. Making use of this, in an investigation into solari- zation and further properties of the latent image, I'') have arrived at the result that the @ subhaloid of the latent image passed, with loss of halogen, into another, the 8 subhaloid. Soon afterwards B. Warrer '®), without being acquainted with my work, concluded from fresh investigations into solarization, that his experiments are best accounted for if two really different ““Zerfallstufen” (stages of decomposition) of silver bromide are assumed. Although, consequently, the existence of a number of subhaloids differing in chemical composition and having different properties, has been proved, KE. Baver’s modification theory may still be maintained by the side of it, for one of these subhaloids might possess the power of forming a modification by the action of light, through which its absorption spectrum might change, as e.g. white P, is transformed by light into orange-coloured P,;. Only if it were proved that this change of colour of the subhaloid was accompanied by a change in the chemical composition of the subhaloid, would the modification theory have to be relinquished. I shall revert to this question later on, and assume for the present that every subhaloid of a definite chemical composition is characterized by its absorption spectrum. During the last few years LüpPo-CRAMER *®), on the ground of a number of chemical reactions and physico-chemical phenomena, has drawn the conclusion that silver subhaloids, with the exception of subfluoride, do not exist, and are nothing but absorption compounds of collodial silver and silver haloid, and he has maintained the direct photo-chemical formation of silver from silver haloid. A short time ago I") succeeded in showing in a paper on silver subhaloids that all chemical reactions and phenomena upon which Lipro-Cramer founded the above-mentioned conclusion, can even be accounted for in a simpler way by assuming that the subhaloids are chemically defined compounds, so that it has not yet been proved that it is correct to deny the existence of silver subhaloids. K. Scnaum*®), too, has pointed out that the direct photo-chemical product of decompo- sition of silver bromide cannot be silver, for if silver bromide is exposed to light by the side of bright, metallic silver, Ag Br becomes dark, but at the same time the silver assumes a violet colour; “es ist thermodynamisch unmöglich, dass in dem nämlichen Systeme gleichzeitig AgBr in Ag und Ag in AgBr iibergeht”. Hereby it was proved, not only that the conception of the silver subhaloid as an absorption compound, was untenable, but also that the formula (732 ) of the stable silver haloid could not possibly be Ag Hal, but must be Ag, Hal,, in which » is a figure as yet unknown. Quite independently of K. Scuaum I *°) have arrived at the same result by the explanation of a few developing phenomena in the case of exposed silver haloid free from binding substance. Let us indicate the sensitiveness to light of a substance generally by placing the letter / before the name of it, thus: lAeJ > / AgBr This is also applicable to the substance of the latent image, the a silver subhaloid : | Ag. Jn a> ACD Now the remarkable case may occur that seemingly l Ag) /(AgJ AgBr) > and | / AgBr Le. the reducibleness, which is pretty generally erroneously identified with sensitiveness, increases by exposure more rapidly in the case of iodide silver bromide than in the case of silver iodide or of silver bromide. This is to be accounted for by the circumstance that iodide silver bromide produces a substance of the latent image less sensitive to light than is yielded by siiver iodide, while at the same time this substance is formed more rapidly than by silver bromide, in other words, while at the same time iodide silver bromide is really more sensitive to light than silver bromide. In the most favourable case the latent image may consist of Ag, Br. As iodide silver bromide was obtained from the synthesis of the subbromide with iodine, it follows that the formula of the silver bihaloid must be Ag, Br, Ja. On the analogy of this Ag, Br, is obtained, if bromine is taken instead of iodine, so that the general formula of silver haloid becomes Ag, Haln. If we now assume that the photo-chemical decomposition of silver haloid, resp. subhaloid, again and again forms a new subhaloid, each time by the loss of one halogen atom, we obtain the following series : Ag, Hal, —Ag,Haly_;—AgnHal,_»— ... ~Ag,Halo—>Ag,Hal—nAg . (I) it being, of course, not impossible that one or more of these sub- haloids do not exist. R. Luruer*’) observed no difference between the oxidation potentials of the subhaloids of the latent and of the visible ‘photo-chemical decompositions, and therefore assumed the formation of the same ( 733 ) subhaloid in both cases. In order to determine the chemical com- position of this subhaloid, he added at intervals to a known quantity of silver a chlorine solution containing */,, of the quantity required for the formation of Ag Cl, and then each time determined electro- motorically the oxidation potential of the remaining chlorine. In the beginning it remained constant at 0,55 Volts, but when the composi- tion Ag, Cl had been reached, it suddenly became 1,45 Volts, and after that remained fairly constant. In the synthesis of silver bromide he also observed a rise of the potential at Ag,Br. E. Baunr*’) con- firmed this in the case of Ag,Cl, and could further demonstrate a slight rise of the potential in proportion as the percentage of chlorine increased, from which he inferred that Ag,Cl forms with silver chloride homogeneous mixture series (absorption compounds). On the ground of the existence of a number of subhaloids it follows from this just as well that the subhaloids with an increasing amount of halogen show a very slight rise of the oxidation potential without there being any need to deny the formation of absorption compounds of these subhaloids with silver haloid. At the same time it appears that for the present a more exact determination of these subhaloids by this method does not promise much success. If, however, these results are considered in connection with series (I), it appears that the first subhaloid formed synthetically, and con- sequent poorest in halogen, is Ag,Hal, and the following improved photo-chemical decomposition series is obtained: Agon Halo, ae Agon Halas Agon Halon—o noon TE ar Agon Hal, Fers Agon Hal, iss 2nAg (II) This series is in harmony with the opinion already expressed by J. M. Eper '®) some time ago: “Vielleicht bilden sich auch Silber- subchloride, welche als Zwischenprodukte von Ag,Cl und AgCl aufzufassen sind.” One of the most prominent peculiarities of the silver subhaloids, by which they were even discovered, is their absorption spectrum. As these subhaloids have the power of rendering the whole visible spectrum (the subchlorides do this best of all), I have endeavoured to determine experimentally the colour sequence of series (II). ‘ Method of Investigation. The method of procedure might have consisted in photographing the sun’s spectrum by the SerBreK process and in subsequently controlling the colour changes by means of a sun spectrum placed 50 Proceedings Royal Acad. Amsterdam. Vol. XI. ( 734) crosswise. If e.g. the red subhaloid originates after the green, the latter will have the power of yielding the red subhaloid, but it will never be possible for the green subhaloid to be photo-chemically formed out of the red subhaloid. This experiment already made by O. Wienrr?’) yields: 1s*, doubtful results, because the primary rendering of the spectrum already leaves much to be desired, and this is, consequently much more the case with the secondary rendering; ged, unreliable results, because in this way the chemically pure subhaloid is never experimented with, but by the side of it there is always an admixture of silver haloid. It is true, the Porrrvin process yields a better colour rendering, but the results are not any more reliable, because, as Liprpo-Cramer*') has proved, in this process photo-oxidation takes places by the side of photo-reduction. I, there- fore, selected another method. According to H. Lvaern **) and R. Luruer **) the photo-chemical decomposition of silver haloids in a closed space gets with a definite light intensity into a state of equilibrium owing to the halogen pressure which sets in. Gunrz**) determined the equilibrium pressure with various light intensities by exposing silver chloride under different, known chlorine pressures, in small glass tubes sealed in a blowpipe flame. This method produces decomposition in these tubes, in which the pressure is too small for the state of equilibrium, but is not suitable for determining the colour sequence, because the colour of the subhaloid can only be observed through that cf the halogen, consequently very inaccurately. Therefore it is better to proceed as follows. If silver haloid is emulsionized in a binding substance (one might also say: “if it is placed in a half-closed space’) the photo-chemically liberated halogen cannot escape immediately, but with a definite light intensity a halogen pressure D per unit of time will set in, which is dependent upon the halogen pressure D, which the photo- chemical decomposition process would produce, diminished in the first place by the pressure D, which is lost by diffusion, and in the second place by the pressure D, which is lost by the chemical combination with a halogen absorbent (chemical sensitizer). DED: (D, AD e e In this formula D, and JD, in the second member, and consequently D too, may be modified. Now seeing that a rise of halogen pressure is accompanied by an increase of diffusion and of the rapidity of reaction of chemical combination, a fairly constant halogen pressure may continue to exist ( 735 ) for a definite period during the decomposition, if the chemical sensitizer is present in not too small a quantity. (Compare the result of the experimental inquiry into the influence of the size of the grain upon JD, below, with the accuracy of this reasoning). Practically it is not even so very simple to take too small a quantity of chemical sensitizer, because the first visible decomposition of the silver haloid already sets in, when a quantity of halogen, too small to be weighed, has been photo-chemically liberated, which is undoubtedly owing to the intense colour of the subhaloids. The cause of the more rapid photo-chemical decomposition of silver haloid in the presence of a chemical sensitizer has been attributed by Guntz**) to the development of heat, which accompanies the chemical combination of the lberated halogen, because this photo- chemical decomposition is attended with heat absorption. If this cause really had such a great influence upon the decomposition process, formula (III) would have to be considerably modified. This, however, is not the case. LürPo-CRAMER **) has demonstrated experi- mentally that the chemical sensitizer only accelerates the decomposition process, if the silver haloid is emulsionized. Silver haloid precipitated in a test tube does not show a trace of accelerated photo-chemical decomposition, neither by the addition of silver nitrate, nor by the addition of ammonia. I must, however, observe that the colours which appear are not the same. Consequently the thermic influence of the chemical sensitizer, referred to by Guntz, is so small that in practice it may be ignored, and may at most be taken as a theoretical correc- tion of formula (III), especially if it is borne in mind that the total amount of halogen that is liberated and reacts in my experiments, is extremely small. On the analogy of the more or less regular changes of properties in the case of other chemical series, as e.g. the fatty acids, it may be assumed that the subhaloids according to series (II) will show an increased or decreased affinity, however small the differences may be mutually. As with a constant light intensity the photo- chemical decomposition of the subhaloid gets, at a certain definite halogen pressure, into a state of equilibrium, we may say that the equilibrium pressure in the ease of the subhaloids according to series (II) undergoes an increase or a decrease. Even the measurements of EB. Baver °°) mentioned before, who observed a rise of the potential with a higher proportion of halogen, point to the fact that with a constant light intensity this equilibrium pressure will decrease if the amount of halogen in the subhaloids is smaller. If now formula (III), in which, consequently, D is practically 50* ( 736 ) constant for some time, is applied to the assumed subhaloids a, 9, y..., in which each subsequent subhaloid is poorer in halogen than the preceding one, and which with the same light intensity have respec- tively the equilibrium pressure D,, Ds, D,,..... then we get Die De > Dy eine eee If D= Ds, then the photo-chemical equilibrium will set in with the 2 subhaloid, recognizable by the colour of this subhaloid, that is to say, the photo-chemical decomposition does continue through tbe loss of halogen, but the state of decom- position will not proceed beyond the # subhaloid, as long as Dz does not decrease. Thus first a definite colour is seen to constantly increase with the photo-chemical decomposition till a maximum has been reached, after which, sometimes after a very long exposure, a change of colour becomes noticeable. In the case of Dz > D > D, a blend of the colours of the 3 and the y subhaloid will appear. Consequently, by giving to D a higher or a lower value, we have it in our power to determine the colour sequence of the subhaloid according to (II); by decreasing the quantity of chemical sensitizer or by preventing diffusion, subhaloids richer in halogen, in the opposite case subhaloids poorer in halogen are obtained. A , p" (eee A EERE ere | : | | | Ee | oeil til | ' \ | DD A | | 4 KH ce A a) | Re —> 9 Tea FF x The graphic representation clearly shows this. If on the abscissa (see fig.) at mutually equidistant points A’, B, C’,.... we indicate the subhaloids a, B,y,...... in equal molecular quantities, and on the ordinate the equilibrium pressures A'A, B'B, C'C,...., corre- sponding to them with a definite light intensity, then the connection of the points A, B, C,.... must yield a line which approaches the abscissa from OQ in a positive direction. If we assume now that the amount of one of the subhaloids, say 3, decreases in such a way from B both in the positive and the negative direction on the il : abscissa, that at — of the distance from £' to A' or C' there is the Nn ( 737 ) th part of the quantity of 3 subhaloid (which is in B), then the quantity of 8' subhaloid will be zero both in A’ and in C'. If now we further assume the same distribution in the case of the other subhaloids, then beginning at the point O, a series of colours will successively appear on the abscissa, among which there are mixed colours composed of at most two components between the points ro ie Se The equilibrium pressure at any point is then deter- mined by the ordinate from this point to the intersection with the line ABC.... Suppose that with the photo-chemical decomposition a constant halogen pressure D= OUP" prevails, then the state of decomposition will not progress beyond P’, because the equilibrium pressure P/P corresponding to P’ is equal to OP". Consequently in P’ a mixture of the p and y subhaloids will appear, the quantities of which are determined by the proportion Ng: VN, = P’C’: P’B’. A mixed colour will then appear, in which, according to the figure, the colour of the 3 subhaloid will be predominant. If D becomes greater, say D’ — OP", then the state of equilibrium will set in at P,’, where, according to the figure, the colour of the 2 subhaloid is predominant. Now by slightly modifying D between the equili- brium pressures of two photo-chemically consecutive subhaloids, it must be possible to obtain all transitional tints from one subhaloid to another, and this could actually be demonstrated in all those cases in which it was experimentally possible. To obtain these results, the subhaloids must, of course, be suffi- ciently sensitive to light. Still with a few experiments, as those with subiodides and subbromides, phenomena are observed which cannot be accounted for as yet, and require further, separate investigation. We shall not enter into them here. From the graphical representation it further appears that D can never become greater than the highest position of the line ABC.... above the abscissa. If D is smaller than the equilibrium pressure of the subhaloid poorest in halogen, then silver begins to separate out photo-chemically. If we assume that the line ABC... is continued still further towards the Y-axis, ie. that the silver haloid is placed in O before the @ subhaloid on the abscissa, then the ordinate belonging to it will correspond to the lowest halogen pressure at which the photo- chemical decomposition of silver haloid still takes place with this light intensity, and which answers to the above-mentioned chlorine pressure determinations by Guntz. Now suppose these results to be incorrect, because the continuation ( 738 ) of ABC....//OX, or from O onward recedes in positive direction from the abscissa, then it will never be possible to obtain by varying (D,+ D,) the colour of one or two of the subhaloids, but the photo- chemical decomposition will always progress more or less rapidly in the same way in all cases. However, the results of the experiments mentioned below cannot be reconciled to this supposition. The influence of the size of the grain. As the photo-chemical decomposition of silver haloids is restricted to the surface of the grains, it is easy to understand that the size of the grain must influence the decomposition process. The finer the grain, the more “saturated” the colour, the coarser the grain, the weaker the colour will be. As in formula (III) D becomes greater owing to the finer grain, D, and D, will therefore also increase. Consequently by exposing coarser and finer silver haloid grains emulsionized under as mneh as possible the same circumstances, it can be determined experimen- tally, to what extent D changes through an increase of D,, and an increase of (D,+D,) depending upon it. I have made this experiment with a fine-grain silver chloride gelatine emulsion, and silver chloride gel emulsionized in gelatine, but found, except in the saturation of the colours, a very small difference, which might even be put down to an inaccuracy in the experiment, viz. a slight difference in the amount of chemical sensitizer. Jontrol. If in formula (III) D, is made smaller, D will become greater, and the colour of a subhaloid richer in halogen will appear. The same thing may, however, be obtained in another way. The equilibrium pressure is smaller with a weaker light intensity (approaches zero in the dark), but in this case D, will also decrease a well as the term (D,+D,) dependent upon it, and, as has been shown by the enquiry into the influence of the size of the grain upon silver haloid, D thereby changes little or nothing. The consequence of making the photo-chemical decomposition take place with a smaller light intensity, will therefore consist in the appearance of subhaloids richer in hatogen. Therefore, by lowering D as well as the light intensity, the same subhaloids richer in halogen, in the opposite case the same subhaloids poorer in halogen, will be seen to appear. Another means of control is afforded by increasing D after the photo-chemical decomposition has been interrupted by introducing halogen into the preparation; in this case the same subhaloids richer in halogen must be obtained back again. This method, however, has the drawback, that owing to the absorption spectrum of the halogen, the colours of the subhaloids are to be observed in a less pure condition. ( 739 ) The colour value. Seeing that side by side with the subhaloid we always have the coloured halogen, although in a less than equivalent quantity, the colours of the subhaloids will never appear so clearly as in the case of the preparations prepared chemically by M. Carry Lra’s and Lipro-Crammr’s**) method. The place of the a-subhaloid. My investigations as to the substance of the latent image?*) and its preparation *°®) show that in the case of iodide as well as of bromide and chloride the a-subhaloid has a green colour. As the assumption of a subhaloid still richer in halogen to account for Bunsrn-Roscon’s photo-chemical induction *') and for auto-sensitation **) is superfluous, I have attempted to trace this possibly extant subhaloid by the photo-chemical method indicated. If silver bromide with an excess of potassium bromide is made to precipitate from a silver nitrate solution, dried, and exposed under carbon tetrachloride to moderately strong daylight, it soon begins to show a green colour. In this preparation the diffusion of the liberated halogen is extremely slight, and potassium bromide is a very weak chemical sensitizer, with which the reaction product KBr, with the subhaloid partially shows a reversed reaction in the dark. Under these circumstances the e-subbromide is, therefore, obtained photo- chemically in visible quantities, and hitherto this has been the simplest method found by me of preparing light-proof preparations from this substance. The halogen pressure can, however, be increased still more by omitting this weak chemical sensitation as well. In a dry, pure collodion coating pure silver bromide yielded a grey discoloration with the lowest light intensity with which decomposition is still to be observed. In the dark the light yellow silver bromide appeared again after some time; consequently an almost complete re-formation of the silver bromide takes place, from which it may be inferred that the loss of bromine has been reduced to a minimum, and the subhaloid richest in halogen may have appeared with the photo- chemical decomposition in a visible quantity. But even now the grey colour need. not be attributed to a subhaloid richer in halogen than the a-subhaloid, for the equivalent amount of bromium may mix its brownish red colours with the green of the «-subbromide so as to yield grey. As, therefore, the green a-subhaloid is to be considered the one richest in halogen we know at present, we may say that every other colour must be ascribed to a subhaloid poorer in halogen. Use of the a subhaloid. If in these experiments we start from the green a-subhaloid instead of from silver haloid, a different sequence of colours is obtained. With silver nitrate the green preparation very rapidly turns grey or black through the formation of mixed colours ( 740 ) with the subhaloids poorer in halogen, and with the exception of a single case mentioned below, it is ‘therefore unfit for controlling purposes. So the best plan is to start from silver haloid only, which owing to its white or light yellow colour has little or no influence upon the colour of the subhaloids that are formed. The binding substance. In most cases gelatine was used for this purpose. In a perfectly dry condition it does not absorb free halogen ; it does, however, in the presence of water. The chemical sensitizers. As it appeared desirable to me, in connec- tion with these experiments, to obtain as much as possible results that could Le compared with each other, I have made as little as possible use of different chemical sensitizers. Besides alkali-haloid and moist gelatine, silver nitrate was selected as being a powerful sensitizer. The photo-chemical reduction in the presence of gelatine has, as appeared from a parallel experiment without silver haloid, no disturbing influence upon the determination of the colours of the subhaloids. Description of the experiments. The silver subiodides. The cause why hitherto nobody has succeeded — in demonstrating the iodine liberated photo-chemically, is no doubt to be put down to the very unfavourable conditions of diffusion (high atomic weight and low vapour tension of iodine), and to absorption by silver iodide with the formation of an absorption compound **). Moreover the visible photo-chemical decomposition is always very slight, notwithstanding the high sensitiveness of silver (sub)iodide, to which I have already referred more than once, in other words, only very little iodine is liberated, which has been associated by H. Lueain**) with the low equilibrium potential of silver iodide. This makes silver iodide together with iodo-silver bromide, mentioned above, the most suitable silver haloids for daguerreotypy, in which the destruction of the «-subhaloïd, upon which the reducibleness depends, would otherwise be much stronger as compared with the emulsion processes. Consequently it is unneces- sary to emulsionize the silver iodide. If silver iodide with an excess of potassium iodide is precipitated from a silver nitrate solution, then a very slight discoloration appears on exposure*’), without a subiodide colour becoming observable. The cause is probably that during the precipitatien of silver iodide potas- sium iodide is separated out as well. Now in photo-chemical decom- position the liberated iodine is absorbed in a less degree by silver iodide, but largely by potassium iodide, potassium tri-iodide being ( 741 ) formed, which with subiodide gets into a state of photo-chemical equilibrium, which is nearer to silver iodide than that of subiodide and adsorbed iodine. The sensitiveness of potassium iodide, by which iodine is photo-chemically liberated, may also have some influence. If silver iodide with an excess of silver nitrate -be precipitated by potassium iodide, it is free from potassium iodide. The photo-chemical decomposition now proceeds much further; a greyish green colour sets in, which points to a high amount of «-subiodide. This seems at variance with the observations of Hrrscnen’s effect, and the solarisation in the case of the silver iodide daguerreotype plate, where with.much shorter exposures a decrease of reducibleness undoubtedly sets in, owing to the photo-chemical splitting up of «-subiodide into B-subiodide and iodine. This contradiction is solved, if it is borne in mind that, to render the exposed silver haloid reducible, the presence of the «-subhaloid alone is not sufficient, but that, moreover, it must be at the surface of the silver haloid, so that through molecular attraction the reduction products of the dissolved silver salt or the mercury vapour may be able to settle on it. The greyish green discoloration may, therefore, be attributable to a high amount of a-subiodide, but it may itself be covered by an extremely thin layer of subiodide poorer in halogen. If we endeavour to remove the halogen pressure of iodine altogether by conducting the photo-chemical decomposition under a silver nitrate solution, then the reductions become very complicated. The acceleration of the visible photo-chemical decomposition appears to be hardly appreciable, while the substance assumes a grey colour, and yet in the silver iodide collodion process silver nitrate behaves as an excel- lent chemical sensitizer. On a former occasion **) I already pointed out that im the light silver nitrate must exercise a strongly oxidizing influence upon subiodide, silver oxide being formed, and accounted for the more rapid increase in the reducibleness (i.e. more rapid rise of the amount of a-subiodide) by this oxidation, which can act only then in such a way that d-subiodide is more rapidly oxidized into a-subiodide than the latter into silver iodide. Consequently when silver iodide in a silver nitrate solution is exposed, two opposite actions take place: a progressive reaction, the photo-chemical decom- position, and a regressive reaction, the oxidation, the former taking place a little more rapidly than the latter, which reactions probably even with the subiodides poorer in halogen perfectly neutralize each other. With oxidation, however, silver oxide is separated out; I therefore surmise that the grey discoloration mentioned before must be ascribed to this silver oxide. ( 742 ) This photo-chemical decomposition might consequently be accelerated dst by taking an iodide more sensitive to light, and 2rd by decreasing the oxidation by silver nitrate (exposure with smaller light intensity). Both these conditions have been complied with in the extremely rapid photo-chemical decomposition of a-subiodide in weak twilight observed by me*’). Beside these reactions others set in, as the probable formation of iodates, oxidation through one of the reaction products of the chemical binding of iodine and silver nitrate, etc., the investigations into which are partly incomplete as yet. As subbromides and subchlorides oxidize far less rapidly, as is well-known, the deviations occurring in the case of subiodides, are much less likely to occur with them. Further a red subiodide of M. Carry Lea?) is well known, which, consequently, must contain a smaller amount of halogen than a-subiodide. Of the colour sequence of series (II) we accordingly know only green; en. red eht i thin: ee CEE The silver subbromides. In the following table the observations have been arranged in such a way, that D constantly increases. Agon Bron Fine-grained. Discoloration in daylight In gelatine. (October, 11 to 2 o’clock) (1). with 10°/, aqueous AgNO, sol. yellowish, reddish or brownish violet (2). with much H,O brownish violet (3). with little H,O red (4). moist reddish violet (5). less moist bluish violet (6). dry bluish Without binding substance (7). with KBr under ether, CCI, green These discolorations were not observed simultaneously, but always when the colour was most intense without passing into another. The decomposition takes place so unequally that, when exposed simultaneously, (1) and (2) already have a clearly perceptible colour after a few tens of seconds, whilst the others show hardly any visible decomposition. Compared with each other, the latter prepara- tions show little difference in the rapidity with which the colours appear. We see here a deviation from Lipro-Cramer’s observation ( 743 ) referred to above, that photo-chemical decomposition is accelerated by chemical sensitation only in an emulsion. If in the ease of preparation (7) the ether or carbon tetrachloride is allowed to evaporate in the dark room after exposure, then the bromine is liberated, which is distinctly perceptible by its smell and the potassium iodide starch reaction. Between the preparations (3), (4), (5) and (6) all intermediate tints from blue to red could be obtained by increase o1 decrease of the amount of water. The preparations (6) and (7) still contain a trace of adsorbed waiter. With the very rapid photo chemical decomposition already referred to, the preparations (1) and (2) show the violet colour anew. This may be accounted for by the fact that D was so small, that no constant equilibrium pressure was to be reached in the photo-chemical decomposition series. When the experiment was repeated a number of times, preparation (1) showed a different colour each time, probably through changes in D and the light intensity. In direct sunlight especially the yellowish brown colour appears; sometimes it was even blue-black through the silver separated out. The yellowish brown mixed colour points to the formation of a yellow subbromide with the ultimate products. Tests have been made by observing the same photo-chemical decomposition with weaker light intensities. Then (2) yielded a red colour, and (3), (4) and (5) showed a distinct shifting of the colour towards blue. Preparation (7) assumed a greyish green colour in direct sunlight. Further the red preparation rapidly developed a blue colour in bromine- water, which colour then entirely bleached into silver bromide, without my being able to observe the green sub- bromide with certainty. Of the green, blue, and red preparations, the size of the grain being the same, and free from binding sub- stance, the red one reacted most rapidly upon sodium thiosulphate, H,CrO,, HNO, and (NH,),5,O,, and especially the green preparation showed considerable resistance. By putting the a-subbromide in a neutral 10°/, sodium thiosulphate solution I was able, even by the light in the room, to observe distinctly a colour change through bluish green to blue. The colour sequence of series II is consequently in the case of subbromides : green, bluish green, blue, violet, red, ... yellow..... . (V) The silver subchlorides. In the following table the same arrange- ment has been observed as in the case of the subbromides. ( 744 ) Agon Clan Discoloration in daylight Very fine grain (October 11 to 2 o’clock) In gelatine. (1) with 10°/, aqueous AgNO, sol. reddish orange (2) with AgNO, dry red. (3) with NH,Cl moist violet. (4) with NH,Cl dry blue. Without binding substance. (5) with NH,Cl under CCl, bluish green. By variation the amount of water all intermediate tints can be obtained between preparations (8) and (4), while by decrease of the amount of AgNO, preparation (2) becomes reddish violet to violet, and has for this reason been placed before (3). The preparations (2), (4) and (5) still contain a trace of adsorbed water. In Porrrvin’s photochromics there also occurs a yellow (more orange-like) subchloride. Preparation (1) already yields a colour which inclines from red to yellow; therefore the yellow subchloride is probably formed after the red. In direct sunlight this preparation assumed a yellow colour. However, | found no indications that this was a subchioride, for neither in ammonia nor in a 10°, sodium thiosulphate solution did it undergo any perceptible change, while the red and the reddish orange preparation became yellow and yellowish brown in these two solutions. So it may just as well be photo-chemically formed collodial silver. However, the yellow sub- chloride cannot be classed anywhere between or before the other colours. In analogy with what is similar in the case of the sub- bromides it may therefore be assumed, that after all, it comes after the reddish orange preparation. The green « subchloride is known to be the subchloride richest in halogen, and preparation (5) already inclines to it, so that in the case of the subchlorides the colour sequence of series (11) is green, bluish green, blue, violet, red, orange, yellow. . . (VI) The silver subjluorides. Silver fluoride is hygroscopic and too insensible to light for these experiments. Still there is a yellow sub- fluoride of Guntz **) with the formula Ag,Fl. The silver subcyanides. Their existence is still doubtful. A chemi- cally pure silver cyanide preparation of E. pr Hain, showed the same bluish violet discoloration both without and with a binding substance, and even under CCI, In the case of silver cyanide the ‘photo-chemical decomposition Te takes place according to he laws than with the foregoing silver (sub)haloids. ( 745 ) The silver subrhodanides. Their existence, too, is still doubtful. On account of its very slight sensitiveness to light silver sulpho- cyanide is unsuitable for these experiments. Results and conclusions. Let us now return to B. Baver’s modification theory, already referred to before, which owes its origin to the assumption of the existence of only one subhaloid, which view at one time obtained universally. As has been demonstrated experimentally, new colours only arise consecutively, if D decreases, in other words, if more halogen is liberated photo-chemically. Every new colour, therefore, belongs to a subhaloid poorer in halogen. The modification theory is not to be associated with this, and the assumption adopted of ascribing a definite absorption spectrum to each subhaloid, can still hold good. If we compare the colours series (V) and (VI), we see that even with the exception of the place of the yellow subhaloid, which has not yet been fixed with absolute certainty, they show the same sequence of colours, while series (IV), as far as it is known, runs parallel with it. If we now bear in mind that, barring slight varia- tions, the a-subhaloid is green, both with subiodide and subbromide and subchloride, we may account for the parallelism of the colour series by the fact that subhaloids of analogous composition have analogous absorption spectra. The colour sequence itself, too, shows regularity: the colours of the silver subhaloids, arranged according to the photo-chemical decomposition series, follow Niersxi’s rule, in which halogen behaves like a bathochromic group, in other words, with the decrease of the molecular weight of the silver subhaloids the maximum of the absorp- tion spectrum is shifted from red to violet. Although the structure formulae of the silver subhaloids are still unknown, it may be inferred from this that they all contain the group Ags, Hal,, which behaves like a chromophore or contains it. The cause of the sensi- tiveness to light of the silver subhaloids is, therefore, to be relegated to electro-magnetic light resonance. As according to Nietski’s rule the maximum of the absorption spectrum of a subhaloid still richer in halogen than the «-subhaloid must be situated in the infra-red, and may therefore be colourless, the above-mentioned experiment for the detection of this subhaloid by increasing D appears to be perfectly worthless. Still this does not alter the conclusion drawn from this experiment, that every ( 746 ) other colour than green appearing in photo-chemical decomposition is to be attributed to a subhaloid poorer in halogen. If this subhaloid actually existed, it would have to have, in analogy with the other silver subhaloids, a maximum sensitiveness to light in the infra-red in accordance with its absorption spectrum. It would then have to make its influence strongly felt in spectrum photography, if the silver haloid plate previously received an exposure below its liminal value. H. Leymann *°) in his experiments as to the infra-red spectra of alkali metals has actually observed that through a previous exposure and an ammonia bath a highly sensitive silver bromide gelatine plate increases its sensitiveness via A into the infra-red near FRAUNHOFER's line Z, but, and this is the great point, a conspicuous rise of sensitiveness in the infra-red appears nowhere from his expe- riments. Consequently what H. LEHMANN observed was nothing else than the well-known auto-sensitation of the plate. So we may say that a subhaloid richer in halogen than the a-sublaloid does not exist. According to Nierski’s rule silver haloid would have to take the place of the colourless subhaloid assumed above, but it shows the deviation that the maximum of its absorption spectrum is not situated in the infra-red, but in the blue or violet, i.e. where the number of vibrations is about twice as high. Consequently in the case of silver haloid, too, the cause of the sensitiveness to light is to be relegated to electro-magnetic light resonance. In the synthesis of silver haloid from silver and balogen, through the series of subhaloids, the electro-magnetic resonator undergoes changes which show a striking resemblance to the foliowing well known phenomenon in aconstics. If we take a small, thin bar with a fixed and a detached end, it will resound at a tone corresponding to its key-note; by constantly lengthening this bar, the key-note is lowered, and the resonance will set in at a lower number of vibra- tions, until, after it has reached its maximum length, the next leng- thening will at the same time fix the detached end, by which the key-note will get one octave higher, and accordingly the resonance will set in at the double number of vibrations. Nietski’s rule points to a yellowish green subhaloid as a subhaloid still poorer in halogen than the yellow one. In numerous experiments repeated under various circumstances, by endeavouring to keep D low and constant, I have not been able to observe a trace of it. It is true, the appearance of the mixed colours of the subhaloids poorest in halogen points to the formation of a yellow subhaloid as the last one formed photo-chemically. Consequently the formula Agon Hal, may be assigned to it. ( 747 ) In the simplest case all the colours that appear, inclusive of the green of the a-subhaloid, may be reduced to 4 different ones, with the subhaloids belonging to them. If these subhaloids are indicated by the letters a, 3,y, and d respectively, then series (II) becomes : AgonHalon SAgenHalen—1 SAgenHalon—2 SAgenHalon—3 SAgonHalon—s >2nAg.(VII) z-subhal. B-subhal. y-subhal. 3-subhal. (green) (blue) (red) (yellow). in which Agon Halon—s = Agon Hal, or 2n-—4 =n nis 4 so that series (VII) becomes: Ag, Hal, — Ag, Hal, — Ag, Hal, — Ag, Hal, — Ag, Hal, — 8Ag. a-subhal. g3-subhal. y-subhal. d-subhal. (green) (blue) (red) (yellow) To summarize we may place the different subhaloids in the follow- ing table, in which Guntz’s solid subfluoride may very well be classed. subhaloid subiodide subbromide subchloride subfluoride a-(green) Ag, Ag,Br, Ag,Cl, unknown >-(blue) Ag.J, Ag,Br, AgCl, Ee y-(red) Ag,J, Ag,Br, Ag,Cl, 2 d-(yellow) unknown Ag,Br, Ag,Cl, Ag, FI, - The g-subiodide (Ag,J,) has been inserted in this table, because the solarization and Herschel’s effect in the case of silver iodide plates point to the existence of this subiodide. In conclusion I wish to point out that I *) have only partly succeeded in observing the high sensitiveness to light of the a-subio- dide. The grey discoloration of the green preparation, which, as was shown by the sodium thiosulphate reaction, is caused rather by silver- oxide than by subiodides poorer in halogen, resp. silver, can hardly be ascribed to the formation of the blue g-subiodide, but may be put down to the formation of the red y-subiodide. If moreover, the regressive reaction by oxidation is taken into account, which, it is true, is less than in full daylight, but all the same has not been neutralized, the a-subiodide appears to be still far more sensitive to light than described by me. (748) LITERATURE. 1) J. M. Eper. Photochemie 1906; 217. 2) Compt. rend. 1891; 113; 72. 3) J. M. Eper. Photochemie 1906; 217. 4) Chem. Zentralbl. 1879 ; 367. 5) WigDEMAN. Ann. d. Physik. 1895; 225. Eper's Jahrb. f. Phot. u. Repr. 1896; 55. 6) Eper’s Jahrb, f. Phot. u. Repr. 1904; 616. 7) Sitzungsber. d. kaiserl. Akad. d. Wiss. zu Wien, mathem.-naturw. Klasse CXIV; Abt. Ua; Juli 1905. Zeitschr. f. wiss. Phot. 1905; IIL; 329. J. M. Eper. Photochemie, 1906; 277. Pkot. Korresp. 1905; 425, 476; 1906: 81, 134, 181, DSL 190170: 8) W. Osrwarp. Lebrb. d. allgem. Chemie. 1893; 2; 1078. 9) Phot. Mitt. 1902 ; 229. Eper's Jahrb. f. Phot. u. Repr. 1902; 476. 10) Phot. Rundschau 1907; 142. Phot. Korresp. 1907 ; 384. 11) These Proc. June 1908; 773. Zeitschr. f. wiss. Phot. 1908; VI, 197, 237, 273. 12) Wien Ann. d. Physik. 1908; 99. 13) Phot. Korresp. 1906; 1907 1908. Liippo-CRAmMER. Phot. Probleme. 1907; 193. LürpPo-CRAMER. kolloidchemie u. Phot. 1908. 14) Zeitschr. f. wiss. Phot. 1908; VI; 364. 15) Zeitschr. f. Elektrochemie. 1908; 14; Nr. 33; 484. 16) Zeitschr. f. wiss. Phot. 1908; VI; 358. 17) Zeitschr. f. phys. Chemie. 1899; 30; 628. 18) Eper's Jahrb. f. Phot. u. Repr. 1904 ; 612. 18) J. M. Eper. Photochemie. 1906 ; 210. 20) EpeR'’s Jahrb. f. Phot. u. Repr. 1896; 89. 21) Phot. Korresp. 1907; 439. 22) Zeitschr. f. phys. Chemie. 1899; 23; 611. Eper’s Jahrb. f. Phot u. Repr. 1898; 162. 23) Zeitschr. f. phys. Chemie. 1899; 30; 628. 24) Phot. Wochenbl. 1905; 102. 25) Phot. Wochenbl. 1905; 93. 26) Phot. Korresp. 1901; 224. Liippo-CrAmer. Wissenschaft]. Arbeiten. 1902. 87. Eprr’s Jahrb. f. Phot. u. Repr. 1906; 648. 27) Eper’s Jahrb. f. Phot. u. Repr. 1904; 612. : 28) Americ. Journ. of Science. 1887; 33; 349. Phot. Korresp. 1887; 227, 344 371, Carey Lea u. LüpPo-CRAMER. Kolloides Silber u. die Photohaloide. 1908. 29) These Proc. June 1908; 789.Zeitschr. f. wiss. Phot. 1908; VI; 197, 237, 273. 30) Zeitschr. f. wiss. Phot. 1908; VI; 438. 31) Zie Zeitschr. f. Elektrochemie. 1908; 14; nr. 33; 488 489. 32) According to new, still unpublished experiments made by me. 33) Liippo-CrAMER. Kolloidchemie u Phot. 1908; 106. 34) Zeitschr. f. phys. Chemie. 1897; 23; 611. 35) J. M. Eper. Photochemie. 1906; 246. 36) These Proc. June 1908; 794. Zeitschr. f. wiss. Phot. 1908; VI; 284, 37) Zeitschr. f. wiss. Phot. 1908; VI; 438. 38) CAREY Lua u. Liippo-Cramer. Kolloides Silber u. die Photohaloide. 1908; 39. 59 Compt. rend. 1890; 110; 1337. 40) Archiv. f. wiss. Phot. 1900; II; 216. 41) Zeitschr. f. wiss. Phot. 1908; VI; 438. ( 749 ) Mathematics. — “An integral-theorem of Gwaenpaunr.” By Prof. KLUYVER. ; (Communicated in the meeting of February 27, 1909). GEGENBAUER has proved a theorem according to which the product of two functions of Brssrr /’*(ax) and /’(bx) with the same para- meter v >— 4 can be given the form of a definite integral '). In a former communication *) I have applied this theorem for the case r =Q(0 when reducing some discontinuous integrals containing functions of Brssen. [ shall now give in the following a direct proof of the indicated theorem and shall use it to extend former results. 1. In order to find the product of two functions of Besser aN On ea 7) 2 Poa) = (5 DME (—1) (=) el me 1)? ba \’ q we can multiply the absolutely converging power series. It is then evident, that (supposing 6< a) we find for the coefficient of an arbitrary power of x a finite hypergeometic series with the fourth b: argument —. The We get’) a \2h 109 lee Ty a es aba Na 2 b? J (ax) J’ (bx) =| —— N ach | —vy-h,-h,y +1, ;) C 4 am HI (v4 LT HAI) 1 To transform the hypergeometric series appearing here I shall use the notation of Riemann for the general hypergeometric function a kt Ea @ ÔrYs A k Paed >| Et PB + y tr ==) Pte er [le A. (2 —a) J- Alea) Arsis |. 1) Nietsen. Handbuch der Theorie der Cylinderfunktionen, page 182. 2) Proceedings 1905. 5) Nietsen. page 20. 51 Proceedings Royal Acad. Amsterdam. Vol. XI. ( 750 ) It is then evident that for two of the singular points the differences of the exponents are equal, so that besides the ordinary substitution of order one also substitutions of order two as 0 —1 +1 0 oost E PE an BN Beal dt BE Nen ' B t a 1 8 : a 2 2 and 0 —l +1 0" ert a DI a Pe ee ot) (US. Zale, = Ue oe ae | ! B 3' a 1 3! a t 9 2 t are possible. The indicated reduction runs as follows: 0 oo 0 b? z b? ne Daf Ng ee he 0 —h 1 ee (ye a | a? | —v —v—h 2v+142h 0 oo if h OC oe ee a | 2 2 a | é BE | D= 3 WW 2 p+ + | 0 al +1 h Bf Ob \e aap h h teh =27(-) es 0 ee a 2 a’? + 5? h h 2 heh. > p+1+ Y 5 Y 5 | 0) ore) 1 Di: bel = = 0 ea) 1 h b\h > hi Aa?b? = Af — B wes 0 0 bie kate == a 9 (a?-+ 57)? fen | 1 F = 2 9 yv | 2 + u | 0 —1 +1 BA 2ab GE Pk ek 0 Mr en ae a a’ +b? —2yv—h vp+i+h vt}h+h | 0 —] 1! a+b \2h 2ab a= EQ —h 0 nl tyra ee a a’? + 5? —2y vtt v+h-+h 0 oo 1 dab —h ee («$y Bap ot epe B he P eae 4ab GE) rees en fe) So finally we have 6? 4ab azh FP (—#. —v—h, v+1, =) = (a+b) F(—A, v4+4, 2v+1, —-— ], | a (a+b) a transformation given by Gauss. We now substitute for the just given hypergeometrical series the integral 1 NL Oe en Varwtd) J” hen ‘(1 at mig and we find if in the integral we put z= cos? 5 ath F( -h, -v-h, v+1 =) = mee) fete 2ab cos 7)! sin?” p d 5 rde) Ami 4) =.) == 2a EE = / } á C . a)” Varetd, Eee In the development found for the product ./’ (az) /(bx) the above mentioned integral is introduced. Putting a? + Bb? — 2ab cos p = Q’, we obtain ( 752 ) Ar 2h a x h=w(—1)h (=) a’ b xv? BAT J’ (az) J” ba = sin?’ pd Lies 2 ee (aa) gg” be 2Ya I (v+3) dine ( -) pay F(v4h1), 0 i. or a’ bP a * J? (Qz) 2 VaToth)) 2 0 J* (az) J* (bz) = sin?’ p dp, by which the indicated theorem of GEGENBAvER has been proved. 2. With the aid of this theorem we can extend some well- known results concerning discontinuous integrals, in which functions of BrsseL appear; particularly do the two following theorems *) aa B i oH (uc) J’ (ua) du = Nae Bee a’ 0 for a >c, A ct? du J*H? (uc) J’ (ua) — = a’ v 0 lend themselves to this extension. The theorem of GEGENBAUER namely allows on the ground of these results to determine in certain supposition the value of the disconti- nuous integrals 4 (c?—a’) fora. DR +l Ee du WZ | J’! (uc) J? (ua,) J’ (ua,) . . . J” (wan) ——— Meren dn” i un; 0 7 or : yo : 3 5 du W ss aar. La, | Use: (uc) Je (ua) B fe (wa,) ae je J (ua) na-D Ek] in which the number of ./-functions is arbitrary and v is > — 4. Let us think the positive numbers a,, a,,...a@, to be successive sides of a broken line OA,A,...A,, let us put 7 OArAri == pt and OA, = sk, then we find successively 7 1 Paya ) 1 J? (uss) os : -— J” (ua) J” (UA) = = sn” pr AP, 5 Ga” 1 { 2 2» Va MG 44 8,” P, Py Be Re ec : TG) si d —— oh (US) ed (Ua 3 KIDS = sau” US, oe 2° Va T(v+3) se Sin Gig OP 0 1) Nietsen, p. 198, ( 753 ) Jl Plu) 1 " F(usn) act ] ———— J*(us,, (ua) = “ cee det. De Er Sn 114? es read, Sn Pn—1 dPn—1 0 so that we get mg 13 = 1 a sin?’ pd sin” —~ dg... | sin?’ ~y—1 dg~y— [2 verend ’p, if p‚dp f Pnt dn X 0 0 etl for (uc) J(us‚‚) du, S’n 0 Ve a eae iat sy foe 9‚dg, fende x x +2 TE ae (uc) Fue) —. Let now in the first place be e>a,+ta,+...-a, then c is certainly greater than s,, and we find a a n—l W=l ———._] sin” a da : 2° /x Pv +4) 0 W.=:(¢ — a,’ 2 il 7 n—l th pias oe eek NS ae a?) Perses sin?’ a de | . 0 As a En Valt) bar eet F(v+1) 0 we find as final result ee a fhe Plus) a tia) saa!" (ue) BCA) IP (la) (a) EDTA) 0 c di du ce —a,?—-a,,7...-Ay? W.= J’+?2 J J” 8 eee jf 3 2 n ; : a tele, Pua). (uaa) et ae FL) 0 (c>a+ta+....+a,). Still in a second case the values of the integrals W, and W, are known. Let a, exceed all other numbers a and let us put a, >cta,+.... + an ( 754 ) It is necessary then for all values of 9,,9.,-+++@n—1 that the closing side of the broken line be greater than c and taking into consideration the two integral-theorems, serving as starting point, we conclude that the integrals IV’, and |W, have both become zero. 3. We might ask whether results as arrived at above are also attained when the functions J (ua) behind the sign of integration have not all the same parameter. The following operation shows that this is partly the case. Supposing that Wo ys. +) fn are numbers greater than rv, then under a definite condition the evaluation of the integrals oo : el pb. s du Wi J’+1 (uc) Js(ua,) J*(ua,)... T° "(uan) i a, a,%2.. A7 ; urn 0 le 2) c+? fb: du W.= y--2 i 7 de sa poe fy (uc) J(ua,) J'2(ua,). . « J" (uay) — EE ke Gj can be reduced to that of the integrals W, and W,,. For the reduction of W, and W, we can repeatedly apply the formula *) ie (wa) J#(ua) = ui i J (ua cos a) cos} a sin®?*-”—1 ada. AI ur We obtain in this way il 2 W ZT 2-1 rn 2 [4 —2v—1 d == ; cos a, sine ada 3 5 VENI AUS P). 1 1 1 = sf cos®*Hl a, sin 2#n—2—1 ade, X Plu) oo F1 du / — | JH! (we) J? (wa, cos a,) … J' (war cos an) — (a, cos @,)’ … (an cos Gy)’ win)" 0 1 = W. —= e 7g 2x1 nm —2¥—1 5 gE ia =), cos a, sin” a,da, pe 2. costa, sin? Pl oden X pee Fa 1) Nrersen, page (81, ( 755 ) c’ +5 (a, cos @,)’ «.. (an du y(rn—1 =D 7 fo J” (ua, cos a). . J? (uar cos Gy) — COS Gn)” If now is given c >a,-+a4,-+ ...+a,, then during the integra- tion the inequality C > a, cosa, + a, cosa, +... + an COS an will continually hold and the results concerning the integrals W, and W, can be applied. Remembering that we have 2 = ap cos?’+! a sin2#-2—| ada = nd ne?) T(1+m)’ 2 2 r(2 ——_—_—. cos’+3 a sin?#——! ada = a ACB) , we finally find y+1 d WE Z an (ue) Ji (ua) J (ua,) … Jin (wan) ——— = GV GPa. Op en um 0 a F(1+?) 25 T(LH-u) Plu) … PLH) „2 e du MES am foe (uc)J (wa, )J%2(ua,) ... J#n(ua,) ran a,t1a,Psantn,) uEu 0 2 v+1 2 vl 2 a E gent Fils) (1 +.) aS (1+un) (e>a, +a, — …… + Gy). In particular we find out of the obtained value for W, for w=t, vr=——} 2? = . . . SUN UC SIN ud, SIN ua, . . SIN Udy Nn du = 9 d, A, « « « Any yn+l —— 0 for p= —}4, p= —} din uc COS Ud, COS fl eee » COS Un Saale ee Te 0 ( 756 ) Mathematics. — “A family of differential equations of the first order.” By Prof. JAN DE VRIES. (Communicated in the meeting of February 27, 1909). 1. The tangent in (x,y) to the integral curve of dy ae P(@)y + QW) is represented by Y —y={P(«)y + Q(#)} (X — 2). For the points of the line «=m we have thus { Qn(X — m) — Y} 4 y{ Pan(X — m) + 1} =0, The tangents indicated by them form therefore a pencil of rays having the point 1 Qn Cie P, as vertex. I call this point the pole of the line «=m. By a projective transformation the linear equation is transformed into a differential equation, determining a pencil of rays of which each ray has a definite pole. Each line, connecting the vertex S of that peneil with a pole, having to touch in San integral curve NS is a singular point, i.e. a point where y’ is indefinite. To confirm this we transform the linear equation by the substitution au+av-a, & butbwvt ob, B oa ee cu Ae ee On account of that y' =P (x) y + Qa) passes into | dv (ay — c,0) (BP* + 7Q*) — (by — «By du (by — 68) — (ay — e,a)(BP* + 7Q*) p=), Qe — o(<). 7 y The pencil «=m is then transformed into the pencil of rays where a=my- F Ei er men or a=0, y= 0 we really fin mn hj Also the points 6 =0, 7 =0 and y=9, P*=0 are singular. ( 757 ) A pencil of rays with the indicated property I call critical. From the following ensues the property : When a singular point of a differential equation of the first order is the verter of a critical pencil of rays, then this equation can be reduced by a projective transformation into a linear one. tata ty — 7, is the vertex of, a ,eritical. pencil, then thé u substitution ji a U Ui en Vv Vv leads to the aim in view. ExampLn I. The equation wv + 2e?y + ay*® + y* dy iy ae + ary? + sy? da has in e=0, y=0 a singular point. The tangent in a point of the line y= me is indicated by (m +. 1) ms + 2m + 1 eg (m+ 1) mx + 1 Y — me = 3y reduction it is evident that the parameter 2 appears here only linear; so the pencil is critical. For the locus of the poles we find the cubic curve 2m + 1 1 ; ee (m + 1) mm z= — varen ra (m+ 1)m? * By the substitution 1 u bs ie Vv Vv y= me passes into w= m, and the given equation into isu “+ = deu du u+ 1 2. Let us treat in particular the equation dy rae P(x)y . Here the locus of the poles of z = 1m is the line y = 0 (polar line). As the homogeneous equation polar line, the question arises whether it is possible to transform the ( 758 ) homogeneous equation projectively into an equation of the form YP (0). If we put then the right line at infinite distance passes into y = 0, the pencil of rays 47—= ms into the pencil =m and the differential equation into the divided equation dy dx y «—f(a) Also the equation dy ee) has for the critical pencil «=m a polar line, namely the line at infinity. So here the vertex of the pencil lies on the polar line. As we can always regulate a projectivity between two point- fields in such a way that a point and a line of the first field are conjugated to a definite point and a definite line of the second, the property holds: If to a singular point belongs a polar line, the differential equation can be transformed projectively into a divided equation of the form dy — = P(«) dz, uy unless the polar line passes through the singular point. In this case we arrive by projectwe transformation at a divided equation of the form dy = Q («) de. ExAMPLE II. The equation dy ye de cyte has two critical pencils of rays: y= me with the polar line r+ — 0, y + 1 =m («@—1) with the polar line x= 0. For the former pencil the vertex lies on the polar line. By the transformation u i 1 ried =— —«,, =— v 7 v 9 v ( 759 ) this line is thrown to infinity, whilst y = nur passes into u = 1 : (m + 1). We find an uh du se u and finally from this ,u—v=—lIgu+C To make use of the second pencil we determine a projective trans- formation, which transforms (4 + 1) : (w — 1) into a linear function of w and «=O into v=0. These conditions are satisfied by the substitution v u—v+1 y+l1 2u+ wa wale ar ute TT hin We now find : dv u du v Tul} 3. When the linear equation y = Pe)jy + Qe) has a polar line we can transform this projectively into the line at infinity. The linear equation is then transformed into a homogeneous one, or, where this is not possible into an equation of the form ors on In the latter case w=:m is the critical pencil; in the former where dv an v du * \u the point «=O, v =O is the vertex of the critical pencil of rays. Let . at + by +e=0 be the polar line of y= Pay + Qa), thus the locus of the pole If we put ( 760 ) then afm + ¢ blm) and the linear equation obtains the form ble —f(@))y = by + af(«) + e. It is clear, that by the substitution ee ‚LZ u u Oi Tr the polar line is brought to infinity and the point of intersection of the rays «=m is brought into the origin. After some reduction we find indeed the homogeneous equation dv ae This reduction is apparently of no use when the polar line has as equation «+c=0O. Then 1 SS SS FP, m and the linear equation has this form: =S + Q mene" (w). By the substitution I v fe. en Eden u U the polar line is transformed into the line at infinity, the pencil “=m into the pencil wu=1:(m-+o), and we find the divided equation L du : dv+Q{——c}]—=0. u u 4+. Let us look at a few more examples. ExamPLeE Ill. The equation " dy an yy +2 da w?—xwy--2 has singular points mest ys Ad; ¢ ='—1, y=1 and inde point at infinity on & = y. The pencil «—y=™m is critical. We find for the tangent (my + m*—2)(¥ —y) = (my + 2)(X—y—m), and out of this the polar line «+ y= 0. ( 761 ) By the transformation esv+u, YU the pencil «— ym passes into the pencil 2 = m, the polar line into v= 0, and we find, as we ought to, the divided equation dv u du one ae Finally we find as integral curves the conics (eg)? + Cet)? =4, touching one another in the singular points «= +1,y== 1. The point of intersection of the lines «— y= m is the double point of the pair of lines (e—y)*? = 4. 2 2v When applying the transformation « + y= —, a—y = — we find u u that #—y =m passes into v= $ mu, whilst the polar line is brought at infinity. We find then, as we ought to, a homogeneous equation, namely dv u du ov For a ray af the pencil y 4+ 1=m(e—1) we find nen wl) y — Y—2) 9+) Lg Se et) may OS 2m 7e) ~ (@—1) (@ +2) —e(y+l)” 24+2—mz (m—1) «—2 , thus for the tangent [(m—1) e—2] Y = [(m—1) me + (1—2m—m’)] X + 2 (m1). Therefore this pencil is also critical. The poles lie on the line y—e=2. ExAMPLE IV. The equation et ae — @—ay—y+ has «= 0, y=1 as singular point. The pencil y—1= me is critical and has y= 0 as polar line. By the substitution y— 1 = wa, 1 v ut v uty ( 762 ) we find the divided equation dv uttl — ==, du. v wt EXAMPLE V. The equation (#? —y) y = ay —1 has a singular point in «=1, y=41, which is the vertex of a critical pencil with the polar line x + y + 1 = 0. By the substitution Su OE ee = 5 i this pencil passes into w == const., whilst the polar line is transformed into v= 0. We then find dv u—2 v os w—u-+l du. EXAMPLE VI]. The equation ayy =a + y¥? has the critical pencil y= me with the polar line z — 0 passing through the vertex. In connection with this the substitution furnishes the divided equation udu +- dv = 0. The integral curves yt oe == Cz? are conics having in QO a contact of four points with z=0 as tangent. d ExAMPLE VII. a ae, — da uty Ty’. This equation of Bernouni has a critical pencil y= me with the polar line «= 0. By the substitution z—1:v, y=u:v it is trans- formed into w-?du--dv=0. Out of this we find #’—y+Cry=0. ( 763 ) 5. When each ray throngh a singular point determines a system of tangents with index two, then the equation is projectively reducible to an equation of the form dy N(a)y? + Ploy + Qa) dz R(«)y + S(x) For, this equation determines for «=m the tangents of a conic and by the substitution ESS, y= i Y (see $ 1) it is transformed into an equation having in a= 0, y=0 a singular point, whilst each ray of the pencil « = my possesses the above indicated property. The equation dy a + y® — 2a°y? — ay ET y”? — 2e y — « is in this case, for each ray y= me furnishes a system of tangents with index two. By the substitution 1 u av Tete 9 y En Vv it passes into the equation (of Riccart) dv . — = 2u— wv v’. du This can be reduced with the aid of the solution v= uw’ to the equation (of BERNOULLI) ae =wuww+t w? du j where w =v —u’. By w=2z—! we then arrive at a linear differen- tial equation. Botany. — Mr. vaN DER STOK presents in behalf of S. H. Koorpers a communication entitled: “Polyporandra Junghuhnii, a hitherto undescribed species of the order of Icacinaceae, found in’s Rijks Herbarium at Leiden by S. H. Koorpers”’ (Plantae Jung- huhnianae ineditae IT)*). (Communicated in the meeting of February 27, 1909). Polyporandra Junghuhnii, Kps n. spec. Hrutee? scandens, ramulis teretiusculis novellis pubescentibus. Folia opposita, oblonga, basi acuta vel obtusa, apice sensim acuminata; 12—13 em. longa et 4—5 em. lata, petiolo 1—14 em. longo, subcoriacea, supra praeter costam 1) Continuation of Plantae Junghuhnianae ineditae I in Proceedings of the Mathematical and Physical Section, of June 27 1908, p. 158—162. ( 764 ) suleatam pubescentem glabra, subtus puberula et trinervia, nervis lateralibus utrinque 5—7 adscentibus in margine exeuntibus, nervis secundariis inter primarios transversis atque vernis reticulatis, subtus distincte prominentibus. Cirrhi in specimine Jungh. desunt. Flores dioict; masc. nondum aperti, cymoso-paniculati; _feminei ignoti. Inflorescentiae axillares laxae folium subaequantes; pedunculi pedi- cellique pubescentes ; bracteae caducissimae (?), in specimine Jungh. dejicientes ; pedicelli alabastris oblongis breviores ; calyx sub-campanu- latus 5-partitus, 2 mm. longus, lacinüs lanceolatis scariosis, erectis, acuminatis 1—1.2 millim. longis, extus appresse pilosis. Petala 5 evassiuscula, calyce breviores, extus pilis longis appressis albis. Stamina 5—6 rarissime 7 filamentis brevissimis, teretiusculis, glabris ; antheris oblongis vel linearibus 8—10-locularibus. Pollen globoso-tetraëdrum laeve 10 u diam. Ovarium rudimentum subnullum. Fructus ignotus. Sumatra: “Hochangkola-Tobing”’ (leg. JuNGnvuN anno? 1839. — Plantae Junghuhnianae ineditae n. 542 in Herb. Lugd. Batav.). The species described above is the third representative of the genus Polyporandra Bece, belonging to the /cacinaceae and related to Jodes Bl. This species, Polyporandra Junghuhnii, was found by me in 1908 in ’sRijks Herbarium at Leiden (Plantae Junghuhnianae ineditae N°. 542) and had been collected by JuncHuun in the Battak country at “Hochangkola-Tobing”. It differs 4. a. from the two already known species of Polyporandra in the structure of the calyx; in PP, scandens Brccart (in Malesia I (1877) 125 tab. 7) and P. Hansemanni ENGrer (in ENererR Botan. Jahrb. XVI, Beiblatt, N°. 39 (1893) 13) the calyx is cup-shaped and has short teeth, whereas in Polyporandra Junghuhnu it is 5-partite, with pointed segments 1 millim. long. Superficially the flowers of Polyporandra Junghuhnit somewhat resemble these of Natsiatwin herpeticum Bucuan, but our Polyporandra is sharply differentiated from their species by the characteristic struc- ture of the anthers, described above. That our species must be included in the above-mentioned genus Polyporandra Bree, seems to me to be highly probable. Since however, all the material, which has as yet been found, consists of a single dried branch with young, not completely developed male flowers, floral bud and three leaves, I consider it possible that afterwards, when the as yet unknown female flower, fruits and seeds of Poly- porandra Junghuhnii shall have been found, this species will prove to be the type of a new sub-genus of Polyporandra or of a new ( 765 ) genus, directly intermediate between Polyporandra Bucc. and Nat- siatum BucHaN. Because the material is so incomplete, I have, however, thought it advisable to refrain even from proposing a new sub-genus and to assign to this species a place in the genus Polyporandra. In the Herbarium of the Royal Botanie Gardens at Kew I last year compared authentic specimens of the only hitherto described species of Polyporandra (P. scandens Buccart and P. Hansemanni Eneier) with JUNGHUHN’S unicum of the Leiden Herbarium. In so doing I have become convinced that Polyporandra scandens and Hansemanni are very closely related but that, as was indicated above, our species (Polyporandra JSunghuhnit) is sharply marked off from these specifically. In conclusion I wish to tender my best thanks to the Director of the Royal Botanic Gardens in Kew, for the facilities given me for the comparison of the above-mentioned authentic specimens of Breccari and of ENGLER. Leiden, February 26%, 1909. Physics. — “On the solid state”. By Mr. J. J. van Laar. (Communi- cated by Prof. H. A. LORENTz.) (Communicated in the meeting of February 27, 1909). 1. In a recently published paper *) I treated the complete theory ‘of association, not only for gases and vapours, but also for liquids. If we assume that only two simple molecules combine to a double molecule, the formula: 5 PO ult ie hl ed, B RER me 4 BRE a ER | ee brats 1—, pt ars holds universally, in which c is a constant to be determined, and further : kr Pee k, +- 2k, we Ab = —b, + 2b,. R . So the quantity yR is the change of the specific heat for infinitely great constant volume, when 1 Gr. mol. of double molecules passes into 2 Gr. mol. of single molecules, while 45 is the change of the volume of the molecules in this transition. The quantity ¢, = — (¢,), + 2(e,), repre- 1) In the Arch. Teyler (2) T. 11, Troisième partie, p. 235—331 (1909). 52 Proceedings Royal Acad. Amsterdam. Vol. XI. ( 766 ) sents the absorption of heat for 7’—O taking place in this transition, and in the first member is 8 the degree of dissociation of the double molecules. Besides various other things I demonstrated that the known discre- pancies between the experimental critical data and those derived from VAN DER WAALS’ equation of state in its usual form can be accounted for by the assumption of an association, which would even be still noticeable at the critical point. I believe that this opinion was already expressed by van per WAALS some years ago (1906) *), and elaborated by van Rw in his Thesis for the doctorate (1908); it was, however, demonstrated by me in the paper mentioned, that quantitative agreement can be obtained only when Ad is not —0. Accordingly this quantity A5 plays an important part in my theory, which I drew up as early as 1902 on the occasion of a paper by the late Prof. Bakuurs RoozrBoom. *) But it is not about the above matters, however important, that I intend to speak now. I may be allowed to return to what IT said on p. 259 of my paper in the Arch. Teyler (p. 25 of the reprint), namely that the isotherms — if Ad is not =0 — present very “remarkable particularities” in the neighbourhood of v= 6. (See fig. 1 of the plate). These particularities now consist in this, that the isotherms, [for Ab positive only below a certain limiting temperature (critical temperature)| after having first risen to comparatively high pressures in the neighbourhood of +» = +, bend back again as far as in the region of low (even negative) pressures, after whicn they rapidiy ascend again for the second time to the highest pressures. We shall namely see that if e.g. Ab is positive, the degree of dissociation ~ of the double molecules approaches rapidly to 0 in the neighbourhood of v—=b=b, + Ab in consequence of the relation (a), which causes v— 6, after having first greatly decreased, to increase again. In consequence of this the pressure will decrease, and it will only increase again when v — 6 begins to diminish again in the neigbourhood of the value B=0O. If Ab is negative, 3 will draw near to the limiting value 1, and we shall evidently see the same course appear in the values of v— 6 and p. 1) In an address delivered in the Meeting of this Academy of Jan. 1906,- which address has unfortunately never been printed, so that only the fact and the subject of the address are known to me. (Cf. also the Thesis for the Doctorate of Dr. van Ris on “Schijnassociatie ete”, p. 3 (1908). 2) “Equilibria of phases in the system acetaldehyde + paraldehyde etc.” (These Proc. Vol. XI, p. 283). Compare also van peR Waars: “Some observations etc, These Proc. Vol. XI, p. 308. ( 767 ) It is clear that this may cause a new phase to appear, and that besides equilibrium between the vapour phase and the liquid phase as in the ordinary theory of vaN per WaAALs, under particular circumstances equilibrium may occur between the vapour phase and the third phase, or between the liquid phase and the new phase. This phase, the molecular volume of which is somewhat less for positive Ad than that of the liquid phase, and for negative Ad somewhat larger, cannot be anything but a solid phase (amorphous or crystalline), which distinguishes itself from the liquid one in this, that the degree of dissociation of the double molecules (see $ 3) is found in the neighbourhood of 0, so that the molecules are for the greater part in the state of double molecules; in the liquid state, however, this degree of dissociation can lie between the values 1 and almost 0, depending on whether we have to deal with so-called non-associating or with strongly associating liquids. It is self-evident that formation of triple or multiple molecules is not excluded in this, but for the sake of simplicity I shall only work out the theory for partially dissociated double molecules in the following pages. The phenomena will not show any quantitative change by the occurrence of multiple molecules. 2. Before proceeding to a fuller discussion of the above formula (a), we shall first briefly repeat its derivation, and also set forth the influence of pressure and temperature on the equilibrium *). In the condition of equilibrium: Fie Se ety. oy en ae eae (BY 0g 0g in which p, = eg and u, =a represent the molecular thermody- n, Ne namic potentials resp. of the double and of the simple molecules, we must substitute the well-known values for u, and u,. Now: 0 ¢ a a Cae a [@ — RT Sn, . log Sn,)] + RT log e, | Ny 9 0 u, = C, — ~— [2 — RT Sn, . log Zn] + RT loge, | 3 On, in which the functions of the temperature C, and C, are given by: C, TA k, td (log her 1) in [(e)o ip KCR | C, are kT (log dt 1) a (€), re T(s)o| while 1) Cf, Arch. Teyler, Lc. 52* Hence substitution in (1) gives: dn, On 4 , © 02! 02’ (CC, + 2C,) —| — — |+ &T (— loge, + 2logc,)— 0, when 2’ — 2— RT Zn, . log =n,. If 8 is the degree of dissociation of the double molecules, we have: 02 0Ddn, | OBdn, _ AZ a’ ican de on Oa) Sane because n, = 1— B, n, = 28, so that we may write: xy Y 0$2' r ase A (— C, + 2C,) — —— + RT log — = 0,7 dg rane or as: 1—p 28 err Es 148 148 also : ; 0&2! = 230) oa nee MG = d a 0p Dap RT - 2 «2° 2 sn de So we must determine the value of Fn From: t ae + 8) RT a a Pv Den) v follows: je (1 + p) RT log (v — b) == ri ) rv so that we find for 2’ (Xn, =1 +9): 1) In this integration the quantity @ must namely be kept constant, because an eventual equilibrium between the components exerts no influence on the determination oe of the values of 2, ay and 5 for the two components. In the calculation n, n 2 of the thermodynamical potentials of the different components of an arbitrary mixture we have namely not to take account of a later eventual occurrence of an equili- brium. Thus we calculate here gj and uz quite independently, and simply introduce for the equilibrium the additional condition — gj +2u,=0. So we must imagine the integration [oa for a perfectly arbitrary ratio of mixing (?, which quantity 6 does not become the degree of dissociation of equilibrium until after the introduc- tion of the condition 4; = 2g. (see also Arch. Teyler p. 4). ( 769 ) | | sab B = (14+ 8) RT log — +. — pv, 1+B wv hence (a independent of 3, see further below): —b 1 3 Ten ) y ) Re RT 3 Ware? (5 Er EN 1-8 oe dp) woe Zop v—b But in consequence of the equation of state all the terms with v . — disappear, hence we get: og Er (x i 2) AOS. ac. OER ap 9 +4) ae db when we represent BS — 6, +25, by Ab. For 6 we may namely write : | b=b(l—B) + 4, . 28 = b, + B(— }, + 26,) =b, + BAB. For a may be written: ai — Ba, + 2 (1 — p28. a,, APG. But in case of simple association evidently a,=*/,4a,, 4,,—='/,4,, so that: . a= (1 — 6)? a, + 2(1 — £)8.a, + Ba, =a,, independent, of 8. For the relation (a) we may now write: 4p? je log = =|¢,—26, ee RT log RL — RT log (p +5) — —Rt—(p+5)40|:27, or after substitution of the values of C, and C,: 22 log = | Pe DDC ho + 2k) — (eo) + 2@,),) + + T (— (s,), + 2(s,),) + RT log R+ RT log T — RT — RT log 4 — — RT log € + 5) — (« + 5) as | BH If we now put: —k, 42, — (8), + Ao), ny ae —~ + . sadam ate kar | k, 4- 2k | B mi a (#,)) + 2e) = % | we get i ( 770 ) 3 a dode” ze log En == loge + y log T — ae + log T — log (» + =) RT Ab. fart or finally: _ % pt “fo Ab 3 opt, BT, RT : 1— nS Pp “Le afs Fee NG ( ) being the most general equation for the binary dissociation in arbitrary state of aggregation. The term with A5 disappears in the gaseous BEE so that then A: verges to 0 BELT DAN zis vrt ss But for liquids (and solid bodies), it is by no means allowed to neglect this term (as was nearly always done up to now). For it would be very accidental indeed, if Ab = —b, + 26,=0. It is just this term with A5, which exerts a very great influence on the value of 8, and is one of the main causes for the occurrence of the state, for then solid state. a For perfect gases p + — may also be replaced by p, and (2) passes p? into Gipss’s well-known formula for the binary gas dissociation, e.g. of N,O, into 2NO,: 1—;? Pp For the further discussion of the equation (2) we refer to the original Paper in the Arch. Teyler; we may only be permitted to make the following general remarks. If we vary the pressure at constant temperature, the second member of (2) will approach to oo for p=0O(v=o) in consequence of the denominator p, hence @ to 1. So in the perfect gas state everything is in the state of simple molecules. But for high pressures the behaviour will be of two kinds, depending on whether Ad is positive or negative. For Ad positive, i.e. when the volume of two simple molecules is greater than of one double molecule, the second member of (2) will evidently approach to 0, when p approaches Cie to oo. For then this member. becomes: = —- — 0. The value of @ le ©) then approaches also to 0, Le. there is complete association for p=. If, however, Ab is negative, so that the volume of the double er» molecules is larger, the limiting value becomes — —== %, and then 8 . GO id ly approaches again to 1, after having passed through a minimum value at a certain pressure (i.e. a maximum association). In the Arch. Teyler 08 (p. 9) L demonstrated that a changes its sign in this case, when p | v has become = 2 (6, —b,)=b, — Ab. (Av given by formula (8) on p. 8 loc. cit. is namely == O then). With regard to the influence of the temperature for constant pressure, it is easy to see that for 7’=o as well as for p=O the dissociation is complete (3—= 1), because y + 1 is always positive. But for lower temperatures the behaviour will again be different, a | Ab (v approaches then 4, so that 2 a/b? may be substituted for a/v*) is positive or negative. If this quantity is positive [where Ab can be as well + as — (q, is always positive) |, the second member of (2: approaches 0 at 7’=— 0, so also 8 approaches depending on whether g, + (7 4. 0 (complete association). But if g, + (» + a Ab is negative (which is only possible for A5 negative), 8 becomes again = 1 for 7’=—0, so that then a minimum value of the dissociation (maximum associa- tion) is passed through. | showed on p. 16 of the Arch. Teyler that a changes its sign, when (see also p. 15 loc. cit.) g = q, + YRT + € + a Av=0. The value of 7’ for this minimum will depend » on the pressure. As for 8B=0, 6=6,-+ 846 approaches to 6,, while for B=1 the limiting value of 6 will be 26,, we may also say that for a Yo +(e as j Ab positive 8 approaches to 0 at 7’=0O, while for 1 do dlp + al Ab negative 8 will approach to 1 at 7=0. For all this compare the figures 1 to 4 on p. 6 and p. 18 loc. cit. 3. Let us now examine the course of an isotherm in a p-v-diagram at a not too high temperature, and that first for the case that Ad is negative. Then, as we saw above, the value of > approaches to 1 for p=, ie. h to 2b,, the volume of the simple molecules. So this is smaller than 6,. The said course is (schematically) represented by fig. 1. (See the plate). Besides the usual maximum at 5 and the minimum at /# of the ideal isotherm, according to the original equation of state of OMEN VAN DER WAALS, another maximum at DY and a minimum at C have now made their appearance. To A as vapour phase corresponds the coexisting solid phase A’. To P' as solid phase the coexisting liquid phase P". [ef. also the p-7-diagram of fig. 4(D, where the line SM runs. back in consequence of the fact that 46 is negative, which implies that also Av is negative for the transition solid-liquid (P'P"). By approximation the quantity Av is namely = (@;— 8;) Ad, in which 8u, is always >> Boud). The pressure of coexistence at PE will always be much greater than that at 44”, when the temperature is considerably lower than that of the triple point S (see fig. 4). For not too low values of 7’ we may also have a metastable coexist- ence gas-liquid QQ" (ef. figs. 1 and 4). If the temperature is very low (as in the following calculation), 3 will draw near to 1 only for exceedingly high values of v, whereas for all values of v between A and D the value of 3 is practically =O (association is complete), as will appear from the following calculation. The minimum value of 8, which was mentioned in § 2, then lies in the immediate neigh- bourhood of 0. Only between the points D and £ does 3 increase rapidly from @=(+)0 to §=(+)1, and this in consequence of the rapid decrease from v—=(+)b, to v—=(+)2b,. From £ to the highest pressures 8 remains then in the immediate neighbourhood of 1, and becomes exactly =1 for p= o \v = 26,). It is self-evident that for higher values of 7’ the values of 8 for P' and P" will approach each other more. At the temperature of the triple point S the three transitions AA’, QQ" (metastable), and PP" will coincide. There is only one pressure of coexistence (three-phase-pressure). For temperatures above that of S (comp. fig. 2 and fig. 4 (ID) the coexistence QQ", which was first metastable, has become stable, whereas AA' has now become metastable, just as P’P". (These last transitions, of course, at not too high values of 7’). Now only a liquid phase coexists with the vapour phase. For higher temperatures the minimum at C and the maximum at D will draw nearer to each other, and they will finally coincide in an horizontal point of inflection C‚,D (comp. fig. 3 and fig. 4 (IID). From this moment we have, therefore, the original isotherm of VAN DER WaaLrs with the simple coexistence gas-liquid. At last at still higher temperature (the critical temperature in K) also the last maximum (4) and minimum (/) will coincide. How all this is modified when Ab is positive, and so the line SM runs to the right in fig. 4, will be easily seen by the reader. We shall revert to this in a following paper, and mention now only that (Sr) then (see fig. 1) v—=2b, lies on the right of v —=h,,and accordingly the solid phase not on the right but on the left of the liquid one. Not C and D, but D and Z will then coincide in an horizontal point of inflection (critical point solid-liquid). 4. After the above digression we return again to the shape of the isotherm in fig. 1, and we shall carry out a simple computation to prove our statements. For this purpose we shall first modify the principal equations somewhat. If we put: we can write equation (2) in the form: 2 7 f ic ER verle, rd ed ca p L €. 2 re GE Ve sb ehehe ee tdi 1 p when cq? de ari A= is put. For me OS ls) ee Ea ee aR may evidently be written: RT a ee ONS ae and the value of v can be calculated from: EEND or as b=b, + 8 AD, from: 4 v=o, —(8-“=")(— ay, . es Oe Me AAG So if we successively assume different values of » for given values of 4,7, 6,,-— Ab and 6(7'), we may calculate for each of them the corresponding values of 3, v and p, If we assume: e= 2(Gr. Cal.); 9, == 3200 (Gr. Cal.); b, =1; 26,—='/,; a= 2700, so that (774% EN a et and [with R= 2 (Gr. Kal.)| __ 2x (3200)% 25/2 XU, = (1600)% X 1/, = 64000 x !/, = 32000 becomes; while for 7’ is taken 9° (absolute), yielding: ae jag 4 9 PPE a A TT ONe the equations (3), (4) and (5) become: 8 1600 | 1600 ek Ff 8) 9 z 9 e Zin 17 Kon et En ee |e B ay di mete soit 1 1+, 2700 ==> | = ¢ — zE) php — —— n° 2 For g =o we get evidently: Bee te el RN 0 For p= 185 we find: ge log™® En OTT 0.4545 A fee = — 76,077 + 80,3843 — 2,267 — 1,999, because : 27 1600 log'* — — X 0,48429 = 1,180 — 77,207 = — 76,077, in consequence of which: ; 1 ue == 9991 Cas e= ee ESL 199,54 So between g =o and y= 185 P is practically —=1. So we find for v: 1 1 2 B in ( zoo” zes) 1 — 0492 = 0,508, ER Ee Et while: Rn 2700 peggy 2700 p = 6660 — (0,508)! = 6660 — cS 3800, k being expressed in Gr. Cal. (2 = 2) in the equation of state, a : : : eg also pv and — are expressed in caloric units, so that if v,b ete. Û u \ . . , . a . are given in cM’ (per Gr. mol), the values of pv and — in ergs Uv CTs) sys 7 a are found by multiplication by 41,74 X 10°. Then p and , we ; expressed in dynes per c.M?. But as 1 atm. = 1,01325 x 10° dynes a per ¢M’., we find the number of atmospheres of p and — by D 41.73 multiplication of the found values by mnt 1 20; The found pressure which will lie in the neighbourhood of the point #7 in fig. 1, amounts therefore to — 3800 x 41,20 = — 156600 atm., from which appears how enormous the distance is of the points D and F at such low temperatures. (At higher temperatures both the term — 76,08 in the equation for 8 and the factor 36 in the equation for p will become considerably higher). In a similar way as above we can now calculate for p== 580: og = — 76,076 + 78,178 — 2,255 — — 0,160 A any vet == 0,692 ; B= 0,640 1? 1 1,640 v= 1——( 0,640 — — 1—0,315 = 0,685 oe 180 6480 pee Es 6480 — 5760 = 720 te EO > ae are) p= 170 log Se = — 76,077 + 73,829 — 2,230 — — 4,478 Lan 0,0000331 ; 8g—= 0,00575 1 1,0058 o= 1 ==(0,0058 = 2 | = 1 -t- 0,0001 — 1,0001 2 170 2700 p= 6120 — ———_ = 6120—2700 — 3420. 1,0002 yp = 1602 log'® a — — 76,0,7 + 69,486—-2,204 — — 8,795 aa = Bp == = : 1,003 dT 2700 33 — 5760 — —_— — 5760 —2680 = 3080. ge: g= 90; toe gal ae g = O05. DE 0,001: (776 ) 1 9 = 0 3) eS P : 7 900 ; aon | p = 3600 — —— = 3600—2670 = 930. 1,110 g—0 hae eae = ao a ma ea * 100 700 Sgn = — 1800-2650 850 1,020 ay LE asta ae g=0 5 vld p= 2700 p = 360 — 360 —2450 — — 2090. 1,103 1 B 4 dn 47 2700 — 36—1200 = — 1160 3—0 Ces ee: =< a == 02 T 2700 By à j= 3,0 = 3.6 — ihk 36 0 (eee Ho Bt Serta: bi eno) pal 2700 p = 0,36 — —_ = 0,36 —1,04 == — 0,68 2601 0 pee en P ERZ Oee se 0,011 =De he eene EN nn log 5 0 Ei B = 0,00838 4 B= 0,0911 1— 2 » = 0,546 X 10 7 p = 36107". CUT) S I So pak; v=o pop, With regard to the course of the values of 8 we see clearly from this calculation, that between the points F'(p =o) and E (p = — 4130) in fig. 1 8 is practically = 1 (8 = 0,983 in Z). This is due to the fact 1— ? tively high positive value (at p = 185 still = 2), in consequence of the value of 0,4848 p, which is then >> — 76,08 (this value holds for 2 7 is large, and ? will lie in the neighbourhood that between p= oo and p—=185, log'° in (6e) has a compara- p T= 9). 30that . of 1 (for gy = 185 g is still as great as — 0,995). But between # and D, i.e. between p =185 and o=170.5 very rapid change in the value of 8 takes place. The value of 0,4343 p becomes then namely < — 76,08, in consequence of which 23 B } Fa ; log’* — — decreases from a positive value to a comparatively large g 1— p p y 8 negative one (for p= 170 already == — 4,5), so that 8 changes rapidly from 1 to 0. In the point D we have 8 —= 0,027, while 3 = 0,006 for p=170. For g=180 at about ?/, of the distance between / and D, we have the intermediate value 0,64. So the entire change of 8 takes practically place between the points # and D. Past D 8 remains practically == 0, till at about p=i0 "*, at an enormous value of v (v= 0,55 < 1075), the value of 3 gradually rises from O to 1. In equation (62) the term — logy’ p begins then to approach the term — 76,08 in absolute value, and it exceeds this term for values of <10. For y=10 7 B is already 0,09, while for p= 0 (w== oo) £ will have become —1. So this transition takes place beyond the limits of the diagram (fig. 1) (for es 9). What was said above, gives a clear idea of the course of the values of 8, and we see at the same time from it, that in the transition solid-liquid (P'P") the value of Buiquid is practically —= 1, and that of Bs practically =O. For the point 2?” lies between # and D, and the point 7” beyond D. The corresponding values of p have been indicated everywhere in the figure, while those of v appear from the above calculation. 5. In connection with what was discussed before, we will finally give the calculation of the different maxima and minima in fig. 1, viz, of the points B,C, D and Z, (778 ) From the equation of state: (l+@)RT a IL ~—— —- 19 v—b v it is easy to derive for 7’ constant (keeping in mind that 6 = 6, + + 8A 5b):3) dp (1—p) RT dg RT OB 26 — 1 Ab ; ‘ dv (v — b)? Ov + v—b 0v a v? (14-8)(—Ad) i.e. with - Wp v—b ee BEE maan ne 2 dn (wv —b) 148 dv But from the equation (2) for 3, viz. in the form (written OE) dk a Oee for pt Ie B VE (14) Ab en EER REI from which p has been eliminated, follows by logarithmical diffe- rentiation (7’ constant) after some reductions (see p. 36—37 loc.cit.): v—b Op UR (LB) (1—@) IER cap oe En b) dp By substitution in the expression for ; obtained above we get now: av dp 2a. (1-8) AT 1 dv o® ED 1+ VBA? This expression passes into the ordinary one for @=O and 1. i ; dps tE ART fligisetrue taboe ss Se wevobtan — = — dy ©? (v—b)? b referring to double-molecular quantities, a—=4a’, v—=2v’, b=2b’ will have to be substituted, in which the accentuated quantities now B ap Za. Nags & refer to single molecular quantities. We get then — =—, TOE v = (7) but a,v and dy' as we should get. | Let us now first examine the points D and F (fig. 1). There vb is small, so p large. In this we notice that the just introduced quantity p is the same as our former quantity g. For in (3) etc. ptn (148) (Ab) Wise ae 5 p. Now p=185 (—Ab) =p was put, so also 1) See also Arch, Teyler, lc. p. 26—27. J.J VAN LAAR „On pr My # ie 0 { „df P Proceedings Royal Acad. At J.J VAN LAAR „On the solid state.” Proceedings Royal Acad. Amsterdam. Vol. XI. Fig. 3 TL. . (779 ) for # and g=170 for D (see § 4), so that we may write '/, 3(1—) g” instead of 1 + '/,8(1—8)(1—g)? — if namely the value of 3 and 1 — B is not too small. It will presently appear that this is not the case. So in the points D and E we have got by approximation: 2a_ (149) RT it ARI oor (LEBARA (Ay (vb)? ge) Now in D (see fig. 1) v is in the neighbourhood of 6,, whereas in £ the volume v is in the neighbourhood of 24,; hence if we put 1 —g8 and 1+ 8=1 for D, where 8 is near 0; and p=—1, 1-+ 8=2 for £, where gis near 1, we get by approximation (R = 2) a lk DD nr PE" 1,8 (1—8) ap, 2D SON ie @=— 2100, by ds 26, Ab VWE find : 18 2 a 1 == = — = 0,027 7 1—se= = 010017. 2700X!/, 75 2700 X8X/, _ 600 So above at D we have neglected 1 by the side of circa 0,0133 X 170° = 400, and at Z£ we have neglected 1 by the side of circa 0,00083 X 185? = 28,5; so that the above values of Bp and 1— fp may be considered as permissible approximations. It is now easy to caleulate the pressure in the points D and F Bp 1 —Ab from the equation of state. With ana ge p the latter becomes (ae (see equation (4) and (6)): BE a 36 2700 Ee Hence we find: pE= 6700 — 10530 = — 3830, 1 as according to (6%) p = 186 and v = 0,506 corresponds to By a Sn: And as g=173 and vp = 0,99 corresponds to Bp = 0,027: Pp = 6230 —- 2750 = 3480. As to the points B and C, for them @=O for 7'= 9, as we saw before. So there is simply : 2a RT v* en (v —b)' ( 780 ) te: withered = by = de — = 300, (v - 1)? yielding vg = 298 and vr = 1,063. Hence from: 18 2700 Br hee OI Vv we find: pB == 09,0606 — 0,0304 = 0,302 pc = 280 — 2400 = — 2120. So reviewing what has been said, we can account for the appearance of the different maxima and minima in the following way. DM A PT In. the expression p= —— ‘the term v—b- iv? v— will increase more a rapidly than — between the points A and B, in consequence of the Vv F . decrease of v, so that p becomes larger. Between B and C, on the . a . . other hand, the increase of — will predominate, so that p decreases. Vv 7 But beyond C, when v comes near 4, Es will again increase more v— a 5 than a and the isotherm will now ascend very rapidly. It would proceed to p= (according to the original theory), but 7 consequence of the abrupt change of the value of 3 between D and E from 0 into 1, the value of “/,2 will increase rapidly from ¢/,2 to afsp. For FE, where this increase of 3 has ceased, » — 6 will begin to decrease again from this moment, so that p can ascend indefinitely. Hence, when only the state of association 3 of the molecules for small volumes is taken into consideration, in connection with the variation of volume 4h attending it, not only the liquid state, but also the solid state (crystallized or not) is quite controlled by VAN DER WAALS’ equation of state. Further particulars, referring to the case A5 positive, to the formation of multiple complexes, to different modifications of the solid state, to pp the ratio ne where 7, is the triple-point temperature, and 7, the ordinary critical temperature, ete. ete, will be discussed in a subse- quent paper. eo ( 781 ) Physics. — “On the calculation of the pressure of a gas by the aid of the assumption of a canonical ensemble.” By Dr. O. Postma. (Communicated by Prof. H. A. Lorentz.) (Communicated in the meeting of February 27, 1909). To the different existing methods for the derivation of the pressure of a gas (or the equation of the isotherm), GisBs has of late added another, namely the method by the aid of the theory of a canonical ensemble. According to this method we take for the force exerted by a system on an external body in the direction of one of the f de de coordinates a of that body, the value — a the mean are taken Ga a over a canonical ensemble. Dr. Ornsrrin was the first to apply this method *). He shows that = J J Zeef" Gu da da da nad ae in which w is determined by e 7 = { e Tdà. Suppose the gas to be in a cylinder, closed by a piston at a height a, A repre- sents the pressure exercised on this piston. The pressure pro Si cM’. will be represented by p= — and by means of this formula Dr. ORNSTEIN arrives at the known result. If, however, we examine the method somewhat more closely, we are confronted with different difficulties. In the first place it should be borne in mind, that if de . A= ie is to denote the force exerted on the piston, the coordi- a nates of the molecules must be thought to be constant in this diffe- rentiation °). Apart from the energy of gravity, the ¢ consists of four parts: €, =the potential energy of the repulsive forces acting between the walls of the vessel and the molecules, ¢, = the potential energy of the repulsive forces between the molecules inter se, ¢, = the potential energy of the attractive forces, ¢,—= the kinetic energy. Only the first is a function of a, and it determines the A, which is 1) Cf “Toepassing der Statistische Mechanica van GiBBs op molekulair-theoretische vraagstukken’”’. Leiden 1908. Further: ‘Calculation of the pressure of a mixture of two gases by means of GrpBs’ statistical mechanics,” These Proc. Vol. XVII, p. 116, and “Statistical theory of capillarity”, These Proc. Vol. XVII, p. 526. 2) When we substitute at once V for a, there is some danger to overlook this. 53 Proceedings Royal Acad. Amsterdam. Vol. Xl. ( 782 ) de . Now we da bi de En really — ae but which is now also represented by a notice that the energy of the repulsive forces is taken into account in the calculation of wp by simply excluding that part of the exten- sion where the molecules would have penetrated into the wall of the vessel or into each other. We may also say «, and ¢, are put = 0 when the molecules do not yet touch the walls, and if they do, put as oo. So the «, and «, have a discontinuous course, and yet it is just ; fr: (ie ae on the assumption that a certain extension exists with finite —, a with respect to which it is integrated after multiplication by the density e 7 , that the method is based. So it is certainly desirable to question the validity of this method of calculation. Disregarding the repulsive and attractive forces between the molecules inter se, we have to Ge: p de T de, ne J 2) Pf Bat when we have integrated with respect to the velocities. So also: 7 oT LW 7 O 7 Op amor! fe da tee: a Is it now allowed to take for this integral fi dz,... dz, integrated with respect to the extension limited by the walls of the vessel? Let us consider the simplest conceivable case of an ensemble of one 2 molecule with coordinate of height x. In fig. 1 e 7 is given as f(z). AC indicates the distance at which the molecule still repulses the wall. When now the piston is moved over a distance Aa, the line AB moves to A’ B’ (with or without change of form). The limit ‚area AB B'A' of 7 now represents — fe Tde. In fig. 2 the distance ZA €) dl AC is reduced to 0, so that ¢, = if c>a,ande,=Oande T=1 if « 2b, which according to the known properties of the isotherm comes to the dp ee —— == 0) | 18-posifive, l same thing as that the pressure in the minima ( av 27 es or the temperature higher than rs of the critical temperature. 11. Let us for the present confine ourselves to the supposition of a quadratic function. So in the case which is still under considera- tion that there is minimum critical temperature and yet a’,, > a,d,, i ( 802 ) dp d : no point of intersection of “ =0-and 0 will occur for very v Vv low temperature in the region on the right of z,. For, as VAN DER Waats has demonstrated (These Proc. June 1908) for such a point of intersection the equations da dai" be ab ae Ye dz dz Wi ee ES v v db db DE ai ; zr a (1) db co dz 9 = — da dx hold, so for very low temperature a point of intersection is only ; da possible in the neighbourhood of an 2, where either — — 0 or the Lv critical pressure is stationary. Both these possibilities, however are not realised here, for as we shall show later on, a minimum critical temperature at the same time with a’,, >> a,a, is only possible for a relative position of the lines a and 5 as indicated in fig. 11, i. e. Fig. 11. da db the point a, on the left of 6, and eN on the right of en x x Now for 6, and h, the critical pressure is infinite, for a, and a, it ( 803 ) is zero; also for v = + mand zr =— o. The equation pj; = ee being of the fourth degree in 2, no more than 4 values of 2 can ever be found for a certain value of py. So no maximum or minimum of the critical. pressure can occur on the right of 5. Here we interrupt the train of our reasoning for a moment, to show that in the case considered a minimum critical temperature must occur. As the equation: | de — iB gives an equation of the third degree, we might expect that 3 values of « which make 7%, stationary, could be found for every system in the complete diagram of isobars. But as for very great values of « da db always 0 oa a,b,z°, one root appears always to lie at v Hij infinity, and there are at most two roots for finite 7. One of them lies between a, and a,, where 7), becomes = 0, the other lies on the right of b,. For in our case we may write the equation for i a and 5: a= 0, (a — 2, — a, 6= be? — b, sO da aa. ( db 21 — Za, (2 — 4 —2b. wv. ‘dx ; ne da > k ; da db —— has the sign of 6 —— a —, and so of: dx da dx 2a, 6, «? (« — x,) — 2a, b, 2 (« — wv) = 2a, b, av @, — 2a, b, xx’, aly . aie : so —— is positive for high values of x, which proves the presence lid of a minimum critical temperature in connection with the value +o for b—= 0. 12. Let us now return to our diagram of isobars. We can now represent it fully for low temperatures, now that we have seen ; ’ ; dp dp that there will be no intersection of ——0O and — = 0 in this case. v av We have only to add tbe observation that the value of the pressure . dp . on the line rs =— 0 approaches indefinitely to zero for very great av value of rz, however small the value of 7’ is, if only not quite zero. ( S04 ) AEN a ee is in inverse ratio to z, and — to a*. So it follows from Vn Uv this that all the negative isobars starting from the point zr =a,, v=0O will have the shape we indicated before. The line p—0 will For dp ed ee ie intersect the line os = 0 at infinite distance. To this isobar, however, v also the branch belongs starting from the point p =O on the line «= .2,, and also the line v=o. For a positive pressure the isobar consists of two separate branches. One of them, starting from the point e=, v =O remains confined to smaller volumes than p = 0, the other starting from a point on the line «=a, arrives some- dp p and 2, has a tangent there parallel to the v-axis, and returns again to the line «—wz,, now on the other side of the maximum pressure. So we get fig. 10 for the complete course of the isobars. where on the vapour branch of =0, with ascending value of v 13. How will this figure be modified with increase of temperature. € Let us consider the temperature which ET of the minimum critical _ temperature. From equation (1) on page 802 follows that we may expect a point of intersection at a volume v = 2/ and a temperature = Ty ; OF Ks db da for the mixture with minimum critical temperature ( where dn 0 5 > da x The line > 0 lying at smaller volumes than the line a= 0 for very great values of #, as we saw in 10, there must be another point of intersection more to the right. It is clear that these points ae orem f ‚dp dp of intersection have arisen by a contact of Je 0 and rl 0, and that the two points of intersection have moved from this point of contact in opposite direction. For as the equations for the points of - intersection are of the first degree with respect to, 7’ and wv, it is not possible that two points of intersection lying beside each other move in the same direction; for then we should find different values of ZY for the same value of 2. In the point of intersection lying most to the left the pressure is 0, from which it already follows that there must be another point of intersection ; for the pressure finally verging again to zero towards the right, there must be a point between where the pressure has reached its lowest value on ( 805 ) dp sah the line “—0. So this point is really a minimum of pressure. Now LU 5 the diagram of isobars has changed in this (fig. 12) that a loop-line x dp.o Vv tte tee srogy per? teeny V Wie. 12. has made its appearance, as we see immediately from the fact that the direction of the tangent of the isobar p= 0, passing through the } : Ì dp dp : f point of intersection of aap 0 and he 0 becomes indeterminate in this point. The two branches of this loop-line start, of course, one from the point «=wz,, v =O, the other from the point on the line w=w,, where p=O. They pursue their course through the double point towards infinity, just as the branches of »=O in figure 10. Now too, the positive isobars have the same course as indicated there. The negative isobars, however, at least part of them, have broken up into two parts, a branch on the left of the mixture with minimum eritieal temperature, which has again the same course as in fig. 10, and a branch on its right, forming a closed curve round ; ; ' 4 dp ‘ dp MAD the second point of intersection of ena O and 0. Only for isobars for a larger negative value than that in the last-mentioned point there exists only one branch. It is clear what will be the course for intermediate temperatures. It follows then again from the given equation that the pressure will be negative in the double point lying then at a value of « where da db ; : ; b—>a =a So we have a loop-line, which itself runs again as a ( 806 ) d d closed line round the second point of intersection of = = Oand ke —0. AX U 27 14. At a temperature higher than =e of the minimum critical temperature, there is positive pressure in the double point. For the double point continually proceeds in the same direction; so da da b ag es constantly decreases during this movement and now the pee Vv da expression : da dan * pee poe 5 MRT 27 de 1 dx 27 1 n\? = == SS SS = = MRT. 8 db db 8 2 a— 2a — dx da 2 has a value O for n=O and n=2, a maximum for n nk be- 2 tween n=—=1 (the 2 of the minimum critical temperature) and x = 5 € ‘ 1 the righthand member is, therefore, greater than 7 and, so MR mn 2 so that the pressure in the double point is positive. For x mn double point lies exactly at 7= 7} or v = 3d, so in the point of d = where this line, which then has split up, has its tangent Vv //v-axis. At still higher temperature the double point of p gets on d the vapour branch of En — 0, and disappears at the temperature at U ee db Heen which = = MKT for x=, as is indicated in fig. 9. So long as & av the double point continues to lie on the liquid branch, that branch of the isobar, which comes from «= «,,v — 0, passes through the : dp double point, meets the vapour-branch of = =0 at larger x, where v its tangent becomes //v-axis, and then runs back to a point of the line «=v, for greater value of v than that of the maximum pressure on the straight line mentioned. The other branch of the loop-isobar comes from a point of this line with smaller pressure than the maximum pressure, and having passed the double point it remains at ( 807 ) d smaller volumes than + —0. The course of the other isobars only av undergoes modification as regards the isobars which intersect the line d : = 0 on the right of the double point. Near the double point posi- ‚U tive isobars are found at this temperature. They come from a point on the line ==, at larger volume than the largest that the loop-isobar has in common with this line. They are // v-axis on the vapour d branch of + — 0, then they run back to smaller z, they are again Vv dp dp // v-axis on the liquid branch of = 0, and // a-axis on a= 0, v after which they proceed towards infinity between the last-mentioned line and the loop-isobar. So we have again got isobars here as in the rigthand part of Van per Waars’ diagram of isobars (Fig. 13). X Fig. 13. ae dp When we ascend above the critical temperature, so that ee 0 Lv breaks up into a righthand part and into a lefthand part, this involves only that part of the isobars no longer possess the retrograde portion, d J Because the two points of intersection with = = 0 have coincided, i?) and then have become imaginary. The last isobar which has the shape described here, is that for p =O, the line at infinity included. : : dp It then runs from the line «=, to the vapour branch of Eel 5 ( 808 ) on the extreme right side of the figure, returns to a point on the liquid branch of 2 0, and then intersects —_ 0 The negative isobars, which have their point of intersection with the liquid branch of d. dl : = 0 still further to the right, again form closed rings round Vv ; d, d the second point of intersection of = = 0 and = = 0. These closed Vv iid rings do not disappear until the temperature has been raised so 1a dre 2a : ae 2 has been reached, which may RE take place either above the minimum critical temperature of the system or not above it, in the first case again either at higher tem- perature or not at higher temperature than that at which the point : ; ‚dp of intersection of ia — 0 high that the value d and = = 0 has shifted from the liquid Hi Vv branch to the vapour branch. So different combinations may present themselves, which, however, do not differ in essential points, and which the reader can easily imagine for himself. 15. At temperatures at which the double point lies on the vapour branch, the loop-isobar, starting from v—0O,2= 2a, passes first ARS ‚dp : ; : , through the liquid branch of En — 0, where its direction is // v-axis, av then through the double point, after which it reaches the line # = a,. The second branch of the loop-isobar comes from the line «= a,, and after having passed through the double point, it pursues its dp dp course always at smaller volumes than a = 0. In the LU v isobars with higher value of the pressure only this change has come that now a retrograde part appears in the branch at the small volumes for a number of isobars, whereas part of the isobars with lower value than the loop-isobar fail to have the retrograde portion in the branch starting from the line «=2,. They do not get it until the value of the pressure has fallen so low that the righthand Oa 2a,, dp je t of — = 0 is intersected. When the temperature — | part of — 1S „Intersecte nen the temperature eh d has been reached, this part of - = 0 has vanished, and so also the UV retrogression of all these isobars. (fig. 1d). The last modification which our diagram may finally undergo, is ( 809 ) da db k when the temperature at which ae MRT ER for «= &,, is reached. at at Our diagram then passes into the usual one, when this is drawn above the critical temperature, and as we have now to deal with the case of fig. 9, into the usual figure after the intersection of X EE \ * +. ¥ Seah CR f ~ : +2 Arn * V Fig. 14, d d B — and =0 has disappeared. dv da 16. Now the course of the isobars for the case that a minimum critical temperature occurs at the same time with a,,* >a,a, is completely determined. Only this complication might possibly be met with — I have at least not succeeded in proving that it is d, impossible — that besides the discussed contact of ee O and ) dp 3 = 0, by which two points of intersection arise, another contact av is found. As we saw before the points of contact which then arise will again have to move to opposite sides. Of the four points of intersection which there are in this case, the two inner ones will again coincide at still higher themperature, and give rise to contact. So the difference is confined to the region between the two temperatures of contact, and it has only influence on part of the isobars at small volumes. Thus of the series of isobars which pass round the ; ae dp ! point with minimum pressure on isa 0 as closed curves, there will av ( 810 ) e:g. be one which assumes the shape of a ©, and the isobars of still smaller value of p will have broken up into two branches each closed in itself (apart from the closed portion starting from vl, v=0). C and D (fig. 15) are then the points of intersection which have newly appeared; the complication vanishes again in Fig. 15. consequence of the coincidence of B and C. If the second point of contact should arise on the left of the point of intersection lying to the extreme left instead of on the right of B a similar result would be met with. 17. At first sight the diagrams of isobars obtained above seem to deviate considerably from the figure given by van DER Waats. This : 1, : is of course partly due to the different course of = == 0. -Pasilyg id however, also to the fact that the figure loc. cit. only holds for 27 temperatures, lying between = of the critical temperature and the critical temperature itself. Therefore we find the closest resemblance with the figure loc. cit. in our figures for higher temperatures. That the resemblance also continues to exist at lower temperatures is imme- diately seen when we examine to what changes the figure l.c. is subjected with lowering of the temperature. First of all we have then the temperature: 27 a, + a, — 2a,, BER 2, dp av As we saw above another point of intersection of ——O and (BS dp dv point of intersection branches of isobars with negative value of the —( is found on the right below this temperature, round which 27 pressure pass as closed curves. For ab of the minimum critical tem- perature the loop-isobar will hold for the pressure 0, as we saw d above. So it reaches the vapour branch of at only at infinity Vv (fig, 16). For still lower temperature also the loop-isobar in the case Fig. 16. of vaN per Waats is no longer closed round the point of intersection d d of ef with the vapour branch of Ze but round the third v v point of intersection. Of course closed rings continue to run round En / = ~ / os rat / eae —| 9 ms Sng - © tape) EREN - + Eke XN ; kg Den ms “ 5 La ~ Fig. 17. ( 812 ) the first-mentioned point of intersection, which continues to distin- guish the figure from the figures given by us. At still lower ese rature the two points of intersection with the liquid branch of ? 0 Vv may coincide. It is true that this clashes with the thesis concerning d; the contact of P — Oana? =0, mentioned in the beginning of & av the previous communication, which gave rise to this investigation, but then this thesis holds only if 6 is a linear function of « and in this case the said point of intersection on the right does not make its appearance. If the two points of intersection have coincided, the loop-line and the closed rings at small volume have disappeared and only those at large volumes remain. (fig. 17). It is, however, also possible that the points of intersection continue to exist down to the absolute zero point, viz. when a minimum anda maximum occurs in the critical pressure. That this is possible for a quadratic function for b is shown by fig. 18, if we bear in mind Fig. 18. that a and so the critical pressure never become zero now. In this case the points of intersection in the liquid branch continue to exist down to the lowest temperatures, their limiting situation is the value for # at which the critical pressure is stationary. With this exception and with those exceptions which arise by the dp Soe modified course of fae this diagram and ours harmonize. 2 In a following communication I hope to show that so long as no maximum critical temperature occurs, no other diagrams of isobars but those discussed are possible in the realizable region (also the unstable one) for whatever values of a, 6 and a,, we combine. ( 813 ) Mathemathics. — “On a class of differential equations of the first order and the first degree.” By Prof. W. Karren. 1. In the last meeting of this Academy Prof. J. pr Vries gave a geometrical criterion for determining whether or not a given differen- tial equation of the first order and the first degree may be reduced by a homographic substitution to a linear equation or to an equation of the form dy _ Ney +P@y+ Qo) da R(w)y + S(z) The object of this paper is to examine the general form of all those equations which by a homographic substitution may be reduced to the equation (1). It is evident that this general form will give at the same time all the equations which are reducible either to the general equation of Riccati, or to the linear form. (1) 2. Let the substitution be __ autayta, a ___ biutbvdb, B 9 me c,ute,vte, eae Pine ae cutevte Y @) where a,b,c are constants, then the equation (4) is dv CBN HBP H7°Q']—Ar [BR +78] 4 du y B[BR*+-yS"]—D[BP?N* + ByP*+y’Q'] where A=b,y—o,8 C=a,y—¢,a@ B= bir 0, 6 Dap oa and var(s) ace ¢ =0(<) r= R(S) s=s(<) Y Y 7 7 Y Transforming now to parallel axes, taking as the new origin o coordinates the point where the lines «== 0 and y=O meet, we find the new equation by substituting EA ke RA (a,c,) = (4,¢,) (4,0;) == Gicg — Oren eae +u,v In this way, we get a=au+av',B=bv+),7 +e=8 Hes, yoouuvt+e,v' e being a constant, and 55 Proceedings Royal Acad. Amsterdam. Vol. XI. ( 814 ) Abt Eke Ce C,) v B= — (be) dep DS (ae N (“) = (un =) —N, , p(<) — P, ete. Y eu tev Y where N, P, ete. are homogeneous functions of w' and v' of degree zero. Hence, if we arrange according to the degrees of w and v' the numerator takes the form (edt [NB 42a By ed ris (5, Cs) Vi LR, By he S, veal + ela, ¢) v [2 N, 8 + Py] — eb, ¢) R, vy Hee [Rey + Sy") + Q° (a, Cy) N, v + ete By and in the same way the denominator may be written (a, ¢,) u LN, B + P, By + Q, v7] — (b, ¢,) v [R, By +S, 77] Hela, ¢) uv [2M, 8 + P, 1] — @ (b: ¢,) Ry wy ij — ec [R, By + 8,1] + 9? (ae) Nou —o’c, R, y. If we examine these values it is evident that the equation (3) reduces to dv! K,+-M,+v(N,4+¢) a du xA,+L,+u'(N,+0¢) where c represents a constant, H, and H, homogeneous functions of the first degree and L, M, N, homogeneous functions of the second degree. From the values EN ir ete, Ry K,= eey ALE o'(a,e) N, we may readily induce that if, in (1) A(z) is absent H, and K, must be zero and if in (1) N(e) is absent, we have c= 0. The preceding considerations furnish the inference that every homographic substitution applied to an equation (1), followed by a transformation to parallel axes through the point a= y =O gives necessarily an equation of the form (4). ( 815 ) 3. Now we will show that where a differential equation of this form (4) is given, there always exists a homographic substitution by which this equation may be reduced to the form (1). For let then we have Kee 1 Hi wek, (=. zerk CE; 2) ane | y me Ee 1 M, = M, (uv) = M, y ’ y ae” ea ‚) etc. Thus (4) reduces to dy tA, (1,2) + ey’ +-L,(1,0)y +N, (1,2) (5) de {#H,(1,x)—K,(1,2)y+e2L,(1,2)—M,(1,c) © ° ~ which is of the same form as the differential equation (1). 4. Therefore we have proved this: Theorem. The necessary and sufficient condition that a differential equation of the first order and the first degree, having a singular point in the origin of coordinates, may be reduced by a homographic substitution to an equation (1) is that it may be written in the form dy K,+M,+y(N.+9 6) de BTL, Fang Corollary 1. The necessary and sufficient condition that a diffe- rential equation of the same kind may be reduced by a homographic substitution to an equation of Rrccarr is that it has the form dy _ M,+yN,+0) : pee aay Corollary 2. The necessary and sufficient condition that a diffe- rential equation of the same kind may be reducible by a homographic substitution to a linear equation is that it has the form dy _M,+yN, (8) Mae DeL ANT. naj REE 5. With respect to the equation (8) we may remark that it is equivalent with dy deren M, +4 N, desen Leste N, as the numerator and the denominator of the second member may 55% ( 816 ) be divided by the same homogeneous function of the first degree. In the special case that L, =a,#-+ b,y, M,=a,¢-+ b,y, N,=er ddy the tangents to the integral curves in the different points of the line y = mez, meet in the pole ea ene rem Kr c, + d, m c, + d, m Hence the locus of these poles for all the rays of the pencil y= ma is the polar line De okey ee he a, —c,| = 0 oe Rea This is the case in the examples II—VI given by Prof. pe Vries. As to the examples I and VII we have respectively TO M =ae + 2y ni 1 i, =9 Wh N= — 1 Physics. — “Contribution to the theory of binary mixtures.” XIV. By Prof. J. D. van DER WAALS. (DOUBLE RETROGRADE CONDENSATION). Before proceeding to the discussion of the significance of negative value of «, and «,, L shall make a few remarks to elucidate what was mentioned in the preceding contribution — and that chiefly on the shape of the surface of saturation in the cases represented by figs. 39 and 40, and the relative position of the three-phase-pressure with respect to the sections of that surface for given value of z. In ease of complete miscibility such a section of the surface of saturation consists of a vapour branch and a liquid branch, which have a continuous course, in which the pressure gradually increases with ascending 7’, and which for certain value of 7, which may be indicated by 7, pass into each other continuously. The pressure must then before have had a maximum on the liquid branch, and then decrease. It passes into the pressure of the vapour branch at 7’. This gradual merging of the two branches into each other con- tinues to exist also for non-complete miscibility. In the case of fig. 39 the upper sheet of the surface of saturation undergoes, however, first of all a modification, which, however, is ( 817 ) confined to values of w between ‘x,), and (v,),, when (,), denotes the value of x for which on the righthand side of the closed dotted curve the tangent is normal to the «z-axis, and in the same way (v,)g the same thing on the lefthand side. For a section between (x,), and (w,), the modification remains restricted between the two values of 7’ which are determined by the closed dotted curve of fig. 39 for the temperature. Outside these values of « and 7’ the upper sheet is, if not quite unmodified, yet of the usual shape. If we call the concentration of the vapour phase 2,, and that of the coexisting liquid phase z,, this usual shape of the upper sheet is determined by the equation : d5 v,, dp =(a#, — «,)| — | de, + 1, dT. Cok. 2 2 And if it were our purpose to determine the situation of the sheet of the three-phase-pressure with respect to the metastable and un- stable sheet of the coexisting equilibrium between vapour and liquid, the above equation might serve this purpose. For along the circum- aa. dT known. Thus to give an example, this quantity is positive in the lower part on the right hand of Q,. And v,, being positive and v,—a@, negative, we find from: dp \ _ Op (Ge) + Gr), d, 0 (F)<(55)- Hence in the righthand part the sheet of the three- 123 Te phase-pressure lies below the metastable sheet — which, however, might be considered as having been known beforehand from the shape of a p‚z-line at constant temperature for the equilibrium vapour-liquid. In the lefthand part the sheet of the three-phase- pressure lies on the other hand above the metastable sheet. In a section normal to the «z-axis just through Q, the sheet of the three-phase-pressure begins at Q, touching the metastable sheet, and in a section just through Q, there is contact at the end. But it would lead us too far to include all the metastable and unstable sheets in our consideration. To examine what the shape is of the stable part of the liquid sheet inside the dotted ring, and so of the upper part of the surface of saturation is, however, very necessary indeed. If we take two points of the dotted ring lying on a level and if we call the value of x for the point lying on the right z, and for ference of the closed dotted curve of fig. 39 the value of is ( 818 ) the point lying on the left z,, then d v,,dp = (#, — 4,) (=) pits + nd © Jp “dp « (òp\ dz, & AT Lee Oss oT ie holds for the equilibrium between these two liquids for the point lying on the right. 2 or In the second member the second term (5) holds for the section Tas for constant value of x, the shape of which section we will determine — in the first term of the second member we have omitted the index 7’ to avoid great complication in our notation. The shape of the line of equilibrium between the two liquids is known of a pse-section of the surface of saturation, if three-phase-pressure occurs. It is a curve which ascends steeply in the points 2 and 3, and. e OE reaches a maximum value between them. In the point 3 | =} hasa Oase . ° . ‚de, . Ore . . high negative value, and if — is positive, which is the case on the dT is smaller d lower side of the righthand half of the dotted curve, S ry 123 le, C dT Op 2 than eS . On the lefthand part of the lower side is negative, Tog 07 . UFR Ea but there A Fe positive, so that we arrive at the same result U) 93 there. On the upper side we come to the opposite conclusion. So if we draw the upper branch of the section of the surface of saturation, such a modification must be applied between the limits of the tempe- rature, which are to be derived from the dotted curve of fig. 39, that the three-phase-pressure lies below this curve — which was, indeed, a priori to be expected. And by a comparison of the two equations: dp __ (0p\ de, Op Ch aR iN aa eh + om : dp (Op de, En Op Cie oe) dT OT yam from which follows: Op Op Ow 13 a Op Op _ da, Ga x oT “13 a dT and ( 819 ) not only the sign, but also the value of the abrupt change of direction in the course of the upper branch is determined. But this modification remains restricted between the two indicated temperatures — and neither of the lower sheet nor of the remaining part of the upper sheet does the shape change in any respect. Only the lower sheet undergoes some modification, but in sections of quite different values of x, namely those which are found between the values of « of the points Q’, and Q’,. And this modification occurs only for one single value of 7, at least so long as the curve Q’,Q’, has the property of being cut only once by a line normal to the «z-axis. The modification in the course of the lower branch consists in this that for certain value of 7’, to be derived from the curve Q’,Q.,, the lower branch begins to rise less steeply. The three-phase-pressure ascends then even more rapidly than the first direction of the lower branch did. Below the temperature at which this change in the course of the lower branch appears, the three-phase-pressure is. already found, but for the considered section it may be considered only as a parasitic branch. The part of the three-phase-pressure that belongs to higher temperatures lies and terminates in the unstable region, i.e. above the lower sheet of the surface of saturation. At least if Q',Q’, is not intersected for the second time at the same value of z. In this case a second, higher temperature occurs, for which the three-phase-curve runs below the lower sheet, and again a part of this curve becomes a parasitic part. But for all the sections between Q', and Q', the upper sheet is not subjected to any change. The proof of what was said here about the modification of the lower sheet, is again found. in the equations, which now with change of the indices, assume the form: dp (Op de, | Op Pee ee ede dT OT J an dp __(Op\ de, Op ere dx al dT a zn If, as is the case on the curve Q’, Q', the three-phase-pressure comes inside the heterogeneous region with ascending temperature then dp Op Op É | 7 —>(s), and 5e): being negative on the whole lower sheet now that the second component is assumed to have higher 7% than and the first component, Dn is necessarily negative, as was drawn for ( the curve QQ. But as we see from the properties of a pr or ( 820 ) Op v,a-section of the surface of saturation at given temperature, ere Lv 0 d, 0 is in absolute value greater than (5: Hence 2 (5) is at 73) On dT’, 23 0 T dp greater than dT Ee & a= 123 So the section of the surface that holds for the equilibrium 3,1 ascends less rapidly than the section 1 for the equilibrium 2,1. And in consequence of the retrogression of « to smaller value at rising temperature the sheet 3,1 comes in the place of the sheet 2,1. If at a second, higher temperature the three-phase-curve should again leave the heterogeneous region, and become a parasitic si below the 1 Op lower sheet, then a <(3) Hence the value of = is positive, ds Ory: Ov d Op because ee continues to be negative. Then both = -( 2 ) and Oa he aT. d1 Ta1 dp Op 4 : i : —_—{—] are negative; but the first difference is greater in BREE NOT p | : Op absolute value than the second. So is then smaller than EE OER There is then another break in the sbomioë of the lower sheet, but yet in consequence of this this section begins to ascend less rapidly, because on account of 2, running to the right with rising temperature the sheet 2,1 takes the place of the sheet 3,1. But though it is good that these circumstances have been examined in detail, all this is not necessary if we only want to know what will be the shape of the line at higher temperatures with regard to that at lower temperatures, if a break appears in the lower branch. The simple remark that the two branches which meet, must have such a position with respect to each other that their continuations repre- senting metastable branches may not lie in the stable homogeneous region, is then sufficient, and evidently this rule could not be satisfied if on the lower branch the line occurring at higher temperatures ascended more rapidly than the preceding one. But for the upper branch this rule leads to the conclusion that on the contrary a branch appearing at higher temperatures, must ascend more rapidly. The systematically sustained application of this rule, therefore, does not present any difficulty in the case of fig. 40, and does not reveal anything new so long as Pe, lies higher than A, as in the case with the mixtures of ethane and alcohols. But in the cases of Bicnner, mixtures of CO, and organic liquids where P,, lies lower than 7, it will have to give rise to what we may call: double ( 821 ) retrograde condensation. So long, namely, as Pa, continues to lie higher than 7% the three-phase-line Q,Q,Q',Q,' can lie entirely on the upper sheet of the surface of saturation, and can then give only rise to the more rapid ascent of the upper branches of the sections at given value of z. Above the temperatures at which this more rapid rise sets in, the section has then its usual shape, merging continuously into the vapour branch, there being only one point where the tangent is normal to the 7'‚z-plane. The temperature at which the upper branch suddenly begins to rise more rapidly is given for every section by the value of 7’ for the points of the dotted curve, which, however, must then be broken off before the point with minimum value of w. Sections on the right side of Q,' and the left side of Q,' present the wholly unmodified shape. Sections between Q,' and Q, get the break in the upper sheet at ever lower temperature as the point Q,' is approached. In each of these sections part of the line of intersection with the three-phase-sheet, i. e. a part that belongs to lower temperatures lies above the upper branch. The higher part of this line of intersection lies and terminates in the heterogeneous region. For the section of Q, this line of intersection lies over its full length in this region, and the initial direction at Q, then touches the upper branch of the section of Q,, as it proceeds unmodified at lower temperatures. For sections between Q, and Q, the point of intersection of the upper sheet with the three-phase-line lies at ever higher temperatures as we approach Q,. The lower part of the tbree-phase-line then lies below this upper branch, and the upper part would project above it; but the abrupt change of direction taking place at the intersection is so great that even the upper part of the three-phase-line remains in the heterogeneous region. So if we want to keep a section between Q, and Q, in homogeneous liquid state, the pressure must always be greater than the three- phase-pressure with the exception of the liquid represented by the point of intersection. For the section of Q, the three-phase-line lies entirely in the unmodified heterogeneous region, and in the highest point it touches the upper branch. For the sections on the left of Q, the upper part of the three-phase-line lies again above the upper branch, and as such it has lost its significance. Then these sections, too, have the sudden change of direction, at which they begin to ascend more rapidly, but at ever lower temperature the more we go to the left. For the section of the point Q', the whole of the three-phase-line lies then above the upper sheet. So when the three-phase-line over its full width intersects only the upper sheet, there is no further complication ; and this may be ¢ 4440) the case if Po > Ts. Also in this case, however, the possibility remains that on the lefthand side the intersection takes place in points of the lower sheet. But the latter must iake place if Pa, > Th. And in the transition of the point of intersection of the upper sheet to the lower sheet we meet with the complication which I am now going to discuss, but which will perhaps be easier to understand, when I shall discuss the properties of the p,a-sections of the surface of saturation, so sections at constant temperature. If we think such a p,«-section at a temperature only little lower than 7, the upper branch has, leaving the line of the equilibrium between the two liquids out of consideration, the known shape with a minimum and a maximum between z, and z,, because the three- phase-pressure exists, P., being < 7. The third phase then lies on the lower branch at 2, < #,