S>) •= 1000" ^ 15,984 9250 V-l +Y^2 we find, at the critical point, 6*87, 74 atm., 30° C. This presents a pretty fair agreement with Andrews’ data, and is not liable to the objection raised against the former tentative formula, probably because j3 has been taken greater than either a or y. As the same may be said for very different sets of constants, such as P(V-1)=1000^ 62,390 55,829 Y + 06 + V + 0-4 it is clear that nothing definite can be asserted as to the true values of the constants until the labour of deducing them directly from the experimental data has been successfully undertaken. The result might give us some real information as to the range and intensity of the molecular force. If we assume the critical values V0, E0, p0, from experiment, the general equation of the Isothermals takes the form MV-/5) = E-^{ (V°-y)3 V-y (V0-*)3 l V-a j in which the disposable constants are reduced to two. For there is a single relation among a, /?, ana y, viz., 3Vo = a + /8 + y+-0. po This, like the values of A and C, is given by the condition that the three values of V are equal at the critical point. 1888-89.] Dr Thomas R. Fraser on Strophanthus hispidus. 73 Strophanthus hispidus : its Natural History, Chemistry, and Pharmacology. By Dr Thomas R. Fraser. (Abstract) (Read February 4, 1889.) A. Natural History . In February 1870, the author made a communication to this Society on the Kombe Arrow-Poison of Africa, a product of the Strophanthus hispidus plant. In that communication the nature of its action on the various structures of the body, and the chemical composition of the seeds of the plant, which are the most active part, were described. It was pointed out that the action is chiefly exerted upon the heart and upon the muscles of the body, and that the seeds contain a crystalline active principle of the nature of a glucoside, to which the name Strophanthin was given. From the examination then made of the action of the seeds of this Strophanthus , as well as of its active principle, strophanthin, it was anticipated that Strophanthus would prove to he of great value in the treatment of disease, and especially of disease of the heart ; and a few years later the author employed it for this purpose in a small number of cases. Various circumstances, such as the difficulty in procuring suf- ficient supplies of the seeds, prevented the author from making the number of observations that appeared to he necessary before the value of Strophanthus in the treatment of disease could he properly estimated; and it was not until 1885 that sufficient evidence had been obtained to authorise any public announcement on the subject. In the interval of fifteen years which elapsed between the first communication to this Society and the communication of 1885 to the British Medical Association, the subject attracted so little atten- tion that only two papers were published on it. One of these papers dealt with the physiological action, and con- firmed the statements made in the communication to this Society. The second paper dealt only with the chemical composition of the Strophanthus seeds, but the chief statements it contained, such as that the active principle is not a glucoside, have since been amply shown to be erroneous. 74 Proceedings of Boyctl Society of Edinburgh. [sess. Subsequently, however, to the communication of 1885, upon the therapeutical applications of the substance, the literature of the subject has rapidly increased, and it now embraces about a hundred separate papers, the greater number of which deal with its uses in the treatment of disease. Until 1885, also, Strophanthus, elsewhere than in Africa, was a mere curiosity, represented in a few museums in Europe by speci- mens of its flowers and fruit. Since that time, it has become a not inconsiderable article of commerce, several tons of the seeds having been exported from Africa by London merchants alone, in order to supply the requirements of medical practice. In the present paper it is proposed to give an account of the observations that have been made by the author on the natural history, chemistry, and pharmacology (or physiological action) of Strophanthus, but to-night only the first of the above subdivisions of the subject would be dealt with. In nearly every narrative of exploration in uncivilised tropical regions, accounts are given of poisonous substances, which in many instances are stated to possess remarkable properties. Usually these poisons are of vegetable origin, and nearly all of them may be in- cluded in the two great divisions of Ordeal and of Arrow poisons. Among the most interesting of the Ordeal poisons are the Physo- stigma venenosum and the Akazga or Akaja of West Africa, the Sassy or Muave of wide distribution over Africa, and the Tanghinia venifera of Madagascar : and of the Arrow poisons, the Antiaris toxicaria and Strychnos Tieute of Java, the Aconitum ferox of China, and the famous Wourali or Carare poison of South America. It is to the enterprise and observation of explorers and travellers that we are indebted for the first knowledge of the Strophanthus Kombe-poison. The first specimens of the plant that reached Edinburgh appear to have been a few ripe follicles sent to Sir Eobert Christison early in 1869, by the Rev. Horace Waller, who had been a member of the Oxford and Cambridge Universities’ Mission of 1861-62, superintended by the late Bishop Mackenzie, with whom had been associated, during the operations of the Mission in the country between the river Shire and the lake Shirwa, the famous traveller Livingstone and the enterprising botanist Kirk, at that time H.M. Consul at Zanzibar. Mr Waller informs me that, 1888-89.] Dr Thomas R. Fraser on Strophanthus hispidus. 75 at his suggestion, the follicles were brought to this country by Mr E, D. Young, R.N., when he went to Africa, in 1867, to clear up the story of Livingstone’s murder. Sir John Kirk had previously discovered that the Kombe poison is prepared from the seeds contained in these follicles. In a letter received from him (31st October 1888), he thus describes the dis- covery:— “ I had long sought for it (the source of the Kombe poison), but the natives invariably gave me some false plant, until one day at Chibisa’s village, on the river Shire, I saw the ‘ Kombe,’ then new to me as an Eastern African plant (I had known an allied species at Sierra Leone (1858), where it is used as a poison). There, climbing on a tall tree, it was in pod, and I could get no one to go up and collect specimens. On mounting the tree myself to reach the Kombe pods, the natives, afraid that I might poison myself if I handled the plant roughly or got the juice in a cut or in my mouth, warned me to be careful, and admitted that this was the 1 Kombe ’ or poison plant. In this way the poison was identified.” Livingstone, in his Narrative of an Expedition to the Zambesi and its Tributaries (1858-1864), states that the tribes inhabiting the Mikuru-Madse, a tributary of the Shire river, use this poison for arrows, with which they kill buffaloes and other game. “ Poisoned arrows are made in two pieces. An iron barb is fastened to one end of a small wand of wood, ten inches or a foot in length, the other end of which, fined down to a long point, is nicely fitted, though not otherwise secured, in the hollow of the reed, which forms the arrow shaft. The wood immediately below the iron head is smeared with the poison. When an arrow is shot into the animal, the reed falls to the ground at once, or is very soon brushed off by the branches, but the iron barb and poisoned part of the wood remain in the wound. If made in one piece, the arrow would often be torn out, head and all, by the long shaft catching in the under- wood, or striking against trees.” Mr John Buchanan thus describes the method followed in pre- paring the poison : — “ A man breaks a follicle and puts the seeds with the wool attached into a pot. He then takes a small piece of bamboo, which has two thin splints inserted crosswise in the end, and he revolves this speedily by rubbing it between his hands. The seeds are thus put into motion, and fall to the bottom of the 76 Proceedings of Royal Society of Edinburgh. [sess. pot, and the wool rises and comes oat at the top, and is carried away by the least breath of wind. The seeds are then put into a small mortar and pounded into a paste, which is then ready for use. It is common to mix the milky juice of a Euphorbia with it to make it stick to the arrow.” This arrow poison has also been found at the western side of Africa, where it is known as the “ Inee ” or “ Onage,” and the poison has been traced to a Strophanthus, which is probably the species liispidus , although the flowers do not appear to have been yet obtained. Only a few poisoned arrows have reached this country from Africa, owing, no doubt to some extent, to the difficulties of car- riage, but certainly much more to the reluctance of the natives to place poisoned arrows in the possession of Europeans. The author had, however, been able to examine arrows of eight different forms obtained from various parts of Africa. Two of them were arrows known to be poisoned with Strophanthus. Of the others, either no knowledge of the poison existed, or it was believed to be derived from plants other than Strophanthus. Microscopical, chemical, and physiological examination showed that the poison of six of the eight arrows consists principally, if not entirely, of a substance made with the seeds of Strophanthus ; and it is an illustration of the extensive use of this poison that these arrows should have been obtained from districts so widely separated from each other as the river Gambir, the Tanganyika lake, and the Zambesi river. Of the other two arrows, one, originally poisoned on the arrow- head, was found to be inert ; and the other, obtained in the Wanika country, was found to be poisoned with a substance acting like Strophanthus, but not giving its chemical reactions, nor exhibiting, on microscopic examination, any structure that could be identified with the structures in the seeds of Strophanthus. It is probable that the poison of the last arrow has been derived from a wood or root. Decandolle, in 1802, first described the genus Strophanthus, and gave it this name because of the twisted, thong-like prolongations of the lobes of the corolla (o-rpo<£os, a cord, and avOog, a flower). About twenty species are at present known, eight of which are found in Africa, and the others in India China, Malacca, and Burmah. 1888-89.] Dr Thomas R. Fraser on Strophanthus hispidus. 77 Strophanthus Kombe is not included in this enumeration, as Pro- fessor Oliver, after an examination of further and more complete specimens of the flowers and leaves, now regards it as “a variety, a geographical race, of Strophanthus hispidus .” The species hispidus has been found only in Africa, and is widely distributed over its tropical and subtropical regions. Mr Buchanan has at various times sent specimens of the root, stem, branches, leaves, flowers, and fruit, and has thus supplied materials for a description of the different parts of the plant. These parts were exhibited and described, and also a young growing plant, reared from seed by Mr Lindsay, of the Royal Botanic Garden of Edinburgh. In reference to the fruit, it was pointed out that it consists of two follicles, united at the ventral surfaces in the young state, hut gradu- ally separated, as maturity advances, by a hinge-like movement at their bases, until each separated follicle projects from the fruit stalk, almost at right angles with it. When fully mature, the two follicles form together a nearly straight line, whose extremities are the apices of the follicles. Each ripe follicle contains three separate structures — the placenta, the seeds, and a large quantity of hairs interposed between the seeds and the endocarp. As the follicle matures, its ventral or placental surface enlarges by the inverted edges of the carpels, which project united together into the interior of the follicle in its immature condition, splitting up more and more, and so expanding this surface. At the same time, the dorsal surface of the follicle, consisting of the thick and strong peri- carp, becomes less rounded, and the placenta, with its still attached seeds, is brought close to the expanded ventral surface. By and by, as maturity advances, and the funiculus of the seeds becomes weakened by drying, the seeds break off from the funiculus, and lie loose in the interior of the follicle. The follicle ruptures at the expanded ventral surface, which is its weakest portion, and through this rupture the seeds are extruded. The actual extrusion of the seeds seems to he produced by the separa- tion from each other of the hairs of the comose appendages, and espe- cially of the hairs separating the seeds from the endocarp. These hairs, in the green and moist state of the follicles, are in close contact 78 Proceedings of Royal Society of Edinburgh. [sess. with each other ; hut in the mature dry state they acquire elasticity, and tend to become separated from each other, and they thus press the seeds through the split ventral surface of the follicle. The hairs separating the seeds from the endocarp seem to possess the additional function of preventing fracture of the long and brittle stalks of the comose appendages, by forming a soft and yielding bed for the seeds during their changes of position before they escape from the follicle; and they thus insure that the seeds shall be dis- seminated with the comose appendages attached to them. Drawings and microscopic preparations were exhibited to illus- trate the histology of the different parts of the plant. A description of the results that had been obtained in the chemical and pharmacological examination of Stroplianthus was deferred to a future meeting of the Society. A New Type of Dimorphism found in certain Anti- patharia. By George Brook, Lecturer on Comparative Embryology in the University of Edinburgh. (Read January 21, 1889.) A more or less elaborate system of polymorphism is of frequent occurrence in certain groups of colonial Ccelenterata. For example, in many Hydroids certain individuals perform the nutritive func- tions for the colony, others are specialised for reproductive purposes, and so on. The variously modified individuals are connected together by a general coenenchyma, which enables the nutriment elaborated by the gastrozooids to be utilised by other members of the colony. Perhaps some of the most interesting and complex cases of polymorphism are to be found amongst the Siphonophora. Amongst the Anthozoa dimorphism frequently occurs in certain groups of Alcyonaria , but in these cases apparently the speciali- sation never results in the formation of reproductive zooids. The modified individuals ( Siphonozooids ) differ from those of typical structure in the absence of tentacles, the great development of the siphonoglyphe, and in other points. They are usually but not in- variably sexless, and in certain cases are stated to develop into typical zooids. In the Actiniaria the animal is usually solitary; 1888-89.] Mr G. Brook on a New Type of Dimorphism. 79 colonial forms, however, do occur, hut so far as is known none of them are dimorphic. The Madreporaria include both colonial and solitary forms. Our knowledge of the structure of the zooids is as yet confined to a few species, and so far as I am aware the only known case of dimorphism is that described by Fowler in Madre- pora Durvillei. In this case the specialisation affects certain of the mesenteries in the modified individuals. In the normal zooids all the mesenteries are similar in structure. In the modified zooids six out of twelve mesenteries (alternate ones) become thickened, and contain a ciliated ectodermal canal running through their sub- stance, and opening at both ends into the stomodseum. Apparently both types are nutritive and both are reproductive, hut Fowler is of opinion that the normal zooid is more reproductive, and the modified zooid more nutritive. Thus in this, the only case of dimorphism previously described amongst the Zoantharia, there is no complete specialisation into nutritive and reproductive zooids, and it is of interest to note that in another species of Madrepora (M. aspera ), examined by Fowler, a similar dimorphism does not exist. A study of the “ Challenger” collection of Antipatharia has shown that some of the genera are dimorphic, whilst others are not. Nearly all the species obtained at great depths are dimorphic, hut others come under the same category which occur in shallower seas. The nature of the dimorphism and the manner in which it has probably been produced will be best understood by a study of the general morphology of a few typical forms. Most Antipathidce have the horny axial skeleton more or less branched, and the zooids are usually situated in a single linear series on the branches, and are connected together by a coenenchyma which contains an axial prolongation of their coelentera, neighbour- ing zooids being thus brought into communication with one another. A vertical mesogloeal partition is usually present between adjoining zooids, but is never so complete as to prevent intercommunication. In a typical zooid there are six tentacles and three pairs of well- developed mesenteries, and, in addition, there may he two or three other pairs, which are always more or less rudimentary. The stomodseum is elongated in the sagittal axis, and usually occupies a position at right angles to the axis of the branch on which the 80 Proceedings of Royal Society of Edinburgh. [sess. zooid is placed. Only one pair of mesenteries ever bear reproduc- tive organs ; these occupy the transverse axis, and are situated one on each side of the stomodseum. The transverse pair of mesenteries, on account of their position, have a greater horizontal breadth than any of the others, and they are also usually somewhat longer than those at each end of the stomodseum (“ directives ”). In the species already examined there is a gradual tendency for the zooid to become elongated in the transverse axis, as a result of which the transverse mesenteries show a corresponding increase in breadth. In Girripathes the outline of the zooid is more or less rounded, but the insertion of the sagittal tentacles into the Dody wall instead of into the peristome interferes with the regularity. The diameter of the zooid in the sagittal and transverse axes is, however, practically the same, and the tentacles have a radiate arrangement. In Anti- pathella a slight elongation of the zooid in the transverse axis has the effect of making the tentacles appear in two rows of three each, parallel with the axis of the branch. On account of the oval out- line, there is in this genus a greater disproportion between the breadth of the transverse mesenteries, as compared with the “ di- rectives,” than is the case in Cirripathes. The increase in the length of the transverse axis is, however, not great, and the tentacles do not become removed far apart. In Parantipatliesi however, the transverse mesenteries become enormously elongated, so that the length in the transverse axis is three or four times that in the sagittal. The elongation has the effect of carrying the “ lateral ” tentacles further away from the stoinodsenm, so that they now appear clearly as three pairs some distance apart. The middle pair, as in other genera, are situated one at each end of the stomodseum, and the “ directive ” mesenteries are very narrow. In Parantipathes larix (Esp.) the peristome becomes somewhat depressed on each side of the oral prominence, so that the zooid presents indications of a division into three lobes, a central one containing the stomodseum and the proximal ends of all the mesenteries, and two lateral ones containing the greater part of the transverse mesenteries. It will be remembered that the reproductive elements are borne on the transverse mesenteries only, and in Parantipathes they are confined to those portions of them which are situated within the lateral lobes. It may be mentioned that in this genus the greatest diameter of the U. 1888-89.] Mr G. Brook on a New Type of Dimorphism. 81 stomodseum frequently corresponds with the transverse instead of with the sagittal axis, as is also the case in Amphianthidce amongst the Actiniaria. In Sehizopathes the three lobes of the zooid in Parantipathes become separated from each other by a further depression in the peristome, and also by the formation of a mesogloeal septum, which projects downwards for some distance from the base of the depres- sion towards the skeletal sheath. By these means each lobe of the primitively simple zooid becomes separated from its neighbour in the same manner as the simple zooids of Antipathella, Paranti- pathes,, &c., are separated from each other. The three lobes of the zooid in Parantipathes , having become separated from each other in Sehizopathes by the formation of vertical mesogloeal partitions, may now be considered dimorphic forms. The middle one containing the stomodseum, which opens by the mouth at the apex of an elon- gated tubular projection, may be termed the gastrozooid, whilst the two lateral ones, containing the reproductive organs, may be dis- tinguished as gonozooids. Each of the three dimorphic zooids bears a pair of tentacles, but the gastrozooid is the only one possessing a permanent opening to the exterior. In Sehizopathes the dimorphic individuals are arranged in a single linear series along one aspect of a branch ; all are in communication with one another through the bases of their coelentera, and there is typically no isolation of the zooids into triplets, but all are pressed closely together. In specimens in which the reproductive elements are well developed, the gono- zooids become much distended, and the sequence of the dimorphic forms along a branch is then readily recognised. Using the letter R to represent the gonozooid and S the gastrozooid, the arrange- ment may be indicated in the following manner, the derivatives of a primitive zooid being included within brackets : — In Bathypathes the differentiation is carried a step further, on account of the fact that the individual zooids are separated from vol. xvi. 25/3/89 f 82 Proceedings of Boyal Society o f Edinburgh. [sess. each other by a considerable interval, but are still connected together by axial prolongations of their coelentera. This arrangement may he indicated in the following manner : — fT\ fR\_ r ^ In Bathypatkes the isolation of the dimorphic zooids, each hear- ing a pair of tentacles, might lead one to suggest quite a different interpretation if the intermediate steps in the differentiation were not known. It will, however, he evident from the points already indicated in outline, that the dimorphism in Antipathidse is brought about — firstly, by an elongation of the zooid in the transverse axis ; and, secondly, by the formation of two vertical constrictions and mesogloeal partitions by which the elongated zooid is divided into three portions, one nutritive and two reproductive. These in Bathypatkes are frequently more isolated than are the unmodified zooids of normal types, and have as much claim to rank as indi- viduals. I have thus been led to divide the family Antipathidse into two sub-families, of which the following short diagnoses will serve our immediate purpose : — ANTIPATHIDiE. 1 . Antipathince. — Zooids of the normal hextentaculate type, show- ing a tendency to become elongated in the transverse axis, which corresponds to the axis of the supporting skeleton. Examples — Leiopathes, Girripathes , Antipathes , Parantipatlies , &c. 2, Schizopathince . — Zooids dimorphic and bitentaculate, of which three — viz., two gonozooids and one gastrozooid — are morpho- logically equivalent to one unmodified zooid of the normal type. Examples — Schizopathes , Bathypatkes, &c. In this connection the genus Parantipatlies forms an interesting link between the two sub-families, and shows clearly the mode by which the dimorphism has been produced. 1888-89.] Mr G. Brook on a New Type of Dimorphism. 83 In conclusion, attention may be called to the twofold bearing of these observations. First, there is the interest attaching to the fact of the occurrence of a dimorphism resulting in the formation of specialised nutritive and sexual individuals amongst the Anti- patharia, such a condition having been hitherto unknown, not only in that order, but in the Zoantharia generally. Secondly, the specialisation resulting in dimorphism takes a peculiar course, and is probably connected with the extension of a colony in the direction of its branches. The dimorphism of the Schizopathince , it is to be remembered, is brought about by the division of one primitive zooid into three, and not as in many other cases by a specialisation of different individuals. It thus differs essentially from the dimor- phism of Madrepora Durvillei as well as from that of the modified individuals which perform similar functions amongst the Hydroids. The Change in the Thermoelectric Properties of Wood’s Fusible Metal at its Melting Point. By Albert Camp- bell, B.A. (Read February 18, 1889.) The thermoelectric properties of tin, at temperatures below and above its melting point, were investigated by the writer about a year ago, and the results of the experiments were given in a paper read before this Society.* In those experiments, however, the height of the melting point of tin made it impossible, with mercury thermometers, to obtain anything but an approximate indication of the change in the thermoelectric properties in passing the melting point. In the present experiments, therefore, Wood’s fusible metal was chosen as having a conveniently low melting point. The com- position of the alloy used was approximately the following: — Bismuth, Lead, Tin, Cadmium, 50 per cent. 26 „ 13 „ 11 „ The metals used were not pure. The melting point was found to be 73° C. January 16, 1888. 84 Proceedings of Boyal Society of Edinburgh. [sess. The chief difficulty with this alloy was its expansion after solidify- ing, which invariably shatters any glass vessel in which it is allowed to cool down. This difficulty was avoided by using, instead of a glass tube, a thin indiarubber one, which was filled with the alloy. One end of this tube was bent up at right angles, and into the fusible metal at that end of the tube a thin strip of iron (tin-plate) was inserted. This junction was placed, beside the bulb of a thermometer, in asbestos wool contained in copper cylinders sepa- rated from each other by layers of asbestos, and heated by a small spirit-lamp underneath. The other ends of the iron strip and the fusible metal were joined to copper wires leading through a com- mutator to a sensitive mirror galvanometer with a scale at 12 feet distance, and the junctions, well varnished, were immersed in a large can of cold water. During the sets of observations (which lasted 1888-89.] Mr A. Campbell on Thermoelectric Properties. 85 many hours) the temperature of these cold junctions rose gradually from 7° ‘2 C. to 8°‘l C. The readings were all reduced to 7° ‘2 C. by a suitable correction. For each reading the temperature of the hot junction was allowed to rise till almost perfectly steady, and then four deflections taken alternately to opposite sides of the scale. The mean of the four was taken as the reading. To keep the deflections well on the scale the sensitiveness of the galvanometer was altered when the deflections became large. Thus three sets of readings [(A), (B), and (C) in the Table] were obtained. These were pieced together by making one point in set (B) lie on the curve got from set (A), and then reducing all the other numbers in set (B) in the same proportion. Then set (C) was pieced on to (B) in a similar manner. The points marked * in the table were those by which the joining was effected. Of course, they are omitted in the diagram. The table gives in the second column the temperature of the hot junction, and in the third column the observed deflections reduced to cold-junction-temperature 7° *2 C. It was found that up to at least 68° *5 the deflections (D) agreed very nearly with the formula D = 33*62(£ - 7*2) , t being the temperature of the hot junction; so that the curve is a straight line. From 74° *4 C. up to 150° C. the readings agree very fairly with the formula D= - T348/2 + 79-22* -2879, i.e ., a parabola whose vertex is at t= 293°*8. The last two columns in the table give the values of D calculated from the two formulae above. The curve in the diagram is from the second and third columns. At 8° C. the position of the line of this allpy in the thermoelectric diagram is between the iron and copper lines. In conclusion, we see from the above formulae that from 7° C. up to about 73° C. (the melting point), the line of this specimen of Wood’s metal runs very nearly parallel to the iron line, and that at the melting point it takes a sudden bend away from the iron line, but almost immediately bends towards it again, keeping straight till 86 Proceedings of Royal Society of Edinburgh. [sess. at least 150° C., and probably meeting the iron line at the neutral point 294° C. Set. t (Hot Temp.). D Observed and Corrected. D Calculated (1). D Calculated (2). A 37° A C. 1010 1006 A 41°’6 1142 1157 A 44°-2 1238 1244 A 51°*1 1476 1476 B * 63° -2 (1882) (1882) A 68°'5 2066 2061 A 74°-4 2284 2259 2268 B 84o<0 2812 2825 A 86° *8 3009 2982 B 105° ’8 3948 3993 B 105°‘9 4029 3999 A 107°*4 4041 4075 A 108°*7 4155 4140 B 113°*2 4324 4362 C ni4°-o (4400) (4400) C 127°-5 5018 5030 C 138°-4 5485 5499 c 151°'0 6025 6008 Note on the Relation between the Mutual Distances of Five Points in Space. By Thomas Muir, LL.D. (Read March 4, 1889.) Lagrange, in his paper “ Solutions analytiques de quelques prob- leraes sur les pyramides triangulaires,” Nouv. Mem. de V Acad. Roy. . . . (de Berlin), Ann. 1773, pp. 149-176, gives unintention- ally the following expression of the relationship between the mutual distances of five points in space, viz. — 4A2/= a(a +/- o ;lzi 58,550 1275 1145 130 13-70 1870 187-69 90 | I „ 2, 56,680 1218 1030 188 12-40 454 169-88 30 1 l „ 3, 56,226 1417 893 524 11-10 0 152-07 105 r ,, 4 56,226 594 1040 + 446 13-50 425 182-95 0 1 „ 5, 55,801 779 765 14 12-60 198 172-62 60 hi. d ,, 6, 55,603 1317 / 486 831 5-67 936 77-68 60 ,, 7, 54,667 1374 \ 1100 274 16-67 0 228-38 20 l „ 8, 54,667 1118 683 435 6-10 + 369 83-57 10 | r ,, 9, 55,036 1246 770 476 12-32 57 168-51 20 | „ 10, 54,979 1133 628 505 7-89 + 85 11709 25 IV. \ : „ it 55,064 990 725 265 5-30 255 72-61 0 1 „ 12, 54,809 1331 778 553 11-00 878 150-70 0 l „ 13, 53,931 1331 oo ft 02 O £ „ s ~ .2 '3 "3 o ft — ft ft *3 "o .2 .2 ’3 "G tap © § . ft 2 . .2 ft ' =. riS* I§2° | = — 5§ ^ * ••ii'-S.Gjl be • • u g . - 35 « ftfl *3, = ,25, “‘to a ffl 1 5 ® -ft «r3 C ft ’g - -5 •,2'gM -g § » m ® ,0 9 « i h ft is Ho o3 « A > S3 «s 33 rig, £ £ S« S3 *3 33 *>3 3 S’ c cS oos ft ft ft qj ft ft5 oO[2ft^iu « « «H a a> O OftftCOO O 0 0=0 0»oo*oo7 cP cc o o" o' o" o' TtiCOCOhOOO(NNOO(NWNClOiOOO rH > *0 CO CO CO rH <>T I—T CsT Hour. 10 A.M. 2 P.M. 5 P.M. 11 A.M. 5 P.M. 12 A.M. 1 P.M. 10 A.M. 3 P.M. 11 P.M. 10 A.M. 11 A.M. 10.30 A.M. 1.30 P.M. 3 P.M. 1 P.M. 11 A.M. 11 A.M. 2 P.M. 3 P.M. 1 P.M. 7.45 P.M. 9.30 p.M. 7.45 p.m. 9.30 p.m. 10.45 A.M. 1 P.M. 10.45 a.m. 1 P.M. 1 P.M. 3 P.M. 3 P.M. 3 P.M. 4 P.M. 4 P.M. 5 P.M. 5 P.M. Date. NhhCCCOOOHHH?HOQOClC1^0QOCOCl^^^^TjiiOiOOiOa5C50CJOCiOO >. HHHHriH St: es *a § a """ ~ 03 o > • • * ,, . • Falkirk, ! ! Room, 168 Proceedings of Royal Society of Edinburgh. [sess. Tlie rapidity with which the products of combustion are cleared out of a room is a subject well worth investigation, and the dust- counting apparatus promises to be of considerable assistance in the investigation. Apart altogether from the question of ventilation by means of openings, there will be in most rooms a certain amount of circulation through the plaster of the walls and ceilings, and it might not be a bad plan to make the plaster of our houses as porous as possible. If this were done, and the space behind the plaster were connected with the open air, an insensible circulation of pure air would be secured. The air of our smoking-rooms will probably contain a very great number of particles. As yet, however, I have made no actual determination of the number. I may, however, mention that I find a cigarette smoker sends 4,000,000,000 particles, more or less, into the air with every puff he makes. As this smoke is not very hot, it does not tend to keep near the ceiling like the products of com- bustion, and must therefore make the air breathed by the smokers very full of particles. It will be observed that the smallest number of particles entered in the table is 500 per c.c. This is about the lowest yet observed in nature, though in some tests made in an agricultural part of Dumfriesshire a number slightly less was obtained. When testing such pure air as this, it does not require to be mixed with filtered air ; the receiver is filled entirely with the air to be tested. When the particles are so few they are just sufficiently far apart for easy counting, and all of them fall with one expansion. Though one expansion is sufficient to bring down all the dust particles in pure country air, we must not imagine that therefore a fog could not be formed in air with so few particles. We must not suppose that, because they are so few, they would form rain, and all of them fall as in the test-receiver. The conditions are very different in the two cases. The high expansion used in the testing apparatus gives rise to a considerable cooling, and consequent supersaturation; whereas in nature the load of vapour tending to condense might be small, and each particle only get enough to make it visible. Five hundred fog particles per c.c. are quite enough to make a fog, and as these particles may not be heavy, they may float and fog the air. 1888-89.] Mr John Aitken on Dust Particles. 169 Part II. — Portable Apparatus. (Read March 18, 1889.) When working with the apparatus described in the beginning of this paper, it soon became evident that, though it was suitable for laboratory work, it was yet very inconvenient when observations had to be made on the air of places at a distance — first, because of its size ; and second, because it required a house of some kind in which it could be fitted up and worked. Further, in making tests in a house, there is always the difficulty of local contamination. We may select the place of observation as carefully as possible, yet the house we select can almost never he in a situation towards which the products of combustion from some neighbouring house, or even from the selected house itself, may not he driven by some direction of wind. As a result of this, we may not he able to make any tests of the air when the wind is from certain directions. It therefore became desirable, for testing the pure air of the country, and for examining the air at places at a distance, that some portable form of the apparatus he constructed — one which could he easily carried by one person, and which could be worked in the open air, so that the observer might carry it to a situation some- where to the windward of human habitations or other sources of pollution. With this object in view, I have prepared designs of a portable instrument which can be packed into a small space, and be easily carried. The portable apparatus is shown in Plate IV., which is drawn full size. As will be seen, the instrument is simply a rearrangement of the apparatus already described, but on a smaller scale, and with some slight modifications. All the sizes in this apparatus are reduced to xg-th the size of the one previously described. In Plate IV., R is the receiver, in this case made of brass with a glass cover. Part of the glass cover, the two sides of the diaphragm, and the bottom are covered with blotting-paper cemented on them. This paper is used for holding the water required for saturating the air tested in the receiver. The receiver has a capacity of 35 c.c., exclusive of the space occupied by the wet blotting-paper. The stage, the inlet and the outlet pipes, all occupy the same positions as before ; the dia- phragm, however, in this case, when stirring the air, has to be moved in the space over the counting stage, and not below it. 170 Proceedings of Royal Society of Edinburgh. [sess. The diaphragm, as in the other instrument, is moved by a rod, surrounded by an indiarubber tube. This rod is not shown in the drawing, owing to its position being at right angles to the section ; A is the air-pump, which has a capacity of 15 c.c. ; the piston- rod is graduated in this case to its displacement, as shown ; S is a stopcock, bored as shown ; M is the apparatus for measuring the air to be tested, while F is the filter. It will be noticed, that in this instrument we have adopted the stopcock plan, for measuring small quantities of air, which was de- scribed in the first part of this paper. The method of working has, however, been so far improved that it is not necessary by the new arrangement to change the measuring apparatus when we require to use a larger or smaller quantity of dusty air. As will be seen, the different measures are arranged in series, and we can use any of them without making any alteration in the apparatus. In this portable apparatus, as in the other one, the pump is used when we require to mix large quantities of dusty air with the air in the receiver, and the measure M is used for small quantities. If 1 c.c. of dusty air is required to be sent into the receiver, then, after sufficient time has been allowed for the filtered air to enter the receiver, and when the pressure inside the receiver is the same as outside, but not before, the stopcock is turned quarter a turn from the position shown, by this means the receiver is put in direct con- nection with the outer air. The piston of the air-pump is now drawn down to the figure 1 on the scale ; by this means 1 c.c. of the air is taken into the receiver, where it is mixed with pure air and tested. Two, or perhaps three, cubic centimetres may be measured in this way; but if more is required, then the stopcock S must be closed before the piston is drawn down, otherwise some of the dusty air might be taken out of the receiver by the pump, the piston of the pump should be pushed back to the top before the stopcock is opened to admit the air. When the stopcock is closed while the pump is used as a measure, allowance must be made for the expan- sion of the air, as already explained.* * After a considerable amount of practice with this apparatus, I find it better always to close the stopcock before drawing down the piston, even when only 1 c.c. of air is required. The graduation of the pump ought therefore to be made to show the amount of air taken out of the receiver while the stop- cock is closed. Pro c . Roy. So c . EdirP, Yol . XVI , Pla te LV. CD W P 0 £ < P Eh CD D Q % O Z W £ H < Proc. Roy. Soc.Edin^ Vol. XVI, Plate IV. J. Aitken on Dust Particles. 1888-89.] Mr John Aitken on Dust Particles. 171 For measuring quantities less than 1 c.c., the “battery” of stop- cock M is used. The following are the sizes of the bores in the plugs of the different measures : — Diameter. Length. Capacity. Large plug, . 4 61 mm. 15 mm. 250 c.mm. Medium plug, . 2 53 mm. 10 mm. 50 c.mm. Small plug, . 1 -27 mm. 8 mm. 10 c.mm. That is, their capacities are respectively J, and of a cubic centimetre, so that, in working with these measures, the number counted per c.c. of the air in the receiver requires to be multiplied by 200 for the large measure, by 1000 for the medium, and by 5000 for the smallest measure. When using the portable apparatus, it is supported on a tripod stand. Two plans are shown in Plate IY. for doing this. Pig. 1 shows the design first made, and fig. 2 an alteration afterwards adopted. In fig. 1, T is the “ head'’ of the tripod. The upper end of the pump is provided with a flange and a nut N, by means of which the apparatus is securely attached to the tripod when in use, but can be separated for easy carriage. The objection to this plan is, that the tripod legs require to be at least 4 feet long. Under most conditions this is no disadvantage, but, when much coach or railway travelling has to be done, a bundle of rods that length is a consider- able inconvenience. No doubt, a folding tripod might be adopted, but it was thought better to make the stand like a walking-stick 3 feet long, so that it might be an assistance in walking, and at the same time easily packed for carriage along with umbrellas and sticks. In order to carry out this idea, the plan shown in fig. 2 was designed. A is the air-pump as before ; B is a metal support, to the top of which the air-pump is secured by a movable screw-joint, while its lower end is securely fixed to the tripod “ head ” T. The pump is separated from the support B when packed for travelling. The piston of the pump is moved by means of the collar C, which slides on B. The scale for the pump is placed on B, as shown at lower end, fig. 2. In order to bind the whole apparatus firmly together, one end of the tube D, fig. 2, is screwed into the lower end of the filter F, fig. 1, and the other end is fixed to the tripod. When the length of the tripod legs is no objection, the plan shown in fig. 1 seems to be the steadiest, whilst the other is the most convenient for travelling. 172 Proceedings of Royal Society of Edinburgh. [sess. When the plan shown on fig. 1 is adopted, it may be advisable to lengthen the part E, so as to bring the lower end of the stopcock S in a line with the top of the tripod, and to bind the apparatus together by means of a support attached to S, and fixed to the tripod by the nut and screw N. W7hen working this instrument daylight is used for illuminating the stage, and is found to work well in the open air. A good deal of the glass top of the receiver is left uncovered with blotting- paper to let in as much light as possible ; and a magnifying glass, not shown in drawing, is used for counting the drops on the stage. This glass should have as little brass mounting as possible below the lens, so as not to interfere with the illumination of the stage. When working in pure air, the measuring apparatus M is not neces- sary, and may be removed, and the filter screwed direct into the stopcock S. The whole apparatus weighs with tripod-head only 3 lb. 0 \ oz. and is packed into a tin-lined leather case 8x5x3 inches. The apparatus with case weighs a little over 5J lbs., but this might be reduced by omitting the metal case. It is, however, easily carried by means of a shoulder strap. The legs of the tripod weigh only 15 oz., and when fitted together form a round staff, which makes a good walking stick when provided with top and bottom caps of indiarubber.* The Prolonged Action of Sea-Water on Pure Natural Magnesium Silicates. By Alexander Johnstone, F.G.S. (Read February 4, 1889.) Pure mineral magnesium silicates are amongst the most difficult substances to decompose by naturally occurring agents. Pure water exerts no chemical action on them, neither does water containing carbonic acid gas,f even although the latter body be present to the * After a considerable experience with this apparatus, I have never found it necessary to use the smallest of the measures, even when testing the air of cities. It may, therefore, be omitted. But if retained for special reasons, the centre stopcock should be placed at an angle with the others, so that the handles may be more easily worked, than when they are all crowded in one line. t As far as I have been able to ascertain by experiment, carbonic acid water cannot decompose a 'pure magnesium silicate, such as white talc. 1888-89.] Mr A. Johnstone on Action of Sea- Water. 173 'point of saturation. Fresh spring, river, or lake waters containing alkaline carbonates in solution are also, as my experiments prove, totally unable to decompose pure natural silicate of magnesia. Of course, it must be remembered that the amount of alkaline carbonate in natural waters is very small. In the Loire, near Orleans, Deville was able to find only 1*46 parts of carbonate of soda in 100,000 parts of water ; while in the Garonne, near Toulouse, the same chemist could only detect 065 part of alkaline carbonate in 100,000 of water. Sea-water, however, and also the waters of salt lakes and brinel springs, i.e ., all waters which contain a considerable amount of sodium chloride in solution, act chemically on pure mineral mag- nesium silicates. This fact I have clearly ascertained by the follow- ing experiment. I allowed a litre of a nearly saturated pure aqueous solution of pure sodium chloride to act for two months on a thin fragment (presenting a fairly large surface) of pure and thoroughly clean white steatite (4Mg"Si03.Si02). On the expiry of that period I removed the mineral from its bath, filtered the liquid in which it had lain very carefully, and proceeded to test it chemically, with the object of discovering whether magnesia had been dissolved out of the steatite by the action of the salt solution or not. I acidulated a portion of the clear filtered liquid with nitric acid, and then added excess of ammonium hydrate, and after mixing the whole properly, filtered at once. Into the filtrate I now poured some ammonium phosphate solution, agitated the mixture violently for several minutes, and then put it aside to allow it to settle for about a quarter of an hour. By the end of that time a little, but a very distinct, white crystalline precipitate, indicative of the pre- sence of magnesia, had fallen to the bottom of the vessel. This white crystalline powder was afterwards examined under the microscope, and was found to consist mainly of translucent prisms belonging to the trimetric system. Thus the microscope confirmed the chemical test. The remainder of the clear solution in which the steatite had lain was evaporated to dryness, and the sodium chloride and magnesium salt of the residue were completely dis- solved out by means of dilute hydrochloric acid. After this had been accomplished there still remained a trace of residue which 174 Proceedings of Royal Society of Edinburgh. [sess. could not be removed in solution, and which I subsequently identified as silica. This then I consider is a clearly proved fact, that magnesia and silica are removed in solution from mineral magnesium silicates by the action of water containing sodium chloride. A considerable amount therefore of the magnesia present in the ocean, must, I believe, have been brought into a state of solution by the chemical action of the sea- water on the abundant magnesium silicate minerals of the earth’s crust. I have strengthened my belief in this matter also by experiment. A piece of pure steatite was permitted to remain completely im- mersed in water, brought from the North Sea, for several months. It was weighed before being placed in the liquid, and after being carefully dried, weighed after its removal. It was found to have decreased slightly in weight; its second weight was less than its first, not much it is true, but still decidedly less. ( 1 ) W eight of pure steatite before being placed in sea-water, . . . . .12*421 grammes. (2) Weight of (1) after removal from the sea- water in which it had lain for several months, ...... 12*416 „ Difference, showing amount of magnesia and silica removed in solution by the sea- water, ...... 0*005 „ In an impure magnesium silicate decomposition by sea-water goes on of course more rapidly. It is, I consider, unnecessary to bring forward more evidence to prove that pure magnesium silicates are decomposed by sea-water ; but this question has yet to be answered, How were the magnesium and silica removed ? In what conditions ? What chemical changes took place ? I am daring enough to attempt to answer this question, and although I may be wrong, I am unable to find at present any other possible explanation than the one I now give. It seems to me that the magnesium combined with the chlorine of the common salt, and that the discarded sodium united with a por- tion of the silica freed from the magnesium, the other part of the 175 1888-89.] Mr A. Johnstone on Action of Sea- Water. liberated silica being dropped in the insoluble or almost insoluble silicon dioxide condition. Perhaps the following equation states the changes correctly : — gives Pure Steatite, and Water, and Sodium Chloride. 4Mg"Si0B.Si02 + 8H20 + 8NaCl Sodium Silicate, and Magnesium Chloride, and Silica. 4Na2Si03.8H20 + 4MgCl2 + Si02 Magnesium, as all are aware, exists chiefly in sea-water in the form of chloride. White pure talc is decomposed at about the same rate as pure steatite. The common green variety, however, owing to the presence of ferrous silicate in fair quantity, is more readily altered. One thing must certainly be remembered distinctly, and that is, that the process of decomposition which is undoubtedly promoted in pure magnesium silicates by sodium chloride waters, does not progress at a rate which from a human standpoint can be considered at all rapid. Deductive Evidence of a Uterine Nerve Centre, and of the Location of such in the Medulla Oblongata. By James Oliver, M.D., F.R.S.E. (Read February 18, 1889.) At no time does Nature furnish us with any proof of bodies existing in a state of absolute rest. The whole molecular world, organic as well as inorganic, is, as far as we can ascertain, in con- stant motion. In consequence of this well-established principle, every function of the body may rightly be considered as resulting from a change in the molecular state of the organ manifesting such, and as being the expression of a correlative variation occurring in its representative nerve centre. Through the agency of long con- tinuance the visceral disturbances are now carried on in a somewhat automatic manner, and fail to excite any feeling of their existence, although they may at one time, in the evolution of higher organisa- tion, have produced a conscious sensation. In no organ do we find these revelations so well depicted as in the uterus. In every 176 Proceedings of Royal Society of Edinburgh. [sess. typically healthy woman this organ, so long as it is free from the influences of gestation and lactation, is periodically for a greater or less length of time the seat of a regularly recurring functional variation in its molecular state, evidenced by the emission of a more or less marked haemorrhagic discharge, and which to all intents is its sole manifestation. The disturbance is evolved quite inde- pendently of the will, and apart altogether from any definable excitation. It appears to be induced spontaneously through the agency of an automatic nerve centre, and fails in consequence to produce any conscious sensation. When, however, the uterus be- comes the habitat of a developing ovum, or prior to this occurrence, and whilst segmentation is as yet progressing in the Fallopian tube, the waves of motion radiated by and from the germinal mass affect in a very decided manner the molecular state of the uterus, and determine a cessation of its routine function, and consequently of its regularly recurring manifestation of activity. Impregnation having resulted, other well-defined symptoms, in addition to that of the cessation of menstruation, are engendered, and help not only to guide the woman in arriving at a definite conclusion regarding her state, but aid us very materially in approximately estimating the duration of pregnancy. The symptoms associated with pregnancy to which I wish more especially to draw attention, as evidence of the existence of a genera- tive centre, and of its location in the medulla oblongata, are two — sickness and cough. It has been alleged, through the agency of experiments, that the sexual centre is located in the lumbar region of the cord. This opinion, however, appears to me to be founded on no very substantial basis. The mere fact that all, or nearly all, the sexual phenomena may be witnessed in animals after the lumbar portion of the cord has as far as possible been isolated by section is no very special criterion. The respiratory centre is located in the medulla, yet under special circumstances all the movements asso- ciated with respiration may be carried on after the medulla has been entirely removed. From the earliest period of existence every organism has been endowed with two distinct qualifications — 1st, that of maintaining self ; 2nd, that of perpetuating the species. At first, in the most primitive state, the double function was performed by a uniform 1888-89.] Dr James Oliver on a Uterine Nerve Centre. 177 mass, free from any semblance of structural differentiation. Habitual localisation of function, however, produces eventually a specialisation of structure, and with it the evolution of a nerve tract, whereby inter- dependence is maintained. It is therefore feasible to suppose that the nerve centre, which regulates the process of assimilation (the pneumogastric nerve centre), is either in close apposition, or at least in more or less direct communication, with that centre which presides over the organs of generation. All the visceral functions are now performed automatically, and appear to be regulated by nerve centres located in the medulla oblongata. It is feasible, therefore, to sur- mise that the uterine functions are governed by an automatic centre — a centre which, because of some innate quality, is thrown into a state of trepidation, and produces thereby evidence of associated disturbance in the uterus itself. When the uterus becomes the nidus for a developing germinal mass, the molecular state of the organ is altered, and certain new impulses are generated. The waves of motion, resulting from segmentation of the ovum and further evolution of the chick, are radiated through the agency of the uterus and its afferent nerve fibres to the uterine or re- productive centre. These disturbances occurring in the uterus are in excess of those commonly generated, so too the disturbances corre- latively produced in the uterine centre are greatly in excess of those usually developed. The extra amount of motion must discharge itself in some other direction until time has accustomed the uterine centre to the augmentation. The direction the overflow of energy shall take, is determined according to existing nerve communications, and that centre is likely to be first affected, by such radiations, which is in closest proximity. Considering, therefore, the intimate relation- ship that exists throughout life between the process of assimilation and the process of generation, it is not astonishing that the excess of molecular motion transmitted to the uterine centre should be radiated to and expend itself upon that governing the nutritive processes generally. In consequence of this there occurs early in pregnancy, and for a greater or less length of time, sickness, and in some women cough. At present we know but little of the nervous mechanism of vomiting. The respiratory centre seems, however, to participate in the act. Both phenomena, however (sickness and cough), in the pregnant woman may with some amount of assurance VOL. xvi. 7/6/89 m 178 Proceedings of Royal Society of Edinburgh. [sess. be referred to molecular radiations from the uterine to the pneumo- gastric centre. Usually, however, in the course of a few months, through the agency of habit, the pneumogastric centre becomes tolerant, and the symptoms evidencing disturbance in this centre coetaneously disappear. It is difficult to understand why the sick- ness should be experienced, more especially, although not solely, in the morning. It is quite possible that the change from the recumbent to the erect position may after sleep render the whole nervous system more liable to explosive disturbances. The state of the stomach, too, may also aid in determining this somewhat anomalous phenomenon. Frequently we find epileptic patients who suffer only from their disturbances on assuming the erect position after sleep. The ano- malous phenomenon, therefore, as it occurs in the pregnant woman is not without a parallel. The more highly unstable the nervous system is generally, the more likely is a woman when she becomes pregnant to suffer markedly, and for a lengthened time, from sickness, whether matutinal simply or more or less constant. It is well, however, to remember that the inherent tone of the uterus itself will affect materially the molecular radiations engendered by the developing germinal mass, and transmitted through the agency of the uterine tissue to the nerve centre. The resulting disturbances will be correlatively augmented or diminished according as the tone of the uterus is high or low. On the so-called “ Liver ” of Carcinus mcenas. By Dr A. B. Griffiths, F.RS. (Edin.), F.C.S. (Lond. and Paris), Mem- ber of the Physico-Chemical Society of St Petersburg, &c. (Read February 4, 1889.) “ A true knowledge of biology must be based on a knowledge of chemistry and physics.” — M. M. P. Muir. This memoir details a continuation of the author’s investigations on the physiology of the Invertebrata. At this point we consider the physiological functions of the so-called “ liver ” of the Brachyura. Was it not M. Letourneau, in his La Biologie , who said, “ Does the pancreas exist in the invertebrates % This is a question of comparative physiology which still waits for a reply. We have seen that we do not begin clearly to recognise the pancreas except 1888-89.] A. B. Griffiths on “ Liver” of Carcinus moenas. 179 in fishes, and then only in a rudimentary state.” From the recent researches of Krukenberg, Fredericq, Jonsset de Bellesme, Plateau, Hoppe-Seyler, as well as those of the author, the problem now re- quiring solution is the following Does a true liver exist in the Invertebrata ? The pancreas appears to he the chief digestive organ (other than a true stomach) of the earlier forms of animal life. The Liver of Carcinus moenas. The “ liver ” of Car emus moenas consists of two large glands on each side of the stomach, and extends the whole length of the cephalo-thorax. These organs are of a yellow colour, and consist of numerous coecal tubes arranged in tufts, which are easily seen in a dissection beneath the surface of water. The secretion of the so- called “liver” of Carcinus moenas , when freshly killed, gives an acid reaction. 1. The secretion of the organs forms an emulsion with stearin, yielding subsequently fatty acids and glycerol : — C67H110O6 + 3H20 = 3C18H3602 + C3H803. 2. The secretion acts upon starch paste. The starch granules disappear, with the exception of their celluloid covering; and on treating with water, and then adding Fehling’s solution, a deposit of cuprous oxide was obtained. This reaction shows that there exists in the secretion a substance capable of converting starch into glucose. 3. The secretion dissolves coagulated albumin. 4. Tannic acid gives a white precipitate with the secretion. 5. The action of the secretion upon milk was to render it trans- parent. 6. When a few drops of the secretion of these organs were examined with chemical reagents under the microscope, the follow- ing reactions were observed : — On running in a solution of iodine in potassium iodide between the slide and cover-slip, a brown deposit was obtained; and, on running in concentrated nitric acid upon another slide containing the secretion, a yellowish coloration was produced, due to the for- mation of xanthoproteic acid. These reactions show the presence of albumin in the secretion of the organs in question. 180 Proceedings of Royal Society of Edinburgh. [sess. The presence of albumins in the secretion was also confirmed by the reactions recommended by Dr R. Palm (Zeitschrift fur Analy- tisclie Chemie , vol. xxvi. part i.). 7. The secretion contains leucin and tyrosin, no doubt produced by the metamorphoses of certain albuminous substances. We know from Professor Poster’s Physiology (4th ed. p. 438), that “one result of the action of the pancreatic juice is the forma- tion of considerable quantities of leucin and tyrosin.” Leucin and tyrosin are “ dehydrated in a true liver , forming a series of cyan- hydrins or cyan-alcohols attached to a benzene nucleus, which then pass into the circulation ” — (Latham). 8. The principal mineral ingredient found in the ashes (inciner- ated at a low temperature) of the so-called “liver” of Carcinus was sodium carbonate. In the ash of a vertebrate liver the chief mineral constituents are potassium and phosphoric acid. 9. The soluble zymase (ferment) secreted by the organ in ques- tion was extracted according to the Wittich-Kistiakowsky method (Pfliiger’s Arcliio fur Physiologie , vol. ix. pp. 438-459). The isolated ferment converts fibrin into leucin (a-amidocaproic acid, C6H13N02) and tyrosin (paraoxyphenylamidopropionic acid, C9HuNOs). 10. No glycocholic and taurocholic acids could be detected by the Pettenkofer and other tests. 11. No glycogen was found in the organ or its secretion. 12. The secretion has no action upon cellulose. 13. By using the methods adopted by M. Zaleski (Zeitschrift fur Physiologische Chemie , vol. x. pp. 453-502) for ascertaining the presence of ferrous, ferric, and ferrosoferric compounds in a true liver, the author could not detect the presence of iron in the organ or its secretion. From these reactions the conclusion to be drawn is, that the so- called “ liver” of Carcinus moenas is pancreatic in function, i.e.f its secretion is more like the secretion of the pancreas of the Vertebrata than those of a true liver. Some biologists look upon the vertebrate liver, pancreas, and salivary glands as differentiated bodies of an original pancreas of the Invertebrata. But have not many forms of the lower animals 1888-89.] A. B. Griffiths on “Liver” of Carcinus mcenus. 181 similar salivary glands to those found in the Yertebrata ?* And is not the so-called “ liver ” of the Invertehrata a true pancreas, capable of producing the same chemical and physiological reactions as the pancreas of higher forms ? On the Air’s Resistance to an Oscillating Body (its Influence on Time-Keepers). By Edward Sang, LL.D. (Bead April 15. 1889.) The influence of the air on the going of time-keepers has naturally been the subject of much discussion, particularly in reference to time-keepers used by astronomers. On pendulums the air acts in two ways : — by its buoyancy it lessens the downward tendency of the parts, and so lengthens the time of the oscillation ; and it opposes resistance to the motion. On the chronometer balance, the latter action alone is felt ; to this action we shall confine our remarks. The incitement to motion is thus composed of two parts : the one due to the inclination of the curve or to the flexure of the balance spring, proportional to the distance from the point of rest; the other proportional to the square of the velocity ; so that, if x repre- sent the distance from the point of rest and v the velocity, the civ differential coefficient of that velocity, — or ltv} is represented by an expression of the general form ltv = ~Px + Qv2 , or, in Leibnitz’s notation, The resolution of this equation, which expresses a relation among the function, its first and its second derivative, belongs to what I have called the Third co-ordinate branch of the Higher Calculus, or the Calculus of Primaries. Here we seek the relation of the primary variable t , to some one Of the three functions. The only useful application of this inquiry is to the doctrine of * See the author’s papers in the Proc. Boy. Soc. JEdin., and the Proc. Boy. Soc. Bond., ] 885-88. 182 Proceedings of Royal Society of Edinburgh. [sess. time-keeping, and we may, therefore, first in a general way, consider what may be the relative values of the coefficients P and Q, and also the signs which these may take. In my mean-time clock, the bob is a cylinder of lead cased in brass, having its diameter 3 inches, its length 5, its weight, by estimate, being 101 200 grains; and therefore for each inch of deflection at the mean distance the redressing tendency is 2586 grains. The air’s resistance on a square inch of surface in grains for a velocity v measured in inches per second, varies, as deduced from £7 2 /^2 several formulae in use, from — — to — ; now the bob offers a ’ 1200 1600 v 2 surface of 15 square inches, and thus we may assume in grains as the air’s resistance in our case. Now the oscillation extends to 1*3 on each side, wherefore the greatest incitement amounts to PX = 3362 grains; the maximum velocity is V = 1*3 x tt = 4*08 inches per second, and causes a maximum retardation of ’166, or the sixth part of a grain at the lowest point ; and thus it appears that the maximum resistance by the air is only the twenty-thousandth part of the greatest redressing tendency. Because of the complexity of the parts, it is not so easy to form an estimate in the case of the chronometer balance. In my two-day chronometer the compensating weights have a diameter of *3, or just the tenth part of the pendulum bob ; and they oscillate to a distance, measured along the arc of 1*2 or 1*3, almost the same as that of the pendulum. Hence, if the times of oscillation had been alike, the ratio of disparity would have been 2000 instead of 20000. But the balance makes two complete oscillations per second, while the pendulum makes only one-half. Now, in order to quadruple the number of oscillations per second, we must augment P sixteen times, Q being unchanged ; however, since the velocity is now quadrupled, the product QV2 is made sixteen times greater, and thus the ratio of PX to QV2 remains 2000 to 1. The rim of the balance meets with direct resistance only on its ends ; its motion is impeded by what may be called friction, on the sides, and in all likelihood the ratio of disparity is higher than for the weights; 1888-89.] Dr Sang on Airs Resistance to Oscillating Bodies. 183 we may, then, regard the ratio of 2000 to 1 as not too high. Thus, in all cases of any importance to us, we may regard Q as a very small fraction whose higher powers need not be taken into account ; it may be more than enough for us to retain the second power. We have now to examine the signs of P and Q ; let us consider first that of P. When the oscillating body is moving from A on the side to which we have agreed to attach the sign + x , towards the middle point 0, + x is decreasing and its derivative v must take the sign - , at the same time that velocity is increasing or becoming more - , where- fore ltv is - , and so therefore must be P. When the body has passed O, x is increasing in the - direction, v therefore is still — ; it is, however, now decreasing, its derivative is therefore + , so that Vx must have the sign + , but then x is — , wherefore P must be - in this quarter of the oscillation. When the body has reached the extreme limit B and is returning along BO, its - x is decreasing more and more rapidly; that is to say, both its velocity and the increment of its velocity take the sign + , Vx then is + , but x is - , wherefore P must be - in this third quarter also. Lastly, when the body has passed 0 into the quarter OA, its + velocity is decreasing, and both ~Px and P take the sign - . Thus it seems that, in all the four quarters of the oscillation, P must have the sign — ; we shall then write it as - a2. The air’s resistance, denoted by Q u2, tends to lessen the velocity, to hinder acceleration, to help retardation irrespective of signs. When the body is moving from A towards O, the - velocity is being increased in the — direction by - a?x, and this increase is opposed by Qv2, wherefore Q here takes the sign +, and the total influence is - a2x + Qv2. In the second quarter, that is from 0 to B, the - velocity is being lessened, and the air’s resistance helps this, where- fore Qv2 takes here also the sign + . Again, in the third quarter, from B towards 0, the + velocity is being augmented, and the resistance opposing this augmentation must take the sign - . Also, in the fourth quarter, the + velocity is being retarded, the retarda- 184 Proceedings of Eoyal Society of Edinburgh. [sess. tion is helped, and Qw2 takes the sign - . Thus we see that Q must have the sign + during the half oscillation from A to B, and the opposite sign — during the other half from B to A. We shall there- fore replace Q of the general formula by + fi, observing that the conclusions thence drawn only hold good from the limit v — 0 until the next recurrence of the same condition v = 0. This is one of the many instances in which generalisation by the accepted law of signs is inadmissible. Setting out from the equation uv= -a2x + fiv2} and taking the differential coefficient in regard to the primary variable t , we get 2tv = - a2v + 2fivltv , and substituting for uv its value as above, 2tV = - a2V - 2a2fixv 4- 2 fi2v3 . Repeating this operation, but rejecting all terms containing fi3 and the higher powers, we find o(V = + a4x - 3a 2fiv2 4- 2 a4 fix2 - 8a 2fi2xv2 4tV = + a4V + 1 0a 4fixv - 1 ia2fi2V3 + 1 6a4fi2X2V 5tV = - a?X + 1 1 a4fiv2 - 10a6 fix2 + 84a 4fi2XV2 - 16a 4fi2X* , and so on. If now we place the zero of time at the instant when the oscil- lating body is at the limit A where v = 0, and if we write X for the extreme value OA of x , the theorem of Taylor gives us v = 0-a2x^x (a4x + 2a4fix2)yt-~- - (< vfix + 1 0a6fix2)—i + &C. , 1 . • • 0 and we see that only derivatives of odd orders enter into the result. We shall therefore save ourselves much labour in writing by tracing the order of formation of these odd derivatives alone. Let us then assume {2n_ 1)tV = + a2nX - A a 2n~2fiv2 + Ba27 3x2 - Ca 2n~2fi2xv2 + Da2nfi2xs , and operate twice thereon ; the result is 1888-89.] Dr Sang on Air's Resistance to Oscillating Bodies. 185 (2n+i)tv = — o.2n+2x + (4B + 3)a2n/3v 2 — (4B + 2)a2n+2ftx2 + (10B + 7C + 6D + 8)a 2n/32xv2 - (2C + D)a2n+2p2xB ; and by help of this scheme we form the following table of the numerical coefficients : — X fiv2 jto2 fi2XV2 jS2^ In- 1 + 1 — A + B — C + D 2?i + l -1 + (4B + 3) - (4B + 2) + (10B + 7C + 6D + 8) - (2C + 3D) 1 -1 + 1 0 0 0 3 + 1 - 3 + 2 - 8 0 5 -1 + 11 - 10 + 84 - 16 7 + 1 - 43 + 42 - 792 + 216 9 -1 + 171 - 170 + 7268 - 2232 11 + 1 - 683 + 682 - 65976 + 21232 13 -1 + 2731 - 2730 696052 - 195648 &c. &c. &c. &c. &c. &c. Whence v ) at aHB aHb aJtf ) = "'tX I r-iX3+i^-T^+&c-/ + apx* | -16^ + 216~-2232^ + &c. } . The series by which -aX is multiplied is that for sinctf. The series of coefficients B may be written 2 = 2{l}j 10 = 2{l + 4}; 42 = 2(1 + 4 + 16}, and so on; or in the more concise form, 2 = | (22 - 1) ; 10 = ^(24-1); 42 = J (26 - 1), and so on ; wherefore the term involving /3 may be written 1 OV2 / 2W ^ 3a/3X \ 1727 3 1...5 +&C- -217^3 + 2 a5 tP ...0 &C. I or supplying the defective terms, by adding zero in the form 186 Proceedings of Boy al Society of Edinburgh. [sess. 2ai t ~at 1 + 2ri -g-a/3X2 1 2 at (2 a<)3 T + 1.2.3 + 2 T 1.2.3 + 2 (2aQ5 1...5 (aQ5 1...5 + &c. - &c. or ^ a^X2{2sin(aO - sm(2a t)} . The coefficients D succeed each other in a very complex manner ; I have not succeeded in separating them into series of powers, the nearest approach giving the formula 2 15 {5 sin (at) - 4 sin (2 at) + sin (3a£)} and thus we must be content to represent v by the formula v= -aX. sina/+ i a/3X2{2sina£-sin2a£} o + ay32X3 | 16^ + 216 ^--2232^ + &c ...5 ...7 ...y and, for the reasons already given, we may neglect this term con- taining y82 . Now when at — 7r, its sine is zero, but so also is the sine of 2a t = 27r, wherefore v is zero when t = — ; in other words, the air’s resistance does not influence the time of oscillation. The value of x may be found in the same way, observing that the first derivative of v is the second derivative of x, and so on ; or it may be found by integration from the value of v. Either process x = X cos at + which when a£ = 180° gives 6 COS 2 at - — cosa t | X' = -X + ^-/3X2, that is to say, the length of the oscillation is shortened by-g-/3X2 . Thus we see that, while the air’s resistance lessens the extent of the oscillation, it also lessens the velocity, and in such a way that the body is brought to rest in the same time as if there had been 1888-89.] Dr Sang on Air’s Resistance to Oscillating Bodies. 187 no resistance. The diminution of the space compensates for that of the distance ; hut this compensation is not equally distributed. Thus, in the quarter of the time of an oscillation, the position is got by making at = 90°, which gives X" = +1/JX2, so that the distance passed over has been X- IjSX*. whereas the distance described in the second quarter of the oscilla- tion is only X - ^-/?X2 In order to get the actual effect on the clock above mentioned, we observe that, since the half oscillation is performed in one second, we must have a = T. Xow the extreme distance is 1*3, wherefore the greatest incitement to motion is l‘3x7r2. But the maximum velocity is 1*3 x t, and the maximum air’s resistance /3 x 1*32 x 7r2, so that 1 * 3 x 7J-2 = 20000 x 1 • 32 x , whence £ = — i_. In 26000 this way we find the shortcoming at the end of a half oscillation to 5 1 he — /3 x 1 *32 = Par^ an inch> a quantity very small even in comparison with that due to the imperfect resistance of the suspending spring. In the case of the chronometer, the half oscillation is performed in the quarter of a second, wherefore a = At, and the effect is, there- fore, sixteen times as great as in the case of the clock, on this account alone ; but the ratio of disparity is only 2000 ; wherefore we must augment the above computed shortcoming 160 times, giving the 115th part of an inch. These two examples may serve to give a general idea of the magnitude of the disturbance due to the air’s resistance. In neither case is the time of the oscillation changed; in clock-work the ex- tent of the arc is slightly lessened, so slightly that the diminution is scarcely worthy of attention. In watch-work it is so considerable as to require perceptible greater maintaining force. 188 Proceedings of Royal Society of Edinburgh . [sess. A Contribution to the Chromatology of the Bile. By John Berry Haycraft, M.D., D.Sc., and Harold Scofield, M.B. (Read March 4, 1889.) One of the chief biliary pigments is bilirubin. It has a red-orange colour, and is derived from the decomposition of hsemoglobin. It can be oxidised first into a green, then into a blue, then into a red, and finally into a yellow-brown pigment. Between the blue and red a violet substance is produced, but it is uncertain whether or not this is only a mixture of the blue and red pigment. The forma- tion of the green pigment, according to Stadeler,* is due to oxidation, together with the addition of a molecule of water. These pigments do not always present exactly the same characters. Thus, according to Dr MacMunn, the green pigment present in the ox-bile differs from what is artificially produced from the oxidation of bilirubin, say, from human bile, in that it is soluble in chloroform. f This pigment can, however, he oxidised up into the blue pigment, and so on ; and it belongs, therefore, to what we may term the bilirubin series. Its colour indicates its position in the scale of oxidised products. We shall in this paper use the term biliverdin as desig- nating a green pigment, which is more oxidised than the bilirubin, and bilicyanin as designating a blue pigment still more oxidised. The violet substance (if such exist) we shall term the violet pig- ment, the red oxidation product the red pigment, and, finally, the yellowT-brown pigment, the most highly oxidised product of all, choletelin. Although, by the action of an oxidising agent, such as impure nitric acid, it is easy to pass from a lower to a higher member of the bilirubin series, it is frequently, and we believe truly stated, that no successful attempt has been made to reduce the higher back again to the lower ones. By the reducing action of, say, sodium amalgam, hydrobilirubin (C32H40M4O7) has been produced both from bilirubin and from biliverdin. When, however, ox-bile is placed in a tall vessel, and allowed to remain for some hours, we have observed that a change in colour takes place. If blue pigment is present * Gorup-Besanez, Physiol . Chemie , p. 207. t Jour, of Phys., vol. vi. p. 2. 189 1888-89.] Dr Haycraft on Chromatology of the Bile. it changes to green, and this finally to orange-brown. If some of the brown bile be then treated with a few drops of nitric acid, it is oxidised back into biliverdin, and further additions of acid develop the blue, violet, red, and yellow pigments. Here, then, without going any further, we have an instance of the reduction of the pig- ment. The reduction does not produce hydrobilirubin alone, as that substance cannot be reoxidised in the way we have described. If a bladder, fresh from the slaughter-house, be opened, one seldom or never fails to see signs of reduction within it. Blue or blue-green bile fills the cavity, but thick orange-brown bile is seen next the mucous membrane, which is itself brown in colour. It is not improbable, therefore, that during life, reduction of the biliverdin takes place, due perhaps to the action of the mucous membrane of the gall-bladder or to the mucus secreted by it. This is, indeed, almost certainly the case, and one can recall the fact that the pigment present in gall-stones from the ox consists, not of biliverdin, but of bilirubin. It was obviously a matter of some importance to investigate the reduction processes we have just described, in order, if possible, to ascertain their cause, and the influence of modifying conditions upon them. We have been greatly assisted in this inquiry by Dr MacMunn. We sent to him on several occasions solutions in which we had difficulty in determining whether definite absorption-bands were present, and we have very warmly to thank him for the courtesy with which he was ever willing to help us. Experiment I. — Temperature of the laboratory, 60°. Two test- tubes were filled with fresh green ox-bile, and watched from day to day. After three days the bile in the lower part of one of the tubes had become of a brown-green colour. The change in colour spread upwards, and in twelve hours the whole tube contained bile of a uniform brown tint. Putrefaction, indicated by an unpleasant odour, had set in by this time. In the other tube the bile did not change in colour, nor was there any sign of putrefaction until the following day, when both appeared simultaneously. Later on the bile became by degrees of a light amber colour, giving, how- ever, the play of colours for nearly four months, after which Gmelin’s test failed. The fluid was then examined with the spectroscope. There was slight shading at the violet end of the spectrum, and the 190 Proceedings of Royal Society of Edinburgh. [sess. bands of cholohsematin * were present. There was no distinct hydrobilirubin band. It is seen, therefore, that on exposure to the atmosphere the biliverdin is reduced ; the bilirubin disappears with- out forming hydrobilirubin. Experiment II. — Test-tubes were filled with ox-bile, and pieces of the mucous membrane of the gall-bladder were added. The changes observed were similar to those detailed in Experiment I. The mucous membrane seemed to hasten the reduction of the bile in its neighbourhood. The reduction of the pigment seemed remarkably to coincide in point of time with the establishment of putrefaction. The following experiments were conducted in order to eliminate putrefactive changes from the other conditions present. Experiment III. — Ox-bile was boiled in test-tubes which had previously been plugged with wool. No alteration in colour was produced by boiling, nor did any take place for seven days. It then became of a light brown colour, rapidly fading, until after a fortnight it was nearly colourless. Gmelin’s test failed after three weeks, and two weeks afterwards the fluid was examined with the spectroscope. Cholohsematin bands were absent; there wTas pos- sibly a trace of hydrobilirubin. This experiment indicates that reduction of the biliverdin takes place, and the bilirubin disappears in bile which has been prevented from putrefying. The reduction of the biliverdin seems to be hastened by putrefaction. Experiment IV. — Ox-bile was evaporated to dryness at 50°. It very slowly changed its colour, and only failed to give Gmelin’s test after several months’ exposure. Experiment V. — Some exhausted and sterilised glass tubes were drawn out into capillary points. The points were thrust through the wall of a fresh gall-bladder, and broken off by the fingers 'which grasped them from without the bladder. They instantly filled with bile, and after their withdrawal from the bladder their ends were at once sealed up in the blow-flame. No change of colour occurred for fourteen days ; the weather was cold and dull. By that time, however, they had become brown in colour. No further change occurred, and after a year they gave a distinct play of colours with nitric acid. * Cholohsematin is a pigment which gives absorption bands. It is present and often partially replaces biliverdin, in ox-bile. — MacMunn, loc. tit. 1888-89.] Dr Haycraft on Chromatology of the Bile. 191 Changes, not very dissimilar in their nature, occur when blood is received directly into a sterilised tube from an artery. In this case the oxyhsemoglobin is reduced, and remains for years without under- going further change. In the sterilised tubes filled with bile the change in colour seemed to be influenced in the thick mucus of the bladder, for those that contained most mucus invariably became brown before those that contained less mucus. It will be seen, however, by the results of the next two experi- ments, that reduction occurs quite readily in the absence of bladder mucus. The two experiments were unfortunately performed, the first in dull cold weather, and the second in warm bright weather, so that they cannot be very rigidly compared. One can conclude, however, from the general results obtained, that thick bladder mucus assists the reduction, although its presence is not essential. Experiment VI. — Exhausted and sterilised tubes were filled with bile drawn directly from the hepatic duct of a freshly killed ox. Reduction took place in three days. Experiment VII. — In order to ascertain whether reduction would take place in bile absolutely free from mucus, this substance was precipitated by means of alcohol. The filtrate subsequent to the separation of the mucus was evaporated over a water-bath until the alcohol was expelled. The reduction took place, however, in two days ; the weather was hot and bright. The last experiments were made during hot midsummer weather, and it was noticed that, while in all cases reduction took place with rapidity, this was especially the case in the tubes most exposed to light. In the following experiments the action of light was more fully investigated. Experiment VIII. — (a) Bile within test-tubes exposed to light was reduced in twenty-four hours, and after a week failed to give Gmelin’s test ; weather hot and bright. Other portions of the same bile in a dark metal chamber were reduced in three days. ( b ) Bile boiled in plugged tubes and exposed to light was reduced in twenty-eight hours. A portion treated in a similar way, but placed within the dark chamber, still preserved a trace of the original green colour for seventeen days. (c) Bile within the exhausted and sterilised tubes was reduced 192 Proceedings of Royal Society of Edinburgh . [sess- by the light in twenty hours. In the dark chamber they began slowly to undergo change at the bottom of the tube after twenty-four hours. (d) A film of bile about 1 millimetre thick was dried at a low tem- perature. On the fourth day it had completely changed its colour to brown ; while, on the other hand, a film, similarly prepared, but kept in the dark, remained blue-green, the original colour of the bile, for a year and a-half — the time when it was finally examined. It would seem, therefore, that biliverdin readily parts with its oxygen like oxyhemoglobin. It is reduced in the sterilised tubes to bilirubin, but no further. This reduction is hastened by light, and putrefaction, and the presence of thick mucus. It is only pre- vented by drying the bile and keeping it in a dark chamber. When bile putrefies, or when, without putrefaction, the bile has been altered by boiling it, the bilirubin finally disappears, and no play of colour is obtained by the application of Gmelin’s test. This residue contained a brown pigment giving no absorption bands, and differing from hydrobilirubiu in its solubilities. It was insoluble in ether, but readily soluble in alcohol. Appendix. On the Reduction and Oxygenation of Pigments in the Bilirubin Series. Bilirubin, when exposed to the air, under ordinary circumstances, never becomes converted into biliverdin. This takes place, how- ever, if the solution has previously been rendered strongly alkaline by the addition of caustic soda solution. Both nascent oxygen and ozone, we find, are capable of effecting this change in bile of normal reaction. To do this experiment, pieces of blotting-paper soaked in bile should be exposed to the vapour of ozone ; ozonic ether, liberat- ing nascent oxygen, can also oxidise these papers. If human bile is poured into a small beaker, and a stream of electricity passed through it, obtained from some five or six Grove cells — the ter- minals should be of platinum — the oxygen given off from the positive terminal, in the course of three or four minutes, changes the bile in its vacinity, first to a green, and finally to a blue-green 1888-89.] Dr Haycraft on Chromatology of the Bile. 193 colour. The experiment may be performed in, perhaps, a more satis- factory manner by dipping a piece of blotting-paper in the bile, and laying it directly on the terminals. The brown colour changes near the positive terminal, first to green, then to blue, and finally to violet. At this stage, however, much of the pigment becomes bleached, so that the violet is not so distinct as either the blue or the green. The lower oxidation products of bilirubin can be reduced artifi- cially. If some bile be acidulated with impure nitric acid, so as to oxidise its bilirubin to biliverdin or bilicyanin, and if pieces of blotting-paper be then dipped in these, the vapour of ammonium sulphide can reduce them. The blue colour is reduced through green to brown, and it can then be oxidised by nitric acid. If blot- ting-paper is soaked in bile and reoxidised by the positive pole of the battery, the pigment is again reduced on reversing the poles. The brown can be oxidised to green and then to blue, and afterwards reduced to brown, passing through green. If oxidised to red, re- ducing reagents then change the colour to a yellow-brown, but not through any intermediate stages. Oxidising agents restore the original red colour. It is much easier to perform these experiments with bile than with pure bilirubin. Bilirubin dissolved in caustic soda is difficult either to oxidise or to reduce. Pure bilirubin, powdered on blotting- paper, moistened with normal saline solution, can be oxidised and reduced without difficulty by the current given by three or four Groves. On the Identity of Hofmann’s “ Dibenzyl-Phosphine ” with Oxide of Tribenzyl-Phosphine, and on some other Points connected with the Phosphorised De- rivations of Benzyl. By Professor Letts and R, F. Blake, Queen’s College, Belfast. (Read May 20, 1889.) In his well-known researches on the phosphines, Hofmann has shown (or believes that he has shown) that when an alkyl iodide (or other haloid derivative) is heated with phosphonium iodide and oxide of zinc, primary and secondary phosphines alone result; whereas, when an alcohol is heated with iodide of phosphonium, tertiary and quaternary phosphines are formed exclusively. Thus VOL. xvi. 4/7/89 N 194 Proceedings of Royal Society of Edinburgh. [sess. the two reactions are complementary to each other. Among the series to which he extended his investigations was that of benzyl, and in a paper published in the Berichte of the Berlin Chemical Society* he describes the preparation of mono- and dibenzyl-phosphine, and gives their properties. Dibenzyl-phosphine he isolated as a crystal- line substance perfectly tasteless and odourless, insoluble in ether, but soluble in alcohol. Its melting-point he found to be 205° C. In a paper read before this Society (19th December 1887), one of us, in conjunction with Mr W. Wheeler, describes further investiga- tions on this body, and shows that it forms a series of compounds of a somewhat remarkable nature for a secondary phosphine. This fact, and some other properties of the substance, led to the suspicion that it was not dibenzyl-phosphine at all, but the oxide of tribenzyl- phosphine. Accordingly the investigation was re-opened with Mr R. F. Blake, and its course has been as follows : — 1. On carefully re-crystallising the substance from alcohol, its corrected melting-point was found to be 216-216*5° C., while that of two specimens of oxide of tribenzyl-phosphine (prepared by two different methods) was found to be the same. 2. Very little difference exists in the percentage amount of carbon and hydrogen in dibenzyl-phosphine and oxide of tribenzyl-phos- phine, as the following numbers show : — (C7H7)2HP (c7h7)3po Carbon =78*50 78*75 Hydrogen = 7*01 6*56 Consequently it would not be possible to decide with absolute precision between the two substances by a mere combustion. On. the other hand, there is a considerable difference between the two bodies in their percentage of phosphorus : — (C7H7)2HP (C7H7)3PO Phosphorus =14*48 9*69 Unfortunately, however, as we have again and again found, the pro- cesses for phosphorus determinations in ordinary organic substances are absolutely untrustworthy when applied to phosphines. A new method was therefore necessary, and after many trials we believe we have found one which is perfectly accurate, trustworthy, and capable * Berichte (1872), v. 100. 1888-89.] Prof. Letts and R. Blake on Dibenzyl-PAosphine. 195 of general application. It is extremely simple, though somewhat tedious in carrying out.* It consists in making an ordinary com- bustion of the substance with pure oxide of copper, and afterwards dissolving the contents of the combustion tube in nitric acid, and determining the phosphorus with molybdate of ammonia, &e. Applying this method to the analysis of the supposed dibenzyl- phosphine, we obtained the following results (IV. and V.). We give at the same time the determinations of phosphorus made both by Hofmann (I.) (by a method not described), and by one of us and W. Wheeler (II. and HI.), in the same substance (by burning with lime in a stream of oxygen) : — I. II. III. IV. V. Phosphorus:— 13*6 14*35 15*00 9*86 9*98 I. Hofmann. II. and III. Letts and Wheeler. IV. and V. Letts and Blake. 3. The following compounds of Hofmann’s body were prepared and analysed f : — Bromide. — Obtained by adding bromine to a solution of the body in glacial acetic acid. It crystallises usually in yellow needles. It is unstable, and loses bromine when boiled with water or glacial acetic acid, and possibly on drying also. Prepared from Hofmann’s Dibenzyl-Phosphine. Obtained : “ {rinat iontl2” Carbon, Hydrogen, 56*4, 56*5, 56*5 5*31, 5*57, 5*04 Calculated: {(C7H7)2HP}2Br2 Bromine, 27*2 Carbon, 57*1 Hydrogen, 5*1 Oxide of Tribenzyl-Phosphine. 28*4 56*9 4*9 7(C7H7)3P0.5Br2 26*31 58*00 4*83 5(C7H7)3P0.4Br2 28*5 56*3 4*7 * Details of this method will be given in another paper. + It will be seen that a formula can be devised in every case, both for a com- pound of (C7H7)2HP and (C7H7)3P0, which corresponds with the results obtained; and it is remarkable how closely most of the results obtained agree with those required for a compound of the former. We also give in some cases the analyses of compounds prepared in a similar manner with what was known to be the oxide of tribenzyl-phosphine. Most of the compounds are unstable, and their composition often varies according to the method or conditions employed in their preparation. 196 Proceedings of Boyal Society of Edinburgh. [sess. Platinum Salt. — Prepared by mixing alcoholic solutions of chloride of platinum and of the substance. The compound crystal- lises out in minute leaflets. The composition of the substance varies with the conditions under which it is prepared. Obtained from Hofmann’s Dibenzyl-Phosphine. I. II. III. Carbon, 59 *5 56*6 58*5 Hydrogen, 5 ‘8 5 ’3 5*9 Chlorine, Platinum, 12'8 13*1 13*1 I., II., III., IV., and Y. were all separate preparations, obtained under slightly different conditions. Calculated for 5(C7H7)2HP. PtCl4 4(C7H7)3P0. 2HCl.PtCf Carbon, 59 5 Hydrogen, 5*3 59*5 5*0 Chlorine, 11*7 Platinum, 14*0 12*5 Iodide. — Prepared like the bromide. It crystallises in minute red crystals of the same colour as ferricyanide of potassium. Obtained. Calculated for Iodine, 36*86 {(C7H7)2HP (2I2 I 7(C7H7)3PO. 5I2 I 5(C7H7)3P0.4I2 37*24 I 36*18 " | 38*84 Chloride. — Obtained by passing chlorine into a solution of the body dissolved to saturation in warm acetic acid. It crystallises when the solution cools in pale yellow crystals, much like penta- chloride of phosphorus in appearance. The compound is most unstable, and loses chlorine rapidly in vacuo% and probably also when air-dried. Obtained. Calculated for Chlorine, 12*00 (C7H7)2HPC1 . 7(C7H7)3P0.5cT2 14*23 | 14*68 Hydriodate. Obtained by saturating a solution of the body in glacial acetic acid with hydriodic acid gas, and separated as the solution cooled in colourless crystals. Obtained. Calculated for )xide of Tribenzyl-Phospliine. 58*5 59*4 5*3 5*4 1l2' Iodine, 2(C7H7)2PH.HI I 3(C7H7)3P0.2HI. 22*8 I 21*05 1888-89.] Prof. Letts and R. Blake on Dibenzyl- Phosphine. 197 Hydro-bromate. — Obtained as the hydriodate : — - Obtained. Calculated for Bromine (1) 19 '5 2(C7H7)2HP.HBr (2) 20-5 ) i5.9 (3) 16-3 \ 10 y Nitro- Compound. — Obtained by dissolving fuming nitric acid, and then precipitating wit] Obtained. Calcula fAVikTi TTnfm oTm,ci x 3(C7H7)3PO. 2HBr 14*44 y the body in cold ti water. Amorphous. ted for Dibenzyl-Phosphine. ) (C7H6N00)2HP Carbon, 55 *23 55*26 Hydrogen, 4*30 4*27 Double Salt with Iodide of Zinc. — This co when fairly strong alcoholic solutions of tb zinc are mixed, in tufts of characteristic need! Obtained from (C7H6N02)3P0 55*38 3*95 mpound separates out Le body and iodide of es. Hofmann’s Dibenzyl- Phosphine. Iodine, 26*57 Calcula Oxide of Tribenzyl- Phosphine. I. II. 26*0 25*9 ted for 3(C7H7)2HP.ZnI2 Iodine, 26*43 1 2(C7H7)3PO.ZnI2 26*48 4. Action of fused Potash on Hofmann's “ Dibenzyl-Phosphme .” — - In tbe paper by one of us and W. Wheeler already alluded to, the statement is made that when Hofmann’s dibenzyl-phosphine is heated with caustic, potash, or soda, “ it fuses and floats on the surface of the melted alkali. Ho violent action occurs, but on cool- ing the mixture and treating it with water the greater portion dis- solves, and acids then precipitate a flocky crystalline substance, which is dibenzyl-phosphinic acid.” In corroboration of this statement, the melting-point of the acid and analyses of its lead and barium salts were given, all in accordance with the required numbers. After we had satisfied ourselves that Hofmann’s dibenzyl-phosphine was oxide of tribenzyl-phosphine, and nothing else, this reaction recurred to our minds as a further and very striking excuse for the mistake we (and Hofmann) had fallen into, and we thought it of importance to verify the previous observation. This we have accordingly done, both with Hofmann’s dibenzyl-phosphine and with a specimen of oxide of tribenzyl-phosphine prepared by a different 198 Proceedings of Boyal Society of Edinburgh. [sess. method. The phenomena observed were exactly the same as those previously described, and the melting-point of the acid obtained after precipitation with hydrochloric acid and two recrystallisations from alcohol was found to be 192° C., which is the melting-point of pure dibenzyl-phosphinic acid. Our previous observation is thus fully confirmed. The reaction in all probability occurs as follows: — (C7H7)3PO + KHO = C7H8 + (C7H7)2KP02. The occurrence of oxide of tribenzyl-phosphine among the pro- ducts of Hofmann’s sealed tube reaction led us to suspect that tribenzyl-phosphine had been formed in the first instance, but was subsequently oxidised by atmospheric oxygen. It became necessary, therefore, to search for the tertiary phosphine in the original product. The investigation was attended with considerable difficulties, as we had already proved that the products of the sealed tube reaction are a highly complex mixture, and contain, among other substances, resinous bodies which are exceedingly difficult to get rid of. We succeeded at last in isolating a liquid which grows hot on exposure to air, with production both of oxide of tribenzyl-phosphine, and of dibenzyl-phosphinic acid ; which is precipitated by hydriodic acid, forming solid compounds ; and which in contact with sulphur gives rise to a crystalline compound, which we believe to be tri- benzyl-phosphine sulphide. Moreover, the liquid acts energetically upon crystallised iodide of benzyl to give iodide of tetrabenzyl- phosphonium. The liquid appears, in fact, to be a mixture of the secondary and tertiary phosphines. Continuing our investigations, we have succeeded in obtaining from the liquid two solid substances which are easily separated from each other. The first is almost insoluble in ether, and has either the formula (C7H7)3P02 or (C7H7)3PS.* The second is undoubtedly tribenzyl-phosphine itself. It crystallises easily from alcohol, and unites with sulphur at ordinary temperature and with oxygen also. We have not as yet obtained it in sufficient quantity to thoroughly investigate its properties. * We give this last formula partly because the first is improbable, and partly because the substance in question has the same melting-point and properties as the sulphide of tribenzyl-phosphine. But, on the other hand, we have not as yet detected sulphur in it, and unless that element was present in the crude products used, its occurrence is incomprehensible. 1888-89.] Prof. Letts and B. Blake on Bibenzyl Phosphine. 199 We have, we consider, all the necessary evidence to show that in Hofmann’s sealed tube reaction phosphuretted hydrogen acts upon chloride of benzyl as ammonia does on an alkyl iodide ; that is to say, that all of the following reactions occur : — (1) C7H7C1 + PH3 = (C7H7)PH2.HC1 (2) 2C7H7C1 + PH3 = (C7H7)2PH.HC1 + HC1 (3) 3C7H7C1 + PH3 - (C7H7)3P.HC1 + 2HC1 (4) 4C7H7C1 + PH3 = (C7H7)4PC1 + 3HC1 With regard to the reaction (4) we are disposed to think that the oxide of tribenzyl-phosphine found by Hofmann, and mistaken by him for dibenzyl-phosphine (and by ourselves as well), owes its origin to the action of the potash on tetrabenzyl-phosphonium chloride or iodide, an action which H. Collie and one of us has already proved to occur easily in the following manner : — (C7H7)4PC1 + KHO = (C7H7)3PO + KC1 + C7H8 We may here mention that we have also isolated from the pro- ducts of Hofmann’s reaction all the possible oxidised derivatives which the whole series of benzyl-phosphines can give rise to, viz. (C7H7)H2P02 — Benzyl phosphinous acid. (C7H7)H2P03 — Benzyl phosphinic acid. (C7H7)2HP02-— Dibenzyl-phosphinic acid. (C7H7)3PO. — Tribenzyl-phosphine oxide. 5. We have also investigated the action of monobenzyl-phosphine on iodide of benzyl, with the view to obtaining pure dibenzyl- phosphine. The reaction occurs readily at ordinary temperatures, though no sensible heat is evolved, and a solid crystalline product results. Although some of this was found to contain the correct percentage of iodine for the formula (C7H7)2HP.HI, it is either a mixture of three substances at least, — viz., dibenzyl-phosphine hydriodate, tribenzyl - phosphine hydriodate, and tetrabenzyl - phosphonium iodide, — or readily decomposes and give rise to them. On decompos- ing it with potash a colourless liquid results, together with a solid — the latter being undoubtedly oxide of tribenzyl-phosphine. The liquid when exposed to the air oxidises readily, and yields both dibenzyl - phosphinic acid and oxide of tribenzyl - phosphine. 200 Proceedings of Royal Society of Edinburgh. [sess. When it is distilled under diminished pressure with great care, a liquid passes over which contains the primary phosphine, as well as some of the secondary body, and possibly a little tribenzyl- phosphine also. Our investigations on this very interesting and apparently extra- ordinary reaction are proceeding — the chief difficulty which we have to contend with being to obtain sufficient of the primary phosphine in a pure condition for the experiment. So far the results of our investigations show that — 1. Hofmann’s “dibenzyl-phosphine” is undoubtedly the oxide of tribenzyl-phosphine. 2. Dibenzyl-phosphine is probably a liquid combining easily with hydracids to give solid products of normal composition; oxidising in contact with air to form dibenzyl-phosphinic acid. 3. Tribenzyl-phosphine is a solid crystalline substance combining with hydracids, and both with sulphur and oxygen at ordinary temperatures to form solid products. 4. In Hofmann’s sealed tube reaction (i.e.} action of phosphuretted hydrogen on benzyl chloride) all the phosphines (as well as the quaternary compound) are obtained, and also all their possible products of oxidation (some of the latter may possibly not pre-exist in the crude product of the reaction, but be formed by subsequent treatment). Ia addition, other substances are obtained. The reaction is in fact highly complex, and is, we venture to think, extremely interesting, as it is certainly different from all similar reactions observed by Hofmann with fatty derivatives. 5. Monobenzyl-phosphine is acted upon by crystallised iodide of benzyl at ordinary temperatures, and probably gives rise to free hydriodic acid, and to the secondary, tertiary, and quaternary compounds. If this be the case, the action is also comparable with that which occurs between ammonia and an alkyl iodide. Our researches on the above subjects have involved a large expenditure of time, energy, and material, and have been tedious and troublesome in the extreme. We have, however, the satisfaction of believing that the work, which has extended over several years, is nearly at an end, and we trust in a few months to be able to give a detailed account of the whole of our experiments to the Society. 1888-89.] A. MAulay on Differentiation of a Quaternion. 201 Differentiation of any (Scalar) Power of a Quaternion. By Alexander MAulay, Ormond College, Melbourne. Communicated by Professor Tait. (Read February 18, 1889.) Nowhere, I think, does Hamilton, or any other author, attempt the very fundamental problem of finding the differential of qn where q is a quaternion and n any scalar. It was by noticing an oversight in Tait’s Quaternions , § 182, where he considers d . q%, that I was led to a consideration of the subject. In that section Tait says that the equation dq.r -rdq = dr.q- qdr . .... (1), where r — ff (2), is sufficient to determine dq as a function of dr, hut this will be found not to be the case. Equation (1) will be found to be equiva- lent to but two equations among scalars, whereas the equation from which it is derived, viz., q2dq + qdq. q + dq.q2 = dr (3), is equivalent to four such equations. Equation (1) may be written VVrV dq = VYqY dr , and thus it only gives the component of Ydq, viz., V ^r.YYrYdq, perpendicular to the axis of r. There are thus two scalars involved in dq (viz., S dq and the resolved part of Ydq parallel to the axis of r), which, so far as equation (1) is concerned, are left perfectly arbitrary. In fact, a , b being given quaternions and q a sought one, the equation aq-qa = b , in which the conditions S5 = 0, Sab = 0, must be satisfied, gives as solution q = x + %(y + b)Y-1a, where x and y are arbitrary scalars. If we use the method and result Tait suggests, we are led to dq = oo .* [* See Note appended. — P. G. T.] 202 Proceedings of Boyal Society of Edinburgh. [sess. Before considering the problem in hand — that of finding d.qn explicitly — we must consider the properties of a certain linear quaternion function of any quaternion. The form of the function depends on q and n, and we shall denote it by ( q , n ). Suppose a is any quaternion. Split it up into two parts, a' a quaternion co- planar with q, and a" a vector perpendicular to the axis of q. There are several useful forms of a and a”. Notice that we have a' = Sa + component of Va parallel to Yq . a " = component of Ya perpendicular to . This gives a' + a" = a (4), a' — Sa + Y^S. Y-1^ (5), a" = YqVY-1qVa = Y-1qnYYqnYa . . . . (6), or 2af> = Y~1q(qa- aq) = Y~1qn(q^a - aqn) . . . (7). These are some of the simple forms of a ' and a ", and we shall employ more than one of them. The quaternion function (q, n) is defined by the equation (q, n)a = nqn~la' + • . (8). We will give some of the simpler forms of ( q , n) in full, though equation (8) is what we shall use in the present investigation. Putting a' = a - a", and substituting for a " the last value in equa- tion (7), we get (g, = + 'jifa-aif) . . . (9)- Again, substituting the first value in equation (6), (q, n)a = nqn^a + (V qn - nqn~vVq)YY-lq\a . . (10). Another important modification is obtained from the fact that if X is a vector perpendicular to the axis of r, r\r = TV. A (for by Tait’s Quaternions, § 354, r~x\r = T2r.r_2A). n— 1 n — 1 2 2 a"q 2 = T'-'q.a". (11), 1888-89.] A. M'Aulay on Differentiation of a Quaternion. 203 n— 1 n—\ Again, v a' and q are coplanar q 2 a' q 2 —qn~la'. .*. from eq. (8) n~l( YUow \ — (2>»)“ = 2 2 \na' + TDg J2 2 ’ Let us here substitute a - a" for a' ; and for a" V- W^VVU^Va = i y-1U^(U^. a - aUgra) . Thus (q, n)a = q~^~{na + ^(~- a-a\Jqn)}q~ (12). This again may be written n— 1 7i—l 1/1 (q, n)a = ?iq 2 aq 2 4- 1 - — - 2T^\VU^ YUq ^(gioKflS-K 2M) (13). We might with ease write a number of other forms. It is to be observed that in equations (9), (10), (12), (13) the long second term is in every case a vector, for it consists of a vector perpendicular to the axis of q operated upon by some quaternion which is coplanar with q. Hence S{(q,n)a}=nSqn~1a (14). We now proceed to those properties of (q, n) which we require. First notice that (q, n) is commutative with any quaternion co- planar with q\ and also with (r, m) (which if not sufficiently obvious will appear incidentally immediately) if r is a quaternion coplanar with q. How we have (q, n)(r, m)a = (q, n)(r, m)a + (q, n){r , m)a" YqnYrm - mnq V»- V + by equation (8) [ = (r, m)(q, n)a ]. Putting then r = qn we get (q,n)(qn,m) = (q,mn) = (q, m)(qm,n) . . . (15). Putting m = - ° n (q, n)(qn, ^ = ( q , 1) = 1 by equations (8) and (4) , 204 Proceedings of Eoyal Society of Edinburgh. [sess. [This means that if ( q , n)a = b, then a — Remembering the meaning of equation (11), we see that (q,n)q-”( )q~n= - (?, - n) .... (17). The last property we propose to prove is that when n is a positive integer (q,n) = ( )qn~l + q( )qn~2 + q2( )qn~3 + . . . + qn~\ ) (18). Calling, for brevity, the linear function on the right Q, we see that • a' and q are coplanar Qa' - nqn ~ la'. Again 2Q a" = Q Y~1q(qa - ^[equation (7)] = V" lqQ(qa - aq) Yan = Y~1q(qna - aqn) = 2^-a" [equation (7)] . Qa = Qa' + Q a" = nqn~xa' + \q a " = (q, n)a , which proves the proposition. [Notice that equation (16) combined with (18) solves the equation aqn~x + qaqn~2 + . . . + ^7l_1a = 5]. We can now prove that for all scalar values of n d.qn = (q,n)dq (19). Equation (18) proves this for n a positive integer. Next suppose n = ljm where l and m are positive integers. Let qn = r , i.e. ql — rm . Differentiating, and using the first case, ( q , l)dq = (r, m)dr d. qn = dr = (r, m) ~ 1(g, T)dq = ()'“ k) t,dq [eiuation e 1 6)] = (?> n)dq [equation (15)] . 1888-89.] A. McAulay on Differentiation of a Quaternion . 205 Lastly, suppose n is negative, and = - m. Thus d.qn — - qnd.qm.qn = - (q, — n)qndqqn = (q, n)dq [equation (17)] , which proves the proposition for all cases. The various forms given above for (q, n) thus give so many forms for d.qn . Notice the meaning of equation (14). This gives Stfl.q^^nSqn-'dq (20), which gives the ordinary form for the differential of a power of a scalar. If we put q = a vector p, a = dp , we have a = pSp ~Jdp a" = pVp~1dp, and equation (8) gives d.pn — npnSp- hip + Ypn. Yp ~ xdp . . . (21), and this, be it remembered, is true not merely for integral values Of 7L Not© on Mr M‘Aulay’s Paper. By Professor Tait. (Read February 18, 1889.) There is, undoubtedly, an omission in § 182 of my Quaternions (2nd ed.), but it is by no means so serious as Mr M‘Aulay asserts. In fact the solution there given is merely an unfinished one, not in any sense erroneous. I sketch briefly the completion of it, as pre- pared for the new edition of my book, which is now being printed. The equation qn~1dq + qn~2dqq+ . . , . -bdqq^-1 = 4>(dq) = dr . . (1) gives, as in my book, qndq - dqqn = qdr - drq , but this does not make dq infinite. In fact it gives 2V.VqhVdq = 2V.VqVdr ..... (2). Now it is easy to see that Vqn = QnVq, where Q» = »( S2)»-i - ”^-]-”^2(Sg)B-8(TV qf + &e. 206 Proceedings of Royal Society of Edinburgh. [sess. Thus (2) gives so that QPJdq = Ydr + xYq , Q ndq = (y + xYq) + dr . (3), x and y being undetermined scalars. Substitute in (1), and again use (3), and we have Q„dq = dr + ^ri(Q „dr - 4>dr) , which is the complete solution. Note that this gives, by means of (1), for an equation satisfied by the linear function , Q„) = 0. The fact that this equation is of the second, instead of the fourth degree, is of course due to the very special form of as shown in (1) above. In fact the first factor kills any scalar, or any vector in the plane of q ; while the second kills a vector parallel to the axis of q. Additional Remarks on the Virial of Molecular Force, By Prof. Tait. (Read March 18, 1889.) (Abstract.) In my paper, read Jan. 21, I stated that I would not “for the present, insist on this point [the value of j3] further than to say that the main effect is merely to alter the value of the disposable quantity A, below.” The present paper contains the more complete investigation here promised, and shows that the Virial equation takes the form p(Y - f3) = kt- v(v-y) which, as a and y are noio at least nearly identical, is practically the same form as that previously given. 1888-89.] Dr T. Muir on the Theory of Determinants. 207 The Theory of Determinants in the Historical Order of its Development. By Thomas Muir, M.A., LL.D. Part I. Determinants in General (1829-35). (Continued from p. 544 of vol. xv.) KEISS (1829). [Memoire sur les fonctions semblables de plusieurs groupes d7un certain nombre de fonctions ou elemens. Correspondanee math, et phys., v. pp. 201-215.] In Beiss we have an author who starts to his subject as if it were entirely new, the only preceding mathematician whom he mentions being Lagrange. Like Cauchy he opens by explaining a mode of forming functions more general than those of which he afterwards treats, the essence of it being that an expression involv- ing several of the n.v quantities, aa at ay . . . aP b« W by . . . bp ca c/3 cy ... CP r°- n 3 ry ... rp is taken, and each exponent (“exposant”) changed successively with all the other exponents, a, or each base changed with all the other bases, a, b, ... . Only a line or two, however, is given to this, the special class known to us as determinants being taken up at once. His notation for alb^cs - ax63c2 — a261c3 + a^c1 4- a^b1^ - a^b2c^ is (abc , 123) , (vn. 7) a line being drawn above the exponents to indicate permutation. His rule of formation of the terms and rule of signs are combined after the manner of Hindenburg. Like Hindenburg, he arranges the permutations as one arranges numbers in increasing order of magnitude ; but, unlike Hindenburg, after the arrangement has 208 Proceedings of Royal Society of Edinburgh. [sess. been made be determines the sign of any particular term. On this point his words are (p. 202) “ Cela fait, determinons gdndralement le signe dn Mme pro- duit (soit M) de la maniere snivante. Le nomhre M sera renferme entre les produits 1.2.3 ... I et 1.2.3 ...£(£+ 1); soit M=m + X x 1.2.3 . . . Z, de sorte que A 0 et <1 + 1.2.3 . . . 1. Cela etant, faisons M=m(- l)Vs (hi. 24) This apparently means that if the sign of the 23rd term in the expansion of (abed, 1234)* be wanted, we divide 23 by 1.2.3, getting the quotient 3 and the remainder 5, and thence conclude that the sign wanted is got from the sign of the 5th term by multiplying the latter by ( - l)3. Of course 5 has then to be dealt with after the manner of 23, the quotient and remainder this time being 2 and 1, so that we conclude that the sign of the 5th term is got from the sign of the 1st term by multiplying by (-1)2. And the sign of the 1st term being + , the sign of the 23rd is thus seen to be (-1)3+2 le. It would seem at first as if the case where M is itself a factorial were neglected. This however, is not so, the condition m .< 1 + 1.2.3 . . . I being corrective of the opening statement that M must lie between 1.2.3 . . . I and 1.2.3 . . . I (£+1). Lor example, the term being the 24th, we put 24 in the form 3 x 1.2.3 + 6, and thus learn that the sign required is different from the sign of the 6th term : then we put 6 in the form 2 x 1.2 + 2, and thus learn that the sign of the 6 th term is the same as the sign of the 2nd term ; finally, we put 2 in the form 1 x 1 + 1, which shows that the sign of the 2nd term differs from the sign of the 1st: the conclusion of the whole being that the signs of the 24th and 1st terms are the same, or that they are connected by the factor (-l)3+2+1. Though interesting in itself, a more troublesome form of the rule of signs for the purposes of demonstration it is scarcely possible to conceive, and, as might therefore be expected, it is on the score of logical development that Eeiss’ paper is weak. Through * Or ( abode ,12345) , or indeed (cqa2 • • . an) 123 . . . n). 1888-89.] Dr T. Muir on the Theory of Determinants. 209 inability to use tbe rule later in the demonstration of the so-called Laplace’s expansion-theorem, he is forced to supplement it by another convention. His words are (p. 203) — “ Avant d’aller plus loin, faisons encore la determination suivante. Soit « une fonction quelconque dans laquelle les k quantites A,B,C, . . . As entrent d’une maniere quel- conque. Supposons que ces dernieres soient les k premieres de 1’echelle (f > Qu’on fasse avec ces s 'A B C . . . A* . . . S 2 3... k . . elemens toutes les combinaisons sans repetition de la classe k , et qu’on les substitue successivement an lieu de A,B, . . . A* dans la fonction co ; c’est-a-dire le premier element de chaque combinaison a A, le second a B, etc. Nous obtiendrons par la autant de fonctions semblables a w qu’il y a de combinaisons de la classe k de s elemens. Or, entre toutes les combinaisons qui en precedent une quelconque, il s’en trouvera une qui aura k- 1 elemens communs avec elle, tandis que les deux elemens qui restent isoles dans Tune et Fautre se suivent immediatement dans 1’echelle. Donnons a la fonction qui contient la derniere de ces combinaisons le signe oppos4 a celui de l’autre fonction ; par consequent les signes de toutes les fonctions semblables a co seront parfaitement determines, et dependront du signe de la premiere fonction (/(A,B,C, . . . A*) ). Soit, par exemple, s = 5, k = 3 ; nous aurons successivement, en remplagant A,B,C, . . . S par 1, 2, 3, 4, 5, et en donnant le signe ( + ) a / (123), + /(123) , -/( 124) , +/(125) , +/(134) , -/(135) +/(145) , -/( 234) , +/( 235) , -/(245) , +/(345) . Yoici comment on determinera le signe de chaque fonc- tion semblable a w d’apres celui d’une antre quelconque. Qu’on cherche les nombres qui se trouvent dans l’echelle 'A B C . . . A* . . . S\ ,12 3... I . . .s) sous les Clemens de l’une et de l’autre de ces fonctions. Si l’on nomme h et li' leurs sommes respec fives, on trouvera le signe de l’une des fonctions = ( - \y~h x le signe de l’autre.” VOL. xvi. 5/7/89 o 210 Proceedings of Eoyal Society of Edinburgh. [sess. Four theorems he considers fundamental, viz., those known to us as (1) Bezout’s recurrent law of formation, in all its generality ; (2) Vandermonde’s proposition that permutation of bases leads to the same result as permutation of exponents ; (3) Laplace’s expan- sion-theorem ; (4) Vandermonde’s proposition regarding the effect of making two bases or two exponents equal. The two most important, viz. (1) and (2), he leaves without proof, and the 4th he says he would at once deduce from the 3rd, — doubtless by choosing the ex- pansion in which the first factor of every term would be of the form (aa , a/3) and therefore equal to zero. The proof of the 2nd theorem, viz., (abc . . . r , a/3y . . . p) = ( abc . . . r , a/3y . . . p) , is by the method of so-called induction, and may be illustrated in a later notation by considering the case al a2 a3 oq h Ci = a2 h C2 C1 C2 C3 a3 C3 From theorem (1) we have c2 cz a* (to a. aY a2 az h h h = a, ci C2 C3 = -\ — Ci — a2 h + Cts \ cx C3 Cl C2 % ai a2 + b2 ci % b3 C1 C2 % ai a2 - c9 4- Co A \ V 6 \ b2 But by hypothesis all the determinants on the right here may have their rows changed into columns ; and this being done we have by addition and the use of theorem (1) — £?1 a2 a3 ai ci \ h = 3 a2 C2 C2 C3 a3 h C3 and thence the identity required. (IX. i) 211 1888-89.] Dr T. Muir on the Theory of Determinants. To this proof the following note is appended (p. 207) “Cette demonstration quoiqu’assez simple semble reposer cependant sur un artifice de calcul : mais en cherchant nne demonstration direde , j’ai rencontre une difficulte d’un genre particulier. En effet, on trouve facilement que lmQ terme de l’une des fonctions en question est aussi egal oil au meme terme de l’autre, ou generalement au mme, et que, dans le dernier cas, le rnme terme de la premiere est aussi egal au Zme de la seconde, abstraction faite des signes. (ix. 5) Mais l’identite de ces derniers (qui est de rigueur) exige des explications tres-longues et beaucoup moins elementaires que la demonstration que je viens de donner.” The remaining six or seven pages of the paper are more interest- ing, and concern the subject of vanishing aggregates of products of pairs of determinants. The theorems were suggested by taking, as we now say, a determinant of even order having its last n rows identical with its first n rows, e.g., the determinant (abab , 1234) , and using theorem (3) to expand it in terms of minors formed from the first n rows and their complementary minors. When n is even, a proof is thus obtained, as we have seen in the footnote to the account of Bezout’s paper of 1779, that the first half of the expan- sion is equal to zero. When n is odd, the method fails, although the proposition is still true.* Keiss’s enunciation is as follows (p. 209) * It is worthy of note in passing, that a common method does exist for establishing the two cases, — a method quite analogous to Reiss’s, but difficult of suggestion to one who used his notation, or indeed to any one who had no notation suitable for determinants whose elements had special numerical values. All the change necessary is to make the last n elements of the first column each equal to zero. This causes no difference in the result when n is even, e.g., from the identity a4 a2 a3 a4 \ b2 b3 b4 _ ^ • 6^2 ^3 $4 • b2 b3 b4 we have, as before, \(hb2\.\a3b4\ - \a1b3\.\a2b4\ + | axbA [.| a2b3 | = 0 ; and when n is odd, the second half of the terms which previously gave trouble do not occur. 212 Proceedings of Boy al Society of Edinburgh. [sess. “ Theoreme V. — Soient les echelles (ab..r, a , b , . . . r \ (a p y . . . an , a*+1 , . . . p \ \1 2 . . n , n+1 , n+2, ... 2nJ \1 2 3 . . . n , n + 1 , . . . 2 n) qu’on fasse avec les elemens /3,y, . . . , p tontes les combinai- sons de la classe (n- 1), et qu’on les substitue successivement dans le premier facteur du produit (ab. . . r , a/3y ... a n) . (ah ... r t an+1 . . . p) an lieu de fiy . . . an ; qu’on remplace maintenant dans l’autre facteur les exposans an+1 . . . p par tous ceux qui ne se trouvent pas dans le premier, en ayant soin de les ecrire suivant l’ordre indique par les echelles. Si l’on donne au premier produit le signe ( + ), et qu’on determine les signes de tous les autres d’apres (II), la somme algebrique en sera = 0, que le nombre n soit pair ou impair.” (xxm. 8) An example of it is (abc, 123 )(abc, 456) - (abc, 124 )(abc, 356) + (abc, 125)(a5c, 346) - (abc, l2Q)(abc, 345) + (abc , 134 )(abc, 256) - (abc, 135 )(abc, 246) + (abc, 13 6)(abc, 245) + (abc, 14 5)(abc, 236) -(abc, 14 6)(a6c, 235) + (ale, 15 6)(abc, 234) = 0, the left-hand side being nothing more than the first ten terms of one of the expansions of the vanishing determinant t?2 flg 0!^ flg b2 b3 54 b5 bQ C1 C2 C3 C4 C5 C6 ax a2 a3 a4 a5 aQ b\ b2 b3 54 Iq C1 C2 e3 C4 C5 C6 ’ or the other ten terms with their signs changed. Keiss’s proof is lengthy and troublesome, the method being to expand each factor in terms of the a’s and their complementary minors, perform the the multiplications (e.g., in the special case just given the multipli- 1888-89.] Dr T. Muir on the Theory of Determinants. 213 cation of aY\ b2c3\ - ct2\< h&j + a^b^ by afb5c6\ - a5\\c6\ + a6|&4c5|, &c.) and show that the terms of the final aggregate occur in pairs which annul themselves. The next theorem is of still greater interest, because it is that peculiar generalisation of the preceding which in later times came to be known as the Extensional. The way in which it is estab- lished is also noteworthy, viz., by deducing it as a special case from the theorem of which, as we have said, it may be viewed as a generalisation. The authors words are (p. 213) : — “ Ce th^oreme nous conduit a une relation qui existe dans le cas le plus general, savoir si v - n est un nombre quelconque ou positif ou n^gatif. Supposons v>n, et v - n — ^ ; soient les echelles, /«&... r , a , b , . . . r , A , B , . . . R\ U2...N, N + l, N + 2, . . . 21ST , 2N + 1, 2N+2, ...v) Qu’on fasse avec les elemens /?, y, . . . aN, aN+1. . . p toutes les combinaisons de la classe N - 1 ; qu’on les substitue succes- sivement au lieu de ft . . . aN dans le premier facteur du produit qu’on remplace dans l’autre facteur les exposans aN+1 . . . p par tous ceux qui ne se trouvent pas dans le premier : qu’on determine enfin le signe de chaque produit d’apres (II) : la et /a (3 ... a? j aN+1 VI 2 ... 1ST , N+l P , A , B , . . . P' 2N, 2N+1, 2N + 2, ... Vt ). (ab. . . rAB . . . B , a/3 ... aNAB . . . P) x (ab . . . rAB . . . R, aN+1. . . p AB . . . P) ; somme algebrique en sera = 0. “ En effet, supposons les echelles (xxiii. 9) (xlv. 6) 214 Proceedings of Royal Society of Edinburgh. [sess. Formons avec ces elemens la fonction decrite dans le dernier theoreme : la somme totale en sera done = 0. et le premier terme aura la forme (ab . . . rAB . . . R , a/3 . . . pA . . . Av~ 3N) x (ab . . . rAB . . . K , A""3N+1 ... PA ... P) . Or, on voit facile ment que tous les termes qni ne contiennent pas dans chaque facteur tous les exposans A,B, . . . P, s’evanouiront separement, parce qu’il y aura des exposans identiques dans l’un ou l’autre des facteurs. 11 ne restera done que les termes qui, contenant a dans le premier facteur, y 4puisent successivement toutes les combinaisons de la classe N-l des elemens /3, y , p. Mais les signes de ces termes sont ^videmment determines comme ils devaient l’etre ; partant la somme algebrique de tous les termes est = 0, ce qu’il fallait demontrer. This will be best understood by considering a special example. Going back to the previous theorem, and selecting its simplest case, we have \aA\-\aA\ - \aM\a2h\ + K^l-Wsl = °- Now what the new theorem asserts in regard to this is that we may with impunity extend each of the determinants occurring in it, provided the extension be the same throughout. For example, choosing the extension £6 rj7* we can, in virtue of the new theorem, assert the truth of the identity \ail)2^hV6^\'\a3^4^5rl6^l\ ~ That the two may be viewed as cases of the same theorem will be apparent when it is pointed out that just as the first is derivable from \ b2 -0, * In Reiss’s notation the extension is A a Bb . . . Rp . 215 1888-89.] Dr T. Muir on the Theory of Determinants. so the second is derivable in exactly the same way from a perfectly similar identity,* viz. al a2 % % a6 Cty ^6 72 % >?4 V5 % % >75 >76 >77 £1 £2 £3 £4 £5 £6 £7 £5 £6 £7 • a2 % «5 % % % «6 • h ^3 *5 67 • £2 £3 £4 £5 f. ft £5 £e £7 • % % % % V7 >75 >76 >77 . £2 £3 £4 £5 & £7 £5 £e £7 Many more products than three (126 in fact) arise in the latter case ; but, for the reason stated by Reiss, only three of them do not vanish. JACOBI (1829, 1830). [Exercitatio algebraica circa discerptionem singularem fractionum, quae plures variabiles involvunt. Crelle’s Journal , v. pp. 344-364]. [De resolutione aequationum per series infinitas. Crelle's Journal , vi. pp. 257-286.] By such memoirs as these, in which Jacobi continued to use determinants, the functions were kept before the mathematical * It is perhaps a little more readily seen to be derivable from tribusque variabil- ibus x, y, z : du bux bu2 du bux du2 0«q du2 du ^ bx' by ' bz dx ° dz by by ' dx bz bu2 bu 0«q bu b ux bu2 bu bux bu2 bz % by' bx by' bz ’ bx bz bx ' by 3 quam patet expressionem casu, quo u, uv u2 sunt expressions lineares, in expressionem ipsius A supra exhibitam redire.” MINDING (1829). [Auflosung einiger Aufgaben der analytischen Geometrie vermit- telst des barycentrischen Calculs. Crellds Journal , y. pp. 397-401.] Unlike Jacobi, Minding was unaware, apparently, of the ex- istence of a theory of determinants. The functions occur at every step of his investigation, yet he makes no use of their known properties to obtain his results. He deals with four problems in his memoir, the second two being the analogues, in space, of the first two. Nothing noteworthy 1888-89.] Dr T. Muir on the Theory of Determinants. 217 occurs in connection with the latter save that use is made of the identity, ffy" P'y _ a(ye" _ yy) + _ Jc") + a"{bc' -Vc) , a where ft = la' - 1' a , /3" = l' a" - Taf, y = cal - c'a , y" = c'a!' - c"a' . This identity, it may he remembered, we have noted under Lagrange as an elementary case of the theorem afterwards well known regard- ing a minor of the adjugate determinant. Strange to say, it makes only its second appearance here fifty-six years afterwards. In the interim, too, no other special case of the theorem seems to have been established. The third is that if P, P', P", P'", be four points in space, given by the equations, q P =a A + q' P' = a' A + q" P" = a" A + q"?"' = a'"A + V"B c C + d D , c' C + d' D, + c" C + d" D , + c'"C + d'"~D ; b B + V B + b"B then for the bulk of the tetrahedron P P' P" P'", we have where PFF'F" A + A' + A" ABC D “ qii'4" 5 A = 0'(/3 "y'"-ry"), and P =a' b -a V , P" = a" V - alb" , P'" = a"'b"-a"V" , A' = S''(/3"y - ffy") , y —a'c -a c' , y" —al'd - a' c" , y =a c - a c , A" = 3 "W-/3V), d' — a! d -ad' , I d" = a" dl — a' d" , d" = a'"d" - al'd!" . The transformation of A + A' -f A" into the form a'a"\ab'c"d'"\ — a transformation all-important for Minding’s purpose — is not made : but in the remark, 218 Proceedings of Royal Society of Edinburgh. [sess. “Man kann den Ausdruck A + A' + A" leicht entwickeln, und wird ihn dann durch a' a" theilbar finden,” there is evidently a foreshadowing of the identity \a! b | , \a' c | , \a' d \ | a!' V | , | a!' c' | , | a" d ' | \a"'b”\ , \a!"c"\, \a'"d"\ - afa"\ab'c"d"'\ . The fourth theorem, concerning the tetrahedron enclosed by four given planes, A + #B + + (a +b x + c y) C , A + &B + yC + (a! + V x + d y) C , A + aB + ^C + (a" +b"x + c"y) C, A + xB + yC + (a"' + b'"x+d"y) C , is made dependent on the third. The intersections II, II', II'', IT" of the four triads of planes are found to be given by q II = (b c' )A + (c a' )B + (a b' )C + (a b c )D , i IT = (&' c") A + (cf a") B + (a' b") C + (a! V d )D , q" n" = (&'V")A + (c" a"')B + q'"IL"'= (b'"c )A + (c'"a )B + (a'"b )C + {a'"b'"d")D , where (bc') = b(c' -c") + Z/(c"-c) + b"(c-c’), (ca') = c(a' - a") + c'(a" - a) + c"(a - a') , (ab') = a(V -b") + a'(b" - b) + a"(b-b'), and ( abc) = a(bd ) + b{ca') + c(ab') , = a(b’c" - b"c') + a\b"c-bc") + a'\bd-b'c). Hence, by the third theorem, nnirn'" A+A' + A" ABC D -qqY4"(Vcl(b''c"’)i where now A = S'(J8'V" - P"Y), A' = 8"(/3"'y - Py'"), A" = 8"'(P'y" - P"y) , 219 1888-89.] Dr T. Muir on the Theory of Determinants. and P = (b'c"){ca')-(bc')(c'a")i /3" = . . . , /r = . . . , y' = (Ve”){ab')-(a'b")(bc' ), y" = . . . , y'" = . . . , 0, = (6'c")(a5c)-(5c,)(aW), 0''=. . . , 0"' = . . . . Minding then continues (pp. 399, 400) “ Man setze a"\bc') - a(0V') + a\b"c"') - a'\b'"c) = M. “ Nach den nothigen Reductionen erhalt man : p = ~(c" -c' )M, y = -(V -b") M, 0' = ~(b'c" - b" d )M , P' = + (c" - c" )M, y" = + (0" - V") M , 0" = + (0" d" -b'c )M P'"=-(C - c"')M, y"= -(0"'-0 )M, 0"'=-(0"'c -0 c'")M “ Hieraus erhalt man weiter : A = - M3(0" c -V c") . (0V"), A' = - M3(0"'c" - 0" c'") . {(0"c"#) - (0"'e)} , A" = -M3(0 c"'-0"'c ).(0'c"). “ Eine weitere Reduction ergiebt : (0c'" - 0"'c)(0'c") - (0"'c)(0'"c" - 0V") = {c'"b' - c'U") . “ Hieraus folgt A. + A' + A" = M3(0'c")(0V"), und als Resultat: nn'irir" m3 „ A B C D q.ff'f" ’ The first point to be noted here is, that since (be') , (ca!) , (ah’) , are in modern notation b V b" c c' c" a a a!' c c' c" a a' a" 0 V b" 1 1 1 } 1 1 1 > 1 1 1 the identity a(bc') + b(caf) + c(ab') = a(b'c" - b"c') + a\b"c - be") + a" (be - be) 220 Proceedings of Royal Society of Edinburgh. [sess. is the same as b V b" c c' c" a a' a" a a' a!' a c c' cu + b a a ' a" + c b V b" = b V b" 1 1 1 1 1 1 1 1 1 c d c" — a disguised special case of Vandermonde’s theorem (xh.), the four elements of one row being each unity. (xn. 11) The next point is, that since the expression denoted by M, viz., a'"(bd) - a(b’c") + a\b"d") - a'\b"'d) is in modern notation the identity is the same as V b" V" d d' d" 1 1 1 a' a" a'" V b" b'" d d ' d" a a / a " a' b V b" b" c d c" c'" 1111, S' = - (&'e"-&V)M b V b" c d c" 1 1 1 a a ' a" b V b" c c' c" V b" a a' a" b V b" c cf c" 1 1 1 V" d" 1 , and therefore is, like its eight companions, a fresh case of the theorem regarding a minor of the adjugate.* (xx. 2) DRINKWATEK, J. E. (1831). [On Simple Elimination. Philosophical Magazine , x. pp. 24-28.] Up to this date, almost 140 years after the publication of Leib- nitz’s letter to De L’Hopital, no English mathematician’s name * Instead of following Minding’s lengthy process, a mathematician of the present time would of course observe that the coefficients of A, B, C, D are the principal minors of M, and using Cauchy’s theorem would at once reach the desired conclusion, viz. , that the determinant of them = M3. 1888-89.] Dr T. Muir on the Theory of Determinants. 221 occurs in connection with the subject of determinants, — a fact most significant of the comparative neglect of mathematical studies in Britain during the 18th century. Apart from the contents, there- fore, some little interest attaches to Drinkwater’s short paper, as being the first sign to us of that revival which, as is well known otherwise, had taken place some few years before. Drinkwater knew of the investigations of Cramer, Bezout, and Laplace; and professed only to put the elements of the subject “ in a more convenient form.” His rule of signs is stated and illus- trated as follows (p. 25) : — 11 Write down the series of natural numbers 1 2 3 4. . . n, and underneath it all the permutations of these n numbers, prefixing to each a positive or negative sign according to the following condition : — ■ “ Any permutation may be derived from the first by con- sidering a requisite number of figures to move from left to right by a certain number of single steps or descents of a single place. If the whole number of such single steps neces- sary to derive any permutation from the first be even, that permutation has a positive sign prefixed to it ; the others are negative. For instance, 4 2 13. . . n may be derived from 1 2 3 4. . . . n, by first causing the 3 to descend below the 4, requiring one single step : then the 2 below the new place of the 4, another single step ; lastly, the 1 below the new place of the 2, requiring two more steps, making in all 4. There- fore this permutation requires the positive sign.” In this there is essentially nothing new : it at once recalls a theorem of Rothe’s (in. 8). In the following paragraph, however, we find the discussion of a point not previously dealt with. The words are (p. 25) : — • “ The same permutation may be derived in various ways, and it is necessary, therefore, to show that this rule is not incon- sistent with itself : thus the same permutation 4 2 13. . . n might have been obtained by first marching 1 through three places, then 2 through two ; and, lastly, 3 through one, making six in all, an even number as before. Without accum- ulating instances, it is plain, if q be the smallest number of 222 Proceedings of Roycd Society of Edinburgh. [sess. steps by which any number p reaches the place it is intended finally to occupy in that permutation, that if p should advance in the first instance m places beyond this, it must subse- quently return through m places : or, which is the same thing, it must at a later period of the march, allow m of those which it has passed to repass it, so that it will regain its proper place after the number of steps has been increased from q to q + 2m, which, by the rule, require the same sign as q. The same reasoning applies to every other figure ; and hence the consis- tency of the rule is evident. (hi. 25) He then establishes four properties of the functions, viz. (1) Vandermonde’s theorem regarding the effect produced on the function by transposition of a pair of letters; (2) Bezout’s recur- rent law of formation ; (3) Scherk’s theorem regarding the partition of one of the functions into two ; and (4) Scherk’s theorem regard- ing the removal of a constant factor from one of the functions. The two latter theorems, which, as we have seen, had been stated for the first time only six years before, are given by Drinkwater in the following form (p. 27) : — (8) If any factor in /{XYZT. . . ( n ) }, as X, be divided into two parts, X = V + W, the function may be similarly divided, so that /{(V + W)YZT . . . (n)} =/{VYZT . . . (n)} +/( WYZT . . . (n)}9 placing each part of X in the same relative position (which in this example is the first) which X itself occupied before the division. (xlvii. 2) (9) If any quantity which does not vary from one equation to the other, and which, therefore, is not liable to be affected with an index, is found under the symbol, it may be con- sidered a constant coefficient of every term of the developed function : and written as such on the outside of the symbol : of this nature are the unknown quantities themselves, so that for instance, f{XYxZT .... (n)}=xf{XYZT . . . (»)}, and so of like quantities.” (xlviii. 2) After these preliminaries the problem of the solution of n linear 223 1888-89.] Dr T. Muir on the Theory of Determinants. equations in n unknowns is taken up. The method followed is essentially the same as Scherk’s. MAINARDI (1832). [Trasformazioni di alcune funzioni algebraiche, e loro uso nella geometria e nella meccanica. Memoria di Gaspare Mainardi. 44 pp. Pavia, 1832.] In his preface Mainardi explains that the algebraical functions referred to in the title are “ funzioni risultanti o determinanti.” But although he thus speaks of them as if they were known to mathematicians by name, and mentions the researches of Monge, Lagrange, Cauchy, and Binet in regard to them, he does not take for granted that his reader has a knowledge of any of their pro- perties. The one theorem on determinants, — the multiplication- theorem, — which forms the basis of the whole memoir, is con- sequently sought to be established without the use of any previously proved theorem. The attempt, as might be expected, is interesting. The first two sections (pp. 9-29) of the three into which the memoir is divided may be passed over without much comment. The first deals with the multiplication-theorem for two determinants of, the 2nd order, and with those applications of it to geometry which arise on making the elements of each determinant the Cartesian co-ordinates of two points in a plane. No proof is con- sidered necessary for this simple case, the opening paragraph of the memoir being ; — “ Rappresentate con xm xnf xaJ xb; ym, yn, yai yb otto quantita qualsivogliano, ed indicati per brevita il binomio Xm-Xa + Vm-Va COl simbolo (xmXa) , il binomio Xn'Xb + Vn-Ub con (xnxb) e simili, si provera facilmente essere ^ (xmVn - xnym)(xayb - xbya) = (Va)(XnXb) ~ {XmXb)(xnXa)P All the seven other paragraphs are geometrical. 224 Proceedings of Royal Society of Edinburgh. [sess. The second section in like manner opens with an algebraical theorem, viz. (p. 13) — {^mfe-2/n)}R(2/c-2/6)} + {xm(zp -zn)}{xa(zc -Z6)} ~ ^n) } ~ %b) } = (xmxa)(xpx c) - (xmxc)(xpxa) + (xnxa)(xmxc) - (Vc)M + (xpxa)(xnxc) - (xpxc)(xnxa) + {xmxb)(xpxa) - (xmxa)(xpxb) + (xnxb)(xmxa) - (%n%a)(vb) + (xpxb)(xnxa) - (xpxa)(xnxb) + (y^)M - (xmxb)(xpxc) + (xnxc)(xmxb) - (xnxb){xmxc) + (xpxc)(xnxb) - (xpxb)(xnxc), (XXIX. 2) where {xm{yp - yn)} and (xmxa) stand for (xmyp xpyf) + {xnym xmytij + (xpyn xnyp) and xmxa + ymya + zmza respectively ; and the remainder is occupied with the applications of the theorem to geometry and dynamics. Each factor of the left- hand side of the identity is evidently a determinant of the third order, and the three pairs of lines on the right-hand side are each the expansion of a determinant of the same order : so that in the notation of the present day the identity may he written xm 2/m i X, X ya l Xm Zm 1 Xa Z y & l + Xn Zn 1 Xb Zb 1 xp yP l X, 2 yc l xp *P 1 Xc 1 ym Zm 1 Va 1 (xmxc) {xmXa) 1 + yn 1 • yb 1 = (xn ,XC) (xnxa) 1 Vp Zp 1 yc K 1 (*i> xc) (xpxa) 1 (xmxa) (xmxb) 1 (xnxa) (xnxb) 1 (xpxa) (xpxb) 1 (xmxb) (xmxc) 1 (xnxb) (xnxc) 1 (xpxb) (xpx0) 1 1888-89.] Dr T. Muir on the Theory of Determinants. 225 There has been no previous instance of an identity perfectly similar to this; the nearest approach to such being, as the numbering shows, a result obtained by Binet in 1811. The exact character of the affin- ity between the two, and the general theorem which both foreshadow, will be most readily brought into evidence by a little additional trans- formation. Taking first the right-hand side of the identity, we ob- serve that the three determinants have only twelve elements among them, being obtainable in fact from a single array of four rows and three columns. Their sum may consequently be put in the form 1 (Va) (xmXb) (Vc) I 1 (xnxa) (xnxb) (xnxc) 1 (xpxa) (xpxb) (xpxc) 0 1 1 1 Secondly, we observe that the first factors on the left-hand side are similarly obtainable from m 1 X» Vn Zn 1 Xp Vp Zp 1 ; and the second factors from Xa Va Za 1 xb Vb % 1 a?c Vc zc i ; and as the so-called product of these arrays is equal to the said left- hand member diminished by xm ym zm xa ya za Xn yn zn xb yb Zb xp yP Zp xc yc Zc Mainardi’s theorem may be put in the much altered form- (xmxa) (xnxa) (xpxa) M (xnxb) (xpxb) (xmxc) (xnxc) (xpxc) 1 1 1 Xm Vm Zm Xn yn Xp y.p zp Xa ya Za Mb yb %b xc yc *c xm ym zm ya Za xn yn Zfi • Xb yb zb Xp yp Zp xe yc Zc VOL. XVI. 10/7/89 226 Proceedings of Royal Society of Edinburgh. [sess. The constitution of the 3rd section is quite like that of the others, the first paragraph dealing with the multiplication- theorem for the case of determinants of the 3rd order, the second paragraph with the same theorem for determinants of the 4th order, and the remaining eight paragraphs with geometrical applications. The mode of proof of the multiplication-theorem is partly indicated by saying that any particular case is made dependent on the case imme- diately preceding it ; but its exact character can only be understood by a somewhat minute examination. The investigation for the case of determinants of the 3rd order stands as follows (p. 29) : — “Si considerino i due polinomj •X'mfyrfip ~~ Vp^n) 4" Xn(Zmyp “ Vm^p) 4" X'pfymZn ~~ V n^m) ^ = 2/w) %>} J ®a(yM -y&) + xh{zayc - zcya) + xc(yazb - ybza) = R, Vb , ^cl- Se ne effettui il prodotto, il quale, mediante l’equazione (a) del primo articolo, si potra disporre sotto la forma seguente xmxa(ynyb)(ypyf) + xnxa(ymyc)(ypyb) + xpxa(ymyb)(ynyc) + xmxb(ynyc)(VpVa) (h) + xnxb(ymya)(ypyc) + xpxb(ymyc)(ynya) + xmxc(jjnyfj(^ypyfj + % (ymyb)(yPya) + xpxc{ymya)(ynyb) Esaminando ora la quantita xmxa{ynyf){ypyf) Va{ymyb)(:ypyc) xpxfymyc){ynyb) xmxb(ynya)(ypyc) xnxb(ymyc){ypya) xpxb(ymya)(ynyc) xmxc(ynyb)(ypy0) xnxc(ymya)(ypyb) xpxc(ymyb)(ynya) . xmxa{xnxb(ypyc) + xpxc(ynyb) + xnxbxpxc - xnxc(ypyb) - xpxb(ynyc) - xnxpxbxcj + xnxa{xmxc (ypyb) + xpxb(ymyc) -i- xmxcxpxb - xmxb(ypyc) - xpxc(ymyb) - xmxbxpxc) + xpxa{xmxb(ynyc) + xnxc(ymyb) + xmxbxnxc - xmxfynyb) - xnxb(ymyc) - xmxcxnxbj, 1888-89.] Dr T Muir on the Theory of Determinants. 227 e le due espressioni che si traggono da questa, cambiando, prima a in b , b in e, c in a; poscia a in c, c in 5, b in a) con facilita si scorge che la somma di questi polinomj e nulla identicamente, per cui si potra aggiungere al prodotto (h) senza punto alterarlo. Eatta quest’ addizione, l’aggregato altro non sara che lo stesso polinomio ( h ), ove si supponga che i simboli (ynyb), (ypyc)i ecc. rappresentino rispettivamente i trinomj seguenti Vb + ynyb + + yPVc + Vc > ecc. Se ora si ordineranno le espressioni (l) portando fuori dalle parentesi y ovvero z in luogo di x, formeremo il prodotto delle medesime cosi scritte, ed opereremo come sopra, il risultato sara il polinomio che si desume da (h) cambiando le x che sono fuori dalle parentesi in y ovvero in z egualmente accentate. Se faremo per ultimo la soinma di queste tre espressioni, tal somma si cavera dal polinomio ( h ) scrivendo ( xmxa ) ovvero (ymya) invece di xmxa; (xpxa) in luogo di xpxa ec. ec. e sara eguale al triplo prodotto delle expressioni (Z). Essendo poi quella somma divisibile per tre, effettuata la divisione per questo numero, avremo (1) {a?m, yn, zpJ.^xa) ybi zcJ (^xmxa) (xnx^j (xpx^j + (xnx^j (xpx*j ) (xmx<^ + (xpxa)(xmxb)(xnxc) - - {xnxa)(xmxb)(xpxc) - (xpxa)(xnxb)(xmxX” (xvn. 6) That the essential points of this method of demonstration may be seen, let us apply it as it would be applied if adopted at the present day. The given determinants being | afbfr | and | a^2y3 1 , we should say | af 2c3 ] = oq | &2c3 I — ^2 I ^lC3 i + | ^lC2 I J and | ai/?2y3 \ = a1\ /?2y3 | — a2 | ^y3 | + a3 | ^y2 \ j hence, using the multiplication-theorem as established for determin- 228 Proceedings of Boyal Society of Edinburgh. [sess. ants of the 2nd order, and (to save on the breadth of the page) denoting in -j Ctf b,C, .. . aa + b^ + cy + . . . by ’ a,(3,y, . . . we should have I a AC3 I • I al/^273 I = «lal ^2’ C2 &’72 ^2? C2 @3>73 a2ttj ^l» cl P2^2 V?i Aj>78 + agCtj ^l> C1 P&72 ci @3’73 ^3> C3 ^3> C3 ^3> C3 ^3» C3 ^2» g2 ^2» C2 $2?l2 ft>73 A’ 72 Ap7b @2)72 ^3>73 - a1a2 ^2> C2 ft>7i ^2> C2 /^3’73 C?20t2 &1> C1 ft>7i ^1> C1 /^3>73 — %a2 hlfl Pv7i /?3>73 ^3’ C3 ^3j ^3> C3 ^3> g3 bjpf 2 ^2> C2 &>7i ft’73 /^l»7l ft’73 £l>7i /^3>73 + ^la 3 b c2 ft»7i 52, c2 ft’72 — $2a3 A»7i $2?l2 + %a3 &i, Cj Al>7i ^1> C1 /^2’72 bsi c3 ^3> ^3 ^3> C3 ^3> C3 ^2> g2 ^2> g2 ft>7i ft’72 ftiTi A’72 Pv7i P2’72 That each line of this result is not altered in substance by writing c2 £or a2’/^2>72 ^2> C2 P&72 9 %>^2> C2 a3’/^3’73 for &c„ P3>73 would probably be shown by expressing the line in the form of a determinant of the 3rd order, e.g ., the first line in the form bv Cj &1, Cj $2*72 03>73 ^2 > g2 ^2? g2 02*72 Ps’7s ^3> g3 ^3’ C3 02>72 03>73 and increasing each element of the second column by a2 times the corresponding element of the first, and each element of the third column by a3 times the corresponding element of the first. The whole result would in this way be transformed into 1888-89.] Dr T. Muir on the Theory of Determinants. 229 av ^1> C1 ^1^2 ^1> ^1 ) ^i» a2>/^2’72 a3’/^3’y3 ai»A>yi a3^3’73 ^2al ^2’ C2 a2, &2, c2 a2a2 a2’ ^2’ C2 a2» ^2» C2 a2>/^2>y2 “jp/Ws ai>/^i>yi aA73 a3 al a3’ ^8’ C3 a3'> ^3> C3 a‘da2 ^3’ C3 a3 ’ ^3> C3 a2’/^2’y2 a3^3^73 avftv7i a3'P3’73 Oh a, 13 tto.a, 3 3 av \,cx «1> ^1’ C1 aiA’yi a2’p2’72 ^2 ®2> ^2> C2 ai»A>yi a2’^2»y2 CJg, Co ®3> ^3» C3 anft5yi a2?^2’y2 jSTow by either of the interchanges /ai j ^2 » a3 J al » a2 ’ a3\ /®1 » ^2 > a8> al > a2 > a3\ > ^2 » ^3 » A f @2 > /^r 'C1 f C2 ’ CS i Tl ) 72 J y8/ the first columns of this, — and the first columns only, — would be affected, the a’s and a’s becoming &*s and /3’s respectively in the one case, and c’s and y’s in the other ; and as neither interchange could affect the left-hand side of our identity, we should consequently note that thus three different expressions would be at once obtained for |a1&2c3| . |a-,jS2y3| . Adding these together, and combining the nine determinants of the sum in sets of three by means of the addition-theorem (xlvil), we should have finally ^1J C1 a\,\,cx ai^nyi a2’Pv72 a3^3->73 ®2» ^2» ^2 ^2’ ^2» ^2 a2'> ^2> C2 ai^i»yi a2^2^72 a3’^3’73 Co ^35 ^35 ^3 ®3’ K ^3 avPv7i a2$2->72 a3^3’73 from which it is only necessary to delete the common factor 3. 230 Proceedings of Royal Society of Edinburgh. [sess. JACOBI (1831-33). [De transformatione integralis duplicis indefiniti f dcfidxf/ A + Bcos <£ + C sin + ( A' + B' cos + C' sin ) cos if/ + (A" + B" cos $ + C" sin<£) sin ^ in formam simpliciorem Z' — • : — . G- G cos^costf- G sm^sin# [ Crelle’s Journal , viii. pp. 253-279, 321-357.] [De transformatione et determinatione integralium duplicium com- mentatio tertia. Crelle’s Journal , x. pp. 101-128.] [De binis qnibuslibet functionibus bomogeneis secundi ordinis per substitutiones lineares in alias binas transformandis, quae solis quadratis variabilium constant ; una cum variis theorematis de transformatione et determinatione integralium multiplicium. Crelle’s Journal , xii. pp. 1-69]. Tbe first two of these memoirs may be viewed as continuations of a memoir with a similar title, which appeared in the second volume of Crelle’s Journal , and to which we have already referred. They are noted here merely in order that the thread of investigation may be preserved unbroken, for the last memoir practically swallows up, by means of its splendid generalisations, all those that had gone before. So long as we confine ourselves, in problems of transformation, to three independent variables, the explicit employment of the theory of determinants may be dispensed with. When, however, a sufficient number of special cases have been investigated, and an alluring glimpse has thereby been got of a generalisation involving them all, he who attempts the establishment of the generalisation must have recourse to the new weapon. In this latter position Jacobi now found himself. He wished to pass from the problem of orthogonal substitution in the case of three variables to the analogous problem in which the number of variables is ?i, or in his own words (p. 7) : — “ Investigare substitutiones lineares huiusmodi y ^ = 4* 0*2 4* • • • • “H CLn 5 y2 — cq" xx + a2" x2 + . . . . + aw" Xn , y* a^n)Xx + a2{n)X2 + .... + an[n)Xn , 1888-89.] Dr T. Muir on the Theory of Determinants . 231 quibus efficiatur VlVl + V2V2 + •••• + VnVn = + Xfa + . . . . + XnXn1 simulque data functio homogenea secundi ordinis variabilium x1,x2, . . . , xn transformetur in aliam variabilium yvy2 , . . . , ?/w, de qua binarum producta evanuerunt.” This being the case he introduces determinants at the outset, fixing upon a notation which is practically Cauchy’s, and imme- diately using properties of them without proof. Much that is contained in the memoir falls to be considered later, as it concerns special forms of determinants, — those afterwards known as Jacobians, axisymmetric determinants, and, of course, determinants of an orthogonal substitution. Indeed, the half-page of introduction is almost all that is of interest at present, but even in this a new and important theorem is enunciated. The first sentence of it stands as follows : — “ Supponamus, designantibus ak(m) datas quantitates quaslibet, ex n sequationibus linearibus propositis huiusmodi ym = a1(m)x1 + a2{m)X2+ .... + an{m]Xn , per notas regulas resolutionis algebraicse haberi sequationes formse : A ^ = ftV1 + A'V2+ • • • • +A(%. Ipsum A supponimus denominatorem communem valorum incognitarum, qui per algorithmos notos formatur : sive fit A = %± a/a2" .... eft signo summatorio amplectente terminos omnes, qui indicibus aut inferioribus aut superioribus omnimodis permutatis pro- veniunt ; signis eorurn alternantibus secundum notam regulam, quam ita enunciare licet, ut termino cuilibet per certam permutationem indicum orto idem signum tribuatur, quo afficitur productum sequens conflatum e clifferentiis numerorum 1. 2 (2 - 1)(3 - 1) .... (n - 1) . (3 - 2)(4 - 2) .... (n - 2) . (4 - 3) etc., eadem numerorum \ permutatione facta.” 232 Proceedings of Royal Society of Edinburgh. [sess. It will be at once observed that Cauchy’s italic letters S, a, b are simply changed into Greek a, (3. The next sentence is : — “ Eadem notatione adhibita, sit B = 3 ± P1P2' • » • • Pnn J ubi ipsam B e quantitatibus /?fc(m) eodem modo compositam accipimus, quo A ex ipsis ak(m) componitur. Quibus statutis, observo fieri : B = Aw-\ ac generalius : 2 ± A'ft" = A-*a ± O • • • • <£’ •” (xx. 3) As for the first theorem thus formulated, the credit of it is, of course, due to Cauchy : the second, however, is new, being indeed the theorem referred to above under Minding as having been fore- shadowed by Lagrange, and left for over fifty years undisturbed. Jacobi evidently knew it in all its generality, for he adds — “ De qua formula generali cum pro variis valoribus ipsius m, turn indicibus et superioribus et inferioribus omnimodis permu- tatis, permultae aliae similes formulae profluunt.” The only other point to be noted at present is contained in the casual remark that the /3’s may be expressed as differential coefficients of A. When dealing later (p. 20), with a special form of determinant, he says — “ Data occasione observo generaliter, si aK ^ et a\tli inter se diversi sunt, propositis n aequationibus linearibus hujusmodi : a\,lUl + al,2^2 + ••••+ al,nUn = V1 j a2,l^l "t" a2,1^2 + •«••+ a2 ~ ^2 > an,lUl + an,2U2 +••••+ <*> n,nUn = Vni statute T= a14a2>2 .... an>n , sequi vice versa : 1888-89.] Dr T. Muir on the Theory of Determinants. 233 IX = IX = 0r dT 0r a — vi + ®ai,i 3 + . . da 2>i • • s — °an,\ ar 0r dT '^1 + 3 v2 + . . (j(X 2,2 . . + ~ 1 0aw, 2 0r 0r 0r 3 Vx + Val,n 3 ^2 + • • °a2,n . . + o — « ^an,n (vi. 6) JACOBI (1835). [De eliminatione variabilis e duabus aequationibus algebraicis. Crelle's Journal , xv. pp. 101-124.] In a memoir having for its subject Bezout’s method of eliminating x from the equations anxn + a^x71'1 + .... + axx + a0 = 0 bnxn + b^x71'1 + .... +51x + 50 = OJ determinants are certain to occur explicitly or implicitly ; and, the author being Jacobi, one is not surprised to find them introduced near the outset and employed thenceforward. It is of course only a special form of them which appears, viz., that afterwards distin- guished by the term jpersymmetric ; consequently, for the present the main contents of the memoir do not concern us. Note has to be made, however, of two points — (1) that while Jacobi does not discard his former notation 21 ± ar „ ar „ . . . ar „ , he introduces and uses another, viz., • • • 5 'I'm S0’ S15 S2’ * * * ? Sm (2) that a page is devoted *to a fuller statement of the above- mentioned theorems regarding the adjugate determinant and a minor of the adjugate. The final sentence of this statement is all that need be reproduced. It is “ Sint igitur r/y\ . . . . , r(w-1) atque s,s',s", , s(w-1 numeri omnes 0, 1, 2, ... , n - 1, quocunque ordine scripti; erit (vn. 8) 234 Proceedings of Boyal Society of Edinburgh. [sess. A Jm) «j(m+l) s i 6 J .(«- 1) ) ( r> /, . , . . . , ^m_1) > = Lw-(1+m)., represent the intensity of excitation at A and B, fE^dt is >J‘E(B)dt for corresponding limits. Considering time- integrals, B may, therefore, he looked on as positive to A during the tetanus. The galvanometer deflection produced by stimulation will be a measure of the difference of these integrals. 236 Proceedings of Royal Society of Edinburgh. [sess. upon the work of which this paper is an account. But from certain experiments on the effect of stimulation on the intrapolar current during the flow, and on both extra and intrapolar currents after the opening of the polarising stream, I suspected that, if one of the galvanometer electrodes were placed very near the polarising circuit, and the strength of the current increased sufficiently, a positive electrotonic variation ought to appear on the side of the anode, but not on that of the cathode. For the explanation of those experiments it was assumed, and the assumption was supported by direct experiments on muscular contraction, that during the flow of the polarising current the conductivity of the nerve for the excitatory change is less around the cathode than around the anode, and that, with increasing strength of current, complete block occurs sooner at the former than at the latter, although eventually it prevails at both. Whether, when this last stage is reached, the whole intrapolar area has lost its conductivity, was left an open question, and need not be considered here. Going back now to fig. 1, let us inquire what the effect would be on the side of the anode, i.e ., with descending current, at a time when complete block was established there, and at the same time let us suppose that the galvanometer circuit is brought quite close to the anode, so that the lower galvanometer electrode is within the non-conducting region. If stimulation be now made at I, the excitation will pass A with a certain intensity, but will altogether fail before reaching B. B will, therefore, be strongly positive to A. We leave out of account for the moment any possible effect of the excitation on the electrotonic currents as such. There will be a current of action developed in the descending direction through the nerve — that is, in the same direction as the anodic electrotonic current. If this true action current be not masked by an over- whelming negative electrotonic variation, it will appear as a positive variation of the electrotonic current. Now let us take the case of the ascending current in fig. 1, Here the lower galvanometer electrode is in the cathodic region, and we know that even with comparatively weak currents the cathodic block appears. B will therefore, above a low limit of current density, be positive to A when the nerve is excited, and the true action stream will be descending. The cathodic electrotonic current, 1888-89.] Mr G. N. Stewart on Electrotonic Variation. 237 however, is ascending, and the action stream will appear as a negative variation of it. These are the considerations which led me to expect that a positive variation, if it existed, would he found with strong currents upon the anodic side, hut not upon the side of the cathode. It was not overlooked that the ordinary electrotonic negative varia- tion might he so large as to reverse the action current. Still it was hoped that, even in this case, indications might he found in the curve of the stimulation effect to show that the expected true action current was really in play. Method of the Investigation. The first one or two observations were made without compensat- ing the electrotonic currents. They, indeed, give the same general results as when compensation was used. But it was obvious that it would not do to accept a positive variation on the evidence of an uncompensated anodic current. For it would he necessary to show Fig. 2. — A, B, C, D are electrodes ; I, stimulating electrodes ; G, galvano- meter; P, P', Polil’s commutators; Com., compensator; Bat., polarising battery ; K, cell connected with commutator ; F, is a paraffin double key by which the polarising and galvanometer circuits were closed at the same time. that the apparent positive variation was not analogous to that which the intrapolar current undergoes when the nerve is stimulated, the so-called “ charge of resistance effect.” It was found that after compensation the positive variation continued in undiminished or scarcely diminished amount. Hay more, over-compensation did not abolish, nor begin to abolish it. Fig. 2 shows the arrangement 238 Proceedings of Royal Society of Edinburgh. [sess. which was at first used ; a , b, c , d , represent the lengths of nerve IA, AB, BC, and CD, respectively. Experiments 1 and 2 are examples of the first method with- out compensation; Experiments 3 and 4, with compensation. It will be seen that on the cathodic side, i.e ., with ascending current, the stimulation effect has the negative sign with reference to the direction of the polarising stream. On the side of the anode the same is true up to an electromotive force of about 3 Daniells working through 9 mm. of nerve. Above this the effect becomes positive. This is so only when the distance C is small. In Experiment 4 it is seen that, with C = 6J mm., the positive effect does not appear with 7 Daniells, nor even when C -3J mm. When C is reduced to 1 mm., it comes in even with 3 Daniells. Experiment 1. Distances — a, 9 mm. ; b, 10 mm. ; c, 2 mm. ; d, 9 mm. I Polarising Current. Stimulation Effect. Polarising Current. Stimulation Effect. 1 D Rh. 100 cm. * 1 D i -12 4 - 3 1 D 4 -15 4 - 3 3 D t -55 1 D Rh. 2000 cm. 4 -25 3 D 4 + 50 t -20 5 D t -42 1 D 4 -20 5 D 4 + 53 Experiment 2. Polarising Current. Stimulation Effect. lDf -14 Galv. shunt 10. 1 D 4 -16 3 D 4 + 45 4 D 4 + 29 No compensation in Experiments 1 and 2. Experiment 3. — Here two sets of observations were taken on the same nerve, the distance between electrodes B and C being altered. * The total resistance of the Rheochord was 2000 centimetre units. 1888-89.] Mr G. N. Stewart on Etectrotonic Variation. 239 ls£ set. — Distances — mm.; b , 9 mm.; c, mm. ; d, 9 mm. Polarising Current. Stimulation Effect. 5 D l + 45 Shunt 10. 1 D l - 6 5Dl + 27 Not compensated. 5D1 + 129 Compensated. No shunt. 5 D f - 51 99 99 2 nd set. — Distance- — c, i\ mm. Other distances the same as before. Polarising Current. Stimulation Effect. 5 D l -22 Compensated. No shunt. 5 D| -10 )) 99 Experiment 4. Distances — a , mm. ; b , 9 mm. ; c, 6| mm. ; d, 9 mm. Polarising Current. Stimulation Effect. Same nerve ; distance c made 3 J mm. lDl */ -47 \ -42 Polarising Current. Stimulation Effect. 2 D l -41 (?) 7 D Jr -37 3 D 4 / -68 \ -69 lDfr - 5 2D| / -65 \ -82 Same nerve ; distance c made 1 mm. 1 D l / -28 t -32 Polarising Current. Stimulation Effect. 5 D 4, / -45 \ -50 1D| -22 3D| + 34 7 D 4 -28 5D| + 26 * The bracketed numbers represent double readings. r In Experiment 4 only half of the galvanometer was in circuit. The deflections given must he doubled in order to compare with the preceding experiments. 240 Proceedings of Boy al Society of Edinburgh. [sess. These results suggested that it might he still better to put elec- trodes B and C in contact, so as practically to make them one elec- trode. Of course it was here necessary to attend to compensation even more strictly than before ; for the danger of a direct escape of current was greater than before ; hut so long as the galvanometer circuit was fully compensated, even such an escape would introduce no error. Experiment 5 gives an example of this method. Experiment 5. Polarising Current. Stimulation Effect. 1 D Rh. 1000 cm. f - 80 1 D Rh. 100 t + 6 After 30" closure. - 18 „ T „ - 22 „ 2' „ - 28 } 5 ^ , , - 35 „ 4' „ - 38 „ 5' „ - 43 15" after opening polarising current. 2Df -110 -133 30" after opening. 3 D t ? Owing to unsteadiness, difficult to read amount, but certainly less than - 100. 3 D l -117 20" after opening. + 255 -215 Another reading. Current kept closed for 5' before readings taken. + 48 30" after opening. With 1 D Rh. 100 f a small positive deflection was got. I have a good many times observed that when the nerve is perfectly fresh, the polarising current very weak, and the reading taken very soon after closure, a positive stimulation effect is got on the side of the cathode. This would suggest that the conductivity around the cathode is not reduced immediately on closing such a current, hut may even he increased. This agrees with what I saw occasionally when stimulating in the middle of the intrapolar area, with the muscle attached. Sometimes with weak currents the descending was more favourable than the ascending for. getting Contraction. This never happened when the currents were fairly strong. Werigo also, in his experiments on intrapolar stimulation, quite 1888-89.] Mr G. N. Stewart on Electrotonic Variation. 241 different in purpose from mine and essentially different in method, found that the cathodic block took time for its establishment, and that, when it appeared, it appeared suddenly. In the example given in Experiment 5 the initial positive effect is seen to change in 30" into a negative effect thrice as great, and this negative effect then gradually increases with still longer time of closure. In order to diminish, as far as possible, the irregularities in- the deflection, which are always a source of trouble with strong electro- tonic currents, especially on the anodic side, I thought of using the currents led off to the galvanometer from two separate nerves of the same frog to compensate one another, a method resembling some- what in principle that which Hermann has used in some of his polarisation work. Then, on exciting one of the nerves, one ought to get the stimulation effect, weakened, of course, by the extra resistance of the second nerve. The same battery was connected with both nerves, so that irregularities in the battery itself might be eliminated. The result was very satisfactory. Figs. 3 and 4 show the arrangement. In the arrangement of fig. 4 two nerves were placed on two separate sets of electrodes A, B, D ; A', B', D', a compensator (Com.) being introduced into the galvanometer circuit. Fig. 3. — G is the galvanometer ; Bat., the battery ; I, the stimulating electrodes. The pieces of nerve BD, B'D' were made as nearly as possible equal in length, and therefore the current would have nearly the same density in each. The electrotonic currents in AB, A'B' would be nearly equal, and they would pass through the galvanometer in opposite directions. The balance was completed by means of the compensator. In the arrangement of fig. 4 the polarising current passed to vol. xvi. 16/7/89 Q 242 Proceedings of Royal Society of Edinburgh. [sess. both nerves through the same electrodes C,D, and the density would therefore be more nearly equal in the two than with the arrangement of fig. 3. As before, a compensator was put in the galvanometer circuit. B was not an electrode, but only a movable i bridge of clay. If we stimulate at I, it will depend upon the distance of B from C whether the anodic effect will be positive or negative. Experiments 6 and 7 are samples of the results got by this method. Experiment 6. Distances — a, 10 mm.; b, 10 mm.; c, 2 mm.; d, 13 mm. Polarising Current. Stimulation Effect. 1D| -184 3D| - 38 5D! + 58 8 D| + 68 1 D + 138 1 D Rh. 90 cm. 4 - 53 5D! + 30 8D! + 63 ID! - 79 2D! - 76 Experiment 7 shows the change of sign on the anodic side even with 2 D. The negative effect on the side of the cathode seems here to diminish with increase of current, and this might suggest that with still stronger currents a positive phase might be found. I cannot say that I have found any trace of such an effect, and it is only in exceptional cases that the diminution in the negative effect appears. 1888-89.] Mr G. N. Stewart on Electrotonic Variation. 243 Experiment 7. Distances — a, 7\ mm.; b, 7\ mm.; ^,10 mm. Polarising Current. Stimulation Effect. 2 D f - 40 B and C in contact. 1 D Rh. 90 cm. t - 8 3 D t - 28 5 D t 1 r - 24 I - 22 1 D f - 9 2 D f - 14 5 D t - 3 lDj 1 i -127 | -114 2D j i C + 28 t + 22 5 D l \ r +126 1+119 1 D 4, i > - 46 \ - 49 It was now desirable to compare the amount of the intra- and extrapolar stimulation effects ; and in order to avoid the uncertainty which must always exist when one tries to get quantitative com- parisons from experiments made on different days with different nerves, I determined to control the other observations by means of a set in which the intrapolar and extrapolar regions of the same nerve were led off alternately to the galvanometer. It was parti- cularly important to notice how the ratio between the amount of the two effects varied with varying density of polarising current when the latter was nearly strong enough to suppress the intrapolar effect altogether. The arrangement is shown in fig. 5 for the case where the two extrapolar regions compensate each other, and the two intrapolar regions are placed one in each coil of a differential galvanometer. A, B, C, D are, as before, the electrodes of one of the nerves ; A', B', C', D' those of the other. G, G' are the two coils of the differential galvanometer ; P is a Pohl’s commutator without cross wires, by means of which either AB or CD may be joined on to G : P' is a Pohl with cross wires, to alter the direction of the polarising current ; Com. is a compensator to complete the compensation when the extrapolar areas are led off; R is a rheostat to equalise the 244 Proceedings of Royal Society of Edinburgh. [sess. intrapolar currents. By an arrangement not shown it could be thrown either into CD or into C'D'. The balancing arrangement was used for the intrapolar currents in order to diminish the irregularity in the deflection, which is much more troublesome than even in extrapolar experiments. Of course, only one nerve was stimulated. i The circuit of G' was broken by a simple key, whenever the extrapolar regions were to be connected with G. The current was then passed through CD, compensation completed in the galvano- meter circuit, and the stimulation effect read off. After the nerves had recovered, the two intrapolar regions were thrown on to G and G', the extrapolar being off. The same current was now passed again in the same direction, for the game length of time, and the stimulation effect again taken. A given number of cells would give practically the same current density in CD, whether the alterna- tive circuit C'D' was open or closed, since the internal resistance of the battery is very small compared with the resistance of the nerves. Experiments 8, 9, and 10 (pp. 246, 247) are examples of this method as applied to currents near the limiting intensity. An electromotive force of about 5 Daniells working through 9 mm. of nerve gives the density corresponding to the disappearance of the intrapolar effect. This limiting electromotive force will be inversely as the length of nerve included in the circuit, if we assume that the specific resistance 1888-89.] Mr G. N. Stewart on Electrotonic Variation. 245 of nerve in the longitudinal direction is a constant. Of course it will vary slightly even for the two nerves of the same frog, as it will depend mainly at least upon the amount and kind of the dissolved crystalloids. In my former results on the intrapolar effect I found that the limiting electromotive force varied from 8 to 9 Daniells, when the length of nerve was from 12 to 14 mm. The two sets of experiments therefore agree as well as one is entitled to expect in observations of this sort. The strength of stimulus, of course, has also to be taken into account. Experiments 12 and 13 show how the effect in the extrapolar region reaches a maximum, while in the intrapolar it declines to a minimum. This of itself is quite enough to dispose of the pos- sibility that the suppression of the intrapolar effect is due to the decline of excitability at the point of stimulation through the spread of anelectrotonus. Experiment 14 is an example of stimulation on the cathodic side. Fig. 6. Fig. 6 shows the curve of the effect in Experiment 12 plotted to scale. The number of points are too few to determine the details. Almost the whole of the ascent lies very nearly in a straight line. 246 Proceedings of Royal Society of Edinburgh. [sess. Summary of Results. 1. With weak currents there is the ordinary negative variation both on anodic and cathodic side, however close the galvanometer and polarising circuits may he brought to each other. 2. As the strength of the polarising current is increased, the negative variation on the anodic side passes into a positive variation, which increases and apparently reaches a maximum. 3. The maximum of the positive anodic variation corresponds to a density of current which is not far from that for which the intra- polar variation is at its minimum (zero). 4. On the cathodic side the variation is always negative with currents above the very weakest. (With very weak currents, fresh nerves, and short period of flow, sometimes a small positive varia- tion seems to he got.) 5. The greater the distance between the polarising and galvano- meter circuits, the stronger must the polarising current be for which the positive anodic variation first appears. All these results hold when the electrotonic currents are com- pensated. Experiment 8. Distances — a , 9 mm.; 5, 7\ mm.; c, 1 mm.; d, 9 mm. Polarising Current. Extrapolar Stimulation Effect. Intrapolar Stimulation Effect. 4 D 4r 38 0 4D! 57 5 4 D l 39 Experiment 9. Distances — a , 9 mm. ; b, 7£ mm. ; c, 1 mm. ; d, 9 mm. Polarising Current. Extrapolar Stimulation Effect. Intrapolar Stimulation Effect. 4 D 4, + 214 + 50 | 4 D 4 + 81 + 55 J 4 D| + 230 l 4 D | + 170 Here before passing current there was a stimulation effect of 38 in same direc- tion as intrapolar effect. Here there was a stimulation effect of 87 in same direc- tion before passing current. 1888-89.] Mr G. N. Stewart on Electrotonic Variation. 247 Experiment 10. Polarising Current. Extrapolar Stimulation Effect. Intrapolar Stimulation Effect. 5 D 4, + 180 0 Experiment 11. Distance — d, 12 mm. Polarising Current. Intrapolar Stimulation Effect. 1 D t Rh. 2000 + 66 Stimulus 85. 5 D t 2000 0 Even with strongest stimula- tion. 5 D 4, 2000 + 55 Coils close up. } ) 1 D 4 2000 + 87 5D| + 30 7D| + 14 5? 1 D Rh. 2000 r+851 t +90/ ? y Experiment 12. Distances in Experiments 12 and 13 — a, 9 mm. ; b, 9 mm. ; c, 1| mm. ; d , 9 mm. Polarising Current. Extrapolar Stimulation Effect. Polarising Current. Extrapolar Stimulation Effect. 1 D 4 -26 7 D 4 + 126 2 D 4 + 25 10 D 4 + 106 3D| + 62 4 D 4 + 115 4 D 4 + 135 4 D 4 + 75 5 D 4 + 138 248 Proceedings of Royal Society of Edinburgh. [sess- Experiment 13. Polarising Current. Intrapolar Stimulation Effect. 1 D Rh. 50 l + 22 Same strength of stimulus. 1 D Rh. 2000 l + 41 2D 4, + 28 3 D 4 + 17 4D| + 24 5D| 0 1 D Rh. 2000 + 36 Experiment 14. Polarising Current. Cathodic Extrapolar Effect. 1 D t -27 2 D f -29 3Dt -31 5 D t -43 ? 10 D t -44 I do not propose to discuss here, further than I have done, the real significance of the results stated, as I hope soon to have an opportunity of doing so in connection with a more extended research, embracing the effect of stimulation on the whole of the polarisation phenomena of nerve and muscle. I should just like to say, that it is by no means impossible that a real positive electrotonic variation may be mixed up with a true action current in the positive direction. The work was done partly in the Owens College, and partly at Edinburgh in the Laboratory of Professor Kutherford, whose great kindness I take this opportunity of acknowledging. 1888-89.] Dr E. Sang on Fundamental Tables. 249 Notice of Fundamental Tables in Trigonometry and Astronomy, arranged according to the Decimal Division of the Quadrant. By E. Sang, LL.D. (Read June 3, 1889.) Canon of Sines. In January of 1878, there was laid on the Society’s table the Canon of Sines to each fifth minute of the decimal division of the quadrant, computed to thirty-three for thirty places ; along with a detailed record of every step in the process. During the years 1880-81, this work was continued for each single minute, but only to eighteen for fifteen places, and the record thereof to fifteen places is now submitted. When the rejected figures were from 497 to 503 a mark of interrogation is recorded, and it is believed that not a single error exists in the work. The arrangement of the sines with their first and second differences in position enables us instantly to detect an error. That fifteen places suffice for all possible practical purposes, is made clear by this consideration, that the Earth’s distance from the Sun, measured in inches, is represented by the number 6, twelve removes from the unit’s place, that is 6 000 000 000 000 ; and that, if we take this as the radius of our circle, the figure 1 in the fifteenth decimal place will represent ’006 or the 170th part of an inch. Thus the present canon gives, on this circle, the co-ordinates of the ten thousand points in the quadrant, each true to within the three- hundredth part of an inch. The process followed in this work differs, in one very important respect, from that used by previous computers. The sine of the smallest tabular arc has hitherto been found indirectly by help of repeated bisections ; in the present work the quinquisection of the arcs has been accomplished directly by the solution of the appropriate equations of the fifth degree, according to the method described in my treatise “ On the Solution of Equations of all Orders.” The ease and rapidity of this method are well shown by the recorded details of the work for the various equations, to thirty decimal places. A table of one thousand multiples of 2 ver. V having been pre- 250 Proceedings of Royal Society of Edinburgh. [sess. pared, the rest of the work was carried on in the usual manner with verifications at frequent intervals. Logarithmic Sines. The logarithmic sines were deduced from the sines themselves by help of my fifteen-place table of logarithms of numbers from 100 000 to 370 000, using the auxiliary table. Beginning at 100° 00', the computations were made on scroll paper for each single minute down to 50° 00', each step being verified by first, second, and third differences. The third differences were then copied into their places on the actual manuscript, and the others were thence reconstructed. In this way all errors of transcription were avoided, and any mistake in the previous work detected. From this half of the canon, the other half, namely from 50° 00' to 00° 00', was deduced according to the formula sin a = \ sin 2 a .sec a ; the method of proceeding being to compute each tenth log sine directly, and to fill in the single terms by differences easily got from the differences already written in the first part. This operation supplied a check on all the previous work. Logarithmic Tangents. The logarithmic tangents were computed in the same way, that is to say, each tenth term was found directly, and the single terms by means of the preceding differences, thus furnishing another verifica- tion of the whole. But, seeing that the log tangents of the one half are the arithmetical complements of those of the other half, it was enough to write out the log tangents of arcs from 50° 00' to 00° 00'. With the exception of the arrangement for computing by differ- ences, and for assuring exactitude, this is the very process used by JSepair in the construction of his Canon Mirificus ; and, indeed, this volume of Logarithmic Sines and Tangents might, with all propriety, have been entitled : — “ John Repair’s wonder-working Canon, changed by his express desire, to suit the Denary System of Arithmetical dotation.” After a labour which must have occupied his leisure time for more than the quarter of a century, Repair had published his 251 1888-89.] Dr E. Sang on Fundamental Tables. Canon, had experienced its utility, had received the approval of the scientific world; and yet, foreseeing the advantage of accommodating his plan to the notation in common use, he at once recommended the putting of it aside for a far better plan. No stronger evidence can he adduced in favour of discarding the time-honoured division by 90 and by 60, and substituting the decimal division throughout. Kepler’s Problem. While the Canon of Sines was still in progress, circumstances led to a repetition of the often fruitlessly made attack on Kepler’s famous problem, and this time an unexpectedly simple solution presented itself. The Royal Academy of Sciences of Turin did me the very great honour of giving that solution a place among their memoirs. The subject, however, may be treated more generally and even more simply, thus : — Let us suppose ourselves to be studying the apparent relative motion of a binary system of stars; each one seems to describe round the other an ellipse, and the areas passed over by the radius vector are proportional to the elapsed times. But, since the actual orbit may be inclined anyhow to the plane on which it appears to be projected, the one star does not appear to be in the focus of the orbit of the other ; nor is the diameter drawn through its apparent place, necessarily be the major axis. If we divide the periodic time into equal portions, the corresponding vectors will similarly divide the area of the ellipse, and hence the problem virtually comes to be this, — “ to subdivide the area of an ellipse by lines diverging from some point within it.” The line from the eye to the revolving star defines the surface of a cone, in our imaginary case sensibly of a cylinder, and the planes passing through the eye, and along the vectors, subdivide this cylinder into wedges. If now this system be cut by any plane, the section so made will have its area also subdivided ; now we can always cut a cylinder so that its section may be a circle, and thus, ultimately, the problem becomes this, “ to subdivide the area of a circle by lines diverging from a point within it.” If S represent the point given within the circle described round the centre O, the diameter ASOa will represent the line of apsides, A being the perihelion, a the aphelion. Let now Q correspond to 252 Proceedings of Royal Society of Edinburgh. [sess. some position of the planet, then the surface comprised between AS, SQ, and the arc AQ, is proportional to the time elapsed from the planet, being in the position A, till its reaching the position corresponding to Q, so that this surface is the planet’s mean anomaly. Draw now ESF perpendicular to AQ, then the arc AE, which has the excentricity OS for its cosine, may be called the arc of excentricity ; we shall denote it by e ; while AQ, the arc defining the planet’s position, may be denoted by p. Having joined EQ, FQ, it is seen, by mere inspection, that the mean anomaly AEQS is half the sum of the two circular segments cut off by the chords EQ, EQ, or that mean anomaly = |{segm(£> + e) + segm(jo - e)} , and so a table of circular segments would enable us to determine the mean anomaly when the position is given, and conversely the position when the mean anomaly is given. In order to avoid the frequent multiplication and division by the 1888-89.] Dr E. Sang on Fundamental Tables. 253 number tt, we measure the areas of the segments not in parts of the square of the radius, but in parts of the surface of the circle ; a superficial degree being the sector standiug on a degree of arc. For the construction of such a table, it was necessary to compute the canon of sines measured in parts of the quadrant. The sines for the single degrees were therefore computed by simple multi- plication of the ordinary sines, to serve as verifications of the subsequent work. Afterwards those for each quarter degree were obtained by using the previous multiples of 2 ver. 25' for second differences ; these two operations completely checked each other. Again the sines for each fifth minute were got by help of the multiples of 2 ver. 5'. But, as the computations were carried only to the tenth decimal of the quadrant, the products by 2 ver. V were not needed. Sines Measured in Degrees. In this way the “ Canon of Sines measured in degrees ” now presented was completed, the actual volume contains the whole details of the work, and it is hoped without any error exceeding two units in the tenth place. Canon of Circular Segments. Since the number which expresses the area of a segment in degrees of surface is the difference between those which express the arc and its sine, it follows that the second differences in the table of circular segments are identic with those of the sines ; and there- fore the canon of segments was constructed directly from those second differences. In the present volume it is extended to the entire circumference, that is to forty thousand minutes, and shows the value of each segment true to within two or three units in the ten-thousandth parts of the centesimal second. Its accuracy, thus, is very far beyond any requirement in actual astronomy. This work furnished another check on that for the canon of sines measured in degrees. This table of circular segments enables us very easily to discover the mean anomaly when the angle of position is known. The con- verse problem, “ to find the angle of position from the mean anomaly, ^ has to be solved by approximation ; which is sufficiently rapid if the first assumption be not very wide of the mark. When, for the 254 Proceedings of Royal Society of Edinburgh. [sess. orbit of some particular planet, we are computing the positions at equal intervals of time, attention to the differences reduces the labour to little more than that of writing out the results. It is only when a sporadic case is presented that the approximation is attended with any difficulty. Mean Anomalies. In order to obviate even this difficulty, a table has been con- structed of the mean anomalies for orbits of each degree of excen- tricity, and for every degree of the angle of position, up to 200° in each of these orbits. This table enables us, in every possible case, to get at once a first assumption so near as to make the subsequent approximation quite easily. This table is presented in two forms. In the volume marked mean anomalies A, the values are given to four decimal places of a second. In the corresponding volume marked B, they are written only to the nearest second ; but the differences and the variation from one orbit to another are inserted. Hence, by the ordinary method of interpolation for two variables, we can solve both the direct and the inverse problem with precision sufficient for all the purposes of practical astronomy. My intention was to have computed also the radii vectors and the true anomalies. For this, however, the only available trigo- nometrical tables were those to seven places printed in a most inconvenient form, by Callet, in his Tables Portatives. The work was scarcely begun when it became apparent that the precision attainable was not commensurate with the labour. Therefore, putting that work aside, I preferred to undertake the hopeless- looking task of computing the logarithmic sines and tangents to a greater number of places. This work is fortunately accomplished, although there still remain the transcription in the order usually adopted for convenient reference. The application of these tables to the computation of the true anomalies, is a task far too great to be undertaken at the close of a long life, and, not without reluctance, it is left to the zeal of other computers. Enough, that I have been enabled to place within the reach of mathematicians some contributions to the progress of exact science. 1888-89.] Dr E. Sang on Fundamental Tables. 255 The convenience of having the true anomaly, and the planet’s distance alongside of the time-argument, would be so great as to dwarf altogether that of having merely the angle of position ; this last mentioned forms, indeed, only a step toward the obtaining of the others. The remaining operation, implying only the solution of a right-angled triangle, is easy though laborious. It may, there- fore, be not inopportune to indicate here the course most convenient to be followed in the subsequent work; particularly because that which may appear to be the most rapid in an isolated case, may not be the best for systematic work. Distances. If P be the planet’s place in the ellipse AsA's', having S and S' for its foci, and SS' for its minor axis, and if the ordinate HP be a7 continued to meet the circle described on AA' as a diameter in Q, the arc AQ is p, the arc of position, and we have SIT = cosj9 - cos e ; PH = sin^>. sin e. 256 Proceedings of Royal Society of Edinburgh. [sess. From those we easily get the tangent of the true anomaly ASP, and thence the distance SP. Here the great part of the labour is in finding the logarithm of SH, the angle HSP from its log tangent, the log secant from the angle, and SP from its logarithm ; that is to say, in using the tables of the logarithms of numbers, and of circular functions to a considerable number of decimal places. This labour, repeated for each of the twenty thousand cases to be tabulated, rises to a formidable total. But if, on the perpendicular diameter AOA', we describe another ellipse having SS' for its minor axis, and consequently s and s' for its foci, and if from Q we draw the ordinate Q ph, we have, accord- ing to the properties of the ellipse, SP = 0 A =?ph, and conversely SP = OA ± PH. Thus the computation of the ordinates in the one of the two orbits gives us, with only the labour of writing the numbers in their places, the vectors of the other orbit, and we are now enabled to compute the true anomaly from its log sine. When following this course, it will be convenient to begin with the orbit e = 50°, and to take the others in couples, e = 49°, e = 51°, and so on. Our working formula then stand thus : — ( 1 tcos p. cos e ) distances < # > whence, log distances , [ 1 =p sin^? . sin e J log sin anomaly = log ord. - log dist., whence anomaly . If it were proposed to make these computations with all the precision obtainable from our fifteen-place tables, it might be economical, even for this single piece of work, to interpolate the logarithmic sines for each hundred-thousandth part of the quadrant. log ordinates log sin jp + log sin e log cos + log cos e whence ordinates 1888-89.] Prof. Tait on the Relation among Integrals. 257 On the Belation among the Line, Surface, and Volume Integrals. By Professor Tait. (Read April 1, 1889.) The fundamental form of the Volume and Surface Integral is fff Vuds = ffXSvuds . Apply it to a space consisting of a very thin transverse slice of a cylinder. Let t he the thickness of the slice, A the area of one end, and a a unit-vector perpendicular to the plane of the end. The above equation gives at once Y(aV)u.tA = t /V.aTJvudl, where dl is the length of an element of the bounding curve of the section, and the only values of UV left are parallel to the plane of the section and normal to the bounding curve. If we now put p as the vector of a point in that curve, it is plain that V. aUv = JJdp , dl = Tdp , and the expression becomes (after division by t) Y(dY)uA = fudp. By juxtaposition of an infinite number of these infinitely small directed elements, a (now to be called Uv) being the normal vector of the area A (now to be called ds), we have at once ff V(UVV)wcfe = f udp , which is the fundamental form of the Surface and Line Integral. In fact, as the first of these expressions can be derived at once from the ordinary equation of “continuity,” so the second is merely the particular case corresponding to displacements confined to a given surface. VOL. xvi. 12/8/89 n 258 Proceedings of Royal Society of Edinburgh. [sess. The Development of Diarthrodial Joints in Birds and Mammals. By David Hepburn, M.B., M.B.C.S. (Eng.), Senior Demonstrator of Anatomy , University of Edinburgh. Communicated by Professor Sir W. Turner. (From the Embryological Laboratory, University of Edinburgh.) (Read May 20, 1889.) After giving a summary of recent literature on the subject, the author then proceeded to state the nature of the material which he had employed in the present investigation. The bird selected was the common fowl ( Callus domest.), and he had examined a series of microscopic sections through the limbs from the fourth day of incubation to the day of hatching. The mammalian embryos examined were mice and rabbits, and the fingers of the human foetus from an embryo approaching the full period of uterogestation. Method of Preparation. — The embryo chicks were prepared, partly by hardening in picrosulphuric acid and partly in dilute solutions of nitric acid. The human embryos were also hardened in nitric acid. The embryos were then dehydrated with alcohol, stained in borax-carmine, and cut with the Cambridge rocking microtome, the average thickness of the sections being *006 mm. The author expressed his indebtedness to Mr George Brook, Lecturer on Embryology in the University of Edinburgh, under whose guidance and in whose laboratory the investigation had been conducted, and then proceeded to give a summary of the results which he had attained. At the end of the fourth day of incubation the wing of the chick - is in the form of a bud, '8 mm. long, and consists of a mass of mesoblast cells enveloped in a covering of epiblast. At this stage the cells of the mesoblast present no differentiation into separate structures, but at the end of the fifth day the free extremity of the limb has assumed a bulbous form, and horizontal sections show that along certain lines a process of condensation has occurred, apparently presaging the positions of future bone matrices. The individual cells in these portions present no great change from the rest of the 1888-89.] Mr D. Hepburn on Dicirtlirodial Joints. 259 surrounding mesoblast, except such variations in outline as can be accounted for by compression. As growth proceeds, digits emerge from the bulbous extremity, and a section at the end of the ninth day shows that a process of differentiation has taken place in the condensed portions, the result of which is the formation of cartilaginous rods separated from each other by masses of undifferentiated cells termed the articular disc or inter-tissue. It is in connection with this articular disc that the future joint cavity and its various appendages are developed. Here the joint cavity makes its appearance, and may be seen in the wing of the chick at the end of the ninth day. The cleft commences within the circumference of the articular disc and extends towards the axis of the disc, so as to divide it into two segments, each of which is applied to the end of a cartilaginous rod, and the segments are held together by its undivided periphery. The sides of the cleft are bounded by a layer of flattened cells. When this cleft does not extend across the axis of the disc, material is left for the formation of an interarticular ligament, as may be seen in sections taken from the leg of a chick about the middle of the second week. In the case of some joints two cavities appear, having between them a portion of the disc, which ultimately develops into a meniscus. Again, when the two cavities fuse in the axis of the disc, we have an incomplete meniscus. Even at this early stage there is a certain amount of moulding of the ends of the cartilaginous rods which foreshadows their future shape, and as this occurs at a time when the muscular system is in abeyance, it cannot be the result of movement, and neither can we ascribe the formation of the cavity to this cause. Tracing the changes which take place in connection with the "various parts of the now partially divided articular disc, we find that the segments applied to the ends of the cartilaginous rods gradually become differentiated into hyaline cartilage, until this process has affected the whole thickness of the segment, with the exception of the row of flattened cells next the cavity of the joint. In the chick these are found still persisting at the period of hatching. The undivided circumference of the disc has meanwhile under- 260 Proceedings of Royal Society of Edinburgh. [sess. gone differentiation into fibrous tissue, and in it may be found the rows of cells and wavy fibres characteristic of that structure. This fibrous capsule also continues to be lined by a row of flattened cells continuous with those on the surface of the articular cartilages. A similar series of changes may be traced in connection with inter- articular fibro-cartilages and ligaments. In mammals the formation of the articular disc and the appear- ance of the joint cavity, as seen in embryo mice and rabbits, are practically identical with those just described. There is therefore no reason to doubt that the subsequent changes in the component parts of the disc are also similar. There is, however, one striking difference in connection with the layer of flattened cells which line the cavity, and this was observed on the articular cartilages taken from the phalangeal joints of a human foetus approaching the full period of uterogestation. Here the usual flat cells were found lining the interior of the capsular ligament, but on tracing them towards the articular cartilages they were seen to be replaced by a narrow band, staining somewhat more freely than the hyaline cartilage, but presenting no cell structure. The free surface of this band was slightly ragged, and it appeared to be undergoing degeneration. Thus it would appear that the flat cells lining the primitive joint cavity have a double fate. Those in relation to the ligamentous structures, and thus within reach of a direct blood supply, become specialised into a synovial membrane; whereas those in relation to the articular cartilages, although present in the chick at the period of hatching, probably disappear as the result of friction ; while, in the case of the mammal, they undergo degenerative changes, which lead to their early disappearance from the same cause. Summary of Conclusions. 1. The bone matrices and the articular disc possess a tissue con- tinuity, and are derivative of a common blastema of which the articular disc is at first the undifferentiated form. 2. The articular disc may conduct itself as follows : — (a) It may develop into a plate of cartilage and form a synchondrosis, e.g., the articulation between basi- occipital and basi-sphenoid bones. 1888-89.] Mr D. Hepburn on Diarthrodial Joints. 261 (b) It may differentiate into fibrous tissue and form a syndesmosis or synarthrosis. (c) It may partially cleave and form a joint cavity. 3. The joint cavity appears within the articular disc at a period when the process of chondrification is at some distance from the cavity. 4. If the cavity remain of small size, and the surrounding articular disc develop into fibrous tissue, an amphiarthrosis is formed, e.g ., the joint between vertebral bodies. (This is specially well seen in some Cetacea, and probably the epiphyseal plates on the bodies of vertebrae are also derived from the articular disc.) 5. The cavity may enlarge to form a diarthrosis. 6. When the joint cavity is single we have a simple diarthrosis ; when there ^re two cavities we have a diarthrosis with an interposed meniscus ; when the two clefts unite in the centre we have an incomplete meniscus ; when the cleft is single, but does not extend across the axis of the disc, an interarticular ligament is formed. 7. The proximal and distal segments of the articular disc develop into the articular cartilages of the joint, and probably form part, if not all, of the epiphyseal ends of the bones. 8. The circumference of the articular disc develops into the capsule of the joint. 9. Interarticular fibrocartilages and ligaments are derived from the articular disc as the result of modifications of the joint cavit3r. 10. The cells lining the joint cavity have a double fate : — (a) Those applied to the ligamentous structures are specialised as synovial membrane. (b) Those upon the articular cartilage persist until the period of hatching in the chick, but undergo degeneration in the mammal ; in both cases dis- appearing by friction as the result of movement. 262 Proceedings of Royal Society of Edinburgh. [sess. Electrification of Air by Flame. By Sir William Thomson. (Read July 15, 1889.) In continuation of experiments on the electricity of air within doors, which I made twenty-seven years ago, and which are described in §§ 296-300 of my Electrostatics and Magnetism, a series of ob- servations was commenced under my instructions at the end of April last, within the Natural Philosophy Class Room and Laboratory, the Bute Hall, the University Tower, and other places inside and outside the buildings of Glasgow University, by Mr Magnus Maclean, official assistant to the Professor of Natural Philosophy, and Mr Goto of Tokio, Japan, for the purpose of endeavouring to find a relation between the electrification of air within a building and the atmospheric potential in its neighbourhood outside ; and of finding causes which produced or changed the electrification of any given mass of air. A large number of series of observations have been made by Mr Maclean and Mr Goto on the potentials of water-dropping collectors within the building, and at different points outside the building. Hitherto no definite relation has been discovered between the ex- ternal potential and the potential at different points within large enclosures, such as the Bute Hall and smaller rooms of the Uni- versity Buildings. The weather has been for the most part very settled and the external potential almost always positive in all posi- tions from a few feet above the ground to the top of the University Tower; while the potential within doors here, in the new University Buildings, on the top of Gilmour Hill, just as in the old College, down in the densest part of Glasgow, was always negative except sometimes in the Natural Philosophy Lecture Room and Apparatus Room, where there were considerable disturbances, undoubtedly due to the electric light wiring. In one ordinarily unused room (the Physical Apparatus Museum) 31^ feet long by 24 feet broad and about 20 feet high, practically quite free from any sensible dis- turbance by electric light wires, or by electrical operations being performed in the Laboratory, a remarkable result has been observed within the past fortnight. The electric potential of a water-dropper 1888-89.] Sir W. Thomson on Electrification of Air. 263 having its nozzle at the centre of the room and about 7 feet above the floor, was always found about 2 volts negative at the commence- ment of the observations, and always increased to about 9 volts in the course of the first twenty minutes of a series of observations last- ing generally forty minutes. During the last twenty minutes of the series the potential remained somewhat nearly constant at 9 volts. Within the room, two quadrant electrometers, each with an ordin- ary paraffin lamp and scale, were used ; one of them for the outside water-dropper, and the other for the water-dropper within the room. Towards ascertaining the cause of this change, an observation was made on the 4th July, between 10 and 11 a.m. The lamps were both extinguished, and one of them was lighted by a lucifer match every five minutes for the purpose of reading the electrometer de- flection. It was found that in these circumstances there was not the increase of negative potential which had been found in every previous series of observations in the same place, and with all other circumstances the same, except the burning of the lamp. This single observation seemed to prove conclusively that the burning of the lamp produced a negative change of the air of the room. Subse- quent experiments made by Mr Goto, with the electrometer and its lamp and scale outside, and with paraffin lamps burning or not burning within the room, have confirmed this result, and are being continued to discover whether corresponding effects are produced by other kinds of flame, or by the presence of eight or nine people in the room. Mr Maclean and Mr Goto will also continue their observations on natural atmospheric electricity, in various localities, indoors and in the open air, and will, I hope, give a paper to the Royal Society of Edinburgh early next session on the subject. 264 Proceedings of Royal Society of Edinburgh. [sess. On the Placentation of the Halicore Dugong. By Professor Sir William Turner. (Read July 1, 1889.) {Abstract.) The only observations hitherto recorded on the placentation of the Dugong are by Paul Harting, of Utrecht, in 1878, who ex- amined the foetal membranes of a foetus 27*8 cent. long. He stated that the placenta was diffused and non-deciduate.* The gravid uterus described in this communication was pre- sented to the author by C. W. de Yis, Esq., M.A., curator of the Queensland Museum, Brisbane, through Professor Anderson Stuart of the University of Sydney. The uterus was bicornuate, and contained a single foetus, 5 feet 4 inches long. The foetus and its membranes occupied the left cornu, and there was no extension of the membranes into the right cornu. The chorion was an elongated sac, upwards of 5 feet long from pole to pole. The placenta formed a zone a little on one side of the equator of the chorion. The zone was 11 J inches broad in its widest part and 6 inches at its narrowest. The rest of the chorion was smooth and free from villi. The villi were closely crowded together in the foetal placenta ; as a rule they were short, though longer villi were interspersed amongst them ; they were cylindriform and filamentous in shape, and branched seldom except near their free ends. The allantois was very extensive, and reached to the opposite poles of the chorion. Connected to the outer wall of its sac, formed by the endochorion, were a number of plate-like allantoic bodies. The amnion was very capacious, and was completely surrounded by the allantois, except for a limited area in the region of the placenta. Ho amniotic corpuscles projected from its inner surface. The uterine mucous membrane had a zone which formed the maternal placenta, and which corresponded in form, size, and * See abstract of his paper in the Journal of Anatomy; vol. xiii. p. 116. 1888-89.] Prof. Sir Wm. Turner on the Halicore Dugong. 265 position to the zone on the chorion. This zone contained multi- tudes of short cylindriform crypts, in which the villi of the chorion were lodged. Longer and more deeply placed crypts were also present for the lodgment of the longer villi. The non-placental area of the mucous membrane was smooth, and corresponded to the non-villous part of the chorion. Uterine glands were seen both in the placental and non-placental areas of the mucous membrane. In the placental area they opened amidst the crypts by special orifices; in the non-placental area they opened obliquely on the smooth surface of the mucous membrane. Owing to the shortness both of the chorionic villi and the uterine crypts and their simple form, it is believed that the placenta, when shed in normal parturition, would be generally non-deciduate, in the sense of the vascular walls of the crypts not being shed along with the villi; it is not unlikely, as the author showed some years ago to be the case in the sheep and cow,* that the epithelial lining of the crypts may separate more or less, and pass off entangled between the villi. It is also possible that the longer villi may carry away with them parts of the vascular wails of their crypts. Should the placenta be non-deciduate in the sense that the vascular part of the maternal mucous membrane is not shed, then the placenta of the Dugong gives a new type of placenta — one which is both zonary and generally non-deciduate. The diffused character of the placenta in the specimen described by Paul Harting was due to its comparatively early stage of develop- ment, for the villi had not as yet limited themselves to a denfinite zone. The paper concluded by a comparison of the placentation of the Dugong with that more especially of the Cetacea, Carnivora, and Proboscidea, and by remarks on the bearings of the form and structure of the placenta on the classification of the Sirenia. * Proc. Roy. Soc. Edin., May 1875. 266 Proceedings of Royal Society of Edinburgh. [sess. On the Geographical Distribution of some Tropical Diseases, and their Relation to Physical Phenomena. By R. W. Felkin, M.D., F.R.G.S., Lecturer on Diseases of the Tropics and Climatology , Edinburgh Medical School. (With 16 Plates.) (Read July 15, 1889.) The subject of the present paper has occupied my attention for some years, but I may state that what follows is the outcome of notes prepared for lectures to my students and is only a pre- liminary attempt to focus our present knowledge of the geo- graphical distribution of some tropical diseases, and to indicate as far as possible the knowledge which we at present possess of those physical phenomena which influence the production of these diseases and the area of their distribution. Why, for instance, some diseases are confined to limited areas of distribution, whereas others are endemic in extensive districts, and others again periodically extend their ravages throughout clearly defined, though wide- spreading, regions. A map was published by A. Keith Johnston in 1856 representing the geographical distribution of health and disease, chiefly in con- nection with natural phenomena; but although it gives a great deal of information, it does not indicate with sufficient clearness the definite areas of the various diseases referred to. Since its publication, too, our knowledge both as to the distribution of tropical diseases and the cause of disease has made considerable progress. More recently Mr Alfred Haviland, M.R.C.S., pub- lished three very valuable and instructive maps dealing with the geographical distribution in England and Wales of cancer in females, of phthisis in females, and of heart disease. These maps show very strikingly the influence exerted on these diseases by locality. Two maps, illustrating the distribution of diphtheria and scarlet fever in England and Wales, were published by Dr E. G. Barnes in the British Medical Journal , July 28, 1888. In the same Journal for January 19, 1889, there is a report of the collective Investiga- tion Committee of the British Medical Association, prepared by Dr 1888-89.] Dr R. W. Felkin on Tropical Diseases . 267 Isambard Owen, on the geographical distribution of ricketts, acute and sub- acute rheumatism, chorea, cancer, and urinary calculus in the British Islands; and various maps, illustrating the geographical distribution of ricketts, chorea, and cancer, have been prepared but not published. In 1886 the report on the mortality and vital statistics of the United States, as returned at the tenth census (June 1, 1880), was published; and it contains some most instructive maps and charts showing the distribution of deaths from various diseases in the United States, which indicate to a certain extent the geo- graphical distribution of those diseases, as also the localities in which they are most prevalent. None, however, of the publica- tions to which I have referred, nor any with which I am acquainted, attempt to depict graphically the geographical distribu- tion of tropical diseases in a manner which would give at a glance the areas throughout the world which are affected by them, and their possible or probable connection with physical phenomena. It must be remembered, too, that in the works above mentioned the various authors were dealing with civilised districts, where authentic statistics were obtainable, and where it was possible to localise the distribution of disease in a minute form, which, how- ever desirable, it is at present completely out of my power to attempt to imitate. I propose to treat my subject on a definite plan. I intend first to give the name of the disease and its various synonyms; secondly, a short definition of the disease, and a very brief description of it. I will then sketch out with some attempt at detail the geogra- phical distribution of the disease, and finally point out as far as possible its relation to various physical phenomena as affecting both its causation, its area, and its epidemic spread. I may state at the outset, that I have taken the definition and description of each disease from the Dictionary of Medicine edited by Richard Quain, M.D., F.R.S., because the definitions and descriptions there given are the generally accepted ones, and in a paper of this character, which deals more or less with broad outlines and general facts, it would be out of place to enter into any personal views I may have as to either the cause, the definition, or the description of the diseases referred to. 268 Proceedings of Royal Society of Edinburgh. [sess. With regard to the data concerning the geographical distribution of the diseases, I am indebted for my principal facts to the Hand- book of Geographical and Historical Pathology , by Dr August Hirsch, and to the copious bibliography which accompanies his various chapters. I have not, however, hesitated in any case where my own information, or information gathered from other sources, modifies or supplements his data, to make use of the same, and I have as far as possible verified the facts I have given. The maps which illustrate this paper have been specially prepared for it, and in order to ensure as great accuracy as is possible on such small maps, the areas of the distribution of the various diseases have been drawn upon large maps used for lecturing purposes, and sub- sequently reduced by photography to the scale suited for publication in the Proceedings of this Society. The maps illustrating some of the physical phenomena with which my paper deals are adapted from various sources. The chart of the world depicting the mean annual temperature of the tropical and sub-tropical zones is taken from the Challenger Reports , and is based upon the most recent investigations on the subject. The chart representing the mean annual rainfall throughout the world is after the most recent map published, namely, that in the Contributions to Meteorology , by Professor Elias Loomis (1889). Plate No. 16 gives the isoclinal lines with reference to pandemic waves of disease, and the prevailing winds upon the ocean. It has been compiled from two maps; one published by Dr Robert Lawson in 1888, and the other by Dr W. S. Wilson in 1881. I trust that this paper may be of special interest to members of my own profession, who will be able to see at a glance the diseases infesting the various districts in the tropics, and who will therefore be the more readily able to give the necessary advice to patients. It should be useful to intending emigrants, and of special service to insurance companies, as indicating the areas of comparative safety or risk for the residence of their clients. The diseases treated of are — 1. Malaria; 2. Dengue; 3. Asiatic Cholera; 4. Yellow Fever; 5. Oriental Boil ; 6. Endemic Hsema- turia; 7. Beri-beri; 8. Oriental Plague; 9. Dysentery; 10. Leprosy (Elephantiasis Grecorum); 11. Yaws; 12. Fungus Disease of India; 13. Elephantiasis Arabum (Barbadoes Leg); 14. Guinea-Worm; ROY. SOC. PRO. Vo I . X V I Robert W. Felkin, del. PLATE. I. Scott & Ferguson, Edinburgh. ROY. SOC. PRO. Vo I . X VI PLATE. I 1888-89.] Dr R. W. Felkin on Tropical Diseases. 269 15. Filaria Sanguinis Hominis ; 16. Scurvy; 17. Tropical Abscess of the Liver. In a future paper I hope to continue the subject with a more extended range of diseases, which, although to a certain extent met with in the tropics, are also to be found in the temperate zones. I. Malarial Diseases. (See Plate I.) It has been hitherto the custom to subdivide the diseases due to malaria, or, in other words, to the malarial process, but I am so convinced that the same cause produces the various types or manifestations of malaria that I include them all under one heading. Malaria ( Ital .) — 8 ynon. — Marsh Miasm.; Fr. Mauvais air ; Intoxication des Marais ; Intoxication telurique ; Ger. Malaria. Definition. — An earth-born poison, generated in soil, the energies of which are not expended in the growth and sustenance of healthy vegetation. Ey almost universal consent, this poison is regarded as the cause of all the types of intermittent and remittent fevers commonly called malarial, and of the degeneration of the blood and tissues from long residence in places where this poison is generated. Malaria therefore includes — A. Intermittent Fever — Synon. — Ague; Fr. Fievre Intermittente; Ger. Kaltes Fieber. Definition. — A fever of malarial origin, characterised by a sudden rise of temperature during the paroxysm, by an equally sudden fall at its termination, and by the regularity of the times of accession and apyrexia. B. Remittent Fever — Synon. — Bilious Remittent ; Marsh Remittent, The Jungle Fever of the East Indies ; The African, Bengal, Mediterranean, Persian, or Walchern Fever; Fr. Fievre Remittente ; Ger. Bosartiges endemisches Fieber. Definition. — A paroxysmal fever of malarial origin, in which the paroxysms do not intermit, but only, as the name implies, remit. C. Pernicious Malarial Fever including — (a) Febris Algida; (b) Febris Comatosa. 270 Proceedings of Royal Society of Edinburgh, [sess. Geographical Distribution. — The distribution of malaria is very extensive, although the intensity of the malarial process varies in different regions. Commencing with Africa, we find that on the west coast malaria is very prevalent in the basins of the Senegal and Gambia, and on the Guinea coast from Sierra Leone to Cape Lopez, especially in the basins of the Niger and Gaboon, on the Ivory and Gold coasts, at Fernando Po, and St Thomas. It diminishes rapidly in intensity until lat. 18° S., where it disappears. On the East coast, the malarial region commences in the south at Delagoa Bay, and ex- tends northwards as far as 5° N. lat., including the islands of Zanzibar, Madagascar, Mauritius, Bourbon, and Seychelles. Areas of endemic malaria are also found in the lowlands of Abyssinia and in the Somali district, in the valley of the Nile, from Khartum to the Great Lakes, west of the Nile between Dongola and Khartum, and all over Tropical Central Africa, up to an altitude of 3000 feet. Egypt, Tripoli, Tunis, and Algiers are likewise affected. The malarial process is most intense upon the equatorial African coast- line until an altitude of 500 feet is reached, and it also extends in its gravest form for about 300 miles up the banks of the Zambesi, the Congo, and the Niger. Malaria is met with in Arabia, on the east coast of the Ked Sea, especially on the coast of Hedjaz, and in Yemen from Jisan south to Moccha. It is also found at Muscat, on the shores of the Persian Gulf and its islands, and in the valleys of the Euphrates and Tigris. It also exists in Syria, especially in the damp valleys of the Lebanon, in the valley of the Jordan and along the shores of the Levant; and this malarious area extends to Asia Minor, from Adana and Tarsus to the Dardanelles, including Smyrna. The disease extends all round the Caspian Sea, overspreads Persia, Beloochistan, and Afghanistan ; and is met with all over India, with the exception of places having a high altitude. Ceylon too and Burmah, Siam, Sumatra, Borneo, New Guinea, the Phillipine Islands, Japan, the Andaman Islands and Australia as far south as lat. 17°, are all affected. The disease also prevails in a severe form in the tropical and sub-tropical parts of China. In Europe we may commence by tracing malaria in the steppe lands of the Caspian. It follows the course of the Volga through Astrakan, includes the central Caucasian plain, and the countries 1888-89.] Dr R. W. Felkin on Tropical Diseases. 271 bordering the Black Sea to the north, the basins of the Dnieper and Dniester as far as Ekaterinoslav, the Crimea, Wallachia, and Bulgaria. There are also endemic areas of malaria in the marshy plains of western Russia, in many places in the Baltic provinces, and in the district of Novgorod. In Galicia malaria is only endemic in Cracow, Wadowice, Zolkiewo, and Zloczow; and in Poland only in the province of Agustusowo. One of the largest malarial areas of Europe follows the course of the Danube and its tributaries, from the plain of lower Austria, over a great part of Hungary. It is also endemic in the marshy districts of Croatia, in the damp valleys of Servia and Montenegro, and in the valley of the. Save. In the Balkan peninsula there are various endemic areas in Roumania, on the shores of the Black Sea and of the Sea of Marmora, in Albania, and northwards along the coasts of Dalmatia and Istria. In Greece malaria occurs at many points in Boetia, Zeituni, Naupantos, and Yonitza ; at Chalcis, Corinth, Mistra, Navarino, Modon, and many other places on the coast. In Crete, Cephalonia, St Maura, and Corfu malaria is also endemic. In Italy there are two areas of endemic malaria, occupying the plain of the Po and its tributaries, and the west coast from Pisa, as far as and including most of Calabria. In the Iberian peninsula, malaria is most severe in the southern and western coast regions, in the low-lying country of Andalusia, on the marshy banks of rivers, especially the Guadiana and Guadalquiver, on the flooded plains of the Tagus, Sado, Mondego, and other coast rivers of Portugal, on the level coasts of Granada and Murcia, and on the plains of Algara and Alemtejo. In France malaria is most prevalent in the western and southern parts of the country, "especially from the basin of the Loire as far as the Pyrenees. Another area stretches along the coast of Languedoc and Provence. There are several other small malarial areas, of which the plain of Auvergne and the marshy country around Lake Indre may be noted. In Switzerland endemic malaria is only found in the southern part of the canton Ticino, and in the canton of Valais along the Rhone. In Austria the chief seats of the disease are along the Danube, and there are smaller areas in the river valleys of upper Austria, Salzburg, Styria, and Carinthia. Where the river widens at Krems we meet with a great region of malaria, which extends as far as the Black Sea. In south-west Germany 27 2 Proceedings of Royal Society of Edinburgh. [sess. there are small and isolated areas where malaria is endemic on the marshy banks of the rivers or lakes, e.g ., the valleys near the Neckar in the Black Forest. The more extensive malarial regions are on the banks of the Rhine, in lower Alsace, in the Palatinate, in the Rheingau, and in the low grounds of the Danube, and its side valleys in Wurtemburg and Bavaria. In central Germany it is only endemic in a few small districts. In North Germany it is found in the basins of the Vistula, Oder, Elbe, Weser, and Rhine. Holstein and Schleswig (west coast districts), the coast belt west of the Elbe, the moorlands of Hanover and Oldenburg, the low grounds of Westphalia, and the plains of Rhenish Prussia are also infected. In the Netherlands malaria is mostly found in the provinces of Gronland, Friesland, and Zealand, on the coast belt of the provinces of north and south Holland, and in the provinces of Drenthe Overyssel. Belgium, West Flanders, East Flanders, and Antwerp are affected. The disease also occurs in Laaland and Falster, on the Hvaloen islands, and in the neighbourhood of Fredericstadt. In Sweden it is endemic at three principal points — around Lakes Maler and Werner, on the east coast of Torhamn, and at the mouth of several coast streams, such as the Angermanna-elf, the Dal-elf, and Gotta-elf. In Britain, the East Riding of Yorkshire, the Fen district, Essex and Kent, the banks of the Thames in Surrey, and the south marsh of Somersetshire, are slightly infected by malaria. In the western hemisphere, endemic malarial fever of the severest type has its principal seats in the West Indies, on the Mexican Gulf coast, and in Brazil, but considerable regions of fever, though of a less intense kind, are met with in the northern parts of the Pacific coast of South America, and in the southern, central, and prairie States of the Union. All the West Indian Islands are affected, save Antigua, St Vincent, and Barbadoes, the Bahamas and the Bermuda group, in which islands it is rarely seen. In South America the worst centre of malaria is the east coast, including the ports of Carthagena, Maracaybo, and Puerto Cabello, and the country of Guiana. Another extensive area covers the whole of the north of Brazil as far as Rio de Janeiro, the banks of the Amazon, Rio Madeira, Maranhao, Paranahyba, San Francisco, Parana, Rio Doce, and their tributaries ; also the island of Santa 1888-89.] Dr R. W. Felkin on Tropical Diseases. 273 Caterina and the marshy districts of the provinces of Piauhy, Para, Mato-Grosso, Goyas, and Minos Geraes. Endemic malaria is also found in the prairie lands of Paraguay and Bolivia, particularly in the provinces of Tucumana, Salta, and Santa Cruz. Chili and Peru are also affected, and the disease extends thence along the coast to Ecuador, probably also to New Granada. The disease is found all over Central America, hut it is less severe on the Pacific coast, except at Corinto. In Mexico malaria is most prevalent on the Atlantic coast, and is less frequently seen everywhere up to 3000 feet in height, except on the tableland proper (Anahuac). On the Pacific coast of Mexico malaria is confined to Acapulco, Tepic, and the strip of coast from San Bias to Mazatlan. In the United States from the Rio del Norte a great malarial field extends all over the Gulf coast to Cape Florida, spreading far into the interior along the Colorado, Brazos, and Mississippi and their tributaries. In Texas the malarial region stretches from the coast into the highlands as far as Fort Duncan in Eagle Pass, and Fort Makavit. In New Mexico malaria is very widely diffused, the limit of its prevalence being Santa Fe. From the western part of the Louisiana coast, between the Sabine and the Mississippi, the malarial region extends across the zone of bluffs in that State, over a great part of Arkansas, especially along the hanks of the Mississippi and Arkansas rivers, over the marshy plains of the north-east of the country towards Missouri, and over the eastern part of the Indian territory, including Fort Gibson and Fort Still. In the peninsula of Florida malaria is only met with on the Gulf coast, especially in Escambia and Tambabay, and at Fort Meade. In Georgia we find it in the creeks along the coast. In the Central States of the Union malaria is met with in South Carolina, North Carolina, Virginia, and Maryland, and to a moderate extent in Tennessee and Kentucky. It is also found within the prairie States proper, e.g ., in Ohio, Indiana, Illinois, Missouri, Iowa, Minnesota, Wisconsin, and Michigan. In no case, however, does malaria extend farther north than 46° 10' N. (Fort Ripley). In the southern parts of the State of Michigan malaria skirts both shores of Lake St Clair to its junction with Lake Huron, and the southern shores of Lakes Erie and Ontario as far as the St VOL. XVI. 12/8/89 s 274 Proceedings of Royal Society of Edinburgh. [sess. Lawrence. Severe malarial fever is found at Fort Gratiot, Detroit and Plymouth, on the United States side of the Lake ; and at Amkurstbury, Fort Maldon, and Sandwich, on the Canadian side. On the northern side of Lake Ontario malaria extends from Hamilton to Kingston, and up the ridge of hills which runs along the shore from Burlington to the mouth of the Trent, attaining in some places an altitude of 600 feet. Endemic malaria extends also to the north- western parts of the State of Hew York, although there are many localities now free from fever. It is most frequently met with along the banks of the Hudson and on a narrow strip of coast. During recent years the disease has increased in the mountain districts of New York and also in Pennsylvania. In the New England States the disease is endemic at only a few points, and it is not endemic in the greater part of British North America. Malaria, as an epidemic, is met with on the banks of the St Lawrence and its tributaries, and on Lake St Peter, as well as at Montreal and Quebec, and at various coast places, such as Halifax (N.S.) and Miquelon (N.F.). In Nova Scotia (except at Halifax), in New Brunswick, and in Green- land, the disease is quite unknown. In western North America the limit of malaria reaches somewhat higher latitudes. It is met with chiefly on the slopes and valleys of the Bocky Mountains and in the territories of Wyoming, Utah, and Colorado. In Cali- fornia there are considerable malarial regions, especially up the valleys of the Sacramento and San Joaquin, and in the inland southern part of the State of Arizona. The incidence of malaria throughout the world has been very well summarised by Mr W. North, whose classification I will quote.* First Category. Highest Degree of Intensity. Class I. Senegal; Coasts of Gulf of Guinea; West Coast of Africa, as far as the 20th parallel of S. latitude ; Madagascar ; the Guianas. Class II. India ; Cochin-China ; Ceylon ; Afghanistan ; Burmah ; Siam ; the whole of the Malay and Philippine Archipelago ; New Guinea ; Nubia ; parts of Abyssinia and the Soudan, and Central America. * See Nineteenth Century , June 1889, p. 867. 1888-89.] Dr R. W. Felkin on Tropical Diseases. 275 Class III. The East Coast of Africa ; Egypt ; the coast-line of Arabia ; Mexico ; China Proper ; the Brazils and Peru. Second Category. Class I. Tripoli ; Algeria ; Morocco ; the Cape de Verde Islands ; and the Oases of the Sahara. Class II. Turkey, in Europe ; Greece ; the Islands of the Archi- pelago ; Sardinia ; Malta ; Sicily, and parts of Italy. Class III. Roumania ; Hungary ; Italy ; Corsica ; Spain ; Portu- gal; Southern Russia, and a large part of the United States. Third Category. Southern Sweden ; Denmark; Belgium and Holland; Germany; France ; La Plata ; Chili, and the Islands of Madeira ; Bourbon and at Helena. Fourth Category. No Malaria or Insignificant. The British Islands ; Norway ; the southern parts of Sweden ; Finland and Russia; all North America above the 50th parallel of N. latitude ; Uruguay ; the Argentine Republic and Patagonia ; Northern China ; almost all Siberia and the greater part of Japan ; New Zealand and the southern part of Australia. Remarks. — Of the malarial fevers, the intermittent is the most widely distributed type, and it will be noticed that the remittent and pernicious fevers are only met with in comparatively small areas, and that they are, as a rule, confined to tropical or sub-tropical countries. A glance at the map will show, without further specifica- tion, their distribution. It may be noted, however, that the quo- tidian and quartan types of intermittent fever are those which are most frequently met with in the tropics, and that the Tertian type is that form which is most widely distributed in the more temperate zones. In fact, the type of fever stands in a definite relation to the intensity of the malarial process ; thus we find that the Tertian type prevails in those regions within the tropics where the milder malarial fevers are indigenous. Again, the frequency of the occur- rence of the quotidian type of fever in endemics or in epidemics is in direct proportion to the severity of the disease. When an epidemic 276 Proceedings of Royal Society of Edinburgh. [sess. wave of malarial fever passes over a district, the Tertian type is seen at its outbreak, whereas at the height of an epidemic, or whenever it assumes a severe character, the quotidian type obtains, and as the outbreak of sickness abates one meets with a return to the types of fever having a longer interval between the paroxsyms, so that in tropical and sub-tropical countries the fever takes on the Tertian type, and in the higher latitudes the quartan type makes its appearance. These remarks apply also to outbursts of the disease where it is endemic. In temperate zones remittent malarial fevers are exceed- ingly uncommon, in fact, so uncommon as to be regarded as a departure from the ordinary type and as due to exceptional causes. All races may suffer from malaria, although the Negroes are less prone to it, always provided that they do not migrate. Indeed, it is very generally acknowledged that in all parts of the world strangers suffer more severely from malaria than does the indi- genous population. The incidence of malaria is to a certain -extent governed by the seasons. In those places where it is endemic it occurs all the year round ; but where it is only slightly developed there are two maxima — one in spring and one in autumn, and a considerable decrease of the disease in the months between them. In regions with strongly developed malaria, there is a maximum beginning in summer, which reaches its height at the end of summer or the beginning of autumn, lasting not rarely into winter, and which so far exceeds the spring maximum, that the latter not unfrequently disappears altogether, so that there is only one minimum, winter and spring, and one maximum, summer and autumn. In tropical countries, in the worst malarious districts, the disease is most rife during the rains. The relation which malaria bears to heat is as follows — the greater the mean summer temperature, other things being of course taken into account, the more malaria, and the amount of malaria decreases with the mean annual temperature of the place, ceasing altogether with the summer isobar of 60° T. But, as Hirsch says, “ in higher latitudes, the malarial fevers which have prevailed endemically or epidemically in spring undergo for the most part a considerable re- mission on the setting in of summer heat, and they do not revive until the cooler weather of autumn,” and again, “in the regions of severe malaria the disease shows itself, and attains wide diffusion, 1888-89.] Dr R. W. Felkin on Tropical Diseases. 277 not at the height of summer, but only when the high temperature is declining in late summer and in autumn, and, for the tropics in particular, at the end of the hot season. As many observers state, this is directly due to the great diurnal range of temperature that occurs at that season.” The influence of rain or moisture has undoubtedly much to do with both the production and spread of malaria. With regard to the rains, we may say that the malarial poison is most virulent either when they set in after a long period of heat, or when the rains cease and give place to warm, dry weather. An endemic outbreak of malaria, and its epidemic spread are both notably diminished at the height of the rains, if they are very abundant, and it has been proved over and over again that the malarial process is developed more abundantly in wet than in dry years. These remarks hold good both as to tropical and temperate climates. But it is not with rainfall alone that we have to do, for moisture must be present in the soil in order for the production of malaria, and this saturation of the ground may be produced in various ways apart from atmo- spheric precipitations. Drainage from rivers, lakes, and pools, may constantly saturate the soil, and so may inundations either periodic or irregular. The irrigation of the soil exerts an undoubted influence on the production of the poison ; its effect is very marked in Egypt, and in the irrigation districts of India. Lastly, the soil may be saturated by sub-soil water. This point is of importance, because it explains the occurrence of malaria in localities remote from river basins, and where the soil cannot become saturated in other ways. The occurrence of malaria in the Sahara, in Spain, Greece, Algiers, Tripolis, and Darfour, is in all probability due to this sub-soil water, arising either from springs or in other ways and resting upon impermeable strata of rock or marl. Apart from the moisture in the soil, it must possess other physical characteristics, although the geological characters of the country would appear to exert little or no influence on the production or non-production of the disease, except in so far as they affect the physical nature of the soil. Clay, loam, clayey marl, and marshy soil are most favourable to the production of malaria; a porous chalky soil is less favourable and sandy soil least so, provided always that the chalk or sand does not rest either upon clay or firm 278 Proceedings of Royal Society of Edinburgh. [sess. rock. The exemption from malaria of some of the islands in the West Indies with a chalky soil is remarkable when contrasted with its special prevalence in islands of volcanic formation. The amount of organic material too, contained by the soil is bound up with the production of malaria. All other circumstances being equal, the greatest amount of malaria will be found where the amount of organic matter in the soil is greatest, the prevalence of the disease diminishing as the organic matter is found in less abundance. Changes in the soil produced by cultivation, the neglect of cultiva- tion, and by excavations, also affect the presence or absence of malaria, but space forbids us entering into these details. In reference to marshes, however, it may be noted in passing that malaria will disappear from a marshy district if it is completely drained and dried, or if a marsh is converted into a pool or lake. The amount of malaria, as also its severity, is affected by altitude and the configuration of the ground. The altitude at which malaria can be produced varies in different regions, being higher in the tropics than in the temperate zones. Thus in Central Africa we find that a height of some 3000 feet must be attained before one reaches a district free from malaria, whereas in the Apennines a height of 1500 feet only is required, and farther north only 500 feet. This, however, must be explained more by variation in temperature than by mere altitude. On the other hand, however, examining the configuration of the ground in plains, it is found that the disease is distinctly more virulent the lower the level of the country. This fact is so marked that often even 50 or 100 feet less makes a considerable difference to the salubrity or otherwise of a given spot. Although winds do not exert any direct influence upon the pro- duction of the malarial poison, they act indirectly, as, for instance, by moderating temperature, &c. They act, however, directly in the diffusion of the poison or in preventing it exercising its potent effects. Wind may carry the malarious poison from a marsh to a healthy district, but it is probable that it can only thus convey it for a distance of some two or three miles. Malaria may rise to a height of some 700 or 800 feet in a calm atmosphere ; wind will prevent this vertical diffusion. Probably on some islands, where from analogy we should expect to find malaria present, the constant Robert W. Felkin , del. PLATE II 1888-89.] Dr R. W. Felkin on Tropical Diseases. 279 winds rapidly changing the atmosphere carry away the morbific elements before they have time to do harm. Water can convey the malarial poison, but it is unknown at present how far it can carry it or how long the poison can remain unimpaired when carried either by a stream or a current. From what has been said, some idea may be obtained of those factors requisite for the production of the malarial poison. It is in all probability due to a micro-organism which may find entrance into the body by means of the air, by drinking water and possibly also by the consumption of food contaminated by it. It is certainly ponderable, as is proved by the effect of altitude, of barometrical pressure, and by the action which winds have in its dissemination. It is also miscible with water. We may sum up our definite knowledge of the disease by saying that it requires for its production a specific germ, suitable soil, a certain amount of moisture, a sufficiently high temperature, and a certain time for development. An examination of the map, bearing what has been said in view, will I think show how the presence of malaria in the various quarters in which it exists is to be accounted for by physical phenomena. II. Dengue. (See Plate II.) Synon. — Dandy Fever (West Indies) ; Third Day Fever ; Red Fever; Leg Fever; Breakbone Fever; Scarlatina Rheumatica (Aitken) ; Aburukah, or Aburuka-bar, or Father of the Knee (Arabia) ; Nadak-Mariata, or the Deity (Southern India) ; Tootiah (Bengal) ; Kidniga pepo, i.e ., Spasmodic pains (Zanzibar); Fr. and Ger. Dengue. Definition. — An infectious, eruptive fever, commencing suddenly, and characterised by severe pain in the head and eye-balls, swelling and pain in the muscles and joints, prone to shift suddenly from joint to joint, catarrhal symptoms, sore throat, congested con- junctivae, and affection of the sub-maxillary glands. The disease may remit, and is liable to relapse. Geographical Distribution. — Dengue has been known since 1780, in which year it attained a considerable diffusion in the tropical and sub-tropical parts of both eastern and western hemispheres. It 280 Proceedings of Royal Society of Edinburgh. [sess. is now known to have visited Egypt, Senegambia, and Tripoli, and the valley of the Nile as far as Khartum ; also Arabia along the coasts, and Mecca. It is epidemic all over India and Further India, and in Batavia; and has appeared in Shanghai, Amoy, and the island of Formosa. It has also visited Keunion, Tahiti, Zanzibar, and the Canary Islands ; and epidemics have overspread the West Indies and a great part of the Southern States of the Union, as well as the northern shores of South America. Space wrill not permit of indicating the various areas of diffusion during the several great epidemics of Dengue, but it must suffice to state that its greatest area of diffusion lies between 33° N. lat. and 23° 30' S. lat. Remarks. — The period of incubation of Dengue is probably about six days. It is a highly infectious disorder, spreading with extreme rapidity. Summer and early autumn are undoubtedly the Dengue season, and the disease appears to depend on a high tempera- ture for its production. In the tropics as well as in the more tropical zones, nearly all the epidemics of Dengue have been in the hot weather, and as soon as a great fall of temperature takes place the disease declines rapidly. It is probable that the moisture of the atmosphere has little or nothing to do with the production of Dengue ; as a rule it is chiefly confined to coast districts, to the courses of great rivers, and to places having a low altitude. It has been noticed in various epidemics of Dengue that it spreads in a curious way amongst various classes of the community. Every race, nationality, age, and sex may be attacked by the disease, although in separate epidemics a remarkable immunity has been noticed on the part of certain classes. Sometimes Europeans will be attacked, and natives enjoy comparative freedom from the disease; again, in other cases, natives will be almost solely attacked; sometimes children suffer more than adults, or the reverse may obtain. For instance, Pasque says, speaking of the epidemic at Benghazi, that it was noticeable for the decided immunity experienced by the blacks, but they are attacked as much as anyone else in Egypt and Senegal. Christie remarks that in one or two of the epidemics which he witnessed in Zanzibar, the natives suffered less than the Europeans. In the epidemic in Mauritius in 1873, hardly any children were attacked by the disease. In the Deccan epidemic of 1872 there was noticed a peculiar predisposition to ROY. SOC. PRO V o I . X V I PLATE. Ill 281 1888-89.] Dr R W. Felkin on Tropical Diseases. Dengue in persons suffering from, any surgical complaint. In Goojerat in 1824, and in Amoy in 1872, it was found that all the severe cases were limited to the native population, that the Euro- peans suffered to a far less extent, and that those who were attacked by the disease had it in a remarkably mild form. As to the specific nature of Dengue, there can be no doubt, but whether due to a parasite or not is at present unknown, nor is it yet ascertained whether the poison springs up de novo at all points where its potency is manifested, or whether it is only epidemic at a few places, and spreads from these under favouring circumstances. Ab first Dengue was not thought to be contagious, but it is both infectious and highly contagious, and the diffusion of the disease could be traced in the epidemic of 1871-73, in the East from port to port, and from country to country along the highways of land and water traffic. One attack of Dengue does not confer absolute immunity from subsequent attacks, although it does so to a certain extent. III. Asiatic Cholera. (See Plate III.) Synon. — Serous Cholera; Spasmodic Cholera; Malignant Cholera; Ft. Cholera asiatique ; Ger. Asiatische Cholera. Definition. — Asiatic cholera is a specific disease, characterised by violent vomiting, rice-water evacuations, cramps, prostration, collapse, and other striking symptoms ; tending to run a rapidly fatal course ; and capable of being communicated to persons other- wise in sound health, through the dejecta of patients suffering from the disease. Geographical Distribution. — With regard to the distribution of cholera, it is not advisable to proceed on exactly the same lines as with other diseases, for, although the epidemic area of cholera is very vast, its endemic area is very limited. It will be well therefore, in the first place, to specify those regions which have hitherto escaped the ravages of the disease, next to define its endemic area, and finally to make some general remarks as to its pandemic diffusion. A glance at the map will render this division clear. Up to the present time, Australia and the islands of the Pacific 282 Proceedings of Royal Society of Edinburgh. [sess. Ocean have remained unaffected "by the disease. So too has the whole of Africa, with the exception of a strip of coast-land commencing at Delagoa Bay and running all round the eastern and northern coasts of the Continent as far as lat. 14° N. The islands of St Helena and Ascension have also escaped. In South America, Terra del Fuego, Patagonia, Chili, the higher slopes of the Andes and the Falkland Islands have been hitherto exempt. In North America, the whole of the country north of 50° 1ST. lat. has remained free from cholera, as well as the highest slopes of the Rocky Mountains and the Bermudas. Greenland and Iceland have not been visited by the disease. In Europe, we find that the Faroe Islands, the Hebrides, the Shetlands and Orkneys, Lapland, the Russian territory north of the 64th parallel, Switzerland, and the northern part of Scotland have escaped. In Asia, the northern districts of Siberia and Kamtschatka have remained free, as also probably Mongolia and Manchuria. We see from these exceptions, what an immense area of the earth’s surface has been visited from time to time by the terrible scourge of cholera. It must be remembered, however, that, although the area of distribu- tion of cholera has been so vast, yet certain isolated districts in the various countries visited by it have remained unaffected. For instance, some mountainous districts in the south-west of France, the south-west of Germany, notably Baden and Wiirtemburg, and the greater part of Greece. Even in India itself there are places where its ravages are unknown ; for instance, it has not attacked the hilly regions of Bengal, but the reason for this is apparently to be found in the fact that the hill men have little or no communication with indi- viduals from the affected area. Should they descend from their mountains, or have any communication with the inhabitants of the plain, they suffer from the disease very severely. With regard to the home of cholera, we must define it as situated in the delta and valley of the Ganges, from which point it receives an impulse which enables it at times to become pandemic in character. There can be no doubt that cholera existed long before 1817 ; of its previous epidemic spread no very certain informa- tion can be given, but that it overspread India and the adjacent 1888-89.] Dr R. W. Felkin on Tropical Diseases . 283 countries is certain. It appears to be equally certain that, apart from the original impetus given to cholera in 1817, the commerce of the world has been the means of spreading it over such a world- wide area as indicated on the map. The first pandemic, of which we have authentic data, occurred during the years 1817-1823, in which period the disease devastated an area from Nagasaki, 147° E., to the coast of Syria, 52° E„, and from Bourbon, 21° S., to Astrakan, 46° 21' N. The second pandemic, which took place during the years 1826- 1837, overspread nearly the whole of the countries marked on the map as being affected by cholera. From 1837-1846 Europe, Africa, and America were free from further outbreaks of the disease. The third great pandemic occurred from 1848-1863, with a remission, as far as the eastern hemisphere is concerned, between 1850 and 1852 ; and during this pandemic it visited the whole of the northern hemisphere, and reached lat. 25° S. in the Old world and 30° S. in the New. The fourth pandemic occurred during the years 1865-75, and is noticeable for the rapidity with which cholera was introduced into Europe by sea from the coast of Arabia. In the former pandemics it had always come from the east by way of Afghanistan, Persia, and Asiatic Russia. From 1875 until the limited outbreak in 1883, when cholera appeared in Italy, Spain, and the south of France from Egypt, the disease has been confined to Asiatic soil, and in the year mentioned the spread of cholera was very strictly limited. Remarks. — Before referring briefly to some points in the origin and spread of cholera, it may be well to remark that, however the disease is produced, the researches of Koch, which have been very recently confirmed by Drs Milles and Macleod, who investigated the subject in 1875 at Shanghai, and an abstract of whose researches was read before the Royal Society of Edinburgh on Dec. 17, 1888, have almost definitely proved that the comma bacillus has a causal relation to cholera. This bacillus is constantly present in Asiatic cholera, and it has never been found anywhere but in Asiatic cholera. Granting, as we think we are right in doing, that cholera is produced by a morbific microbe, we must next refer to various facts with regard to the conditions necessary for an epidemic of 284 Proceedings of Royal Society of Edinburgh. [sess. cholera to arise. Cholera may he introduced into a given area, and yet it does not necessarily follow that it will thrive in that area, and some definite local condition must exist for its propaga- tion. With regard to the influence of altitude, it may be said that, although cholera can penetrate to a considerable height, yet, unless in very severe epidemics, the number of cases are fewer the higher it ascends; and Farr adduced the law, that “ the proportion of deaths among the inhabitants from cholera is inversely as the elevation of the ground.” This law holds good, not only in high altitudes, hut in slight elevations in very limited areas. Ackland, writing on the mortality from cholera, mentioned the fact that in three epidemics of cholera in Oxford, in 1832, 1849, and 1854, the mortality per thousand people, living at a mean height of 30 feet above water-level, was respectively 3*3 and 1’8, as compared with 9-8 and 5 ’3 deaths per thousand which occurred amongst the people living in the low-lying part of the town. There are, however, exceptions to this rule, as in some epidemics hilly districts have suffered more than the lower-lying surrounding country. Again, cholera follows the course of rivers, this probably being due to the fact that the riparian areas possess soil saturated with water and decaying organic matter. There can be no doubt, too, that cholera spreads most rapidly in countries having an alluvial or tertiary soil. With reference to the influence of weather on cholera, we know that the disease is met with under every known variety of climate ; yet doubtless a high temperature aids its spread, and most epidemics are brought to an end, or at least are most markedly reduced in extent, when the temperature falls greatly. The Yienna Confer- ence of 1884, in which were assembled representatives of all coun- tries, stated that there were no facts to show that atmospheric causes alone could bring on cholera, and also, that all facts go to show that in free air the generative principle of cholera rapidly loses its morbific character. With regard to the influence of the moisture of the atmosphere upon the production or spread of cholera, it would appear that a certain amount of moisture in the atmosphere, or perhaps more correctly speaking, in the soil, is needed for its development and spread; it is probable that spring showers or sudden summer ROY. SOC. PRO Vo I . X V I 40 20 Robert W. Felkin, del. PLATE. IV. f. SOC. PRO Vol.XVI PLATE. IV 1888-89.] Dr R. W. Felkin on Tropical Diseases. 285 rains will give to cholera an impetus, whereas continuous rains, by completely saturating the soil, prevent its occurrence. No one knows the cause which gives it epidemic impulse, hut we know that, generally speaking, cholera becomes epidemic during the intensely hot weather following heavy rains. The wind may have a certain action on the spread of cholera, always provided that it he a moist wind, hut the old idea that winds could convey cholera poison for long distances seems now to he obsolete. But indirectly the south-west monsoon certainly aids in spreading cholera, because it brings with it moisture which is necessary for its propagation, and because numbers of vessels take advantage of this monsoon to sail from the endemic area of cholera to other places. As Macnamara justly remarks, “ cholera thus pro- gresses with man along the great high-roads upon which he travels, spreading no faster than he moves, and being generated by wet, hot weather.” Although the disease is thus indirectly spread by the south-west monsoon, it must not be concluded that it cannot travel against wind, for it travels just as fast against the wind as with it. Water is capable of disseminating cholera germs after they have been produced in the soil, or after it has been contaminated by the discharges of cholera patients. The great factor in the distribution of cholera is certainly that of human intercourse. Persons suffering from the disease, though it may be only in a latent form, undoubtedly convey the poison for long distances, and it is a well known fact that troops, pilgrims, and emigrants have spread it far and wide. It is necessary, however, for an epidemic of cholera to arise, that the poison conveyed into a district should find there a fitting soil for its growth. What that fitting soil is, it is impossible yet to say. IV. Yellow Fever. (See Plate IV. A.) Synon. — Yellow Jack; Bronze John; Yomito Prieto; Fr . Fievre jaune ; Ger. Gelbes Fieber. Definition. — A pestilential contagious fever of a continuous and special type. It presents two well-defined stages. The first extends from 36 to 150 hours, and is marked by rapid circulation and 286 Proceedings of Royal Society of Edinburgh. [sess. elevated temperature. The second is characterised by depression of the nervous and muscular powers, and of the circulation, with slow and often intermittent pulse; jaundice; suppression of urine, albuminuria, and desquamation of the renal epithelium; diminution of the fibrin of the blood, capillary congestion, passive haemorrhages from the mucous surfaces and black vomit ; fatty degeneration of the heart and liver ; and convulsions, delirium and coma. As a general rule, it occurs but once during life. Geographical Distribution. — Although yellow fever extends over the areas mentioned below, it must be noticed that there are only three districts where it is really endemic, i.e ., (a) in the West Indies, especially in the Greater Antilles ; ( b ) on the Mexican part of the Gulf Coast ; and (c) on the Guinea coast at Sierra Leone, The area of distribution of yellow fever is at present limited in Africa to the west coast from 19° N. to a point on the mainland opposite Fernando Po. In the Western Hemisphere it occurs along the eastern shores of the United States from lat. 38° JSL, and, skirting the coast round the Gulf of Mexico and Central America, it passes along the northern coast of South America and the eastern coast as far as 32° S. On the western shores of South America yellow fever has appeared in epidemics from 5° S. to 42° S. It is also prevalent throughout the whole of the West Indies. Although in the United States and South America yellow fever chiefly infests the coast regions, exception must be made to the great rivers, such as the Mississippi, the Amazon, and the Rio de la Plata, for it extends up these rivers to varying distances. In the western hemisphere the yellow fever area is bounded on the north by 44° 39' K (Halifax), and on the south by 34° 54' (Monte Video); in the eastern hemisphere by 43° 34' N. (Leghorn), and 8° 48' S. (Ascension); these are its extreme limits. Remarks. — There are some curious facts with regard to yellow fever and its relation to climate which it is necessary to remember. Firstly, Negroes are congenitally exempt from it, unless they leave the tropics for any length of time and then return; if they do this their immunity seems to be lost. It appears, too, that Mongolians escape yellow fever. All other races, however, suffer from it, and it is noteworthy that the further north from its area a person was born the more likely is he to suffer from it, 287 1888-89.] Dr R. W. Felkin on Tropical Diseases. should he come within reach of an epidemic. New arrivals are most exposed to this disease ; should they escape it at first they are the less liable to suffer from it, the longer they reside in one place ; but if they travel about, this acclimatisation appears to he lost. Whatever may he the real cause of yellow fever, its origin seems to he connected with heat, for although it occurs in sporadic and epidemic forms at all seasons in the tropical part of the yellow fever zone, the disease is greatest, and takes an epidemic spread at the hottest period of the year, a temperature of at least 70° F. being required for its production. Frost puts an end to an epidemic at once, and storms, heavy rains, or cold weather check its progress. Although heavy rainfall will stay an epidemic, yet moisture in the atmosphere would seem to be necessary for the production of the poison, for in dry years or during seasons of long-continued drought the number of yellow fever cases are always remarkably few. Winds influence yellow fever by their effect upon the atmosphere, but it does not appear that they are capable of conveying the poison, at least to any distance. On looking at the chart, it will at once be seen that yellow fever is most prevalent on the sea coast and along the courses of the great rivers ; it always spreads from centres of dense population, and the greatest number of cases occur in the dirtiest and most overcrowded parts of large towns. Altitude, certainly, has an important influence on the spread of yellow fever, and it is only in very severe epidemics that it leaves the plains. The protection which altitude confers against the disease is almost certainly due to the lower temperature of elevated spots, because where yellow fever has made its appearance in highly situated regions, it has always been in localities noted for the exceptional heat of the days. A certain saturation of the atmosphere is an essential condition for an epidemic of yellow fever. It is probable that it does not occur until a high dew-point, the minimum being upwards of 74, exists, and it is certain that epidemics cease before the dew-point descends to 58. The geological characters of the soil have apparently nothing to do with the production of yellow fever, and all those conditions of soil which we found to be necessary for the production of the malarial poison, exert no influence in pro- ducing yellow fever. Electricity has a curious influence upon 288 Proceedings of Royal Society of Edinburgh. [sess. persons suffering from the disease, even should they he almost con- valescent. It is said that should a thunderstorm occur, severe symptoms are immediately manifested in persons suffering from the disease, or a relapse occurs in the apparently convalescent. Yellow fever poison clings to the ground, and its diffusion may be barred by streams, walls, and, some say, by much travelled thoroughfares, and it does not appear that the water-supply of cities aids its spread. Its period of incubation is variable, and may be said to be between twenty hours and several weeks ; this varies in different epidemics, and the most common period of incubation Is from twenty to one hundred hours. Preventive inoculations are now being largely practised against yellow fever with very great success, and there seems to be every reason to hope that at last a method of preventing the disease will be firmly established, although up to the present its true origin has escaped scientific inquiry. The disease may be spread by fomites, and its toxic power retained for very long periods. Y. Oriental Sore or Boil. (See Plate IY. B.) Synon. — Aleppo Evil; Mycosis Cutis Chronica (Carter); Lupus Endemicus (Lewis and Cunningham) ; Oriental Sore (Fox) ; Mooltan and Biscara Boil ; Date Disease ; Caneotica ; Liblib ; Yemen and Cochin-China Sores; Scinde Boil; Parangi; Maid’ Alep ; Fr. Bouton d’Alep ; Ger. Yeule von Alep. Definition. — An indurated, indolent, and very intractable sore ; papular in the early, encrusted or fungating in the advanced stages ; spreading by ulceration of the skin, single or multiple ; and often occupying extensive surfaces of the exposed parts of the body, such as the face, neck, and extremities. It is capable, if innoculated, of reproducing the disease, and it also affects dogs and horses. Geographical Distribution. — In Europe the boil is met with in Crete, in Cyprus, and in the Crimea. In Africa there is a consider- able area in Morocco, on the banks of the Muluia, where the disease abounds, as it does also in numerous oases of the Algerian desert and in the Tunisian Sahara. It is occasionally met with in 1888-89.] Dr R. W. Felkin on Tropical Diseases. 289 Egypt between Suez and Cairo. In Asia it is more widely distri- buted; it is seen at Broussa, and in Syria it is endemic, chiefly be- tween Killis and Aleppo. In Mesopotamia it is endemic over the wdiole plain between the Euphrates and the Tigris, extending from Diarbekir to Bagdad and Bassara. In Persia it is endemic at Teheran, Kashan, and Ispahan; whereas in Hamadan it is not so frequently seen. There is also a small endemic focus in the district of Elizabethpol. The sore is met with in Tashkend and skirts the river Tchirtchik, and in all probability it exists in Turkestan, Afghanistan, and Beloochistan. Another Very important endemic area extends along the Indus from the Punjaub southwards through Scinde as far as Goojerat and the Gulf of Cambay, and to the east through Rajpootana and the North-West Provinces as far as Delhi, Meerut, Lucknow, and Gwalior. Remarks. — Objection may be taken by some to classing under one heading a sore having so many names, but it seems to me that, taking all things into consideration, the various designations all refer to one and the same disease, and that their different features are simply modifications produced by varieties of climate; their various manifestations are also most probably, to some extent at least, influenced by racial characteristics and by the habits of the patients attacked. In each locality where these sores obtain, they vary considerably in their appearance with the character of the season. Two theories have been, and still are, advanced as to their cause, some authorities considering that they are a local manifesta- tion of a cachectic condition due to a residence in unhealthy localities or badly drained towns in certain parts of the tropics. Others again, and notably Carter, consider that the disease is distinctly due to a parasite, and this view is supported by cogent facts. Carter has found in the sores spheroids and mycelium ; the disease is localised, and this fact is against it being the outward manifestation of a constitutional state. The disease can be innoculated, and it is contagious. Again, many facts lead one to suppose that the parasite is introduced into the body, either during ablution or by the bite of some insect. The area of the distribution of this disease prevents us enter- taining the idea that its production is influenced by the physical VOL. xvi. 12/8/89 T 290 Proceedings of Royal Society of Edinburgh. [sess. characters of the soil, nor has altitude anything to do with its production. With regard to water, however, it may he that we should consider it as a cause, for there are instances on record where a change of the source of the water supply has exerted a marked influence upon the number of cases seen. When, however, we come to inquire as to what substance in the water produces or might produce the disease, we are met by various statements which are contradictory, or at any rate give no certain clue on which to base a definite opinion. For instance, the gypsum found in the water at Aleppo is given as a cause : the abundance of nitrates in the water of the Punjaub, or the presence of sulphurated hydrogen due to putrefying matters, is blamed. Hard water is suspected by others, and in Algiers the excessive amount of choride of sodium is considered suspicious ; lastly, the amount of earthy salts contained in some waters is said to cause the sore. But there are many other observations which cause grave doubts as to the correctness of any of these views. Meteorological conditions must, we believe, exert a not incon- siderable power in the production* of Oriental sores. It will be noticed that the distribution of the disease is over arid regions, but in these regions it is strictly localised to various foci. In the sub- tropical regions the disease makes its appearance in the late autumn, and it is met with in the winter in the tropical zone. Where it is found at all seasons of the year, the climate is characterised by hot, dry air during the day, sometimes heavy dew at night, and rapid fluctuations of the thermometer. It must, however, be remembered, that in some places the Oriental sore attacks its victims in the season of the year when vital powers are lowest. It is important not to mistake phagedsenic tropical ulcers for the Oriental sore, as the former are due to constitutional debility, induced either by anaemia, malaria, scurvy, fatigue or want, in persons residing or travelling in swampy regions, where the atmosphere is hot and moist. YI. Endemic Hematuria. (See Plate IY. C.) Synon. — Distoma Haematobium ; Bilharzia Haematobia. Definition. — Endemic haematuria is caused by the entrance into ROY. SOC. PRO Vo!. XVI PLATE. V Robert W. Fclkin, dd. 1888-89.] Dr R. W. Felkin on Tropical Diseases. 291 the body of a trasmatoid hsematozoon, which is found in the portal system, the mesentery, bladder, &c. It produces hsematuria and anaemia more or less profound. Geographical Distribution. — The distoma has, so far as we know, a peculiar and very limited area of distribution. It is found in Mauritius, where the disease it causes was first described in 1812 by Chanotin. It is strictly limited to the delta of the Nile and to various points of the White Nile between 6° N. and the Albert Nyanza. It is also indigenous at the Cape, where it is strictly confined to the coast territory and to the banks of streams for a distance of some 10 or 20 miles from the sea. Its chief seat is in the south-eastern districts of Cape Colony near Algoa Bay, especially at Uitenhage and Port Elizabeth, the neighbourhood of King William’s town and East London in Kaffraria, as well as at several places in Natal; for example, on the banks of the Umlasi, the Ungeni between port Natal and Pietermaritzburg, and the Umhloti. It is probably also found in various other places in Central Africa. Remarks. — A knowledge of the geographical distribution of this parasite is of great importance, because, by taking proper precautions when residing in its limited area of production, it is possible to escape its ravages. There can be no doubt that it exists in stagnant pools, in the shallow water of declining rivers, near estuaries, and at the sea coast. It is affected by season, being found in the water in the summer, and it is during the summer too that most people are affected by the disease. It is curious to notice its preference for the male sex, and it is most commonly seen in them between the ages of five and thirty-five years. At Pietermaritzburg the majority of youths are affected by the parasite. As it can only obtain entrance into the body by means of drinking water or in bathing, the necessary precautions should be taken, and Europeans who may contract the disease should immediately remove from the infected area. VII. Beri-Beri. (See Plate V. A.) Synon. — Barbiers; Loempoe (Java); Kak-ke (Japan); Maladie des Sucreries (French Antilles); Sleeping Sickness (west coast of Africa) ; the Bad Sickness of Ceylon. 292 Proceedings of Royal Society of Edinburgh. [sess. Definition. — A disease characterised by anaemia, anasarca, de- generation of muscular tissue, effusion into the serous cavities, de- bility, numbness, pain and paralysis of the extremities, especially the lower; precordial anxiety, pain, and dyspnoea, and in some cases drowsiness or sleepiness. Beri-beri occurs in a chronic and an acute form. Geographical Distribution. — Beri-beri, like other similar diseases, has a wide distribution in many countries situated in the tropical and sub-tropical zones, but, although its area of distribution is so extensive, it is strictly limited in its endemicity. It is to be found both in the eastern and western hemispheres. Looking at the eastern hemisphere first, we find that one of its chief habitats is in Japan. There it was limited, until fifty years ago, to the coast towns, but since that time it has been met with practically all over the islands. In China it is not so frequently seen now as it used to be, but it is endemic in Burmah, in Singapore, and in the Calabash islands. In the Malay Archipelago, we find that most of the islands are endemically affected, especially Sumatra, Banka, Borneo, Labuan, Celebes, and some of the Molucca group of islands. We meet with it, too, on the west coast of Hew Guinea, and on the extreme east of Java, as well as in the prisons of Batavia in Passuruan and Samarang. With regard to India, Beri-beri infests a strip of country, 100 miles broad, on the coast from Gran dj am to Masulipatam, while it is more rare on the Coromandel and Malabar coasts, and in the plain of the Carnatic. It is also met with at various isolated spots in the provinces of Decca and Assam, and it is seen occasionally in Calcutta. It exists in Ceylon, especially at Trincomalee and Candy. As an epidemic, it is met with in Mauritius, Reunion, Nossi Be, Zanzibar, and probably on the Congo. Passing to the western hemisphere, we find that Beri- beri has been undoubtedly imported from the east. Thus, at various times it has been epidemic in Guadaloupe, Cuba, Cayenne, Para- guay, and San Francisco. Beri-beri may be said to be endemic over the greater part of Brazil, where it commenced in the Bahia Province, and it is also met with in Guiana. It must be mentioned, too, that Beri-beri often breaks out on board ship, especially in transports, coolie ships, and vessels tradiug in the Malay Archipelago, Bay of Bengal, and with Japan. 1888-89.] Dr R. W. Felkin on Tropical Diseases. 293 Remarks. — Many have been the theories started to explain the cause of Beri-beri, but, owing to the limited well-defined areas in which the disease is endemic, most of them are unsatisfactory. Its epidemic spread, however, is probably influenced by climate, and seems to coincide with conditions of high atmospheric moisture and extreme thermometric variations. Some parasite will, doubtless, ere long, be proved to be the real cause of the disease, and it is probable that its production will be traced to the soil, for in those places where Beri-beri is endemic the soil abounds in saline materials, such as magnesia, lime, chlorides, alumina, and iron. Although all races and persons of all ages are attacked by Beri- beri, the dark races suffer most, and adult men far more than women and children. Indeed, so great is the disproportion between male and female sufferers, that the cases seen in women are about one to thirty-one in men. Although it is, as a rule, rare for children under fifteen to be attacked by Beri-beri, yet there are epidemics on record in which children seem to have suffered most. Whatever be the cause of Beri-beri, a residence of eight or ten months in an endemic area appears to be necessary before a person can be attacked by it, and it is also a remarkable fact that it attacks by preference persons in an apparently robust condition according to some authorities, others think debility a predisposing cause. VIII. Oriental Plague. (See Plate V. B.) Synon. — The Pest; Inguinal, Bubolic, Glandular, Oriental, Indian, Pali, and Levantine Plague; Oriental Typhus; Septic Pestilence ; Fr. La Peste ; Ger. Die Pest. Definition. — A specific fever, attended by bubo of the inguinal or other glands, and occasionally by carbuncles. Geographical Distribution. — In olden times Oriental Plague had a very wide distribution; now it is met with in much narrower limits, and it is to its present distribution that we refer. In Africa its area is distinctly limited to the northern coast belt, including Morocco, Algiers, Tunis, Tripoli, and Lower Egypt. In Egypt it has never gone beyond the first cataract of the Nile. In Russia it is met with in Astrakan along the Volga, and outbreaks 294 Proceedings of Royal Society of Edinburgh. [sess. sometimes occur in Turkey. In Asia, epidemics of plague arise in Syria, Caucasia, Mesopotamia, and Persia; also in Arabia, on the coasts, and inland as far as Mecca. Epidemics have visited Hindostan, and tli ere are endemic centres of plague on tbe southern slopes of the Himalaya, in the provinces of Kurnaon and Gharwal and in Peshawur. It is probably endemic in the mountain valleys of Yunnan in China, and in Burmah. Description. — In recent outbreaks of plague, three varieties have been described — (a) Abortive or larval plague ; ( b ) Plague proper ; (c) Fulminant plague. The usual characters of plague may be rapidly summed up as follows : — After a day or two’s lassitude, shivering, and vomiting of a black material, high fever is experienced, with great pain in the axillary and inguinal regions, where buboes soon form. Often, too, the body assumes a livid hue, which gave plague the name of “black death.” The aspect of the plague patient is peculiar ; the face is haggard, the eyes retracted, and the conjunctive red. Remarks. — In considering the causes of plague, there can be no doubt that the influence of the seasons is very marked. The disease commences to make its appearance in the winter, and cases become more numerous as spring sets in ; but extreme heat or extreme cold usually puts an end to an epidemic. It is probable that plague has no relation to soil, and with regard to altitude, moderately high situations are more prone to be affected by the disease than are low-lying places, although from this it must not be understood that plains escape. There can be no doubt, however, that want, filth, and overcrowding are necessary to the production of plague, and it is well known that hygienic measures both prevent its appearance and stop its epidemic progress. In con- clusion, it may be noted that the closer the association of healthy people with the sick the more liable are they to contract the disease; hence, persons residing in the same house with plague-stricken patients are much more likely to be attacked than others, and cloth- ing and bedding may carry the infection. Robert W. FelJcin , del. PLATE. VI. Scott <£• Ferguson , Edinburgh. 1888-89.] Dr R. W. Felkin on Tropical Diseases. 295 IX. Dysentery. (See Plate YI.) Synon. — Fr. Dysenterie; Ger. Dysenterie. Definition. — A specific febrile disease, characterised by consider- able nervous prostration and inflammation of the solitary and tubular glands of the large intestine, sometimes ending in resolu- tion, but frequently terminating in ulceration, occasionally in more or less sloughing or gangrene; always accompanied by tormina and tenesmus. Geographical Distribution. — The distribution of tropical dysen- tery is very wide, and to a great extent coincides with the area in which malaria is endemic. Commencing with its existence on the west coast of Africa, we find that it is extremely prevalent in Senegambia, on the Sierra Leone coast, in Upper Guinea, and on the Gold and Slave coasts, as well as throughout the area watered by the Niger. In all these regions it affects natives as well as Europeans. It is not so frequently met with in the Cameroons, and from thence southward to Cape Lopez it is also less frequent. From Cape Lopez along the Congo coast endemic areas of dysentery are only to be found in isolated spots. Fernando Po is severely affected by this disease, as are also the islands of St Iago and Nicolao. In Madeira it is only epidemic. Passing on to the Cape of Good Hope, we again meet with a wide area of its distribution, the natives being especially affected by it ; but it is to be noted that the disease is more severe in the interior of the country than at the coast. On the east coast of Africa dysentery is endemic at Mozambique, Madagascar, Reunion, and Mauritius; also at Zanzibar and along the adjacent coast, but it is much less severe in Mayotte, Nossi Be, and St Marie. It is very prevalent all over Abyssinia, except in the dry open tablelands, and it is met with throughout the whole of the southern Soudan and Nubia; it also passes down the valley of the Nile to the Delta. It is endemic in Algiers and along the coast regions of Morocco, Tunis, and Tripoli. In Asia dysentery is met with in the valleys of Syria, in the plain of Mesopotamia, and in many parts of Persia, but it is most severe on the western and southern coasts of Arabia. It is found in the deep mountain valleys of Beloochistan and Afghanistan, and throughout 296 Proceedings of Royal Society of Edinburgh. [sess. all India, being least severe in the Presidency of Bombay. It occurs in Ceylon, in Further India, in most of the islands of the Malay Archipelago, and on the southern and eastern coast zones of China. In the islands of the Japanese Empire the disease is only epidemic. In Australia dysentery is endemic only on the west coast, although it occurs in slight epidemics in Melbourne, Sydney, Tasmania, and New Zealand. Endemic dysentery is also met with in New Caledonia, the Fiji Islands, Tahiti, the Mangareva group, and in the Hawaiian Islands. Passing to the western hemisphere, we find that in South America dysentery is endemic in French and Dutch Guiana, in Brazil, especially on the coast of the provinces of Maranhao, Piauhy, and Parahiba, and in the northern and central parts of the country. It is seen in Paraguay and on the coast of the Argentine Bepublic, as well as in the Provinces of Tucuman and Salta. The coast of Chili is also infested by dysentery, and the disease extends north- wards along the coast of Peru; it is, however, most severe in the forest region on the eastern slopes of the Cordillera, on the Peruvian Pampas, and in the marshy country bordering the Amazon. This area ends at the coast-line of Ecuador, Granada, and Venezuela. Dysentery is also endemic in Central America; that is to say, in Panama, Costa Pica, Nicaragua, the Mosquito shore, San Salvador, Guatemala, and Mexico. It is found all over Mexico until the Anahuac Plateau is reached. In the West Indian islands, Cuba and Hayti are its principal seats, Jamaica being less subject to it, and it is probably met with in the other islands. . With regard to the United States of America, it is difficult to say where the disease is endemic, but it occurs more or less all over the States, including California, and is most frequent along the Atlantic and the Gulf coasts. In British North America, Prince Edward Island and Vancouver Island are affected. In Europe, endemic dysentery is confined to a few spots chiefly in the southern peninsulas and islands. It is common in Andalusia, on the tableland of Estremadura and New Castile, in Aragon, and in the southern parts of Galicia and Catalonia. It is prevalent throughout Italy, especially in the southern provinces and in Sicily ; also in Malta. It is found in Greece in the Peloponnese, and in Constantinople, Boumelia, and Asia Minor. In France it is 297 1888-89.] Dr R. W. Felkin on Tropical Diseases. endemic in some parts of Guyenne and Provence, in Lyonaise and Auvergne, in a few valleys of the Yosges, in the marshy districts of the Rrenne, and in Sologne and Guer. In Sweden it is probably endemic at Jonkoping, Skaraborg, Elfsborg, Wermland, Goteborg, and Bohus, and in the island of Gottland. In Russia it casually occurs in the Baltic Provinces, hut more especially in Trans-Cau- casia. Remarks. — That endemic dysentery is due to some microbe, there can be little or no doubt ; at least the researches of Prior and Cartullis almost definitely prove it. It is, however, difficult to say with exactitude what gives rise to the morbific microbe. Although, as above stated, endemic dysentery occupies very nearly the same geographical distribution as malaria, yet it has some points of difference. For instance, its endemic spread can proceed to far higher latitudes than does malaria. Various facts prove that the disease is influenced by climate, for it is endemic at all seasons in hot tropical regions, and it is also found that in more temperate latitudes outbursts of the disease occur chiefly, one might almost say solely, during the late summer and early autumn season. Extreme cold puts an end to epidemic dysentery just as it does to yellow fever. All this shows that heat is necessary for its production. But, again, we find that fluctuations in temperature exert a marked influence in its pro- duction and spread. "Where hot days and cold nights obtain in tropical regions dysentery is most prevalent. This fact has been observed again and again in various wars, as, for instance, in Ashantee, Abyssinia, China, and more recently still in the Soudan. In Sweden the disease rarely attains any considerable epidemic diffusion, unless the summer has been remarkable for great heat ; and even the severe epidemics confined to limited portions of the country mostly coincide -with extremely hot summer weather. It is difficult to decide what influence the moisture in the atmosphere has upon the production of dysentery, as authorities differ very much on this point, but we cannot help believing that moisture does play a part in either the production of the disease or in predispos- ing persons to suffer from it, and that marshy districts and the neighbourhood of large rivers where morning fogs are of frequent occurrence, are certainly injurious, even if the moisture itself does 298 Proceedings of Royal Society of Edinburgh. [sess. not aid in producing the dysenteric poison. It may, of course, be that swampy districts exert their influence more indirectly than directly, for, doubtless, malaria does predispose individuals to attacks of dysentery, and the marshes may give rise to malaria which is only too often followed by intestinal disease. It has been stated by Annersley that dysentery rages in Bengal in the rainy season, and it is well known that the disease is most prevalent in lower Egypt at the time of the overflow of the Nile. It does not seem that elevation or configuration of the ground, nor the geological formation or the physical characters of the soil, have any connection with the production of dysentery. Although, as we have just remarked, malaria may predispose to dysentery, we do not find, in considering the geographical distribu- tion of the two diseases, that the points in which they are severally most virulent coincide, which one would naturally expect to he the case were the diseases produced by the same factors, or if they had a very intimate relation the one to the other. It can only he said, in conclusion, that contaminated drinking water may frequently serve to introduce the virus of dysentery into the system, and that it appears to he proved that a person suffering from the disease may introduce it into a previously healthy com- munity. X. Leprosy. (See Plate VII.) Synon. — Elephantiasis Grecorum ; Lepra ; Lepra elephantia ; Black Leprosy ; Bed Leprosy ; Elephantiasis tuberosa, anses- thetica, nodosa, mutilans, leontina, satyria; Joint Evil; the Myckle Ail or Great Disease ; Fr. La Lepre ; Ger. der Aussatz ; Scand. Spedaklalskshde ; Norway, Likpra. Definition. — A specific disease, endemic in many parts of the world, characterised by the slow development of nodular growths in connection with the skin, mucous membranes and nerves, and (in the last case) by the supervention of anaesthesia, paralysis, and a tendency to ulcerative destruction and gangrene (Bristowe). Two types of leprosy are described — the tubercular and anaesthetic varieties; the first variety is more frequently seen in temperate climates, the latter in the tropics. ROY. SOC. PRO. V O ! . X V i PLATE VII 20 40 \ 80 100 120 140 160 80 60 40 20 0 20 40 60 Scott & Ferguson , Edinburgh. 1888-89.] Dr R. W. Felkin on Tropical Diseases. 299 Geographical Distribution.— Leprosy is endemic at the present time in Egypt, and throughout the whole of the basin of the Nile, as well as on the shores of the Mediterranean and Red Seas. It is very prevalent in Abyssinia, on the coast, on the plains, and in the hill districts. It is also endemic at Zanzibar, Mozambique, Madagascar, St Marie, Mauritius, Reunion, St Helena, in the Canary Islands, and on the west side of the island of Madeira. It is met wTith in Algiers and Morocco, but is rarely seen in the Azores, and Tripoli and Tunis are said to be free from it. On the west coast of Africa leprosy is endemic in a very extensive area, extending from Senegambia to Cape Lopez. In this region it exists all over Sene- gambia and Sierra Leone; it overspreads the districts watered by the Niger and the Binue, as well as the whole of the Cameroons district. Leprosy is not met with on the Loango coast, and Angola as well as the Congo are free from it, as also is the province of Natal, but it is endemic to a considerable extent at the Cape and in Zululand. Passing on to the endemic area of leprosy in Asia and the Archi- pelagos adjoining it, it is to be noticed that India and the eastern parts of Asia are the most affected. In Nearer Asia the disease is endemic in a few limited areas; e.g ., on the southern coast of Arabia, especially at Muscat, as well as in the centre of the country. The mountainous districts of Persia, Syria, and Cyprus are affected, as are also some parts of Turkestan, especially Samaracand, Miankal, and Hissar. In Asia Minor it is now only endemic in isolated spots — at Smyrna, in the neighbourhood of Sinope, and on the shores of the Black Sea. It is met with all over India, but is least prevalent in the Madras Presidency. The disease is fairly common in Ceylon, but chiefly on the south and south-western coasts, and endemic areas are found in British Burmah, in the peninsula of Malacca, in Siam, and Cochin-China. In the East Indian Archipelago the most import- ant areas of endemic leprosy are on the west coast of Java and in its mountainous regions, the disease being more rarely met with on the southern and eastern coasts. Leprosy is also endemic in the Anda- mans and Nicobars, in the elevated and inland regions of Sumatra, on the west coast of Borneo, in Celebes (province of Menehasse), in Elores, in the interior of Timor, in Banda, and in the Philippines. In the Chinese Empire the areas of endemic leprosy are chiefly confined to the southern and eastern coasts ; it is rarely met with in the 300 Proceedings of Royal Society of Edinburgh. [sess. interior, except towards tlie northern part of the empire. In Japan leprosy is very prevalent, only the Loo-Choo Islands being free from it. It exists in New Zealand, in the Hawaiian group, and some- times in Tahiti. In Europe leprosy only occurs endemically, in small and, for the most part, clearly defined areas; in Spain it is confined to the provinces of Catalonia, Andalusia, Galicia, Asturia, and Granada. In Portugal it is met with in the provinces of Beira, Estremadura, and Algarve. In Italy it is endemic in two places — viz., at the Riviera- di-Ponete and at Comacchio. It is also found in Sicily. In the Balkan Peninsula leprosy still exists in small centres on the coast of the Ejalet of Salonica (Thessaly and Macedonia), and a few cases occur in Constantinople, which are probably imported. In Greece it is sometimes seen in the neighbourhood of Parnassus, and on the islands of Samos, Rhodes, Chios, and Mitylene, hut the chief seat of leprosy in this region is Crete. In Roumania and Hungary occasional cases of leprosy occur, hut it has ceased to he endemic there. In Sweden leprosy is met with in the districts of Anger- manland, Mendelpad, Helsin gland, Upland, and Bobus, hut it has diminished very greatly in recent years. The most considerable area of endemic leprosy in Europe is the west coast of Norway, from Stavanger up to Tromsoe, most of the cases belonging to the depart- ments of Sondre and Nordre Berghus. The disease is also met with in Iceland. Turning to the western hemisphere, there are three endemic areas of leprosy in North America — Mexico, Louisiana, and New Brunswick and Nova Scotia, and it is found amongst the Chinese immigrants in California. It also occurs in Costa Rica and in the West Indies, particularly in Cuba, Jamaica, St Bartholomew, St Kitts, Nevis, Antigua, Guadaloupe, St Vincent, Barbadoes, Trinidad, and the Bahamas. Leprosy is endemic in the elevated regions of Ecuador, and in various parts of Guiana, but its headquarters in South America are undoubtedly in Brazil, the provinces of Maranhao and Rio Grande being almost exempt. Erom the southern pro- vinces of Brazil the area extends over Paraguay and the northern parts of the Argentine Republic, especially throughout the pro- vinces of Entre Rios and Salta, and it stretches across the continent as far as the eastern frontier of Bolivia. Remarks. — There seems to he no doubt now that leprosy is ROY. SOC. PRO. Vo I . X V ! 160 140 120 lOO 80 60 40 20 Robert W. Felkin , del. PL ATE. VIII Scott da Ferguson, Edinburgh. Vol.XVI PLATE. VIII 301 1888-89.] Dr R. W. Felkin on Tropical Diseases. both contagious and hereditary, and that it is caused by the bacillus leprae, but for the present we must confess ignorance as to its origin. It should be stated, however, that the production of leprosy has been ascribed to extremes and frequent and rapid transi- tions of temperature, associated with high degrees of atmospheric moisture, but a glance at the chart will disprove this idea. Again, it has been said that leprosy must bear a special relation to the sea coast; but although in isolated cases it does occur chiefly near the sea, its area of distribution completely disproves this theory. Various articles of diet have been blamed as its cause — fish diet, salt or rotten fish, immoderate use of pork, and the use of decom- posed rice or maize. It is, however, impossible, after studying the subject, to arrive at the conclusion that any of these causes is the true one. The mere fact of the very definite isolation of the areas of endemic leprosy goes against these theories. XI. Yaws. (See Plate VIII. A.) Synon. — Framboesia; Button Scurvy; Verruga-Peruviana; Peru- viana Wart; Buba or Boba, and Patta (West Indies); Framosi (Calabar); Tetia (Congo); Tonga or Coco (Fiji); Lupani and Tono (Samoa); Fr. and Ger. Pian. Definition. — Yaws consists of an eruption of yellowish or reddish- yellow tubercles, which gradually develop into a moist exuding fungus without constitutional symptoms, or with such only as result from ulceration and prolonged discharge, namely, debility and prostration. It is epidemic, and contagious by actual contact. The period of incubation of the poison varies from three to ten weeks, and as a rule it only occurs once in a life-time. Geographical Distribution. — In Africa, yaws is to be met with on the west coast, from Senegambia in the north, as far south as Angola, together with the westerly Soudan, where it is especially frequent in Timbuctoo and Bornu. It is occasionally seen in the Xiie Valley, as well as on the northern and north-eastern African coast -line. It is very frequently met with in Madagascar, Mozambique, and the Comoros. In the East Indies it is chiefly seen in the Moluccas, Java, Sumatra, and Macassar; it is also 302 Proceedings of Poyod Society of Edinburgh. endemic in Ceylon, New Caledonia, Fiji, and Samoa; and it is met with amongst the Hindoo population of Pondicherry. In the West Indies it is endemic in San Domingo, Jamaica, Barbadoes, Martinique, Guadaloupe, Sta Lucia, and Dominica. It is found all over Brazil and in Guiana, and is said to he rather frequent at Punta Arenas in Central America. Remarks. — Yaws is distinctly a tropical disease, and the poison, whatever it may he, depends for its production on extreme heat and moisture; but although these factors would appear necessary for its production, there must be other causes, for in some countries, as in India for example, where the same temperature and moisture exist, yaws is unknown. Negroes chiefly suffer from its ravages, but no race is exempt, although it must be admitted that it attacks Europeans with comparative rarity. XII. Fungus Disease op India. (See Plate VIII. B.) Synon. — Madura Foot ; Mycetoma ; Morbus tuberculosos pedis ; Ulcusgrave ; Podelkoma ; Fr. Degenerescence endemique des os du pied; P^rical; Keerenugra. Definition. — A diseased condition of the hands and feet, occur- ring in India, characterised by enlargement and distortion of the affected extremity, due to thickening of the cutaneous tissues, with degeneration and subsequent fracture of the osseous structures. There are two varieties of this malady — one, the pale or ochroid form ; the other, the melanoid or dark form. This disease has been recognised since 1712, when Kampfer first called attention to it, but little definite was known about it until Goodfrey in 1846, and Vandyke Carter in 1860, investigated it thoroughly. In all probability, as Carter assumes, the disease is due to a parasite, but authorities still differ as to the nature of the parasite, and also as to the nature of the method by which it finds entrance into the hand or foot affected. It seems to be clear that it is not a constitutional disease, but the various theories which have been put forward as to its precise cause cannot be reconciled. Hindoos of all classes are affected by the disease. Mahommedans ROY. SOC. PRC Vol.XVI PLATE. IX ROY. SOC. PRC Vo I. XVI PLATE. IX 303 1888-89.] Dr R. W. Felkin on Tropical Diseases. are rarely attacked by it, and as yet there is no case on record of a European or half-breed suffering from it. Geographical Distribution. — Broadly speaking, this disease is met with in its dark variety in Madras, Bombay, the west and north-west of India, whilst cases of the pale variety occur all over India. The Malabar coast and inland places near it are chiefly affected, and the disease is reported as being present at Pondi- cherry, Bellary, Tanjore, Guntoor, Madura (whence one of its names), Cuddapah, Trichinopoly, and Combacumum. It is also met with on the slopes of the western Ghauts, in Rutnagherry, Poonah, and other parts of the Bombay Presidency, as well as in Kattivar, Goojerat, and Cutch ; in Kurrachee and other places in Scinde ; in Bawalpur, Bicanir, and other parts of Rajpootana ; and in the Punjaub at Jhelum, and the North-West Provinces at Sarsa and Hissar. It is very rare in Bengal, and the cases met with in Calcutta are all imported. Remarks. — It is most difficult to refer the cause of this disease to definite physical phenomena. It appears, however, that it is associated with certain definite local conditions, although what these conditions are it is hard to say. At the places where it occurs there is a heavy rainfall, the altitude of the district is not high, and as a rule the soil is moist, dolometic, and rich in vegetable matters. At the same time, it must be noticed that at Cuddapah the soil is clayey limestone, that at Pondicherry it is clay, and at Tanjore and in the places where the disease is known on the Malabar coast, the soil is alluvial. It is highly probable that the disease has an intimate relation to the soil; those most affected by it are persons employed in agriculture, and who go barefooted, exposing themselves thereby to wounds on the feet, which would readily permit the tissues to be invaded by a parasite, if a parasite, as we believe, causes the disease. XIII. Elephantiasis Arabum. (See Plate IX.) Synon. — Barbadoes Leg ; Cochin Leg ; Bucnemia indica ; Pachy- dermia ; Fr. Elephantiasis ; Ger. Elephantiasis. Definition. — A non-contagious disease, characterised by recurrence of febrile paroxysms, attended by inflammation and progressive 304 Proceedings of Royal Society of Edinburgh . [sess. hypertrophy of the integument and areolar tissue, chiefly of the extremities and genital organs ; occasionally by swelling of the lymphatic glands, enlargement and dilatation of the lymphatics, and in some cases by the co-existence of chyluria, and the presence in the blood of certain nematoid haematozoa ; together with various symptoms indicative of a morbid or depraved state of nutrition. Geographical Distribution. — Although elephantiasis may be occasionally seen in all parts of the world, it is endemic in cir- cumscribed areas, in tropical and sub-tropical countries ; in these areas of distribution it is not uniformly present, but is almost always limited to well-defined foci. In India, the disease is frequently met with along the littoral of Lower Bengal, in Pondicherry, and at a few other points on the Coromandel coast. It is especially frequent in the district of Tanjore, but most, of all on the Malabar coast (principally in Travancore and Cochin). It occurs at Ramghar, Chota-Nagpore, Sirgooja, and in the district of Tirhoot. In Ceylon the headquarters of the disease are on the coast, especially between Colombo and Matura. In the East Indies the places most severely affected are Sumatra, Banka, the Nicobars, and Philippines. In Further India it is met with in Penang and Cochin-China. In China elephantiasis is principally seen on the southern and south-eastern coast districts, especially at Canton, Amoy, Shushan, and Shanghai. Some of the worst regions of endemic elephantiasis are to be found in the Polynesian Archipelago, e.p., the northern part of New Caledonia, the Tonga and Fiji groups, the Samoa group, Wallace Island, the Society Islands, especially Tahiti, and Raiatea, and the Gambier group. Elephan- tiasis is less frequently seen in the Marquesas and Hawaiian islands. In equatorial and sub-tropical Africa and the adjoining islands, elephantiasis is very common, especially in Reunion, Mauritius, Seychelles, Madagascar, Nossi-Be, the Mozambique and Zanzibar coasts, the coasts of Senegambia and Liberia, and the Guinea coast as far as the equator. Further inland elephantiasis is met with in the Cameroons, in Bornu, and Sego ; and a few cases occur in Tunis, Algiers, and Egypt, not far from the sea coast, and in the swampy valleys of the interior of Abyssinia. Throughout the whole of the upper Nile valley and the adjacent districts isolated cases only are met with, save in the Bari and Madi districts, where 1888-89.] Dr R. W. Felkin on Tropical Diseases. 305 it is more frequently seen. It is said to be often found to the west of Lake Nyassa. In the western hemisphere elephantiasis is met with in New Granada, Venezuela, and Peru, in those parts of Brazil which are mostly tropical in character, on the coasts and marshy levels of Guiana, on the Gulf coast of Central America and of Mexico. Elephantiasis is also seen in the following islands of the West Indies: — Barbadoes, Martinique, Guadaloupe, Trinidad, St Vincent, and St Bartholomew. Remarks. — Although sporadic cases of elephantiasis are met with occasionally in Turkey, the south of France, Lisbon, and the south of Spain, as well as on the east coast of Scotland and in some parts of the south of Ireland, the endemic area of elephantiasis is from 35° 1ST. to 25° S. in the eastern hemisphere, and 25° jST. to 30° S. in the western. We must, therefore, consider that the disease for the most part depends upon high temperature and much atmospheric moisture for its production. Where cases occur outside the limits indicated above, it is in connection with moist soil and humid atmosphere, such as is met with on sea coasts and the banks of rivers. Climate not only appears to influence its pro- duction, but variations in temperature undoubtedly bear some relation to its growth, and some observers have maintained that it has a lunar periodicity. At any rate, as Hirsch says, “ the more flat and damp the ground is in a tropical or sub-tropical piece of country, the more suited does it seem to be for the endemic existence of elephantiasis.” Various theories have been advanced to explain the production of this disease. It has been said to be due to fish forming the staple of diet, to the drinking of water rich in saline constituents or tainted by organic matters ; and others have thought that it is a form of malarial poisoning, but there are numerous facts which prevent these views from obtaining general assent. It is, however, agreed by nearly all observers that the disease attacks principally the male sex, of dark races, over twenty years of age. Before concluding this note, it may be remarked that there is an increasing number of observers who believe that the cause of elephantiasis is the filaria sanguinis hominis • the maps show how far the distribution of this parasite is identical with that of elephantiasis. vol. xvi. 12/8/89 u 306 Proceedings of Royal Society of Edinburgh. [sess. XI Y. Guinea- Worm. (See Plate X.). Synon. — ^Dracunculns ; Pilaria medinensis. Definition. — The Guinea-worm is a nematoid parasite, usually- measuring from 1 to 3 feet in length, and having a breadth of -i- of an inch. It infests the feet and legs, as well as other parts of the body that are much exposed. Geographical Distribution. — As a general rule, the Guinea-worm is only found in the tropical parts of the eastern hemisphere, and even there the area of its endemic distribution is limited. It may, however, he conveyed from place to place, and, although rarely, become propagated in a fresh locality. In Africa, the principal area in which the Guinea-worm is found extends from Senegal as far as Cape Lopez. In Senegambia it is met with, not only on the coast, but in that more elevated region which extends from Bakel to Galam, but the parasite does not infest the banks of the Casa- mance. The Sierra Leone coast is less extensively infested by Guinea-worm than the Grain coast, Ivory coast, Gold and Slave coasts; it is met with on the shores of the Niger and Gaboon. It is to be noticed that on these coasts various places, such as Cape Coast Castle, Elmina, Cormantia, and Accra, are especially affected, whereas the surrounding country is very often free from the parasite. The Guinea-worm is found throughout Sennaar, the southern district of Kordofan, and the whole of the Bahr-el-Ghazel, the district between Dem Suliman and the Sobat being very extensively affected. It is doubtful if it exists south of latitude 6° N., and in Abyssinia the parasite is limited to the sea coast. In Asia the endemic seats of the Guinea- worm are Arabia- Petrsea, a few points on the coast of Hedjaz and Yemen, and the south coast of Persia. It is said to have been met with in the Bay of Skanderoom, and it is known in some parts of Turkestan, in Khiva, Bokhara, and Kokaun, on the shores of the Sir-Daria and on the northern shore of the Caspian (lat. 47° N). In India the Guinea- worm is most widely diffused in the northern division of the west coast, the Bajpootana States, and the western parts of the Deccan. It rarely occurs in the North-West Provinces, where it is only known at Delira Dhun, Sirsa, and Hansi. The parasite is also rarely met ROY. SOC. PRO. Vo I. XVI Robert W. Felkin, del. Scott &c. = &c. (pn+ 1 - a-n)(pn ~ an) = 1 + = An . These equations ensure that if the tensor of any one of the ps be unit, those of all the others shall also be units. Thus we have merely to eliminate p2 , . . ., pn ; and then remark that (for the closure of the polygon) we must have Pn+l — Pi • That this elimination is possible we see from the fact already mentioned, which shows that the unknowns are virtually mere unit-vectors ; while each separate equation contains coplanar vectors only. In other words, when p1 and cq are given, p2 is determi- nate without ambiguity. We may now write the first of the equations thus : — (p2 ~ a2)(pi ~ ai) = + (ai - a2)(Pi “ ai) = 9.1 > suppose. Thus the angle of qx is the angle of the polygon itself, and in the same plane. By the help of the second of the above equations this becomes ^(Pi — ai) = (Pb _ a2)9i 3 whence 9.2 ~ ^(Pi “ ai) + (2 — as)9i = (pb — aB)9i • By the third, this becomes (p4 - a 3)^ = Ag^i y whence (P4 — ai)92 = A-b9i "b (a3 — a4)92 = 9s * The law of formation is now obvious ; and, if we write 9q — Pi~ a] , = ai - ®2 ’ P2 = a2 “ a3 > ^c,» 1888-89.] Prof. Tait on a Geometrical Problem. 317 we have 9\ ~ Ai + fii9o ; 22 = A22o + A#l 9z — A32l + {$2,92 \ &C. (1) We have also, generally, Pm~ <*7 9m- 1 9m— 2 or _ 2m-l + am9m- 2 _ Am-l2m-3 + am-l9m-2 _P m- 2 „11tinADfl /Q\ pm = = = - — 5 suppose. . (i). 2m— 2 2 m— 2 i/m— 2 Prom (1), and the value of 2o> we see that all the values of q are linear functions of /q of the form 2rn = rm + SmPl (3). By (2) Pm—1 = %2m-2 + «m2m-l = (1 + ali)qm_ 2 + am{Am_1gm_3 + (am_x - am)gw_2} 9m— 2 P l9m— 3 P ^m— l2m— 2) 9m— 2 P ^mPm—2 1 Similarly 2m-i = j9m_2 - amqm_J But the first equations in (1) give at once 1+“iPi j. whence 20 ~ “ ai P Pi ' Po~ ~ 9oPi 1 2h = a2 — ai P P a2al)Pl ) 21 = l + a2al — (a2 — al)Pl J 2i= -PiPi (• This suggests that } 2» = (-)”2V>i Pm=(~ T+1imPl By (4) we have Pm— 1 = 9m— 2 P amPm— 2 > 9m— \ Pm— 2 ®m2m— 2 • Let m he odd, then we should have by (5) j?m_2 = ApBp1? 9m— 2 = B — Ap^ j (5). whence Pm- 1 = B - APl + am(A + Bpx) , 9m- 1 = A + Bp: - am(B - APl) ; 318 Proceedings of Royal Society of Edinburgh. [sess. or Pm-i = B + dmA - (A - amB)p1 , Qm— 1 R “I” (B 4" CtmA jpj • These agree with (5), because m - 1 is even. And similarly we may prove the proposition when m is even. If now, in (2), we put n+ 1 for m, we have C + T)pi . ^ - p«+i=pi=i j-Cp lf n be evenj = ~ if n be odd, D + Cpj C and D being quaternions to be calculated (as above) from the data. The two cases require to be developed separately. Take first, the odd polygon: — then pf) 4- piCp1 = C - D/q, or pfd 4- 8) 4- pfc 4- y)p1 = c + y- (d + S)p1 , if w^e exhibit the scalar and vector parts of the quaternions C and D. Cutting out the parts which cancel one another, and dividing by 2, this becomes dpY 4- S Sp1 4- /qSy/q -,c = 0, which, as p, is finite, divides itself at once into the two equations Sy/q + d = 0 , S 8p1 — c = 0 . These planes intersect in a line which, by its intersections (if real) with the sphere, gives two possible positions of the first corner of the polygon. For the even polygon we have Pi-D - PiCPi = C 4- DpT , or V /qS - pfiypi - y = 0 ; which may be written v.Pi(s-yw)=o. This equation gives pi = (* + y) !(s + ~f) i 319 1888-89.] Prof. Tait on a Geometrical Problem. where a? is to be found from Z2-V2 = ®¥_82 The two values of x 2 have opposite signs. Hence there are two real values of x, equal and with opposite signs, giving two real points on the sphere. Thus this case of the problem is always possible. The Solubility of Carbonate of Lime in Fresh and Sea Water. By W, S. Anderson, Chemist at Marine Station , Granton. (Read May 20, 1889.) At Dr Murray’s request, 1 have during the past winter continued the investigation of Messrs Irvine and Young on the solubility of carbonate of lime in its different forms in sea water (the results of which they submitted to this Society in May 1888); and the following notes of the work done and the results obtained by me, under Mr Irvine’s guidance, in the laboratory of the Marine Station, Granton, may be of interest. At his request, I have satisfactorily checked the results already laid before you. This has also, I understand, been done by Professor Thoulet of Haney. In that paper special attention was given to the solubility of amorphous and artificially crystallized carbonate of lime, and the various forms of coral in sea water. The later experiments with Iceland spar show it to be much less soluble than the above-named forms of carbonate of lime in sea water. As shown in the Table, calcite is less soluble in sea water than in pure water, the former dissolving of the impalpable powder only 0*0082 grammes per litre; while distilled water, free from carbonic acid, dissolved during the same time 0*0251 grammes per litre (more than three times as much). There is hardly any notable difference in the solubility of calcite, whether in the form of im- palpable powder, or in the condition of coarse powder, or large crystals, in sea water ; the solubility being only a trifle less with massive than with the more finely divided variety (see Table). With distilled water there is a very marked difference, the powdered 320 Proceedings of Eoyal Society of Edinburgh. [sess. spar dissolving to about double the extent. A very important factor to be taken into consideration in conducting these experi- ments is the time of exposure. When the same sea water stands over carbonate of lime for a lengthened period a curious and interesting reaction sets in, and the carbonate of lime it has dissolved appears gradually to diminish in quantity and be thrown out of solution again. This was observed by Professor Dittmar, who, in his paper on the composition of sea water (“Challenger” Eeport), says — “It seems that under certain abnormal conditions sea water dissolves lime largely in addition to what it contains normally, and subsequently will redeposit even more than the surplus lime in crystals of carbonate.” This result was also found by my predecessor, Mr A. Drysdale. Of course such a condition as this last may not occur in nature, as Dr Murray states, where the sea water is in continual circulation by tides, currents, &c., but it will help to explain the gradual petrifaction of the porous masses of dead coral reef, which being constantly supplied with salt water saturated with amorphous car- bonate of lime on standing, depositing, will gradually fill up the interstices, and produce the massive condition all old coral formations exhibit. This would take place in comparatively shallow waters and while in contact with carbonate of lime, but in deeper waters, and under greater pressure, any carbonic acid present might be called into play, as shown by Mr Eeid in his paper to the Society in February 1888. As is well known, carbonic acid water has a powerful solvent action on calcspar ; the more finely . divided it is the greater the solubility. One litre of water, saturated at ordinary temperature and pressure, dissolved in twenty-four hours, of the massive 0*0815 grammes, and of the powder 0*472 grammes per litre, or nearly six times as much (see Table). In order, if possible, to throw some light upon the condition in which the carbonate is present in sea water, a series of experiments was undertaken on the solubility of carbonate of lime in solutions of the different salts said to enter into the constitution of sea water. A hard variety of coral skeleton ( Oculina coronalis) was finely powdered, and the solutions allowed to act upon portions of it separately for four days. 1888-89.] Mr Anderson on Solubility of Carbonate of Lime. 321 As in the case of Iceland spar, the solubility was greater in distilled water than in sea water (some experiments conducted by Professor Thoulet, Nancy, confirming this result in the case of crystalline varieties of carbonate of lime). As shown in the Table, the magnesium salts dissolved the largest quantity of carbonate of lime, the solution of sodium chloride coming next, the calcium sulphate solution dissolving the least. Calcium sulphate appears rather to retard the solubility. It was to be expected that the magnesium salts would dissolve a compara- tively large amount of the carbonate and the sulphate of lime very little, for such is their action towards the amorphous, or non- crystalline, form of carbonate of lime. A solution of calcium sulphate dissolves very little amorphous carbonate, but a solution of magnesium chloride, holding the same amount as is present in normal sea water, takes up a large quantity of it, forming a clear solution, which on standing throws out the greater part of the carbonate of lime in a crystalline form. If a stronger solution of magnesium chloride be used, rhombohedral crystals of carbonate of lime are obtained large enough for their form to be seen distinctly with the eye. This experiment is indirectly important, as if there is an inter- change between the sodium chloride and the carbonate of lime entering the ocean, as is held by Tornoe, we might reason by analogy that since magnesium chloride dissolves more carbonate of lime than sodium chloride (although there is seven times the amount of the second than of the first in sea water), an inter- action between these salts would also take place, as — MgCl2 + CaC03 = MgC03 + CaCl2 . But such is not the case. Magnesium carbonate is not thrown down along with the calcium carbonate, as would be expected if such a reaction took place. It seems to be nothing more than a question of solubility. Sea water acts in very much the same manner as a solution of magnesium chloride, as amorphous carbonate of lime is soluble to the extent of 0*6 grammes per litre, which may be taken as the greatest amount of carbonate of lime in its most soluble form that sea water can dissolve without the help of free carbonic acid, in this case acting vol. xvi. 30/8/89 X 322 Proceedings of Royal Society of Edinburgh. [sess. as a vehicle for the carbonate of lime, as shown by the gradual crystallising out of the greater part that at first dissolved. After standing twenty hours, this solution only held 0*186 grammes CaC03, and after four days 0*162 in place of 0*6 grammes above referred to. This soluble action of sea water on amorphous carbonate of lime has nothing to do with carbonic acid. An artificial sea water, free from carbonic acid and carbonates of any kind, will dissolve up quite as much. It is distinctly confined to the soluble action of the salts present. In regard to the solubility of the coral skeleton in the various salts of sea water, it will be seen from the Table that if all these salts be acting together on the substance in one solution, the soluble action of the mixture is about the same as that of sea water. That is to say, normal sea water, and that artificially prepared by adding in proper proportions the salts present in sea water to distilled water free from carbonic acid, will dissolve practically the same amount of carbonate of lime. But with solutions of the individual salts, the results are in some cases higher, as in sodium chloride and magnesium chloride, and lower, as in sulphate of potassium and sulphate of lime. This difference is most probably owing to the sulphate of lime, which, as before said, seems to have a deterring action on the solubility, although when the carbonate of lime is once dissolved, the subsequent addition of the sulphate has no precipitating effect. Curiously enough, if these results of the solubilities of the coral he added up as before mentioned, the resulting figure comes to very nearly the amount of carbonate, taken as lime, given by Professor Dittmar as being present in sea water : — NaCl . = 0*0525 MgCl2 . = 0*0746 MgS04 . = 0*0712 k2so4 . = 0*0296 CaS04 . = 0*0209 0*2488 Subtract amount dis- solved by 4 litres Sea water contains — 0*1210 extra water used, And will dissolve of = 0*0285 x 4 = 0*1140 same CaC03 0*0237 0*1348 0*1447 0*1348 0*0099 1888-89.] Mr Anderson on Solubility of Carbonate of Lime, 323 Leaving only a difference of 1 centigramme per litre in favour of sea water. From this it appears to be a reasonable conclusion that the carbonate of lime present in sea water as such, is there simply owing to its solubility in the river water flowing into it, the salts present helping or retarding the solution as the case may he. It would seem, according to this view, that (except in special cases) the whole of the soluble carbonate of lime in sea water can be accounted for without the help of carbonic acid as a solvent, although, doubtless, its local action at great depths and in presence of decaying organic matter is notable. (See Reid’s paper, and Irvine and Young’s paper, Table II., Proceedings of Royal Society.) It will he seen by referring to the Table accompanying this paper that the amount of carbonate of lime dissolved by the various salts present in sea water amounts to 0T348; whilst the total amount of carbonate of lime figured as present in sea water, added to what it can yet dissolve, is 0T447, making the difference of 1 centigramme per litre in favour of sea water. The carbonic acid is therefore free to perform its true function, which is to support the enormous flora present in the ocean. Messrs Irvine and Woodhead, in their paper read before this Society, May 1888, indicate this as follows : — “ The relation between plant and animal life in the ocean is much the same as that between plant and animal life on land, so far as inter- change of carbon is concerned, considering the requirements of marine plant life in the form of carbon, which it can only obtain from the sea in the condition of carbonic acid.” The behaviour of a solution of carbonate of lime in sea water on standing in a closed vessel, where it is impossible carbonic acid could escape, seems to prove beyond question that its solubility has nothing to do with the existence of free carbonic acid or bicarbonates, as the major portion is thrown out of solution. Again, all the solutions of the various salts present in sea water, referred to in the Table, were made up according to the proportions in which they exist in sea water, with distilled water absolutely free from carbonic acid. [Table 324 Proceedings of Royal Society of Edinburgh. [sess. Table. — Solubility of Carbonate of Lime in Distilled Water, free from Carbonic Acid, in Sea Water, and in Solutions of Salts entering into the Composition of Sea Water ; 1 litre Water or Solution being used at Temperature 10°-15° C. Results in Grammes per Litre. Distilled Water. Sea Water. Carbonic Acid Water, Atmos- pheric Pressure. G « g is O O 3 £ ^q .3 h i« «w a O O ,2 c lg as q o o S CO .3 © 05 II X q ° .2 O) C3 ,© C |'s © Q -w -3 c £ o o 3 8 f the azimuth. Hence, from the formula cot a sin b = cot A sin C + cos b cos C 4 360 Proceedings of Royal Society of Edinburgh. [sess. we have at once tan (azimuth) = sin h sin X cos h - tan 8 cos X * Capt. Weir, in his diagram, virtually puts x = sin h sec X y = cos h tan X so that tan (azimuth) = - — , (i) x and y being found by the intersection of the confoeal conics /£#2 + , = 1 , the latitude ellipse, sec2X tan2X and sin2/i cos Vi = 1 , the hour-angle hyperbola. The Amplitude is the value of the azimuth at rising or setting, so that the corresponding hour-angle is to he found from cos h + tan X tan 8 = 0. With this value of h, equations (1) become x = sec X ^1 - tan2 X tan2 8 y= - tan2 X tan 8 Elimination of 8 gives, of course, the latitude-ellipse as before. But elimination of X gives, instead of the confoeal hyperbola, the curve x2 + \y - J(tan 8 - cot S)]2 = ^(tan 8 + cot 8)2 , or x2 + (y + cot 2S)2 = cosec2 28 , which is a circle passing through the common foci of the ellipses and hyperbolas. The construction of the “ Diagram” by means of (1) is, theoreti- cally, a very simple matter. Thus, take OA as unit length on the 361 1888-89.] Prof. Tait’s Note on Capt. Weirs Paper. axis of x, and draw AP parallel to y. Make AOP = X, and yOH = h. Draw the circles whose centre is 0, and radii OP and AP respec- tively. Let OH meet them in p , q. Prom p and q draw lines parallel to 0 y, Ox, respectively. Their point of intersection, Q, belongs obviously to the ellipse X, and to the hyperbola h. A somewhat similar, simple, construction can easily he given for the circle. On the Coagulation of Egg and Serum Albumen, Vitellin, and Serum Globulin, by Heat. By John Berry Haycraft, M.D., D.Sc., and C. W. Duggan, M.B. {From the Physiological Laboratory of the University of Edinburgh. ) (Read July 15, 1889.) A large number of proteid substances, when in solution, are coagulable by heat. As the temperature of such a fluid is raised, faint opalescence at first appears, and then, at a higher temperature, masses (flocculi) of albumen separate out, in most cases, suddenly, from the fluid. It is generally held that each coagulable albumen is so affected at a definite temperature peculiar to itself ; thus, egg albumen is said to become opalescent at 60° C., and to separate out in flocculi at 63° C. Unfortunately, hardly two observers agree as to the exact temperature at which opalescence and coagulation 362 Proceedings of Royal Society of Edinburgh. [sess. occur; thus, keeping to the example, egg albumen, Wurtz puts the coagulation point at 73° C., and Henrijean at 60° to 61° C. It is hardly possible to explain such differences, either on the assumption that any of the above authors had used imperfect apparatus, or, that they had been guilty of inaccurate observa- tion. It is more probable that the conditions, under which the experiments were performed, were not always the same. What are the conditions which are capable of modifying the coagulation point of albumen % It seemed to us a not unimportant point to investigate systematically these conditions ; as such investigation is calculated to throw light on the nature of coagulation itself, and may enable one to arrive at the exact specific coagulation points of the more important proteids, heated as they should always he under exactly similar conditions. The conditions modifying coagulation, which we have studied, are, — the rapidity at which the coagulation is allowed to take place, the degree of concentration of the proteid substance itself, the presence of acids and alkalies, and the presence of soluble salts. The Rapidity at which Coagulation is allowed to take place. This is an acknowledged factor varying the indicated temperature of coagulation, and at least one author has alluded to it in the case of the particular albumen studied by himself. If a solution of a coagulable proteid be heated quickly, the proteid will be found to coagulate at a higher temperature than if the heat be applied more slowly. Thus we found that egg albumen, diluted with one volume of water, coagulated at 64° C., when slowly heated, the temperature taking forty minutes to reach this point. Another portion of the same solution coagulated at 66° C., when heated rapidly, the experiment taking in this case only one minute. It is not difficult to explain this fact. If a drop of an albuminous fluid is mounted for microscopical examination, and, if it he heated on the stage of the microscope, the process of coagulation can he readily followed out. When opalescent, the fluid will be found to contain numbers of tiny granules. These granules increase in size, and apparently become adherent, and run together to form granular masses or flocculi. This naturally requires time, and if the fluid be heated 1888-89.] Haycraft and Duggan on Coagulation by Heat. 363 rapidly the temperature may materially increase above the point at which, were the fluid kept for a sufficient time, coagulation would occur. Although our experiments convince us of the general truth of this fact, it occasionally happens that an albumen slowly heated coagulates at a very high temperature, and perhaps never forms distinct flocculi, the coagulation being in the form of a thin jelly. Another portion of the same solution quickly heated coagulates in flocculi at a lower temperature. "We have found this occur with some specimens of serum albumen. We are inclined to explain this occurrence on the supposition that the slow and continuous heating in these cases causes some chemico-physical change in the albumen itself, whereby its coagulation is affected. The Influence on the Coagulation Point of the Degree of Concen- tration of the Albumen itself We find, as the result of our experiments, that in all the albuminous solutions we have investigated, the coagulation point is considerably raised by diluting the solution. A very dilute solution may not coagulate even on boiling, and egg white, diluted, but nevertheless forming a comparatively strong solution, cannot be coagulated, as Sir William Roberts long ago pointed out. In our experiments we invariably proceeded in the same way as regards the rapidity with which the solutions were heated, so as to eliminate any fallacy which might arise on this score. The usual method for determining coagulation points was adopted. The solution was placed in a test-tube containing a thermometer which could be used as a stirrer. The test-tube was immersed in a water- bath consisting of two beakers, one within the other, and each one filled with cold tap water. The water-bath was heated by a Bunsen, the flame of which was kept always at the same height, and so arranged that it took some forty minutes for the fluid in the test- tube to reach the temperature of 80° C. All our experiments were performed in this way, so that uniformity of results was obtained. We are inclined to think, however, that the heating process was unnecessarily slow, not only on account of loss of time, but what is more important, because it permitted changes to take place in the albuminous solution, especially when acids or alkalies were present in the fluid. 364 Proceedings of Royal Society of Edinburgh. The Effect of Dilution on the Coagulation Point of Egg Albumen. Egg albumen was prepared by cutting up the glairy white of an egg and squeezing it through a linen cloth. When this was diluted with water, the dilute solutions were carefully filtered. The egg albumen was always alkaline in reaction, but we decided not to neutralise it. In the first experiments the opalescence of the heated solution alone was observed. (1) Undiluted egg-white became opalescent at 58° C. (2) Egg-white, diluted with one volume of water, became opal- escent at 58°*75 C. (3) Egg-white, diluted with two volumes of water, became opal- escent at 59° *75 C. (4) Egg-white, diluted with three volumes of water, became opalescent at 60° -5 C. (5) Egg-white, diluted with four volumes of water, became opalescent at 610,75 C. In the second experiment the appearance of flocculi was noted as well as the opalescence. Opalescence appeared in the undiluted egg-white at 59° C., but did not appear so soon in the diluted portions, occurring about 1° G. higher for each dilution. (1) The undiluted albumen coagulated with the formation of flocculi at 64° C. (2) With one volume of water flocculi formed at 65°*5 C. (3) With two volumes of water flocculi formed at 69° C. (4) With three volumes of water a few flocculi formed at 80° C., the albumen never completely separating out. (5) Greater dilutions showed opalescence, but flocculi did not appear. The Effect of Dilution on the Coagulation Point of Serum Albumen. Serum albumen is said by Hoppe-Seyler (iii. p. 232) to become opalescent at 60° C., and to coagulate at 72°*C. to 73° C., and Schafer places it at 70° C. (4, p. 181). Serum albumen was prepared in the following way : — The serum from bullock’s blood was saturated by the hand with magnesium 1888—89.] Haycraft and Duggan on Coagulation by Heat. 365 sulphate, the precipitated globulin filtered off ; by this means one obtains a solution of serum albumen in a saturated solution of magnesium sulphate. It would have been useless to dilute this solution with water, for, in that case, both the albumen and the magnesium sulphate would suffer dilution. Dilution was effected by the addition of a saturated solution of magnesium sulphate. Vitellin, Serum alb. (mag. sulph.), Hydrocele fluid, . Serum globulin, . Egg albumen, Fig. 1. — Showing the Temperature at which certain Albumens coagulate when diluted with One, Two, Three, and Four Volumes of Fluid. (a) Undiluted serum albumen, saturated with magnesium sul- phate, becomes opalescent at 77° C., and coagulates at 79° C. ( b ) The same solution, diluted with one volume of a saturated watery solution of magnesium sulphate, becomes opalescent at 79° C., and coagulates at 82° C. (c) When diluted with two volumes, opalescence occurs at 79° C., and coagulation at 83° C. (d) When diluted with three volumes, opalescence begins at 81° C., and coagulation at 84° C. 366 Proceedings of Roycd Society of Edinburgh. [sess. (e) When diluted with four volumes, opalescence begins at 81°*5 C., and coagulation 840,75 C. (/) When diluted with five volumes, opalescence begins at 82° C., and coagulation at 85°'25 C. The numbers quoted do not give us the correct coagulation points for diluted solutions of serum albumen ; they are the coagulation points of diluted solutions plus magnesium sulphate, which raises the coagulation point considerably, as we shall subsequently see. The experiment only serves to show how coagulation varies with dilution of the albumen. In this experiment fine flocculi appeared even in the more dilute solutions, and their presence rendered the determination of the coagulation point quite easy, even in the most dilute solutions. In order to determine the action of magnesium sulphate, serum albumen was prepared in another way. Blood serum was diluted with two volumes of water, and a stream of carbon dioxide passed through it. The precipitate of globulin was filtered off. By this method the albumen was obtained mixed with a small quantity of globulin ; its presence, however, did not prevent the recognition of the point of opalescence and the coagula- tion point of the albumen. (a) The serum albumen became opalescent at 70° C., and coagu- lated in flocculi at 74° '25 C. The coagulation point being raised two or three degrees above the figure given by Hoppe-Seyler on account of its dilution. (b) This solution of serum albumen, diluted with one volume of water, became opalescent at 74°, the opalescence becoming very dense at 78° C. No flocculi appeared. On comparing these figures with those given for serum albumens in a saturated solution of magnesium sulphate, it will be seen that the former are uniformly lower, the presence of magnesium sulphate tending to elevate the coagulation point. The effect of dilution is more marked in the case of serum albumen by itself than in that of serum albumen in the saturated magnesium sulphate solution. In the first place, the coagulation becomes very imperfect in the dilute solutions ; in the second place, the temperature in the dilute solution is very much raised. 1888-89.] Haycraft and Duggan on Coagulation by Heat. 367 The Effect of Dilution on the Coagulation Point of Vitellin. The yolks of several eggs were dissolved in 6 per cent, solution of sodium chloride and filtered. The filtrate was poured into a large volume of distilled water, the precipitate of vitellin redissolved in saline solution, reprecipitated in distilled water, and dissolved in 5 per cent, solution of sodium chloride. In this case the vitellin, prepared from six eggs, was dissolved in 300 c.c. of the solution. In order to study the effect of dilution, a 5 per cent, solution of sodium chloride was added in all cases. (a) The vitellin solution became opalescent when heated to 80° C., and coagulated at 85° C. (b) When diluted with one volume of 5 per cent, solution of sodium chloride, the vitellin became opalescent at 81° C., and coagulated at 85° ’5 C. (c) When diluted with two volumes, it became opalescent at 82° C., and coagulated at 86°*5 C. (d) When diluted with three volumes, it became opalescent at 82° C., and coagulated at 87° C. (e) When diluted with four volumes, it became opalescent at 83° C., and coagulated at 88° C. The experiment was repeated, giving a result almost precisely the same. It will be noticed that in this proteid the coagulation point does not vary to a very considerable extent with dilution. The Effect of Dilution on the Coagulation Point of Serum Globulin. The coagulation point of serum globulin is given by Halliburton as 75° C. (Reference 6, p. 163). In the first experiment the globulin was precipitated from bullock’s blood by magnesium sulphate. The precipitate was, after washing, dissolved in a 5 per cent, solution of magnesium sulphate. It was diluted with a 5 per cent, watery solution of magnesium sulphate. Unfortunately the opalescence was not noted down. The flocculi were well marked. (a) The solution of serum globulin in a 5 per cent, solution of magnesium sulphate coagulated at 74° C. (b) The solution, when diluted with an equal volume of 5 368 Proceedings of Royal Society of Edinburgh. [sess. per cent, watery solution of magnesium sulphate, coagulated at 75° C. (c) When diluted with two volumes, coagulated at 75° ‘5 C. (d) When diluted with three volumes, it coagulated at 75° *5 C. (e) When diluted with four volumes, it coagulated at 76°-25 C. (/) When diluted with five volumes, it coagulated at 77° C. (y) When diluted with six volumes, it coagulated at 77° C. In another experiment serum globulin was prepared by passing a stream of carbon dioxide through dilute blood serum. The pre- cipitated globulin was dissolved in 5 per cent, solution of sodium chloride. The solution of globulin not being of the same strength (a little weaker), and the salt used for its solution being a different one, the coagulation points do not correspond with those obtained in the previous experiment. (a) Serum albumen, dissolved in 5 per cent, solution of sodium chloride, became opalescent at 74° C., and coagulated at 79° C. (b) Serum albumen, dissolved in 5 per cent, solution of sodium chloride and diluted with one volume of a 5 per cent, watery solution of sodium chloride, became opalescent at 77° *5 C., and coagulated at 81° *5 C. (c) Diluted with two volumes, opalescence commenced at 78°*5 C., and it coagulated at 82° -5 C. (i d ) Diluted with three volumes, opalescence commenced at 79° C., and coagulated at 84° C. The albumen at this stage had begun to putrefy, and on repeating the experiments it was found that the coagulation point was raised about two degrees for (a), ( b ), (c), and that ( d ) did not coagulate even on boiling. The Effect of Dilution on the Coagulation Point of Hydrocele Fluid. Hydrocele fluid contains the same proteids as are found in blood plasma, namely, fibrinogen, serum globulin, and serum albumen. In a case of chronic hydrocele there may be an almost entire absence of proteid matter. The proteid substance when present varies in amount, and the coagulation point varies with it. On diluting hydrocele fluid the coagulation point is raised. (a) Hydrocele fluid became opalescent at 65° C.; at 72° C. it assumed the consistence of a thin jelly which thickened, and at 76° C. flocculi separated out. 1888-89.] Haycraft and Duggan on Coagulation by Heat. 369 (b) Diluted with one volume of water, it became opalescent at 67° C., and coagulated at 81° C. ( c ) Diluted with two volumes of water, it became opalescent at 69° C., and coagulated at 86° C. (d) (a) Diluted with three volumes of water, it became opalescent at 73° C., and a few flocculi separated out at 90° C. Another specimen of hydrocele fluid, apparently containing less proteid matter, became opalescent at 70° C., and coagulated with the formation of flocculi at 80° *5 C. A third specimen became opalescent at 70° C., flocculi forming at 78° C. General Conclusions regarding Dilution. In the case of albumens and globulin existing in a natural con- dition within an animal fluid, such as white of egg, serum, or hydrocele fluid, the point of opalescence is gradually and almost uniformly raised by successive dilutions. The coagulation point, on the other hand, rises rapidly, and the more dilute fluids often refuse to form flocculi, or even to coagulate at all. When a globulin is dissolved in an artificially prepared saline solution, both the point of opalescence and coagulating point are uniformly raised on diluting the solution. The same appears to apply to serum albumen saturated with magnesium sulphate. The Action of Salts on the Coagulation Point of Albumen. It is known that the addition of many neutral salts to an albuminous solution has an important action on the temperature at which it coagulates. Some salts are stated to lower and others to raise the coagulation point. It is impossible to explain at present their action, and we have accordingly commenced a somewhat systematic examination of the question. Our results are far from complete, and will subsequently, we hope, be more fully extended. We have at present studied the action of two important salts, namely, magnesium sulphate and common salt, on the coagulation points of egg and serum albumen, vitellin, and globulin, and the action of these salts has been studied in all degrees of strength up to complete saturation. VOL. xvi. 31/10/89 2 A 370 Proceedings of Royal Society of Edinburgh. [sess. Although we feel that it would he quite out of place to attempt general conclusions, yet we believe one or two inferences may be drawn from the facts that we have gleaned. Some of the facts we have already obtained are sufficiently striking to justify us in thinking that a more extended investigation, made on similar lines, may throw some light on the mutual relation- ship existing between the albuminous and saline molecules when in solution together. We are aware of the extreme difficulty of the subject, since so little is known as yet regarding simpler problems, such as the mutual relationships that exist between simple mixtures of inorganic salts. The Action of various Salts on the Coagulation Point of Egg Albumen. Yarenne (Reference 8) finds that many salts by their addition elevate the temperature of coagulation, such are, common salt and sulphate of magnesium; others, such as sulphate of copper and chloride of barium, lower it; while a third series, such as sulphate of sodium and chlorate of potassium, have no effect. Table I. showing the Action of various Salts on the Coagulation Point of Egg Albumen. Salt added. Proportion. Opalescence. Coagulation. Per cent. °C. °C. Original solution of Albumen, . 61 65 Lithium chloride, 10 65 70 Sodium chloride, 10 64 66-5 Potassium fluoride, 10 66 71 Potassium chloride, . 10 63 68 Potassium bromide, . 10 67 7775 Potassium iodide, 10 67 75 Ammonium chloride, 10 64-5 70 Ammonium nitrate, . 10 71 73*5 Ammonium sulphate, 10 67 74 Magnesium chloride, . 10 69 75-5 Magnesium nitrate, . 10 68 70-5 Magnesium sulphate, . 10 65 70 Potassium nitrate, 10 68 76*25 Potassium sulphate, . 10 65 68-5 Bechamp (Reference 5,p. 29) finds, on the other hand, that sulphate of magnesium, alum, and the salts of sodium and potassium lower the coagulation point. He came to this conclusion after working with 1888-89.] Haycraft and Duggan on Coagulation by Heat. 371 very dilute solutions of albumen ; these did not coagulate at all, until after the addition of the salts mentioned. He added very small quantities of the salts to the albuminous solution, viz., less than one per cent. Had he worked with coagulable solutions of albumen, and had he added larger quantites of salt, his result would have been different. While, as we shall afterwards show, these salts as a rule raise the point of coagulation, it is not at all. improbable that dilute uncoagulable solutions of egg albumen may be enabled to coagulate, when they otherwise would not ; in fact, our results point to this conclusion. If so, it is only one of the many facts which indicate how little is at present known as to properties of the albuminous molecules and the factors which determine their solubilities. In the preceding table we have placed some of our own results. In all cases the temperature, at which opalescence and coagulation occur, has been raised, though often, as in the case of common salt, to a very slight extent. The Precipitation of Egg Albumen by Single and by Double Saturation with Neutral Salts. By complete saturation of an albuminous fluid with a neutral salt the proteid may be precipitated at the temperature of the laboratory. Thus Hammarsten has shown that globulin may be precipitated from serum by the addition of magnesium sulphate. In this case the globulin is not converted into a coagulated proteid, but can again be dissolved after the magnesium sulphate has been diluted. . The Action of Magnesium Sulphate. — The egg albumen was diluted with one volume of water and freed as much as possible from membrane. A portion of this was saturated with magnesium sulphate and filtered. The saturated solution contained about 100 per cent, of magnesium sulphate. In order to obtain solutions of albumen containing a lower percentage of the salt, the saturated solution was diluted with portions of the original albumen. The original diluted albumen became opalescent at 65° C., and coagulated, forming flocculi, at 6 6° -5 C. (a) The saturated solution became opalescent at 78° C., and coagulated at 80° C. ( b ) Egg albumen, containing 50 per cent, of magnesium sulphate, became opalescent at 67° '25 C., and coagulated at 68° '5 C. 372 Proceedings of Royal Society of Edinburgh, [sess. (c) Egg albumen, containing 25 per cent, of magnesium sulphate, became opalescent at 65° C., and coagulated at 67° C. (d) Egg albumen, containing 12*5 per cent, of magnesium sulphate, became opalescent at 63° *25 C., and coagulated at 65° C. (c) Egg albumen, containing 6*25 per cent, of magnesium sulphate, became opalescent at 63° C., and coagulated at 65° C. The action of this salt seems a very curious one, for while in large quantity it raises the coagulation point very considerably, small quantities seem to lower it slightly, and no doubt Bechamp Vitellin. Serum alb. Egg alb. Serum glob. Fig. 2.— Showing the Effect of different Strengths of Magnesium Sulphate on the Coagulation Points of certain Albumens. is correct when he states that the dilute uncoagulable albumen can readily be coagulated after the addition of the salt. He is hardly, 1888-89.] Haycraft and Duggan on Coagulation by Heat. 373 however, justified in speaking of magnesium sulphate as lowering the coagulation point of albumen by its presence. It is a point of some interest to discover whether a salt, which, by its addition to an albuminous solution, raises the temperature at which coagulation occurs, will produce the same result on an albu- minous solution already saturated with another salt. This we have determined to some extent. Effect of the Addition of various Salts on the Coagulation Point of Egg Albumen already saturated ivith Magnesium Sulphate. Egg albumen was diluted with two volumes of water, and saturated with magnesium sulphate. The solution was filtered, and it was found on heating to become opalescent at 79° C., coagulating at 810,75 0. The salts added, most of which have already been studied in respect to their action on the coagulation point of egg albumen (Table I.), are seen to lower the coagulation point of egg albumen saturated with magnesium sulphate. Table II. showing the Action of various Salts upon the Coagulation Point of Egg Albumen already saturated with Magnesium Sulphate. Salt added. Proportion. Opalescence. Flocculi. Albumen saturated with magne- Per cent. °C. °C. .• sium sulphate , 6 79 8175 Sodium chloride, 6 72 79 Sodium iodide, .... 6 70 Sodium sulphate, 6 79 81-5 Potassium chloride, . 6 72 79 Potassium bromide, . 6 70 74 Potassium nitrate, Potassium chlorate, . 6 70 73-75 6 71 74-5 Potassium sulphate, . 6 74 77 Ammonium chloride, . 6 62 73 Ammonium nitrate, . 6 i 63 65 On comparing this table with that on page 370, it will be noted, first, that those salts which on Table I. do not raise the coagulation point of egg albumen to any great extent, NaCl, KC1, K2S04, and Xa2S04 (Yarenne), do not lower the coagulation point (Table II.) to any great extent. On the other hand, salts like KBr, K2ND3, and NH4N03, which raise the coagulation point in Table I., depress 374 Proceedings of Royal Society of Edinburgh. [sess. it in Table II. It is possible still more to lower the coagulation point by the addition of larger quantities of the latter salts, until one can precipitate the albumen by double saturation at the tempera- ture of the laboratory. On the other hand, the addition of large quantities of NaCI and Na2S04 exerts very little action. Effect of Magnesium Sulphate on the Coagulation Point of Serum Albumen. Although Dr Halliburton has succeeded (Reference 6, p. 192) in precipitating serum albumen by double saturation by means of sulphate of magnesium in conjunction with such salts as sodium sulphate, sodium nitrate, potassium iodide, &c., magnesium sulphate in itself raises the coagulation point of serum albumen. (a) Serum albumen, containing 100 per cent, magnesium sulphate, became opalescent at 84° C., and coagulated at 89° C., a slight opalescence appearing at 40° C., due to a trace of serum globulin. ( b ) Serum albumen, containing 50 per cent, magnesium sulphate, became opalescent at 77° C., and coagulated at 86° C. (c) Serum albumen, containing 25 per cent, magnesium sulphate, became opalescent at 76° C., and coagulated at 84° ‘75 C. (ft) Serum albumen, containing 12 \ per cent, magnesium sulphate, became opalescent at 76° C., and coagulated at 82° C. (e) Serum albumen, containing 6J per cent, magnesium sulphate, became opalescent at 74° C., and coagulated at 78° *25 C. (/) Serum albumen, containing 3J per cent, magnesium sulphate, became opalescent at 72° C., and coagulated at 76° C. (g) Serum albumen, somewhat diluted in this experiment, became opalescent at 68° C., and coagulated at 75° C., without the formation of well-marked flocculi. Sodium Chloride. — Although Hoppe-Seyler states that this salt lowers the coagulation point of serum albumen, we find that this is only the case when present in large quantity. Small quantities appear, if anything, to raise it. A saturated solution of the same serum albumen as that used for the last experiment coagulated at 72° C., when saturated with common salt. A solution, containing 20 per cent., became opalescent at 74° C., and coagulated at 80°*5 C. 1888-89.] Haycraft and Duggan on Coagulation by Heat. 375 The Action of Sodium Chloride on a Solution of Serum Albumen already saturated with Magnesium Sulphate. In this case the coagulation was lowered as sodium chloride was added in greater and greater quantity. (a) Serum albumen, saturated with magnesium sulphate, became opalescent at 77° C., and coagulated at 79° C. (b) The same solution, plus 10 per cent, sodium chloride, became opalescent at 72° *5 C., and coagulated at 75° C. ( c ) The same solution, plus 20 per cent, sodium chloride, became opalescent at 70° C., and coagulated at 73° C. A larger quantity of common salt was not added, since 20 per cent, did not dissolve readily. The Action of Magnesium Sulphate on the Coagulation Point of Vitellin. Some vitellin was dissolved in a dilute solution of magnesium sulphate. Some of this was saturated with the salt, the precipitate filtered off, and the filtrate tested. (a) Vitellin, dissolved in a saturated solution of magnesium sulphate (100 per cent.), became opalescent at 88° C. Coagulation did not occur even on boiling, a few flocculi alone appearing. (b) Vitellin, dissolved in a 50 per cent, solution of magnesium sulphate, became opalescent at 87° C., and coagulated at 89° C. with flocculi. (c) Vitellin, dissolved in a 25 per cent, solution of magnesium sulphate, became opalescent at 81° C., and coagulated at 86°*5 C. (< d ) Vitellin, dissolved in a solution containing 12’5 per cent, magnesium sulphate, became opalescent at 79° C., and coagulated at 82°*5 C. (e) Vitellin, dissolved in 6 ’25 per cent, solution of magnesium sulphate, became opalescent at 74° C., and coagulated at 79° C. (/) Vitellin did not completely dissolve in 3J per cent, solution of magnesium sulphate. It was not heated. When further diluted until only about 1 per cent, magnesium sulphate was present, a distinct precipitate separated out in the cold. This experiment was repeated with a more dilute solution of vitellin. The coagulation points at corresponding strengths of the 376 Proceedings of Royal Society of Edinburgh. [sess. magnesium sulphate were all higher. The result was otherwise the same, the saturated solution requiring the greatest temperature for its coagulation. Vitellin. Serum alb. Serum glob. Fig. 3. — Showing the Effect of different Strengths of Sodium Chloride on the Coagulation Points of certain Albumens. The Action of Sodium Chloride on the Coagulation Point of Vitellin. Some vitellin was dissolved in 5 per cent, solution of common salt. It was saturated with the salt, and a precipitate of globulin filtered off. (a) Vitellin, dissolved in saturated solution of common salt, became opalescent at 70° C., and coagulated at 76° C. (b) Vitellin, dissolved in 20 per cent, solution of common salt, became opalescent at 83° C., and coagulated at 89° C. 1888-89.] Haycraft and Duggan on Coagulation by Heat. 377 (c) Vitellin, dissolved in 10 per cent, solution of common salt, "became opalescent at 80° C., and coagulated at 86° C. ( d ) Vitellin, dissolved in 5 per cent, solution of common salt, became opalescent at 79° C., and coagulated at 85° C. (e) Vitellin, dissolved in 2 ‘5 per cent, solution of common salt, became opalescent at 78° C., and coagulated at 83° C. This experiment was repeated, and showed that common salt raises the coagulation point of vitellin, but that it is lowered just before the point of saturation, and that it continues to be lowered until saturation occurs. Action on the Coagulation Point of Vitellin of both Common Salt and Magnesium Sulphate dissolved together in the Solution , If, to vitellin in a saturated solution of common salt, some magnesium sulphate be added, the latter dissolves with difficulty, precipitating the vitellin in flocculi ; on heating, other flocculi appear. If, to vitellin in a saturated solution of magnesium sulphate, common salt be added, the coagulation point is lowered. Thus, on adding 15 per cent, of common salt, coagulation occurs at 88° C., and with a little over 20 per cent, it is lowered to 70° C. Serum Globulin. — Serum globulin is precipitated by magnesium sulphate in excess, as Hammarsten has shown. The same observer obtained a precipitation by saturating with common salt. The Action of Magnesium Sulphate on the Coagulation Point of Serum Globulin. Serum globulin was precipitated from the serum of ox’s blood by passing a stream of C02 through it. The precipitate after careful washing was dissolved in magnesium sulphate solution. (a) Serum globulin is precipitated in the cold by saturating the solution with magnesium sulphate. (b) Serum globulin, dissolved in a solution containing 50 per cent, magnesium sulphate, became opalescent at 7 4° *5 C., and coagulated at 79° C. (c) Serum globulin, dissolved in a solution containing 25 per cent, magnesium sulphate, became opalescent at 78°*5 C., and coagulated at 80° *75 C. 378 Proceedings of Royal Society of Edinburgh. [sess. (d) Serum globulin, dissolved in a solution containing 12*5 per cent, of magnesium sulphate, became opalescent at 7 7° *5 C., and coagulated at 80° C. (e) Serum globulin, dissolved in a solution containing 6*25 per cent, of magnesium sulphate, became opalescent at 76° C., and coagulated at 78° *75 C. (/) Serum globulin, dissolved in a solution containing 3*125 per cent, magnesium sulphate, became opalescent at 71° *5 C., and coagulated at 77° C. Effect of Sodium Chloride on the Coagulation Point of Serum Globulin. (a) Serum globulin, saturated with sodium chloride, is pre- cipitated in the cold. (b) Serum globulin, containing 20 per cent, sodium chloride, became opalescent at 77° C., and coagulated at 79° *5 C. ( c ) Serum globulin, containing 10 per cent, sodium chloride, became opalescent at 79° C., and coagulated at 81° C. (d) Serum globulin, containing 5 per cent, sodium chloride, became opalescent at 79° C., and coagulated at 81°*75 C. (e) Serum globulin, containing 2*5 per cent, sodium chloride, became opalescent at 78° C., and coagulated at 80° C. (/) Serum globulin in much smaller quantity does not dissolve to form a clear solution. Tentative Conclusions regarding the Action of Salts. (1) A salt may raise the temperature of coagulation if present in a certain percentage ; at another percentage it may lower it. Thus common salt raises the coagulation points of both vitellin and serum globulin when present in moderately small quantity. Large quantities lower the coagulation point. (2) If a proteid be present in a saturated solution of a salt — such as magnesium sulphate — and, if another salt be then added, which by itself would raise the coagulation point, the coagu- lation point may in this case be lowered. It appears, too, that salts which are most active in raising the coagulation point are most active in lowering it, when added to a solution already saturated by another salt. 1888-89.] Haycraft and Duggan on Coagulation by Heat. 379 Statement as to whether it is possible to speak of the Specific Coagulation Point of an Albumen. From what has been already said, it is obvious that it is a difficult and perhaps a valueless task to attempt to determine what may be termed the “ specific coagulation point ” of an albumen. The coagulation point varies with the rapidity of heating, with the concentration of the fluid, with its reaction, and with the saline substances present. All that one can say is that, under such and such conditions, an albumen coagulates at such a temperature. It is probably hardly possible to obtain even two albumens under such similar conditions that their coagu- lation points may with advantage be compared. The nearest approach to this would perhaps be to dissolve a certain weight, say both of vitellin and another globulin such as serum globulin, in the same volume of salt solution. The coagulation points may, in this case, with advantage be compared. But what would be the value of the coagulation points so obtained for purposes of comparison with serum or egg albumen dissolved in water? The coagulation points quoted by previous writers cannot be taken in any sense as absolute values for the albumen named, modifying conditions having, as a rule, been totally disregarded. The same may be said of our own results, for the percentage strengths of the albuminous solutions used by us were in no case determined with any care. Although the forms of the curves represented in the charts are not affected by this, their altitudes in the scale of temperatures may be so to some, considerable extent. On so-called Fractional Coagulation. So far we have been dealing with albumen in its natural condition, or mixed and possibly combined with neutral salts which we had added. The solutions were alkaline, and, as we found, when dealing with the natural albuminous solution, difficult to coagulate, especially if in a dilute condition. Let us now consider the coagulation point of an albuminous solution to which an acid has been added. On adding an acid to an albumen solution, the coagulation is rendered, as every one knows, more easy, and it occurs at a lower temperature. The very dilute solutions, uncoagulable in the alkaline solution, 380 Proceedings of Royal Society of Edinburgh. [sess. are at once coagulated after the addition of a few drops of weak acid. No one has brought this out more clearly than Dr Halli- burton in a most suggestive paper (Reference 6), which will pre- sently be quoted in relationship to fractional coagulation. He showed that the coagulation point of serum albumen varies with the amount of acid present, the greater the quantity added, the lower the coagulation point, until finally coagulation could be produced at the temperature of the laboratory. If then the co- agulating point depended on the two factors, heat and the amount of acidity, it seemed to him a natural deduction, that, on keeping one of these, the acidity, a constant quantity, it might be possible to separate by fractional coagulation two or more albumens mixed together, and having different coagulation points. He investi- gated serum albumen, and found that if it be neutralised by the addition of some drops of a 2 per cent, solution of acetic acid, and if, further, it be rendered slightly acid by the addition of one drop of the dilute acetic acid to seventy-five drops of the albuminous solution, then it coagulates at 70° to 71° C., and if this coagulum be filtered off, and the solution again brought to the same degree of acidity, a coagulum occurs the second time at 77° to 78° C. If this coagulum be filtered off and the filtrate acidified as before, a third coagulum may be produced at 82° C. Dr Halliburton considers that the serum albumen, originally regarded as one proteid, in reality consists of three. MM. Corin and Berard have followed this process of fractional coagulation, applying it to egg-white. This substance, commonly held to consist of albumen and globulin, they believe to consist of three albumens and two globulins. They neutralise some egg-white, slightly acidify it, and raise its temperature, until opalescence appears; then they keep the tem- perature constant for a considerable time — an hour or even more. They filter off the coagula, re-acidify to the same degree, raise the temperature until opalescence occurs, and then after more prolonged heating flocculi again appear. In this way they have succeeded, as already stated, in fractionating five proteids. Without doubting that it may be possible to fractionate some proteids, nevertheless the results of our own work, and many of 1888-89.] Haycraft and Duggan on Coagulation by Heat. 381 the facts frankly stated by Dr Halliburton, seemed to throw some doubt upon the correctness of his deductions in the case of serum albumen, and this applied with equal force to the experiments con- ducted by MM. Corin and Berard on egg albumen. Our previous experiments have shown that, in alkaline solution, the more dilute a solution is, the higher is its coagulation point, and we have found that we could never completely precipitate any albumen at the temperature at which flocculi first appeared. The reason of this is very simple ; as soon as a solution begins to coagulate, the remnant, still soluble, is practically a more dilute solution of the same proteid, and must be heated two or three degrees more before it will begin to precipitate. In this case, also, the coagulating proteid will leave another soluble remnant, coagul- able at a still higher temperature, and so on. In fact, we may venture to make this general statement — In order to coagidate com- pletely any proteid it must be heated to that temperature at which its most dilute solutions are coagulable. We have not made so systematic an investigation upon the effect, on its coagulating point, of diluting acid solutions of albumen, but we have assured ourselves that the more dilute solutions coagulate at a higher temperature. One out of several experiments may be quoted the following : — Some egg albumen was diluted with two volumes of water and carefully neutralised. It was then brought to the same degree of acidity as is recommended for fractional coagulation, 1 cubic centi- metre of a 2 per cent, solution of acetic acid being added to 75 cubic centimetres of the albumen. This solution was found to coagulate at 53° C. When diluted with one volume of water, acidulated to the same degree, it coagulated at 54° C. With three volumes of water, it coagulated at 58° C. With seven volumes of water, it coagulated at 62° C. With fifteen volumes of water, it coagulated at 68° C. It is seen, therefore, that dilution has the effect of raising the coagulation point a great many degrees, the more dilute albumen requiring a much higher temperature for its separation. This may be shown in the most striking manner by heating some of the acidulated water to between 60° and 70° C. ; and dropping in some acidulated egg albumen it at first dissolves. Now divide the solu- 382 Proceedings of Royal Society of Edinburgh. [sess. tion into two portions, A and B, and heat A to 75° C., and keep B at the original temperature. A will coagulate, showing that although in too dilute a solution to coagulate below 70° C., it could neverthe- less coagulate, provided its temperature he raised. B will remain clear, hut, if more albumen be dropped into it, a point will be reached, at which it will cease to dissolve, and then it will separate out in flocculi. Here then, without going any further, one has come across an observation which, if it does not explain all the facts described under the head of fractional coagulation, must at any rate account for some of them. Both Dr Halliburton and MM. Corin and Berard found that after coagulation the filtrate, which they separated from the clot, was less acid than it was before coagulation had occurred, the latter observers finding that, as a rule, it was actually alkaline. Here, again, is a factor which we cannot afford to lose sight of. If the coagulation point is lowered by acidity, as all persons are agreed, one would expect that, while coagulation is proceeding, and while pari passu the acidity is decreasing, that the decrease of acidity would at last bring the coagulation — at that temperature — to a standstill. In this case one would expect, that on re-acidifying to the same degree, another crop of coagula might fall at the same temperature as did the first crop. Dr Halliburton does not mention any such coagulation, although undoubtedly it occurs, and we have found it on repeating his ob- servations, but MM. Corin and Berard evidently find that one is produced, and in consequence they heat the albumen for an hour or more before filtering off the coagulum. After this time, they found that the albumen never gave a second coagulation at the same temperature. We cannot but conclude from this that their experi- ments clearly indicate that the albuminous solutions with which they worked must have been very materially changed by the temperature, nor is it at all improbable that very material changes may occur in a solution of egg albumen kept in an acid solution at a high temperature for over an hour ; in one of their quoted experi- ments fractionation lasted over six hours. We may, we think, make this statement, and one fully borne out by our own experiments, that during coagulation in an acid medium 1888-89.] Haycraft and Duggan on Coagulation by Heat. 383 the coagulation point is being continually raised, both in virtue of the albumen becoming more dilute and in virtue of its becoming less acid ; these factors bring the coagulation to a standstill, but, after filtering off the coagulum, if the fluid be brought back to its original degree of acidity, and heated to the same temperature, coagula will again form, unless the albumen has undergone some physico-chemical change. It follows, too, that it is impossible to separate two albumens from one another by heat coagulation, unless, during the process of coagulation, the degree of acidity is kept uniform by the addition of small quantities of fresh acid, and unless the coagulation point of the most dilute solution of one of the albumens present be below the coagulation point of the other albumen. We became more con- vinced of this, when repeating in detail the experiments on fractional coagulation. After keeping an albuminous solution, either egg or serum albumen, at the temperature at which flocculi appear, for five or six minutes, and then filtering off the flocculi, we found that fresh flocculi appeared, when the filtrate had been re-acidified, and again raised to the same temperature. Two or three crops might be thus removed in the case of egg-white. Keeping up the same degree of acidity, and raising the temperature, we were able to get other crops of albumen. We were struck, however, by the fact that, while dealing with the more dilute albumen, the coagulation took place with difficulty, and it was longer delayed. This was particularly the case with egg-albumen. If the fluid filtrate from the coagulated flocculi be divided into two parts, and one portion raised gradually in temperature, opalescence followed by the formation of flocculi will appear. If the other portion be raised in temperature and kept for, say, three minutes at a temperature one or two degrees below the temperature at which opalescence appeared in the first portion, it will become opalescent and perhaps form flocculi. We found, in fact, that it was impossible to get the subsequent coagulation at definite points, as indicated by the previous observers, for the coagulation point depended upon the way in which the operations had been carried out. This was especially the case, when dealing with egg albumen, and we have little doubt that MM. Corin and Berard, working with careful method, invariably raised their temperatures to points which perhaps their first experiments 384 Proceedings of Royal Society of Edinburgh. [sess. had suggested. They, no doubt, produced coagula, but, had they tried the experiment, they would have obtained them equally well at a lower temperature provided they had raised the temperature more slowly. It is not difficult to fractionate egg albumen ten or twelve times. Another point that struck us was the smaller and smaller amount of coagulation produced, as the temperature of the solution was raised and successive crops produced. This was noticed by Dr Halliburton in the case of egg albumen. It is certainly the case with egg albumen. This, of course, in itself renders it highly probable that we are dealing in both cases only with one albumen, the more dilute solutions of which are alone coagulated at the higher temperatures. Even supposing that the y serum albumen of Dr Halliburton, of which he “in some case only found a trace,” and which coagulates at 82° C., is different from a and /? serum albumen, found in greater quantity, and coagulating at lower temperatures, yet fractional coagulation could not give us the means of proving this. One cannot compare the coagulating points of a dilute with a strong solution of two albumens, and presuming that y serum albumen is a dilute solution of an albumen differing from a and /?, yet its coagulation point would be lower than 82° C. in a solution of corresponding strength. It is, of course, possible that serum albumen may consist of more than one albumen, although it is probable, from what we have brought forward, that all the albuminous matter present coagulates at the same degree of concentration — at or about the same tempera- ture. Other methods may enable the physiologist to separate these, if they exist, from one another, and no methods have in the past yielded such valuable results as those in which separation has been obtained by the addition of neutral salts. Dr Halliburton has by this means tried to separate the a, /3, and.y serum albumens from one another, and frankly states that he has failed to do so (Reference 6, p. 173). In conclusion, we may. state that the method of fractional coagulation could only be of service when the coagulation points of the albumens present are widely separated from each other. In reality, fractional coagulation has been for a long time in use, and one of the few cases in which, as far as we can see, it is at all 1888-89.] Haycraft and Duggan on Coagulation by Heat. 385 applicable, is tbe separation of serum globulin from serum albumen. Serum globulin is precipitated at the atmospheric temperature on acidifying by a stream of carbon dioxide, or by tbe addition of weak acetic acid. This precipitation is not a complete one, how- ever, as Hammarsten has shown. The reason is, that, at the atmo- spheric temperature, part of the globulin remains in solution. This paper contains some of the results of a research, towards the expenses of which a grant of money was voted by the Scientific Grants Committee of the British Medical Association. Papers referred to in the Text. 1. Wurtz, Dictionaire de Chemie, article “Albumine.” 2. Henrijean, Contributions a V etude de l antisepsie. 3. Hoppe-Seyler, Handbuch der Chemischen analyse. 4. Schafer, Journal of Physiology , vol. iii. 5. J. Bechamp, Nouvelles Recherches sur les Albumines. 6. W. D. Halliburton, on Proteids of Serum, Journal of Physiol., vol. v. 7. Corin and Berard, Travaux du Laboratoire de Leon Fredericq, vol. ii. , 1887-8. 8. M. C. Varenne, Recherches sur la Coagulation de l' albumine Jahresberichte der Anat. v. Phys., 1885, p. 249. 9. Hammarsten, Archiv f. die gesammte Physiol., Bd. xvii,, 1878. Some New Points in Connection with Muscle Contraction. By Alexander James, M.D. (Read July 15, 1889.) The interval which elapses between tapping a muscle or tendon and the resulting movement of the limb has been estimated by many observers — Burckhardt, Tschirjieu, Waller, Brissaud and Francois Franck, Eulenberg, De Watteville, &c. — but the precise signification of these so-called reflexes is not yet fully understood. What follows is intended to add to our knowledge of this subject. The observations were made on a patient in the Boyal Infirmary, aged 26, who, as the result of a blow on the left side of the neck, sustained three years previously, presented (1) greatly impaired voluntary motor power in the left arm and left leg; (2) marked jerkings on tapping the tendons of the left supinator longus, left vol. xvi. 16/11/89 2 B 386 Proceedings of Royal Society of Edinburgh. [sess. quadriceps femoris, and left gastrocnemius ; (3) marked clonus of the left ankle and left knee-joints ; and (4) to a less extent impaired voluntary motor power of the right leg, with increased ankle and knee jerks and clonus. Knee Jerk. — The method of recording this, which I followed, was to connect two recording tambours, the first with a tube having a flexible extremity, which could he held over the patellar tendon, and through which the tendon could he tapped ; the second with a receiving tambour, the button of which was held in contact with the leg. The first recording tambour indicated on a rapidly revolving cylinder the instant at which the tendon was tapped, and the second the moment that the limb began to move. A chronograph, vibrating 100 per second, enabled the interval between the tap and the move- ment to he ascertained. Care was taken that the tubes connected with the tambours were of the same length. In both limbs the time was found to he about ’06 second — a rather longer interval than that stated by most other observers to he the case at any rate in health (see Tracing I.). Ankle Jerk. — To ascertain this the flexible end of the tube con- nected with the first recording tambour was held in contact with the tendo-Achilles, whilst the button of the receiving tambour was held in contact with the ball of the great toe. In the case of both limbs the time which elapsed between the tap on the tendon and the resulting movement was about -08 second (see Tracing II.). Plantar Reflex. — This was ascertained in the right leg only ; the cutaneous sensibility of the left being so much impaired, that its plantar reflex was practically non-existent. The method followed was to substitute for the flexible end of the tube connected with the first recording tambour an ordinary receiving tambour, to the button of which was fixed a pin point. The button of the other receiving tambour was held in contact with the lower end of the thigh. The interval between the prick thus applied to the sole of the foot and the resulting movement was T6 second. Wrist Jerk. — To ascertain this the flexible end of the tube con- nected with the first recording tambour was laid on the tendon of the supinator longus of the left arm, and the button of the receiving tambour was held over the metacarpal region of the hand. The tap applied to the tendon through the tube readily produced the desired Tracing VI. Tracing V. Tracing IV. 1888-89.] Dr A. James on Muscle Contraction. 387 movement, and the interval was found to be • 05 second. It was observed, further, that with each tap of the supinator longus tendon a contraction of the biceps muscle occurred. This was also timed by means of tambours employed in a similar way, the button of the receiving one being held in contact with the middle of the muscle. The interval between the tap on the supinator longus tendon and the contraction of the biceps was found to be -045 second (see Tracing III.). Ankle Clonus. — The method by which this was timed was the ordinary one, and is so well known that it need not be described. The rate of the clonus was found to be in both legs about 7 per second (see Tracing IV.). Knee Clonus. — This could be easily induced in both legs by drawing down the patellae, and was timed in the ordinary way. The rate was found to be about 8 \ or 9 per second (see Tracing IV.). From these data the following conclusions may, I think, be drawn : — I. That (as has been pointed out by other observers) the interval between the tap on the tendon or muscle and the resulting move- ment is too short to enable us to regard these jerks as being ordi- nary reflexes in which sensory nerves, nerve centres, and motor nerves are together concerned. Thus in this patient the plantar reflex was found to be T6 second, and the Achilles tendon jerk *08 second. Were both of these similarly produced reflexes, the latter would have taken as long, or even a longer, interval to occur instead of a much shorter one. II. That yet these muscle or tendon jerks cannot be regarded as direct contractions. In evidence of this is to be noticed — (a) That the interval between the tap and the resulting contrac- tion differs in different muscles, being greater in the case of the gastrocnemius than in the quadriceps femoris, and greater in the quadriceps femoris than in the supinator longus. Were they direct we should expect the interval to be the same or nearly the same in all. (I further found in this patient that with single induction shocks, as stimuli applied directly to the muscles, the contraction of the gastrocnemius took place more rapidly than that of the quadriceps. In the tracings given the interval was in the case of the quadriceps 388 Proceedings of Royal Society of Edinburgh. [sess. about ‘05 second, and of the gastrocnemius about *03 second (see Tracings Y. and VI.). (b) That when the tendon of the supinator longus was tapped contraction occurred in the biceps as well, and that the contraction of the biceps preceded that of the supinator longus. This could only occur as the result of some reflex in the cord. From these special conclusions the general one which I think may be drawn is, that these muscle and tendon jerks are really reflexes, but reflexes of a nature much more simple than the ordi- nary ones, in which sensory nerves, nerve centres, and motor nerves are concerned. Looking upon muscle, motor nerve, and central nerve cell as being composed alike of masses of irritable protoplasm, and remembering that the masses of irritable protoplasm which compose these can conduct equally well in either direction, we can suppose that the stimulus of the tap applied to the muscle, directly, or indirectly through its tendon, produces its contraction only after the impulse so generated has traversed through muscle and motor nerve fibre to nerve cell, and down again to muscle along the same nerve fibre. In this way, ceteris paribus, the longer the distance between a muscle and its nerve centre the longer will be the interval between the tap and the resulting contraction. This is, of course, borne out by these observations, the ankle jerk taking the longest and the wrist jerk the shortest time to occur, the knee jerk occupying an intermediate position. But the fact that contraction of the biceps occurred when the tendon of the supinator longus was tapped, and that the contraction of the former preceded that of the latter muscle, denotes that the impulse generated in muscle, motor nerve, and nerve cell, as the result of the tap, may be reflected from the nerve cell along other nerve fibres. We must conclude, how- ever, that for this reflex, as for a reflex along the same nerve fibre, a much shorter time is required than for one in which the afferent impulse travels along an ordinary sensory nerve. The phenomena of clonus bear out this view. Clonus may be regarded as being a series of jerks or contractions, each jerk or con- traction acting as the stimulus to the one which follows. The fact, then, that (as shown by the observations made on this patient, and as demonstrated by myself at greater length in a previous paper)* the * “ Clonus and Tendon Reflex Phenomena,” Edin. Med. Jour., Aug. 1880. 1888—89.] Dr A. James on Muscle Contraction. 389 rapidity of clonus in a given muscle varies inversely with its distance from its centre in the cord, the ankle being slower than the knee, the knee than the elbow, &c., indicates that in the production of each jerk, nerve cell and nerve fibres are concerned equally with the muscle, and that the length of the motor nerve fibres seems to be the element of most importance in conditioning the rapidity of the clonus. The Theory of Determinants in the Historical Order of its Development. By Thomas Muir, M.A., LL.D. Part I. Determinants in General (1836-41). (Continued from p. 234 of Yol. XVI.) GRUNERT (1836). [Supplemente zu Georg Simon Klugel’s Worterbuch der reinen Mathematik. Art. Elimination (I. Gleichungen des ersten Grades), ii. pp. 52-60.] With Grunert it is necessary to take a long step backward. Although the memoirs of Bezout, Vandermonde, and Laplace were known to him, in addition to those of Hindenburg, Rothe, and Scherk, he advances only a short distance into the subject; his' aim, indeed, is little more than the establishment of Cramer’s rule for the solution of a set of simultaneous linear equations. His mode of presenting the subject, however, is fresh and interesting, the method of “undetermined multipliers” being taken to start with. Writing his equations in the form (l)i^i + (2)pr2 + (3)^ + •••• + (n\xn = [1]1 (l)2aq + (2)2x2 + (3)2x3 + .... + (n)2xn = [1]2 (l)3aq + (2)3x2 + (3)3x3 + ....+ ( n\xn = [1]3 [ (1)^1 + (2)n^2 + (3)«^3 +>•••+ (»>A = [1], 390 Proceedings of Royal Society of Edinburgh. and taking pv pB, ... , pn as multipliers, lie readily shows of course that if the multipliers can be got to satisfy the conditions (2)iPi + (2)^2 + (2)sft + .... + ( 2)„pn = 0 1 (3)iih + (3)2^2 + (3)3^3 + • • • • + (3 )nVn = 0 (4)iPi + (4 )2y>2 + {^)sPz + • • • . + (1)* = 0 > (”)iP: 1 + (»)2i,2 + 'Wafts + • • • • + (n)nPn = 0 i the value of x1 will be K1P1 + [^2^2 + [l]sPs + • • • • + [1 ]nPn . (UiPi + (O2P2 + (Usft + • • • • + (1 )nVn ’ in other words, that xx can be determined at once if a function (l)ift + (1)2^2 + (OaPg + • • • • + (1)a can be formed of such a character that it will vanish when instead of the coefficients (l)l5 (1)2, (1)3, . . . , (1)M we substitute the members of any one of the n— 1 rows (2)1 (2)2 (2)3 . • • • (2). (3)i (3)2 (3)3 • • • • (3)» (Cl (C2 (Cs • ■ • • (C» Wi (re)2 (n)s .... ( n)„ ; the said function itself being the denominator of the value of x1 and the numerator being derivable from the denominator by insert- ing [l]i> [U2. [1]» [1]» ™ place of (1)1; (1)2, (l)s>. . . ,(!)„, Further, as any one of the unknowns may be made the first, the complete solution is thus put in prospect. “Alles kommt demnach auf die Entwickelung einer Function von der angegebenen Beschaffenheit an.” (xiii. 5) Two rules, Grunert says, have been given for the construction of such a function, one by Cramer, the other by Bezout. The former he states, and illustrates by constructing the desired function for the case where n = 4. The proof of it is then attempted, and is said at the outset to consist essentially in establishing the proposi- 391 1888-89.] Dr T. Muir on the Theory of Determinants. tion that a permutation and any other derivable from it by the simple interchange of two indices must, according to Cramer’s rule, differ in sign. This proposition is therefore attacked. The permu- tation (*). (l).+n A is taken in which the inferior indices are in their natural order 1, 2, 3, . . . , n, and k and 1 being interchanged, there arises the permutation (i)« (*).+/. B The part preceding ( k)a in A is called I., which thus of course also denotes the part preceding (l)a in B : the part between ( k)a and (l)a+/3 in A or between (l)a and (k)a+P in B is called II.; and the remaining part common to both A and B is called III. The number of inversions in both, when 1 and k are left out of account, is denoted by y : the number in both due to k and the division III. is denoted by A : the number in A due to k and the division II. by A' : and the number in both due to the division I. and k by A". The counting of the inversions then takes place for the two permuta- tions. In the case of A there are the inversions due (1) to I. and k , which are X" in number. (2) to I. and II. (3) to I. and 1, . . . . a - 1 . . . (4) to I. and III. (5) to k and II., .... A' . . . (6) to k and 1, . . . . 1 . . . (7) to k and III., .... A . . . (8) to II. and 1, . . . . /5 - 1 . . . (9) to II. and III. (10) to 1 and III., ... 0 . . . and as those not counted here are y in number, the total is seen to be a + /? + y + A + A' + A"-l. Similarly in the case of B the total is found to be a + (3 + y + X- X + X" — 2 . 392 Proceedings of Royal Society of Edinburgh. [sess. But the former total exceeds the latter by 2A' + 1, and this being an odd number, the proposition is proved. (hi. 26) Before proceeding further it is important to note that Grunert here establishes a more definite theorem than he proposed to himself, viz., the theorem of Rothe (III. 7). If he attains greater simplicity it is in part due to the fact that instead of taking any two indices for interchange, k and r say, he takes k and 1. To prove now that the function constructed in accordance with Cramer’s rule will satisfy the requisite conditions, it suffices to show by means of this theorem that on making any one of the n— 1 specified sets of substitutions the function will be transformed into one consisting of pairs of terms which annul each other ; in other words, to prove Vandermonde’s theorem regarding the effect of making- two indices alike. This is done; and then it is shown how xK can be got by interchanging xK and xl in all the given equations, the first step being of course to establish the fact that the denominator of xK and the denominator of xx only differ in sign. Bezout’s rule of 1764 is next taken up, and shown to be identical in effect with Cramer’s. The proof, by reason of the recurring character of the former, is inductive ; that is to say, it is demon- strated that, if the two rules agree in the case of n unknowns, they must also agree in the case of n+ 1. Paraphrasing the proof, but taking for shortness’ sake the case where n— 4, we say that it is agreed that both rules give in this case the signed permutations 1234, - 1243, +1423, -4123, - 1324, +. . . Now for the case where n — 5 Bezout’s rule directs that to the end of each of these permutations, e.g., the permutation - 4123, a 5 is to be put, and asserts that the result - 41235 will be one of the desired permutations with its proper sign. That it is a permuta- tion of the first five integers is manifest, and since the number of inversions in 41235 is necessarily the same as the number in 4123, its sign is correct according to Cramer’s rule. In order to obtain four other permutations, Bezout’s rule then proceeds to bid us shift the 5 one place and alter the sign, shift the 5 another place and alter the sign again, and so on. The result is + 41253, -41523, +45123, -54123. In regard to this, it is clear as before that permutations of the 1888-89.] Dr T. Muir on the Theory of Determinants. 393 first five integers have been got, and that the altering of the sign simultaneously with the shifting of the 5 is in accordance with Cramer’s rule, because every time that the 5 is moved one place to the left the number of inversions is increased by unity. The only question remaining is as to whether all the permutations are thus obtainable ; and as it is seen that each of the 24 permutations of the first four integers gives rise to 5 permutations of the first five, we have at once grounds for a satisfactory answer. (iii. 27) LEBESGUE (1837). [Theses de Mecanique et d’Astronomie. Premiere Partie : For- mules pour la transformation des fonctions homogenes du second degre a plusieurs inconnues. Lioumllds Journal de Math., ii. pp. 337-355.] This simply-worded and clear exposition is a natural outcome of a study of Jacobi’s memoirs on the subject. Like these it mainly concerns determinants of the special form afterwards individualised by the term axisymmetric ; and, indeed, it is notable as being the first memoir in which a special name is given to a special form, the expression u determinants symetriques” being repeatedly used for the particular determinants referred to. His general definition is (p. 343) : — “ Si l’on considere le systeme d’equations T ^ 1,2^2 + + AltJn = mi > i j ^-2,1^1 + ^2,2^2 + + A2ntn = W2 5 I * [ Anf i + Anf2 + + AnJn = mn , le denominateur commun des inconnues tly t2, ... , tn est ce que l’on nomme le determinant du systeme des nombres ' Au Ah2 • • • A^,! A2)2 • • ■ i 1 ^ An> i An>2 • • A 394 Proceedings of Royal Society of Edinburgh. Com me ce denominateur peut changer de signe, selon le mode de solution qu’on emploiera, on conviendra de le prendre de sorte que le terme A1(1A2(2A3)3 . . .A„ w> qni en fait partie, soit positif.” (vih. 3) No use, however, is made of this for the purpose of establishing the properties of the functions, results being for the most part taken from previous investigators and merely restated. A notation for what are nowadays called the minors of a determinant is given in the following words (p. 344) : — (xli. 7) “ Ceci rappele, si Ton represente par D le determinant du systeme (17), par \_g , i\ le determinant du system e qui se tire du systeme (17) par la suppression de la serie horizontale de rang g et de la serie verticale de rang i, et semblablement par la notation ft 3 le determinant du systeme qui resulte de l’omission des series horizontales de rangs g et i et des series verticales de rangs i et k dans le systeme (17), on pourra, . . REISS (1838). [Essai analytique et geom^trique. Correspondance math, et phys., x. pp. 229-290.] Reiss’s memoir, the first part of which appeared in 1829, was never completed. In the course of some remarks introductory to the present essay, he says by way of excuse : — “Je m’apergus bientot, et plusieurs savans me l’ont fait remarquer, que ces recherches, fussent-elles tres-fecondes en r^sultats elegans, etaient trop abstraites pour interesser le public qui n’apprecie les theories que selon le degre de leur ^ utilite. J’ai done tache de montrer, par un exemple, de quelle maniere on peut se servir de ces fonctions dans la geometrie analytique ; et j’ai choisi le tetraedre qui, par le V concours de plusieurs circonstances qu’on aura occasion de reconnaitre plus tard, permettait. une application tres-facile et presque immediate des premieres consequences auxquelles j’etais parvenu.” 395 1888-89.] Dr T. Muir on the Theory of Determinants. The analytical portion of the essay is to a considerable extent identical with the original memoir. In so far as there is a differ- ence, the change is towards greater simplicity, less seemingly aimless plunging into widely extensive theorems, and in general a better and more attractive style of exposition. Less space too is given to it, — not even half what is occupied by the portion on the tetrahedron, the main aim now being to urge on mathematicians the capabilities of the analysis in its application to geometry. The matters falling to be noted as not having been given in the original memoir are few in number and of little importance. In restating the theorem ( abc ...?*, a/3y . . . p) = ( abc . . . r, aj3y . . . p) the remark is incidentally made that the order of the terms on the one side is never the same as that on the other except when the number of bases is 1, 2, or 3 ; for example, the number of bases being 4, we have (abed, 1 234) = af^eft^ - af2cft^ - af>zcfl± + aAC4^2+ • • • > whereas (abed, 1234 ) = a1b2csd^ — a1b2d3c4^ — alc2b3d^ + . . . + aYc2df>± + . . . , the difference first appearing at the fourth term. (ix. 6) Bezout’s recurrent law of formation, formerly merely enunciated, is now accompanied by a demonstration. This is not without its weak point, the cause of which, as might be expected, is the awkwardness of Reiss’s rule of signs. The first paragraph, which will suffice to show its character, is as follows (p. 233) : — “ Portons notre attention d’abord, seulement sur la fonction (abc . . . r, a f3y . . . p). Si l’on se represente la maniere dont on fait les permutations des n elemens a,/3,y, ... p, on verra qu’a partir de la premiere, il y aura 1.2.3 . . . (n-1) complexions qui commencent par a, et que, si Ton separe cet element par un trait vertical des autres, on aura a droite toutes les permutations des elemens (3,y, . . . p. Les 1.2.3. . . (n- 1) premiers termes de (abc . . . r, a/3y . . . p) commencent 396 Proceedings of Royal Society of Edinburgh. [sess. done tons par aa, et puisque les signes de ces termes sont de- termines d’apres la maniere exposee plus haut, on tronvera leur somme — aa(bc . . . r, /3y . . . p).” Vandermonde’s theorem regarding the effect, on the function, of interchanging two bases is stated generally, and a demonstration is given. The mode of demonstration, which occupies one page and a half, will he readily understood by seeing it applied in later nota- tion to the case where there are four bases, that is to say, where the theorem to be proved is I aabpcyds | = - | baapcyd8 | . By repeated use of the recurrent law of formation we have | ttabpCyds | = aa | bpcyd8 | - ap \ bacyds | + ay\baCpds\ - a8 1 bac^dy\ = aa{ bp | cyds | - by | Cpd8 1 + bs\cpdy\} ~ ap{ba\Cyds \ - by | Cads | + b8 \ cady |} + ay{ba\ cpds | - bp\cads\ + h\cadp\) - as{ba | cpdy I - bp | c^y \ + by \ cadp | } . By collecting the terms which have ba for a common factor, bp for a common factor, and so on, this result becomes | O/gbpCydfr | ~ ba^ap | Cyd$ j cjy| Cpd§ | cpdy j j- + bp{aa | cyds | - ay\ I + «a| cody 1} -by\aa\ cpds\ - ap\cads\ + as\cadp\} + bs{aa\cpdy\ - ap\cady \ + ay\ cadp |} , = - ba | apCyds | + bp\aabyds\ - by\ aacpds | + &s| aacpdy | , — [ bofipCyds | , as was to be proved. (xi. 5) The suggestion readily arises that this -process would be equally applicable in proving Vandermonde’s theorem regarding the vanish- ing of a function in which two bases are identical, and the process, it may be remembered, was actually so employed by Desnanot. One of the theorems given by Scherk, and later by Drinkwater, appears in the following form (p. 240), the peculiar notation adopted for a determinant with a row of unit elements being constantly employed throughout the remainder of the essay: — 1888-89.] Dr T. Muir on the Theory of Determinants. 397 “ Si une des bases, par exemple a, est telle que la quantite qu’elle represente soit la meine quel que soit l’exposant dont elle est affectee, c’est-a-dire, si aa = a? = av = . . . , on aura (abc . . . r, afiy . . . p) = a°\fbc .. .r, fly ... p) - (be ... r, ay ... p) + (be. . . r, a/3S\ . . p) + La quantite qui se trouve sous la parenthese, peut done etre representee de la maniere suivante : (I be . . . r, a/3y . . . p) ; (xlviii. 3) en admettant une fois pour toutes que le chiffre romain I soit tel que 1 = Ia - IP = Iv = . . . II va sans dire que toutes les propriety qui ont lieu pour (abc . . . r, a J3y . . . p) se rap- portent egalement a (I be ...r, afiy. .. p)." The character of the identities used in the treatment of the tetra- hedron will be learned from a glance at the following examples : — aflbc, 123) - bflac, 123) + cflab, 123) = (abc, 123). (flq -a2)(I6c, 123) - - b2)(lac, 123) + (cx - c2)(Iab, 123) = 0. (ab,l2)(ac, 34) - (ab, 34)(ac, 12) = - afabc, 234) + a.2(abc, 134), = + afabc , 124) - afabc, 123). (lab, 123)(Iac, 124) - (lab, 124)(Iac, 123) = - (a± - a2)(Iabc, 1234). (lab, 123 )(abc, 124) - (I ab, 124 )(abc, 123) = + (ab, 12)(Ia&c, 1234). The first of these we have already seen used by Minding ; the second is nothing more than the manifest identity, 1 1 1 1 1 1 1 ax -«2 a1 a2 a3 or ai a2 «3 \ -h V CO h «1 ~C2 ci C2 C3 ci ci C2 C3 the third is evidently the equatement of two expansions of 398 Proceedings of Royal Society of Edinburgh. [sess. al a2 • • a3 «4 al a2 a3 a4 or a3 «1 a2 «4 ^2 h bl C1 C2 C3 C4 C3 C1 C2 C4 the fourth is a case of the fifth : and the fifth is itself a case of a theorem ( C' ) of Desnanot’s. CATALAN (1839). [Sur la transformation des variables dans les integrates multiples. Memoir es couronnes par VAcademie royale . . . de Bruxelles , xiv. 2me partie, 49 pp.] The first of the four parts into which Catalan’s memoir is divided hears the title “ Valeurs generates des inconnues dans les equations du premier degre , et pr opr ietes des denominateurs communs,” and in the introduction it is said to contain several remarkable new pro- perties of the functions called resultants by Laplace “ et connues aujourd’hui sous le nom de determinants” His method of dealing with the opening problem is to derive the solution of n equations with n unknowns from the solution of n- 1 equations with n— 1 unknowns ; or more definitely, to show that if the multipliers Xv X2, X3 necessary for the solution of the set of equations, aYxY + bpc2 + cpc3 — ax \ apC-^ + bpC2 d" CpC3 — a 2 V i + b3x2 + c3x3 = a3 / , be the determinants of the systems a2 b2 a3 b3 al bx a3 b3i al a2 b2i then the multipliers X15 A2, X3 , A4 necessary for the solution of the set oqaq + \x2 + cpe3 + dpc± = a4 ' a2xx + b2x 2 + c2x 3 + d£x± = a2 d“ b3x2 + c3x3 + d3x^ = a3 + b^x2 + cpc3 -1- dpc^ = a4 _ 399 1888-89.] Dr T. Muir on the Theory of Determinants. are the determinants of the systems «2 G2 a3 C3 «4 C4 aY C1 = I $_1?>0C1^2 I 1 — a determinant in which the index-numbers proceed by the common difference 1, and which is obtainable from | | by diminishing each index-number by 2. Sylvester’s form of the result thus is £ •JS2(«M).^PD(0aM)| = £_2(0 abed).* Following this comes the application to simultaneous linear equations, or as they are called “ equations of coexistence.” The system is represented by the typical equation a,x + bry + crz+ . . . + lrt = 0 , in which r can take up all integer values from - oo to + oo , there being really, however, only n equations, because of the periodicity imposed on the arguments of the bases. One so-called “leading theorem” is given in regard to the system, its subject being the derivation of an equation linear in x, y, z, . . ., t by a combination of the equations of the system. The theorem is enunciated as follows (p. 40) : — “Take /, g, . . ., Jc as the arbitrary bases of new and ab- solutely independent but periodic arguments, having the same * It is rather curious that Sylvester overlooks the fact that the legitimate equatemeut of two zeta-ic products implies an identity altogether independent of the existence of zeta-ic multiplication. Thus, the identity just discussed is essentially the same as the identity a a2 a? a4 a a? a4 a5 b 62 63 64 x {ah + ac + ad + be + bd + cd ) = b b 2 64 b5 c c2 c3 c4 c c2 c4 c5 d d? d 3 dx d d? di d 5 where the index-number denotes a power and the multiplication is performed in accordance with the ordinary algebraic laws. From this point of view the above quoted proposition of Sylvester’s involves an important theorem re- garding the special determinants afterwards known by the name of alternants. 406 Proceedings of Royal Society of Edinburgh. [sess. index of periodicity ( n ) as a, b, c, . . . , l, and being in number ( n — 1), e.e., one fewer than there are units in that index. “The number of differing arbitrary constants thus manu- factured is n[n— 1). “ Let Ax + By + Cz + . . . + lrt = 0 be the general prime deriv- ative from the given equations, then we may make A = £PD(0 afg . . . 7c) B=£PD(0 bfg ... 7c) C=£PD(0 cfg ... 7c) L=£PD(0(/# . . . 7c)” (xiii. 7) As in the case of the other theorems, no demonstration is vouch- safed. In order, however, that the connection between it and previous work may be more readily manifest, it is desirable to in- dicate how it would most probably be established now. Taking the case where the number of unknowns is t7iree and the number of given equations /owr, viz. — a4x + \y + cYz = 0 ' a2x + b2y + c2z = 0 a3x + b3y + c3z = 0 a4x + b4y + c4z = 0J , we should form an array of 4(4- 1), i.e. 12, arbitrary quantities, fi 9i hi f 2 92 ^2 f 3 9% di3 A 9 4 K > from which we should select the multiplier |/2<73/i4| for the first given equation, the multiplier \ffg3h^ for the second equation, and so on. The multiplication then being performed we should by addition obtain il* + Wf$A\y + K/s?AI 3 = 0 . which is what Sylvester would call “ the general prime derivative of 1888-89.] Dr T. Muir on the Theory of Determinants. 40 7 the four given equations,” the process being an instance of what he would similarly term the “ derivation of coexistence.” By proper choice of the arbitrary quantities it may be readily shown, as Sylvester proceeds to do, that the theorem gives (1) the result of the elimination of n unknowns from n equations ; (2) the two equations of condition in the case of n + 1 equations connect- ing n unknowns; (3) the ratio of any two unknowns in the case of n— 1 equations connecting n unknowns ; and (4) the relation between any three unknowns in the case of n - 2 equations connect- ing n unknowns. For example, the equations being axx + b^y + CjZ = 0 a2x + b2y + c2z = 0 a3x + b3y + c3z = 0 the theorem gives the general derivative «1 fl Ol \ fi Oi ci fi 9i «2 fi 9i X + ^‘2 fl Os y + G2 f% 92 this takes the form \a-J)2c^x -i- \b-J)2c(abc. . . ) = 0, 408 Proceedings of Royal Society of Edinburgh. [sess. a formula that from its very nature suggests and proves a wide extension of itself.” (xxm. 10) / It belongs evidently to the class of vanishing aggregates of pro- ducts of pairs of determinants, of which so many instances have presented themselves. There is a manifest misprint in the third product, which should surely be £PD (cfg . . . &)x£PD (ab . . . 1); and there is an error in the signs connecting the products, which, instead of being all + , should be + and - alternately. When the determinants involved are of the third order, the theorem in the later notation is K/a&MViAl - \hifi9z\-\aicid%\ + \cifiS sl-KMsI - l^iMl-KVsH0. which is readily recognised as an identity given by Bezout. With this theorem the paper proper ends, but in a postscript an additional theorem of a curious character is given. As enun- ciated by the author — even his double mark of exclamation being reprinted — it is (p. 43) : — “ Let there be (n - 1) bases a,b,c,. . . , l , and let the argu- ments of each be “ recurrents of the nth order,” that is to say, let Let It* denote that any symmetrical function of the rth degree is to be taken of the quantities in a parenthesis which come after it, and let ^ indicate any function whatever. Then the zeta-ic product, £(£R ,(abc ... 0 x i^PD(0abc . . . 1)) is equal to the product of the number 1888-89.] Dr T. Muir on the Theory of Determinants. 409 K cos . 67 r . 2t r\ ( ^ 7 — ; . 47T' sin — ) , , ( cos . — 4- J - \ . sin — n J \ n v n . 67A sm — J . n J COS -( (2n— l)7r — . 2(n-l)ir v J-l.sm 1 1 v 0 )) multiplied by the zeta-ic phase £e_£PD(0aZ>c. . . Z)!!” Unfortunately the meaning of the proposition is seriously obscured by misprints and inaccurate use of symbols. Instead of u rth » degree we should have Zth degree ; £ preceding R* ( abc ... Z) is meaningless, and should be deleted ; £ preceding ^PD (0 abc . . Z) in the first member of the identity is unnecessary when a £ has already been printed at the commencement ; and the subscript e, although giving an appearance of greater generality, serves no purpose what- ever. Making the corrections thus suggested, and denoting 2tt — - . 2ir cos J-l sin n n 47 T . . 47T cos- — + J - 1 sin — , n n which are the roots of the equation xn~l + xn~ 2 + xn~s + ... +^+1 = 0, by a, /3, y, .... A., we are enabled to put the theorem in the more elegant form £ |R t(a,b,c . . . , Z) . S . PD(0,a,6,c, . . ., Z)]- =Z-,{My. • • • . x).a.PD(0,a,6,c, • • 0} It is readily seen to be a generalisation of the first theorem of the paper, into which it degenerates when instead of being any function of a,b,c, ... Z, is a constant, and R*, instead of being any symmetric function, is one of the series 5a, 5a5, 5a&c, .... As, however, the constant Rf(a,/?,y, ... A.) on the right-hand side will then be one of the series 5a, 5a /3, 5 a/3y , .... and will not therefore be + 1 unless when t is even, there must be an inattention to sign in one or other theorem. The matter can be more appro- priately inquired into when we come to the subject of alternants, because, as has been pointed out in a recent footnote, it is to this 410 Proceedings of Royal Society of Edinburgh. [sess. branch of the subject that identities between two zeta-ic multipli- cations of difference-products really belong. This early paper, one cannot but observe, has all the characteristics afterwards so familiar to readers of Sylvester’s writings, — fervid imagination, vigorous originality, bold exuberance of diction, hasty if not contemptuous disregard of historical research, the outstrip- ping of demonstration by enunciation, and an infective enthusiasm as to the vistas opened up by his work. SYLVESTER (1840). [A method of determining by mere inspection the derivatives from two equations of any degree. Philosophical Magazine , xvi. pp. 132-135.] The two equations taken are anxn + + . . . + ape + a0 = 0 ) bnxn + bn_ itf"-1 + . . . + bpc + 60 = 0 \ f and rules are given for attaining three different objects, viz. (1) a rule for absolutely eliminating x; (2) a rule for finding the prime derivative of the first degree, that is to say of the form Ax - B = 0 ; (3) a rule for finding the prime derivative of any degree. The first of these concerns the process afterwards so well known by the name “ dialytic.” Only part of it need be given (p. 132): — “ Eorm out of the a progression of coefficients m lines, and in like manner out of the b progression of coefficients form n lines in the following manner : Attach m — 1 zeros all to the right of the terms in the a progression ; next attach m - 2 zeros to the right and carry 1 over to the left ; next attach m - 3 zeros to the right and carry 2 over to the left. Proceed in like manner until all the m - 1 zeros are carried over to the left, and none remain on the right. The m lines thus formed are to be written under one another. Proceed in like manner to form n lines out of the b pro- gression by scattering n - 1 zeros between the right and left. If we write these n lines under the m lines last obtained, we 411 1888-89.] Dr T. Muir on the Theory of Determinants. shall have a solid square m + n terms deep and m + n terms broad.” (liv. 1) The rest of the rule deals of course with the formation of the terms from this square of elements, the old and familiar method being followed of taking all possible permutations and separating the permutations into positive and negative. As applied by Sylvester in the case of the elimination of x between the equations ax 2 + bx + c = 0 lx2 + mx + n = 0 that is to say, as applied to the development of the determinant of the system a b c 0 0 a b c 1 m n 0 0 l m n , the method is lengthy. No hint at an explanation of this or either of the two other rules is given. The principle at the basis of them all, however, is essentially that of the preceding paper. A single example will make this plain, and will at the same time serve to give a better idea of the two remaining rules than could be got by mere quotation.* Let the two given equations be ax 3 + bx2 +cx + d = 0 axi + J3xB + yx2 + Sx + e =0 and suppose that it is desired to obtain their “ prime derivative ” of the 2nd (rth) degree, that is to say, the derivative of the form Ax2 +' B# + C = 0 . Taking the first equation followed by m - r - 1 equations derived from it by repeated multiplication by x} and then the second equa- tion followed by n - r - 1 equations derived from it in like manner, we have m + n- 2r equations, ax 3 + bx 2 + cx + d — 0 ax^ + bxB + cx 2 + dx =0 ax 4 + f3xB + yx2 + Bx + e =0 * The third rule is incorrectly stated. 412 Proceedings of Royal Society of Edinburgh. [sess. from which we have to deduce an equation involving no power of x higher than the 2nd. To do so we employ, as just stated, exactly the same method as was used in obtaining the “leading theorem ” of the preceding paper. That is to say, we form multipliers a b . a a a a p > a b effect the multiplications, and add, the result being . a b . a c . a d a b c X 2 + a b d x + a b a P y a ft S a p e 0 . (liv. 2) This is what Sylvester’s third rule would give. His second rule is simply a case of the third, viz., where r= 1 ; and his first rule is another case, viz., where r = 0. Had he followed the order of his former paper, he would have called the third rule his “ leading theorem,” and given the others as corollaries from it. RICHELOT (May 1840). [Nota ad theoriam eliminationis pertinens. Crelle’s Journal , xxi. pp. 226-234.] Just as Jacobi (1835) brought determinants to bear on Bezout’s abridged method of eliminating x from two equations of the ntYl degree, so did his fellow-professor Richelot, in treating of the other method of elimination, Euler’s and Bezout’s, discovered in the same year (1764). Euler’s method, it will be remembered, consists in transforming the problem into the simpler one of eliminating a set of unknowns from a sufficient number of linear equations; and Richelot in a few lines (p. 227) points out that this may, of course, be done by equating to zero the determinant of the system of equa- tions. An investigation connected therewith occupies the main portion of the paper. Sylvester’s method (1840) is described in passing, and the principle at the basis of it given. We have just seen that, when originally made known by the author, it was merely in the form of 1888-89.] Dr T. Muir on the Theory of Determinants. 413 a rule without any explanation. Although no doubt exists as to the mode in which it was obtained, still this first published description of the mode by Richelot deserves to be put on record. The whole passage in regard to it is as follows (p. 226) : — “ Quam aequationem * inveniendi methodi diversae a geo- metris adhibentur, ex quarum numero eius, quae a clarissimo Sylvester in diario The London and Edinburgh Philosophical Magazine and Journal of Science nuper exposita est, mentionem faciendi hanc occasionem haud praetermittere velim. Ibi illius eliminationis problema reducitur ad problema elimina- tion! s m + n - 1 quantitatum ex systemate m + n aequationum linearium. Multiplicata enim aequatione fx = 0 ex ordine per yn~ \ ?/n-2, . . . . , y°, nec non aequatione f2 = 0 ex ordine per ym~\ ym~ 2, . . . . , y°, adipiscimur systema m + n aequationum linearium inter quantitates ym+n~1i ym+n~2} ... f y°} quarum m + n— 1 prioribus eliminate, aequatio inter coefficientes f a' et a!' prodit. Quae eliminatio facillime ita instituitur, ut deter- minantem harum m + n aequationum linearium ponamus = 0. Determinans vero, cum quantitates a' et a" in aequationibus ipsae tantum lineariter involvantur, et quantitates a i in n, nec non quantitates a " in m ceteris aequationibus sobs reperiantur, respectu illarum dimensiones ntae est, respectuque harum mtae. Unde concluditur, earn positam = 0, esse quaesitam illam aequa- tionem finalem X = 0, quae omni factore superflua careat. Notissima enim est proprietas ab Eulero inventa aequationis X = 0, quod eius dimensio respectu quantitatum a' est = w, atque respectu quantitatum a'\ = m, ita ut quaeque functio integra evanescens, inter quantitates a' et a ", has dimensiones quadrans, pro genuina aequatione finali habenda sit.” (liv. 3) Taking Sylvester’s example, ax 2 + bx + c = 0 | * /. 0, A, 0, 0, b, a, B, A, 0, < c, b, C, B, A, d, c, 0, C, B, 0, d, 0, 0, C. On arriverait encore aux memes conclusions en partant de la formule (3). En effet, choisir les coefficients P, Q, p, q , r, de maniere a faire disparaitre de cette formule les di verses puissances /y* /y»2 /y»B 1 tvj CA/ y tA/ j j J de la variable x, c’est 6liminer ces puissances des cinq equations, (6) xf(oc) — 0, /(a) = 0, x2B{x) = 0, xY(x) = 0, F(aj) = 0, ou ” ax4 + bxs +cx2 +dx — 0 , ax 3 + bx2 + cx + d = 0 , (7) ]Ax* + Bx* + Cx2 =0, Ax 3 + Bx2 4- Cx = 0 , [ Ax2 + Bx + C = 0 . C’est done 6galer a zero la fonction alternee formee avec les quantity que presente le tableau, ' a, b, c, d, 0, 0, a, b, c , d , (8) <{ A, B, C, 0, 0, 0, A, B, C, 0, ^0, 0, A, B, C. Or cette fonction alternee ne differera pas de celle que nous avons deja mentionnee, attendu que, pour passer du tableau (5) au tableau (8), il suffit de remplacer les lignes horizontales par les lignes verticales, et reciproquement.” (liv. 4) Bezout’s abridged method for the equations %xn + aqx”-1 + ....+ an_xx + an = 0 bQxn + bpc"-1 + .... + bn_pc + bn = 0 is shown to lead to the final equation S = 0, 1888-89.] Dr T. Muir on the Theory of Determinants. 41 7 where S is “une fonction alternee de l’ordre n formee avec les quantity que renferme le tableau, in which o o <1 -A-0,1 • • • -h-Q,n- 2 Ao,n-l ^0.1 Am • • • ^l.n-2 -A-l.n-2 * • • • •h-n—2,n—2 A n— 2fn— ] ' -^0,71-1 K,n-1 • • . . h-n-2,n-l A A o,l + II — b0al+1 , — b±al+1 + A0(?+1 , ■h-2,1 — a2^l+l — b2al+1 + Av+1 . In connection with this, however, no reference is made to Jacobi’s paper of 1835. The fourth method, which occupies much the largest space (pp. 397-422), is not a determinant method. SYLVESTER (January 1841). [Examples of the dialytic method of elimination as applied to ternary systems of equations. Cambridge Math. Journ ., ii. pp. 232-236.] In returning to extend the method, here and generally afterwards called “ dialytic,” Sylvester takes occasion to say that “ the principle of the rule will be found correctly stated by Professor Eichelot of Konigsberg in a late number of C retie’ s Journal .” It may be noted, too, that he now for the first time uses the word determinant. Only the first and last of the four examples need be given, as the subject strictly belongs to the application rather than the theory of determinants. Even these, however, will suffice to show the masterly grip which Sylvester had of his own method. “To eliminate x, y, z between the three homogeneous equations VOL. xvi. 16/11/89 2 D 418 Proceedings of Royal Society of Edinburgh. [sess. Ay2 - 20! xy + Bx2 = 0 (1), Bz2 - 2A!yz + Cy2 - 0 (2), Cx2 - 2B 'zx + A z2 = 0 (3). Multiply the equations in order by - z2, x2, y 2, add together, and divide out by 2 xy ; we obtain C'z2 + C xy - A'xz - B'yz = 0 (4). By similar processes we obtain A'x2 + Ayz - B’yx - C 'zx — 0 (5), B'y 2 + B zx - C'zy - A!xy = 0 (6). Between these six, treated as simple equations, the six functions of x, y, z, viz., x2, y2, z2, xy , xz, yz , treated as in- dependent of each other, may be eliminated ; the result may be seen, by mere inspection, to come out ABC(ABC - AB'2 - BC'2 - CA'2 + 2A'B'C') = 0, or rejecting the special (hT.B. not irrelevant) factor ABC, we obtain ABC - AB'2 - BC'2 - CA'2 + 2A'B'C' = 0.” (liv. 5) The example, • however satisfactory as illustrating the dialytic method, cannot he passed over without a note in regard to the unaccountable blunder made in developing the determinant in- volved. In later notation the determinant is • C B -2A' c A . - 2B' B A . - 2C' A' A -C' -B' B' -C' B -A' C' -B' -A' C Now neither of the factors given by Sylvester are really factors of this, the truth being that it = - 2(ABC + 2A'B'C' - BB'2 - CC'2 - AA'2)2. 419 1888-89.] Dr T. Muir on the Theory of Determinants. The fourth example concerns the elimination of x, y, z between the three equations Ax 2 + B if + Cz 2 + 2 A 'yz + 2B 'zx + 2C 'xy = 0 Lx2 + My2 + Nz2 + 2L 'yz + TWzx + 2 Wxy = 0 Vx2 + Q.y2 + R z2 + 2P 'yz 4- 2Q 'zx + 2R 'xy = 0 Using each of the three multipliers x> y , z with each of the three equations, we obtain nine equations linear in the ten quantities, xs, y3, z3, x2y, x2z, y2x, y2z, z2x, z2y , xyz . Another such equation is thus necessary for success. Sylvester obtains it very ingeniously by writing . the given equations in the form (A# + B 'z +C'y)x + (By + C'x +A!z)y + (Cz h-A'^ + B'^ =0^ (Dr + M'^ + ^'y)x + (July + Wx + Uz)y + (Nz + L'y + Wx)z = 0 > (Px + Q 'z + R 'y)x + (Q y + R'^r + P 'z)y + (Rz + P 'y + Q 'x)z = o) i and then eliminating x, y , z. The work is not continued further. We may ourselves note, in conclusion, that the fourth example includes in a sense the three others, but that it does not follow therefrom that by giving the requisite special values to the co- efficients in the result of the general example, we should obtain the results for the particular examples in the forms already reached. Indeed, it is on account of this apparent non-agreement that the dialytic method is valuable to the theory of determinants, some very remarkable identities being arrived at by its aid. An explanation is also thus afforded of the trouble we have taken to elucidate its history. CRAUPURD, A. Q. G* (February 1841). [On a method of algebraic elimination. Cambridge Math. Journal , ii. pp. 276-278.] In Craufurd we have an independent discoverer of the dialytic method. A full account of his paper is quite unnecessary : the few * Only the initials A. Q. 0. C. are appended to the article. There can be little doubt, however, that they belong to Craufurd, whose name in full appears elsewhere in the Journal. 420 Proceedings of Royal Society of Edinburgh. [sess. lines dealing with his introductory example will suffice to establish the fact. He says : — “ Let it he required to eliminate x from the equations X 2 +px + q = 0 , x2 +p'x + q' = 0 . Multiply each of the proposed equations by x, and you obtain xs +px2 + qx = 0 , xs -\-p’x2 + fx=0. These two combined with the two given equations make a system of four equations containing three quantities to be eliminated, viz., x, x 2, xs ; and they are of the first degree with respect to each of these quantities. We may, therefore, elimin- ate x, x2, xs by the rules for equations of the first degree. The result is .... ” He enunciates a general rule, and then takes up the analogous subject in Differential Equations, where successive differentiation takes the place of successive multiplication by x. In a postscript he acknowledges Sylvester’s priority which the editor had pointed out to him. He knew nothing of determinants. CAUCHY (March 8, 1841). [Hote sur la formation des fonctions alternees qui servent a resoudre le probleme de l’elimination. Comytes Rendus .... Paris , xii. pp. 414-426; or CEuvres Completes * P Augustin Cauchy , lre Ser., vi. pp. 87-99.] Recalling the fact that the final equation, resulting from the elimination of several unknowns from a set of linear equations, has for its first member “ une fonction alternee,” and pointing out the further fact that the same holds good in regard to the elimination of one unknown from two equations of any degree, “puisque les methodes de Bezout et d’Euler reduisent ce denier probleme au premier,” Cauchy affirms the importance of being able easily to write out the full expansion of such functions. There can be little 1888-89.] Dr T. Muir on the Theory of Determinants. 421 doubt, however, that it was the second fact alone, — in other words, the discoveries of Jacobi, Sylvester, and Bichelot, — which influenced the veteran Cauchy to return to a subject practically untouched by him for thirty years. The opening part of the paper is, of course, necessarily old matter. One thing to be noted is that Cauchy tacitly discards the term determinant , which he was the means of introducing, using uniformly the more general expression fondion alternee instead. Another is that he adopts the rules of signs which makes use of the number of interchanges. From this his own peculiar rule of signs is deduced, and made the starting point for the fresh investigation which forms the main portion of the paper. The exposition of his rule, which differs from that of 1812, is worthy of a little attention, both on its own account and because otherwise the matter following would be scarcely intelligible. In the case of any term (“ terme ” or “ pro- duit ”) of the determinant say the term ^ — a0,Qal,ia2,2a3>3a4cAa5,5a6,G 5 ^0.laL0a2.5^3.3a4.6a5,4%.2 ’ there is an underlying separation of the indices 0, 1, . . , 6 into groups (“ groupes ”), by reason of the system of pairing; that is to say, since an index is found paired along with one index and not with another, there arises the possibility of looking upon those which happen to be paired with one another as belonging to the same family group. Thus, attending to the first a of the term, we see that 1 and 0 belong to the same group, and as on scanning the rest of the term, we find neither of them associated with any other index, we conclude that the group is binary (“ un groupe binaire ”). Again, we see that 2 is paired with 5, 5 with 4, 4 with 6, and 6 with 2 ; this gives us the quaternary group (2, 5, 4, 6). Lastly, 3 is seen to be paired with 3, and thus forms a group by itself. Now, if we wish to find how many interchanges of the second indices are necessary in order to obtain the given term ^O.l^l.O a2,ba3,3a±,QabAaG,2 from the typical term ^O.O^l.l a2,2a3,3a4Aa5,ba6,G > we may do the counting piecemeal, attending at one time to only 422 Proceedings of Royal Society of Edinburgh. [sess. that part of the term which corresponds to one of the groups of indices. In the case of the group (3), the number of interchanges s 0; in the case of the binary group (0, 1) it is 1 ; and in the case of the quaternary group it is 3 — the number of interchanges being “evidemment ” one less than the number of indices in the group. If, therefore, for a given term there be in all m groups, viz. / groups of one index each, g groups of two indices each, h of three, k of four, &c., the number of necessary interchanges will be Of + l.g + 2 .h + 3 .k + which f Jr 2 . g + 3.7?/ + i.k + . . . , — (/ + g + h + k +...), f--n- to ; and consequently the sign of the term will be + or — 1 according as n - m is even or odd. (hi. 28) The first step of the new investigation is to define “ termes semblables ou de meme esp&ce.” Two terms are said to be alike or of the same species when the one may be obtained from the other by subjecting both sets of indices in the latter to one and the same sub- stitution or permutation. Thus recurring to the term above used, tt0.1^1.0a2.5%.3tt4.6ffl5, 4^6.2 > and substituting in both of its sets of indices 6, 0, 1, 4, 3, 2, 5, instead of 0, 1, 2, 3, 4, 5, 6 respectively, — in other words, and with the notation of the memoir of 1812, performing the substitution /0 1 2 3 4 5 6\ \6 0 1 4 3 2 5/, we obtain the like term tt6,0a0.6^1.2a4>4a3,5a'2.3a5.1 • (LV‘) The groups in two like terms are evidently similar, the values of f g , h, . . . for the one being the same as those for the other. Indeed, since it is in this matter of groups or cycles that the terms have any likeness at all, the expression “ cyclically alike ” would have been a better term for Cauchy to use. From the definition there arises the self-evident proposition — Terms which are cyclically alike have the same sign. (hi. 29) 1888-89.] Dr T. Muir on the Theory of Determinants. 423 Also, the full expansion of a determinant may he represented hy writing a term of each cyclical species , and prefixing to each such typical term the symbol 2 icith its proper sign, + or — . (lv. 2) To obtain a term of any given cyclical species, that is to say, corresponding to given values of f g, h, . . . , all the preparation that is necessary is to write the indices 0, 1, 2, 3, \ , (7.-1), enclose each of the first / of them in brackets, enclose in brackets each of the next g pairs, then each of the next h triads, and so on. This gives the groups of the term, and the term itself readily follows. For example, if we desire in the qase of the determinant 2 ± ^ooautt22a33a44a55a66 a term corresponding to /= 2, g = 1 , h= 1* we take the indices 0, 1, 2, 3, 4, 5, 6 ; bracket them thus (0), (1), (2, 3), (4, 5, 6); and with the help of this, write finally a0,0 ah\ a2,3 a3.2 a4.5 %4 ' (IL *0 The number of different cyclical species of terms in a determin- ant of the 77th order is evidently equal to the number of positive integral solutions of the equation /+ 2^+3/.+ . . . + nl — n . (lv. 3) Cauchy’s illustration of this is clearness itself. He says (p. 419): — “ Si, pour fixer les id4es, on suppose n = 5, alors, la valeur de n pouvant etre presentee sous 1’une quelconque des formes, 1 + 1 + 1 + 1 + 1, 1 + 1 + 1 + 2, 1+2 + 2, 1 + 1 + 3, 2 + 3, 1 + 4, 5’ les systemes de valeurs de f 9, h, k, l, se reduiront a l’un des sept systemes * It would be convenient to say, a term of the cyclical species 2(1) + 1(2) + 1(3). 424 Proceedings of Royal Society of Edinburgh. [sess. /= S, II o II O II JO O-J il JO /= 3, 9 = 1, II o' II o' Jl cT II t—H II II JO II O o-j II JO II qo II o II i— < ?S*> II JO II JO II o II h= 1, II JO IT JO /- 1, II JO II JO 7c = 1, OnJ II JO o' II o' II II JO o If i-i; et par suite, une fonction alternee du cinquieme ordre renfer- mera sept especes de termes.” The next question considered is as to the number of terms of a given cyclical species which exist in any determinant of the niYl order. The species being characterised by / groups of one index each, g groups of two indices each, h groups of three indices each, &c., the required number of terms is denoted by N/, • Now all the terms of the species will certainly be got if we write in succession the various permutations of the n indices 0, 1, 2, 3, . . . . , n - 1, and then in the usual way mark off each permutation into the specified groups, viz., first / groups of one index each, then g groups of two indices each, and so on. As a rule, however, each term of the species will, in this way, be obtained more than once. For, if we examine in its grouped form the particular permutation, which was the first to give rise to a certain term, we shall find that changes are possible upon it without entail- ing any change in the term. For example, the set of groups (0) , (1), (2,3), (4,5, 6), instanced above as corresponding to the term ^0, 0^1, 1^2, 3%, 2^4, 5^5, 6^6, 4 ’ might be changed into (1) , (0), (2, 3), (4, 5, 6) or (1), (0), (3, 2), (6, 4, 5) or which, while still corresponding to the term <*0.0 <*1,1 <*2,3 <*3-2 <*4,5 <*5,6 <*6,4 1888-89.] Dr T. Muir on the Theory of Determinants . 425 are derivable from different permutations of the seven indices 0, 1, 1, 3, 4, 5, 6. In fact, the / groups of one index each may be per- muted among themselves in every possible way, so may the g binary groups, the h ternary groups, &c. Further, with like immunity to the term, each separate group may be written in as many ways as there are indices in it, — the group (4, 5, 6), for example, being safely changeable into (5, 6, 4) or (6, 4, 5). The number, there- fore, of different permutations of 0, 1, 2, 3, 4, 5, 6, which will give rise to any particular term, is (1.2.3.../x 1.2.3. m.gx 1.2.3.. .hx ... x 1.2.3...?) x (1/2*3*... w*) , or say, (f\g\h\. . . ?!)(1W. ..»*). There thus results the equation (f\g\h\. . ./!)( 1W. . • nl) N/, , , * = h\, whence != (f\g\ hY. . . Z!)(l'2»3*. . . ' ^LV' ^ Following this interesting result a few deductions and verifica- tions are given. First of all it is pointed out that since the total number of terms of all species is n ! we must conclude that where f+2g + 3h + ...+nl = n. Cauchy says (p. 423) : — “ Cette derniere formule parait digne d’etre remarqueb. Si, pour fixer les idees, on prend n = 5 l’equation donnera 1 . 2 . 3 . 4 . 5 = N5i0, 0,0.0 + ^"3,3 ,0,0,0 + N"l,2, 0,0,0 + N"2>0( 1,0,0 4" -N"o, 1,1, 0,0 + Nl, 0,0, 1,0 + No, 0,0, 0,1 5 et par suite 1 . 2 . 3 . 4 . 5 = 1 + 10 + 15 + 20 + 20 + 30 + 24 = 120 , ce qui est exact.” Again, since the number of positive terms in a determinant is equal to the number of negative terms, and since the terms, whose number ...n has .just been found, have all the sign-factor 426 Proceedings of Royal Society of Edinburgh. [sess. ( _ f )w— (/+0+^+ • • • +0^ we have on leaving out the common factor ( - 1)M the identity o = 2(-i y+9+h+: • +i (flglhl n ! . . I \){lfP?>h . . . nl) ’ which like it’s companion may be illustrated by the case of n = 5, viz., 0 = 1 -10 + 15 + 20-20-30 + 24 * Lastly, attention is directed to the fact that when n is a prime, and therefore not exactly divisible by any integer less than itself, the number n ! (/! g\h \ . . . Z !)(1/2?3A . . . nl) must be exactly divisible by n , except in the case /= n , <7 = 0, h = 0 , . . . 1 = 0 , when it has the value 1, and in the case /= 0, g = 0, h = 0, . . . 1=1, when it has the value (n-1) ! It, therefore, follows from either of the two preceding identities, that the sum of these two values must be divisible by n , — which is Wilson’s theorem. The remaining two pages are occupied with the expansion of a determinant of special form, viz., that afterwards known by the name axisymmetric. JACOBI (1841). [De formatione et proprietatibus Determinantium. Crellds Journal , xxii. pp. 285-318.] The value which Jacobi attached to determinants as an instru- ment of research has already become well known to us : we have * In connection with this and in illustration of a previous remark regarding a mode of expressing the full expansion of a determinant, we have 2 ± &00alla22a33a44 = ®00alla22®33a44 — 2 «00alla22a34a43 + 2a00a12®21a34a43 d" 2 2^(^23^34*42 — 2 av a2, ... . an, while in the order here written, the definition stands as follows (pp. 285-286): — “Vocemus eas indicum 0, 1, . . . , n permutationes, pro quibus P valorem eundem servat, positivas ; eas pro quibus P valorem oppositum induit, negativas ; sive priores dicamus pertinere ad classem positivam permutationum , posteriores ad cla-ssem negativam. ” This implies of course that the original permutation 0,1, 2, ... . , n is to be considered positive; and, such being the case, there seems to be a certain appropriateness in applying the term negative to a permutation whose corresponding difference-product is of the opposite sign from the difference-product corresponding to 0, 1, 2 , .... ,n. The propositions which lead from the definition to Cramer’s rule may he enunciated as follows : — (a) One permutation performed upon another gives rise to a third, and the combined effect produced by performing the second and first in succession is the same as the effect of performing the third. (b) Two given permutations belong to the same class or to opposite classes according as the permutation by means of wdiich the one is obtained from the other belongs to the positive or negative class. (c) If the same permutation he performed on a number of per- mutations which all belong to one class, the resulting permutations will still all belong to one class, viz., the same or the opposite according as the operating permuta- tion is positive or negative. (d) The order of compounding a set of permutations is, as a rule, not immaterial. 429 1888-89.] Dr T. Muir on the Theory of Determinants. ( e ) The permutations which arise by compounding a set of permutations in every possible order belong all to the same class. (in. 31) (/) The interchange of two indices is equivalent to the perform- ance of a negative permutation. (g) The interchange of two indices causes all the positive per- mutations to become negative, and all the negative to become positive. Definition. — Two permutations may be called reciprocal which being performed in succession do not alter the order exist- ing before the operations. (xxiv. 2) (7i) Eeciprocal permutations belong to the same class. In the original, it must be borne in mind, these are not separated and numbered, but appear merely as consecutive sentences in a paragraph. The words “ classem negativam” of the definition above given are followed in the same line by “ Binis propositis permutationibus quibuscunque, certa ex- stabit permutatio, qua post alteram adhibita altera prodit. Pertinebunt duse permutationes propositse ad classem eundem aut ad classes oppositas, prout permutatio, qua altera ex altera obtinetur, ad classem positivam aut negativam pertinet,” &c. — that is to say, by the propositions which have been paraphrased into ( a), ( b ), &c. The most essential point to be considered in connection with them is the probable meaning of the expression “ permutationem ad- hibere,” or the free English translation of it, “ to perform a permutation.” An example will make it clear. To perform the permutation 35412 would seem to be the operation of removing the 3rd member of a series of five things to the first place, the 5th member to the second place, the 4th member to the third place, and so on. With this explanation the proposition (a) is self- evident, an example of it being (if we may improvise a symbolism) (354 12)(41 352) = (32541), where 35412 is the operating permutation. Cauchy’s usage, it may 430 Proceedings of Royal Society of Edinburgh. [sess. be remembered, was to speak of “ applying a substitution to a per- mutation.”* Of the proposition (b) a proof is given, which may be paraphrased as follows : — Let the three permutations referred to change P, the original product of differences, into e jP, e2P, e3P, respectively, the e’s of course being either +1 or - 1. Then as the performance of the first two permutations in succession will result in the change of P into e1.e2P, we must have ei • e2 = e3 > so that e1 and e3 have the same or opposite signs according as e2 is + 1 or - 1 ; and this is virtually the proposition to be proved. (in. 30). A demonstration of ( d ) is also given. The two permutations being A and B, l the first index of A, and m. the first index of B, the performance of A on B implies that the Ith index in B is to take the first place, and the performance of B on A that the mth index of A is to take the first place. The resulting permutations will con- sequently not agree in the first index, unless the Zth index of B is the same as the mth index of A, which manifestly need not be the case, f To prove (/) is of course the same as to prove that the interchange of two indices r and s, r being the greater, alters the sign of the product of differences ; and this is done by separating the product into three portions, viz., (1) the portion which contains neither ar nor as ; (2) the single factor which contains both, ar- as; and (3) the product of all the factors having either one or the other for a term. It is then asserted that the interchange of r and s cannot alter the last of these, because it is symmetrical with respect to ar and as ; also, that no alteration is possible in the first, and conse- quently that the change in the second accounts for the validity of the proposition. (in. 32) '* He says, for example {Jour, de Vfic. Polyt., x. p. 10), “ Si en appliquant suceessivement a la permutation A, les deux substitutions ) et 011 obtient pour resultat la permutation A6; la substitution^1^ sera equivalente au produit des deux autres et j’ indiquerai cette equivalence1 comme il suit + This also is a paraphrase of Jacobi’s proof. 1888-89.] Dr T. Muir on the Theory of Determinants. 431 As for the permutations which are called reciprocal they are, exactly those whose existence we have seen noted by Rothe, and called by him “verw'andte Permutationen.” Jacobi’s definition, however, presents them in a slightly different light, the property involved in it being readily deducible from Rothe’s. The latter’s illustrative example was, as may be seen on looking back, 3, 8, 5, 10, 9, 4, 6, 1, 7, 2 8, 10, 1, 6, 3, 7, 9, 2, 5, 4 Bj . Now the performance of either A on B or B on A* gives rise to 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, the original arrangement : consequently A and B satisfy Jacobi’s definition. The proposition (h) is also Rothe’s. After these propositions, as already intimated, the subject of other rules of signs is taken up, the first rule considered being Cramer’s. Since in the product of differences corresponding to any permu- tation every factor in which an index is preceded by a smaller index would require the sign-factor — 1 to be annexed to it in order that the said product might be transformed into the original product of differences, it is clear that the determination of the class to which the permutation belongs is reduced to counting the number of such inversions. But the pairs of indices in the product of differences corresponding to the given permutation are exactly the pairs of indices to be examined in applying Cramer’s rule. The identity, of the two rules is thus apparent. (hi. 33) To the demonstration Jacobi adds “ quam regulam olim cel. Cramer dedit ill. Laplace demonstravit.” The last assertion is notable for two reasons : first, because the rule like Jacobi’s own is incapable of proof being a definition, postulate, or convention according to the mode in which it is expressed : secondly, because an examination of Laplace’s memoir shows that there is no ground for the statement. The fitness of the rule for the determination of the signs of the numerators and denominators of the unknowns in a set of simultaneous linear equations may of course be demonstrated, and perhaps this was in Jacobi’s mind, but prior to the, statement the abstract subject of permutations had alone been discussed. * In the compounding of reciprocal permutations the order is immaterial. This is the exception hinted at in ( d ). 432 Proceedings of Royal Society of Edinburgh. [sess. The other rule of signs dealt with is Cauchy’s, in which permuta- tion-cycles are counted instead of inversions. The existence of such cycles is the first point to be established, that is to say, it has to be shown that any 'permutation of 1 2 3 . . . n may be obtained from any other by the performance of one or more cyclical permuta- tions. Let 3271654 be the permutation sought,* and 2647513 the permutation from which it is to be derived. Placing the former under the latter, thus 2647513 3 2 7 1 6 5 4, we see that 2 has to be changed into 3, then seeking 3 in the upper line we see that it has to be changed into 4, similarly that 4 has to be changed into 7, 7 into 1, 1 into 5, 5 into 6, and 6 into 2, the element with which we started. Now the proof turns upon the simple fact that the elements in the two lines being exactly the same, by following a string of changes like this we are bound sooner or later to reach in the second line the element we started within the first. It may be that as here one cycle suffices for the second transformation ; but if not, as in the case of the two permutations 2647513 4 1 5 7 2 3 6, where the short cycle 245 is obtained, we turn to the remaining elements, and knowing that those in the first line are of necessity the same as those in the second, we see that the application of the same process to them must, for the same reason as before, lead to a cycle. The possibility of arriving at any permutation by means of cyclical permutations alone is thus made manifest. The next point to be established is that a cyclical permutation of r elements can be accomplished by r- 1 interchanges of pairs of elements. Little more than the statement of this is necessary. For if the elements of the * This is a paraphrase of Jacobi’s demonstration, which is not so simple as it might have been. The notation of substitutions, which Jacobi did not follow Cauchy in using, is here a great help toward clearness. 1888-89.] Dr T. Muir on the Theory of Determinants. 433 cycle be av a2, a3, ... , ar, it is clear that to change aY into a2, a2 into a3 j &c., has the same effect as to interchange ax and a2, then ax and a3, then a1 and and so on, the final interchange being that of ax and ar ; and there are in all r - 1 interchanges. This being proved, the final step is taken as in Cauchy’s Note of 8th March, (iii. 34) This rule of Cauchy’s Jacobi deservedly characterises as beautiful. It is important, however, to take note that it possesses the other quality of usefulness in as marked a degree ; and such being the case one is surprised to find that it has not received the attention which was its due. Any reader who will make a comparison of it and Cramer’s by actual application of them to a number of examples will soon find that Cramer’s is more lengthy and requires more care to be given to it to avoid errors.* The preliminary subject of permutations having been thus dealt with, determinants are taken up. In the first section regarding them there is little noteworthy. Cauchy’s word “terme” is supplanted by the fitter word element , and term (“terminus”) is put to a more appropriate use ; that is to say, a® is called an element of the determinant 2 ± aa\a"2 . . . and ’aka a"k aff a term. Further, the word degree is employed in place of Cauchy’s more suitable word order , “ipsum R dicam determinans n + lti gradus .” A section of two pages is given to considering the effect produced upon the aggregate of terms by the vanishing of certain of the elements. The propositions enunciated, with the exception of one made use of at an earlier date by Scherk, are as follows (pp. 291, 292):— “ I. Quoties pro indicis k valoribus 0, 1, 2, . . . , m - 1 evanescant elementa , ... . , determinans "L±aaxa a^ 1 A n abire in productum a duobus determinantibus 2 ± aal .... «<::;> . 2 ± a(y:^ (XIV. 6) * The best way perhaps of applying Cauchy’s rule is to write the primitive permutation, 128456789 say, above the given permutation, 683192457 say, draw the pen through 1 and the figure below it, seek 6 in the upper line and draw the pen through it and the figure below it, and so on, marking down 1 on the completion of every cycle. VOL. XVI. 16/11/89 2 B 434 Proceedings of Royal Society of Edinburgh. [sess. “ II. Evanescentibus elementis omnibus. m (m+l) ak > ak ’ a(n) > ak in quibus respective index inferior Tc indicibus superioribus m, m + 1 , . . . , n minor est, fieri (vi. 7 ) y + an ’n." fn) - . y + ' A™-1) Zj±aa1a2 . . . . a n - a m a m+1 . ... a n Zj±aax. ... a m_x. “ IY. Evanescentibus elementis omnibus, „(»») _(ot+1) /») > in quibus indices inferiores superioribus minores sunt, si insuper habetur, a(m) = a(m+ 1) . _ . _aW_l m m+l fit 2± Ipi . ... a As immediate deductions from tbe definition these are somewhat out of place, the trouble of demonstrating the first of them being virtually thrown away. The trouble taken by Jacobi, too, was less than required, the question of sign, for example, being inadequately discussed. In the course of the next section which deals with what we have called the recurrent law of formation, and with the vanishing aggregate connected with this law, Jacobi gives an expression for the complete differential of a determinant, the elements being viewed as independent variables. The passage is (p. 293) : — “ Determinans R est singularum quantitatum a respectu expressio linearis, atque ipsius a® coefficientem, qua in deter- minante R afficitur, vocavimus ; unde adhibita differen- tialium notatione ipsum A^ exhibere licet per formulam, 3. A®- 3B Hinc si quantitatibus incrementa infinite parva tribuimus, da<%, simulque B inerementum <7B capit, fit 4. = ®= * + aT\ °> !) + ^(0.2) . . + ^(0, A(l/) = «(/')(l,0) + * + <4° (1.2) • • + w (1. ii. \ ; A^) = a (2, A^ = a(/)(w, 0) + ckp (n, 1) + AV. 2) . . + * Similes formulae e (9) derivari possunt. In aequationibus (11) ipsorum a[f\ a(f'\ etc. coeflicientes in diagonali positi evanescunt, bini quilibet coeflicientes diagonalis respectu symmetrice positi valoribus oppositis gaudent. Quae est species aequationum linearium memorabilis in variis quaestionibus analyticis obveniens.” (lvii.) The simple step from the expression of A^ as a differential co- efficient to the similar expression for A^’^, is next made (p. 301) : — “ Ex ipsa enim aggregati Ag,fgl definitione eruimus formulas 1 32R _ 02R g'* daffWp ddpda

0.A(/> 0.A(/:> 0.A(/,) 0.A(/) l _ A^, 0 = «A. + OjA; + . . . . + al A(J, E = “*A* + • • • • + aA'l’ 0 — a A, + a A, + a (n) * («) ; using the multipliers A v' O,*' » A *’• > a M' * ’ > ‘ and adding, there is obtained i,i' n, V » 4. R. A A^A^ - A(;?A(p, fC tC rC fC J — a result at once recognisable as a case of the theorem regarding a minor of the adjugate. Next by starting with Bezout’s identity connecting any eight quantities, the particular eight taken being A*>, A®, A®, A®, A« i®, A« A(0 and making six substitutions of the kind A?A<£> - A«A« - B.A#, just seen to be valid, there arises the identity A*’* A -i- A ^kjc'^k"k'" + A Ic,k" k"',k' + Kx*Kr = °- (xxm. 11) This clearly belongs to the class of vanishing aggregates of products of pairs of determinants ; but in order that its true character may be seen, and comparison made possible between it and others of the same class already obtained, a more lengthy notation is necess- ary. Taking for shortness the case where the primitive determin- ant is of the 8th order, but writing it in the form and making i, i = 3, 6 and A, A:", k"' = 5, 6, 7, 8, 438 Proceedings of Boyal Society of Edinburgh. [sess. we find the identity to be \af2d^gff\af.2d^gbh&\ - \af2d^fis\.\af2d^gfi^ + = 0, a glance at which suffices to show that it is nothing more than the extensional of \gjh[\9jh\ - \9A\Wh\ + \9 the very identity of Bezoufc which was taken as a basis for it. As the same extensional has already been found among those of Desnanot, any new interest in it is due to the peculiar way in which Jacobi obtained it. By the same method, viz., by substitu- ting for secondary minors an expression (4) involving primary minors and the primitive determinant, he shows that + Axt = o. (xxm. 12) This being translated in the same manner as the preceding, becom'es |ai^2^3e4 f&9f& 1*1 ^1^2^3e4^5^8 I I af>iA$Aif b9f% I’ I af>2(^'Ze^9 + KVs^/^'sl^Ms^AI = and is thus seen to be another of Desnanot’s results, viz., the exten- sional of l/e0Vks - \frth\-9s + I fsffelto = °- (XXIIL 12) The deduction 9 A? ‘A? a®a;::. Af "A? CD II AfAf ’ 8 a(g (<") A *’ i k’ (0 is made from it by substituting appropriate differential coefficients for the primary and secondary minors involved in it. (lviii.) The eleventh section is devoted to the establishment of the general theorem which includes the theorem «-AA = AfA-A?Af of the preceding section, and which, as we have seen, Jacobi had first enunciated in 1833. To start with it is repeated that the system of equations • 1888-89.] Dr T. Muir on the Theory of Determinants. 439 a t + ax 4 + . . . + aktk + ak+l tk+1 + . . . + a. t —u. n n ’ a' + a/ tx + . • • + % V + A + + . . . j*> t . • • + 7c + ayfc+l Tfc+l + • * . + =w7 , ran k ’ o(B)i + + . , An)/ i /y(W) / i • • + ak h + ak+ 1 **+i + ’ ' . + n n n i gives rise to the system A u + A Ul + . . ■ + \ uk + \+ 1 uk+ i + • • ■ ■ + Aw^n = R . t A' u + A^ ux + . . • + V uk + A’t+i uk+ 1 + ■ • • + A u — R . t n n A + A<\ + . . • + Ai ’“i + A*+l “i+1 + • • • + A 9^U = R . t. n n k A” M + AJ°«] + . . ■ + Ak\ + A*+i“*+: 1 + - • • + Aww =Y..t n n n in which R = 2 ± aa'x , 4’°> An) = ^±aa\ a(w_1) a(n-\y Then taking only the first k + 1 equations of the first system and eliminating t, tv . . . , 4-u there is obtained CA+CWA+1 + • • • • +CJn = I)u + Dl“l + • • • + (X) where the multipliers D, D1? . . . , D*, by which the elimination is effected, are (-1 ( - l)7m2 ±aar"ak-i , + 2±«a/1a"2...a(^J), and consequently by denoted , > » (*-i) (fe) 2,±aa1a 2- a\ > 2 ± aa\a'\. . > 2±aa>//2...«(^J)a(*). 440 Proceedings of Royal Society of Edinburgh. [sess. Similarly, taking only the last n - k + 1 equations of the second system and eliminating uk x ? Ujc+2 • • • . , un there is obtained E u + Kiui + • • • + \\ = RF^ + RF,+1^+1+ . . . . + RF ntn, where the multipliers Ffc , F&+1 , . . . , Fw by which the elimina- tion is effected are 2±A<*+fA(*+*)...A(:), -E±A^X|..A« /_ l\n-Jc'Z + ak,c+L)a{ { 1/ Zj±JrVk Ai+l •••‘“n-l* and consequently by E, Ej , . . . , E^ are denoted 2±A,A%?. . .a? . .a? bs±a^a»+J...aW- This is the keystone of the demonstration. The simple continua- tion of it may for sake of historical colour be given in Jacobi’s own words (p. 304) : * — “ In hac formula generali ipsi k tribuendo valores n - 1 , n - 2 ? n- 3 , . . . ,1, prodit : The demonstration in the original is considerably disfigured by misprints. 441 . 1888-89.] Dr T. Muir on the Theory of Determinants. ' 2 ± aa{ a(n~2) ' • * ftre-2 2 + A(M~1)A<”) — re— 1 re e +1 re — 1 RAW ’ re 2±aoq' . . . . re-3 y . a(»-2) a(w_1) a(w) ^±Are-2 A re— 1 re \ 2 ± aeq' . (re -2) . . . a> i re -2 BS±A^:?A« a s±A,'V- • • 2 ± aoq' R2 ± A2"A3"'. . . . A(”> Hamm aequationum prima suppeditat, 2±a<::;,a(:) = ke±««1' • a in- 2) re— 2 = RA! re — 1, re re- 1, re > quae cum formula (4) § pr. convenit. Deinde aequationum (10) duas, tres, quatuor etc. primas inter se multiplicando, prodit formularum systema hoc : f 2 + A(”_i)A(',) = — re — 1 re E 2 + adj. . a(w_2) • ' a n- 2 » 2 + A(w-;?A<”-1)A(”) = R22 ± aa-[. . ^ — re— 2 re— 1 re re-3 5 2 ± A/ A2" .... = “ Quas formulas amplectitur formula generalis, 2±A(*+}>A(*+*\ . . . A(”; = R’*-*-1 "V ' (ft) Z±aa1 ... . a k ( xx . 5) Cauchy’s theorem 2 ± AA,' .... Aw = R” , which may he viewed as the ultimate case of this, Jacobi arrives at by expressing S±AA/. . . A(f in terms of A, A1? . . . , An and their cofactors, substituting for the said cofactors their equivalents as just obtained, viz. dR71-1 , cqR"-1 , a2RM_1 , . . . . , awRM_1 , and then using the identity A a + Ajcq + . . . + A nan = R . Passing over the twelfth section, which relates to certain special 442 Proceedings of Royal Society of Edinburgh. [sess. systems of equations, we come to two sections devoted to the multi- plication-theorem. Of the five formally enunciated propositions which they contain, two, the second and fourth, need not be more than referred to, as their substance comes from Binet and Cauchy, and as the mode in which they are established will be sufficiently understood from the treatment of one of the others. The general problem of the two sections is the investigation of the determinant 2±celC2"....c« where (fc) (0 (*) .(OJfr) a a + cq ax +....+ ap ap Taking a single term of the determinant, we have of course cci'cf . . . c(^= (a a +a1a1 + . . . . +%ap) x (a! a! + oq'aq' + . . . . + ap af) x-(.V + + ««<£>), and we see that if the multiplications indicated on the right be performed there must arise a series of (p+ l)n+1 terms of the type arar • asas * at"ai" or by alteration of the order of the factors (n) , f aw •arasat W (») w In) r s t where each of the inferior indices r,s,t, . . . , w may be any member of the series 0,1,2, . . . , p. If we bear in mind the meaning which we thereby assign to the summatory symbol S we may write this in the form ccJcf .... = S(a a V" . . . . a a 'a" . . . a^n)) . 1 ^ n \ r s t w r s t w ' The next point to consider is the transition from the single term cc{cf . . . c^ to the full aggregate 2 ± ccfcf . . . c^. A glance at the sum of terms denoted by cj® shows that by permuting the superior indices of cc{cf . . . , the superior indices of the a’s are subjected to the same permutation, and that, on the other hand, when we permute the inferior indices of cc{cf . . . it is the a’s 443 1888-89.] Dr T. Muir on the Theory of Determinants. that are affected, the like permutation being given to the superior indices. Making the choice of the superior indices of the c’s, let us permute them in every possible way, and to each term thus derived from cc-[cf . . . prefix the sign + or — according as its superior indices constitute a positive or negative permutation. By so doing the left-hand side of our identity becomes 2 ± ce{cf . . . c^> and, owing to the consequent permutation of the superior indices of the as, each term on the right-hand side gives rise to 1 . 2 . 3. . .(n + 1) terms whose signs are the same as the signs of the terms correspond- ing to them on the left hand side ; — in other words, each term a a 'a" . . . a(n) . a a 'a'/ . . . dP gives rise to the compound term r s t w v s t w ° *■ «AV • • ^ • 2 ± ar°.'a/' ••••“' (n) We thus reach the result 2 ± cc/c . . . c(n) == S (a a 'a” . . . a'f . 2 ± a a 'a" . . — l z n \ r s t n — r s t (n)\ * ) Although the number of terms on the right is the same as before, viz. (^9 + l)w+1, arising from giving to each of the n+ 1 indices r, s, t, . . . , vj any one of the p + 1 values 0, 1, 2, . . . , p, it has now to be noticed that a goodly proportion of them must vanish because of the fact that 2 + a a 'a" . . . a ^ = 0 when any two of its inferior indices are alike. The right-hand side will thus not be altered in substance if the summatory symbol be now taken to mean that r, 8, t, ... , w are to be any n + 1 of the p + 1 indices 0, 1, 2, . . . , p. If p be less than n it will be impossible to have r, s, t, ... t w all different, so that in that case the right-hand side must be 0. This is Jacobi’s first proposition, and it constitutes his addition to the multiplication- theorem. His formal enunciation of it is (p. 309) : — Sit 4 W*V*> + + . . . + a quotiesjp<% evanescit determinans 2 ± ccxV2 Sn) » (xvm. 6) 444 Proceedings of Royal Society of Edinburgh. [sess. The consideration of the case when p — n leads to his second pro- position. The natural addendum is then made regarding the multi- plication of more than two determinants of the same degree (p. 310):- “ Datis quotcunque eiusdem gradus determinantibus, eorum productum ut eiusdem gradus exhiberi posse determinans, cuius elementa expressiones sint rationales integrae elementorum determinantium propositorum.” (xviii. 7) The equally natural transition to the subject of the multiplication of two determinants of different degrees results in the proposition (p. 311):- “ Sit pro indicis i valoribus 0, 1, 2, . . . . , m, c® = d(%W + a®af + . n n ’ pro indicis i valoribus maioribus quam m, _./*), (}) aQc) , (0 aW , ck~ai + ai+ 1 i+i + ai+2ai+2 + • • • n n erit 2 + acq' . ...a«2±aa1'....a#2±CClV. M » ' * * • Cn ‘ Proposition IV. concerns the case where p >n. Proposition Y. is but a corollary to the combined propositions L, II., IV., its subject being the effect of the specialisation ak ~ak * The enunciation is as follows (p. 312) “Posito Cf=(®=«V + afaf + tC Z 11 sit determinans = P; ubi p>w fit P = S{2±o a' i ' ampa.o-Tos, unexpected. 47 0 Proceedings of Royal Society of Edinburgh. [sess. border. Central space and rosette absent. Markings subpearl y, robust, pentagonal or hexagonal, each with a faint central dot, 2J to 3J in ’01 mm., decreasing but little towards the border. Border sharply defined, subopaque, broad ; striae evident, 3 \ to 4 in *01 mm.; a series of closely placed concentric lines sometimes visible, the outer edge rugose. Girdle aspect,* with smaller areolae, 4 to 5 in *01 mm., forming straight or slightly flexuous rows parallel to the edge of the girdle; girdle narrow. — Cleve, Bih. k. Sv’Vet.-Ak. Handl. Stockh ., 1873, No. 11, p. 4 ; Sch., Atl ., pi. lxii. fig. 8; C. con- cavus, Ehrb., pro parte, Abh. Ber. Ak., 1841, p. 412; Mikrog., pi. xxi. fig. 4 f (excl. pi. xviii. fig. 38) ; Endidya oceanica , Ehrb., Mon. Ber. Ak., 1845, p. 76; Mikrog., pi. xxxv. A, 18, figs. 6, 7 ; Balfs in Pritch. Inf., p. 831, pi. v. fig. 70; Moll., Typ. PI. 4, 4, 8, Cleve and Moll., Diat., No. 110, 259; Baben., Alg. Europ., No. 2556 ; H. L. Sm., Diat. Sp. Typ., No. 148 ; Sch., Atl., pi. lxv. figs. 10, 12, 13, 15. Coscinodiscus concavus, var. africanus. Kiitz, Sp. Alg., p. 125; C. oceanicus , Kiitz., ibid., p. 126; Melosira cribrosa , de Breb., W. Sm. in Ann. and Mag. Nat. Hist., 1857, p. 11, pi. ii. fig. xv.; Orthosira oceanica, Brightw., Quart. Jour. Micr. Sci., 1860, p. 96 ; Endidya minor, Sch., Atl., pi. lxv. figs. 14, 16; Melosira oceanica, Habirsh, Cat. Diat., ed. 2, § Endidya. The specimen named C. concavus by Ehrenberg (Mikrog., pi. xviii. fig. 38), from Richmond, approaches the unnamed organism figured by Gregory from the Glenshira Sand (Trans. Micr. Soc. Lond., 1857, p. 85, pi. i. fig. 52), but is distinct from the present species. C. concavus, var., Sch. (Atl:, pi. lix. fig. 16) is C. antiguus, Grun. W. Smith followed de Brebisson’s determination of Melosira cribrosa provisionally, separating it from Coscinodiscus only because he believed that the frustules might occur concatenated. C. concavus, var. africanus, from Oran, was first differentiated by Ehrenberg as C. concavus africce (Mon. Ber. Ak., 1844, p. 79), with 3J to 4 markings in '01 mm., but this is inadequate on which to establish a variety. Cork specimens, authenticated by Grunow as Endidya minor (Sch., Atl., supra), agree with Gregory’s C. concavus. Specimens sometimes named Endidya oceanica differ from C. con- cavus only in showing the markings somewhat more irregular. * This applies to specimens hitherto named Endidya. + Specimen not typical. 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 471 Habited. — Oran deposit (Ehrenberg) ; Oamaru deposit (Grove and Sturt); Mejillones guano (Deby! Hardman! Firth!); Peruvian guano (Cieve ! Johnson !) ; Chincha Island guano (Arnott !) ; Arica and Saldanha Bay guanos (Ehrenberg); off Bermuda, 1075 fathoms (Kae !) : Loch Fyne (Greville !) ; Lamlash (Gregory !) Biarritz, Bay of Biscay (Brebisson !) ; Black Sea (W. Smith) ; Java (Cieve) ; Amboina shell scrapings (Kinker !) ; Ascidia, Roundstone Bay, county Galway (O’Meara !) ; S. America (Moller !) ; * Lough Hym, county Cork (Grove!); Kirkwall, Orkney (Grove!); Edible sea- weeds, India (Macrae!); Locality? (Barnett ?f Weissflog !) ; Valparaiso (Schmidt); Villefranche, Trinidad (Cieve and Moller !); Campeachy Bay (Grove !) ; Monterey Stone (Cieve !) ; Balearic Islands, Pabillan di Pico guano, Bolivian guano (Cieve !) ; Port William, Falkland Islands (Rabenhorst and Schwarz !). C. bisculptus, sp. n. C. labyrinthus, Roper, var.? Sch., Atl., pi. lix. fig. 14. — Diam. *035 mm. Surface somewhat convex. Central space and rosette absent. Markings large hexagonal or pentagonal, unequal areolae, 2 to 2J in *01 mm., somewhat smallest towards the centre ; within these more minute, faint, angular areolae — the larger in obscure radial, but more evident oblique substraight decussating lines, the smaller without order. Border sharply defined, about \ of radius broad; striae coarse, 3 in *01 mm. Habitat. — Peruvian guano (Schmidt). C. labyrinthus. Roper, Quart. Jour. Micr. Soc., 1858, p. 21, pi. iii. figs. 2 a, 2 b. — Diam. *0625 to *0875 mm. Central space absent. Markings hexagonal towards the centre 4, decreasing gradually to the border to 6 in *01 mm., punctate, forming straight or slightly curved oblique decussating rows and distinct secondary subequal hexagonal, areolae from ”0025 to *0085 mm. broad; minute apiculi sometimes present at the border. Border indistinct ; striae 7 to 8 in ”01 mm. — Ralfs in Pritcli. Inf., p. 831 ; Cieve and Moll., Hiat., No. 276 (excl. C. labyrinthus, Roper, var.? Sch., Atl., pi. lix. fig. 4). The smaller hexagonal markings recall those of C. excentricus , Ehrb., and of C. sol., Wallich. At the centre a faint stellette is * In the collection of Julien Deby. t In the collection of Dr Griffin. 472 Proceedings of Royal Society of Edinburgh. [sess. sometimes found. Not a Pyxidicula , as suggested by Grunow ( Denk. Wien. Ak., 1884, p. 73). Habitat. — Californian guano (Norman!)*; stomach of Ascidia, Hull (Greville !) ; Lamlash (Greville ! ); Caldy, Pembrokeshire (Roper !) ; Hull (Firth !) f ; Humber (Dickie !) ; California (Cleve and Moller !). C. bipartitus , sp. n. Sch., Atl., pi. lix. fig. 35. — Diam. about •0875 mm. Central space absent, rosette large, surrounding a single small circular areola. Markings hexagonal, 2J subequal, for about J of radius, on outer \ 6 in *01 mm., forming a distinct band. Central papillae absent, radial rows on inner J of radius obscure, the oblique decussating straight rows manifest, on outer \ the radial rows evident, the secondary oblique decussating rows uniformly curved. Border narrow, hyaline ; beyond the border 4 large unsilicified subrugose blunt protuberances. Habitat. — Java (Griindler). C. blandus. Sch., Atl., pi. lix. figs. 36, 37. — Diam. about ’07 mm. Central space small, rosette large, at the inner angles of its component areolae distinct minute round granules. Markings hexagonal, 3 in *01 mm., somewhat smaller towards the periphery ; the central papillae faint, a distinct band sometimes present adjacent to the border, upon this the markings rounded granular and irregular apiculi numerous, inserted at inner edge of border. Border narrow ; striae evident, 6 in *01 mm. Habitat. — Gulf of Mexico (Schmidt). C. lineatus. Ehrb., Abh. Ber. Ak, 1838, p. 129. — Rarely angular. Diam. *05 to *15 mm. Surface towards the centre flat, slightly convex near the border. Central space and rosette absent. Markings hexagonal, 2J to 4 in '01 mm., subequal, or sometimes at border 6 in -01 mm., their central dots distinct ; apiculi small, sometimes absent. Border distinct, consisting of a few concentric rows of contiguous granules, 8 or 9 in ‘01 mm. — Ehrb., ibid., 1841, p. 371, pi. i. 3. fig. 20 ; pi. iii. 7. figs. 7, 8; Mikrog., pi. xviii. fig. 33; pL xxii. figs. 6a, b; pi. xxxv.a, 16. fig. 3; pi. xxxv.a, 17. fig. 7; In the collection of Dr Greville. + In a slide prepared by Mr Norman. 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 473 Raben., Alg. Europ ., Nos. 2481, 2482, 2483, 2484, 2485, 2486 ; Van Heurck, Syn. Diat. Belg ., p. 217, pi. cxxxi. fig. 3; Janisch, Gazelle Exped., taf. iv. fig. 8; taf. xx. fig. 14; Sch., Atl., pi. lix. figs. 27-30 ; H. L. Sm., Diat Sp. Typ., No. 98 ; Cleve and Moll., Diat, Nos. 57, 114, 148, 150, 162, 207, 276. C. lineatus, var.? Sch., Atl, pi. lix. figs. 31, 32. C. Ehrenbergii, O’Me., Proc. Boy. Irish Ac., 1875, p. 264, pi. xxvi. fig. 24. Sp. n. Sch.. Atl., pi. cxiv. fig. 13. Coscinodiscus lineatus (Weisse, Bull. Ac. Imp. Sci. St Petersb . 1855, p. 276, pi. i. figs. 2 a, &)is perhaps Didyopyxis subtilis, Ehrb., according to Grunow (Denk. Wien. Ak., 1884, p. 92). Schmidt separates the specimen figured in his Atl., pi. cxiv. fig. 13, because of its convexity. Habitat. — Richmond, Ya. (Ehrenherg, Bailey, Hardman ! Cleve and Moller !); Caltanisetta, Peruvian and African guanos (Ehrenherg); Patos guano (Kinker !) ; Moron (Schmidt) ; marine deposit, Fiji Islands (Grove !); Sta Monica deposit, Sta Maria deposit (Grove !); Cambridge deposit, Barbados (Hardman !); Barbados (Cleve and Moller !); Californian guano (Norman !);* Rappahannock (Bailey!);* Nancoori (Cleve and Moller! Cleve! Hardman !);* Ningpo (Kinker!); Mascara (Cleve and Moller!); Kamtschatka Sea, 1700 fathoms (Bailey !); Indian Ocean, sounding by Capt. Pullen, 2200 fathoms (Greville !) ; Japan (H. L. Smith !) ; Singapore (Hardman !); Yokohama and Brazil (Schmidt); Mejillones (Cleve! O’Meara); Cambodia (Hardman !); f Monte Gubbio (Grove !); edible sea- weeds, Indian Ocean (Macrae !); Campeachy Bay (Cleve and Moller ! Grove !); Cannibal Islands (Greville !); Andaman Islands (Macrae !); Cuxhaven (Bailey); Malahide and Dollymount, county Dublin; Ascidia, Roundstone Bay, county Galway (O’Meara) ; Patagonia, California (Cleve and Moller !) ; west coast, Florida, U.S. Survey (Febiger!); Yeddo; Patagonian guano; near Elbing, West Prussia; Mors deposit; Labuan; Kusu; between Aden and Bab-el-Mandeb (Cleve !) ; Archangelsk (Cleve); Yera Cruz, among Sertularia ; Laguna, Mexico, on stones among Algae ; anchor ground, Laguna Harbour, 20 miles N. of Laguna in the sea (Rabenhorst and Schwarz !); Simbirsk Polirschiefer (Hardman !) ; f Faeroe Channel (Grove !). * In the collection of Dr Greville. t In the collection of Julien Deby. 474 Proceedings of Royal Society of Edinburgh. [sess. C. marginato-lineatus. Sch., Atl., pi. lix. fig. 33. — Diam. *0335 mm. Central space absent. Markings hexagonal, equal 3J to 4 in *01 mm. Border about \ of radius broad; strise distinct, 6 to 8 in *01 mm., the inner half separated from the outer by a distinct narrow line. Distinguished by the regularity of the markings and border. Habitat. — Campeachy Bank (Schmidt). C. peruanus , Grun. Sch., Atl., pi. lviii. fig. 43. — Diam. *0425 mm. Central space absent. Markings polygonal, decreasing slightly from the centre outwards ; towards the centre 5, towards the border 6 in '01 mm.; the oblique decussating rows straight or slightly curved outwards, well-marked ; apiculi numerous, distinct, close to the border. Border hyaline. — Grun., Denk. Wien. Ah., 1884, p. 85. Distinguished from C. excentricus by the size of the markings, the apiculi, and border. Habitat. — Peru guano (Schmidt). C. sublineatus. C. {excentricus, var.1?) snblineatus, Grun., Denk. Wien. Ak., 1884, p. 85, pi. iv. (D), figs. 21, 22.— Diam. -032 to *053 mm. Central space and rosette absent. Markings hexagonal, gradually decreasing towards the border ; at the centre 5, at the border 9 in ‘01 mm.; the oblique slightly bent, decussating rows distinct, non-apiculate. Border narrow, hyaline. Distinguished from C. excentricus by having the markings at the border smaller in proportion to the others, and from C. lineatus by the less uniform markings. Habitat. — Pranz Josef’s Land, White Sea (Grunow) ; Simbirsk Polirschiefer (Grunow). C. august e -lineatus. Sch., Atl., pi. lix. fig. 34. — Diam. *0275 to •0455 mm. Central space and rosette absent. Markings polygonal, subequal, 6 in *01 mm. Apiculi minute, sometimes indistinct, at the border. Border narrow, hyaline. — Janisch, Gazelle Exped., taf. iii. fig. 6. C. lineatus, var. tenera, Tru. & Witt., Jeremie Diat., p. 14, pi. ii. fig. 2 ; Cleve and Moll., Diat., No. 154. Truan and Witt’s C. lineatus, var. tenera, differs chiefly, according to the figure in the somewhat more distinct appearance of fasciculi 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 475 between the radial rows. It is not strictly defined by the authors. Habitat. — Yokohama (Schmidt), Cambodia (Hardman !) * Mejil- lones (Hardman ) ;* Los Angelos (O’Meara !) ; marine deposit, Fiji Islands (Grove !) ; rice fields, Georgia (Greville !) ; Zebu, Philippine Islands (Weissflog !) ; Jeremie deposit, Hayti (Truan and Witt) ; Indian Ocean, sounding by Capt. Pullen, 2200 fathoms (Greville !); Balearic Islands (Cleve and Moller !). C. jpseudo-lineatus. Pant., Fossil. Bacil. Ung ., p. 73, pi. ix. fig. 77. • — Diam. ’08 to ‘125 mm. Centre occupied by a siugle circular areola, *0025 mm. broad. Markings hexagonal, 8 to 9 in *01 mm., decreas- ing somewhat outwards ; within the border a narrow punctate hand, somewhat irregular on its inner side. Apiculi minute, numerous, forming a circlet at the border, one larger inserted somewhat further inwards. Border striae, delicate, 12 in *01 mm., merging into the adjacent band on the inner side. Habitat.— Dolje deposit (Pantocsek !). C. cristatus , sp. n. — sp. n. ? Sch., Atl., pi. lix. fig. 4. — Diam. *0305 mm. Central space and rosette absent. Markings regularly angular, 10 in *01 mm., a narrow hyaline hand adjacant to the border; apiculi numerous, their outer ends obtuse, placed at the inner edge of the hyaline hand. Border distinct, hyaline. Habitat. — Peruvian guano (Schmidt). Yar. distans. (?) Sch., Atl., pi. lix. fig. 5. — Diam. ’02 mm. Markings 10 to 12 in *01 mm., secondary oblique rows distinctly curved outwards ; no hyaline band adjacent to border ; apiculi similar, hut more distant. Habitat. — Kings Mill (Schmidt). C. tumidus , Janisch. Sch., Atl., pi. lix. figs. 38, 39. — Diam. *164 to *2 mm. Surface convex towards the centre. Central space absent. Markings hexagonal, towards the centre 4 to 4J in -01 mm., increasing outwards to the border to 3 in *01 mm.; oblique de- cussating rows, straight, or with slight bendings. Border striae, 4 in *01 mm. — Cleve and Moll., Diat., Nos. 125, 207. Habitat. — Table Bay (Schmidt, Weissflog !) ; surface, Antarctic * In the collection of Julien Deby. 476 Proceedings of Royal Society of Edinburgh. [sess. Ocean, H.M.S. Challenger (Rae! Cleve and Moller !) ; Patagonia (Cleve and Moller!); Patagonia, 1375 fathoms, H.M.S. Challenger (Cleve !). Yar. fasciculata, nov. — Diam. ‘2 to -28 mm. Similar to the type, but markings in fasciculi, the rows in each parallel to that at its centre, the fasciculi few and wide, or numerous and narrow. The fasciculi when wide resemble those of C. polyradiatus , Cstr., but the border is quite unlike that of the latter. Habitat — Surface, Antarctic Ocean, H.M.S. Challenger (Rae!); Gazelle Expedition (Weissflog !). G. leptopus, Grun. Van Heurck, Syn. Diat. Belg ., pi. cxxxi. figs. 5 and 6. — Diam. *1 mm.. Central space absent. Markings hexagonal, decreasing but slightly at the border, and showing numerous round granules ; towards the centre 4, towards the border 5 in *01 mm.; on a narrow irregular zone adjacent to the border minute round, granular ; the oblique decussating rows sub-straight, evident ; apiculi well defined, forming a circle close to the border at subregular intervals of about *005 mm., one larger inserted somewhat farther inwards. Border sharply defined ; striae delicate 8 to 10 in *01 mm. — Cleve and Moll., Diat., No. 114; Janisch, Gazelle Exped ., taf. v. fig. 4; C. lineatus , Sch., Atl ., pi. lix. fig. 26; C. macraeanus , Grun. (non Grev.), fide Sch., Atl., ibid. The large apiculus is not unlike that of Podosira oliverana, Grun. (Yan Heurck, ibid., pi. cxviii. fig. 5), found abundantly at Kerguelen by H.M.S. Challenger. Habitat. — Off Ascension Island, S.S. Buccaneer (Grove!); Mascara (Cleve and Moller!); Californian guano (Greville!); Patos Island guano (Greville !); Mejillones guano, Balearic Islands (Yan Heurck); Santa Marta deposit (Doeg!); Los Angelos (Cambridge!); Macassar Straits (Grove!). Trawled by H.M.S. Challenger, lat. 34° 37' N., long. 140° 32' E. (Rae!); Indian Ocean soundings, Capt. Pullen, 2200 fathoms (Greville !). Yar. discrepans, nov. — Diam. 475 mm. Markings hexagonal, at the centre of the valve an inequilateral 4-sided area, bounded by a narrow irregular band of dissimilar angular areolae, which are con- tinued outwards unequally from the angles towards the border; 1888-89.] Mr John Battray on the Genus Coscino discus. 477 the rows in the intervals slightly hent, oblique, and decussating ; the hand adjacent to the border sub-regular, with punctiform striae 8 to 10 in *01 mm. Apiculi on border at intervals of ‘005 to *0075 mm.; the single large apiculus with a knob-like free end, inserted at inner edge of band adjacent to the border, about *0055 mm. long. — (PI. II. fig. 3.) Habitat. — Gazelle Exped. (Weissflog !). § Y. Fasciculati, Grun., Denk. Wien. Ak ., 1884, p. 80; Pant., Fossil. Baeil. Ung., p. 71. Markings fasciculate, the fasciculi sometimes indistinct, or re- cognisable only on outer portion of valve, the rows composing each parallel to that at its centre or side ; apiculi frequent. C. vetustissimus. Pant., Fossil. Baeil. Ung., p. 71, pi. xx. fig. 186. — Diam. ‘075 to ‘1 mm. Central space and rosette absent. A small slightly excentric nodule, distinct. Markings hexagonal, increasing slightly to about the semiradius, again decreasing somewhat to the border, towards the centre and border 5, at the semiradius 4J, in *01 mm., central papillae distinct; irregular, on a small subcircular somewhat excentric area, about *006 mm. broad, and surrounded by an indistinct narrow, irregular, hyaline hand; elsewhere in obscurely fasciculate suhstraight or in suhradial rows, those in each fas- ciculus parallel to that at one of its edges, and most obvious when the papillae are in pairs ; non-apiculate. Border narrow, indistinct ; striae delicate, 8 to 10 in ‘01 mm. — Cleve and Moll., Diat ., Nos. 57, 155, 162, 164; Grun., Bot. Centralbl ., Bd. xxxiv. Nos. 2, 3, p. 35 ; C. incequalis, Grove and Sturt., Jour. Quek. Mic. Cl.} 1887, p. 68. —(PL II. fig. 17.) In the fasciculation this species recalls C. curvatulus , hut is distinguished by the excentric arrangement of its markings. It approaches C. africanus, var. wallicliiana , Grun., in the latter respect, but Grunow’s var. is non-fasciculate. In the excentric arrangement of the much smaller markings, as well as in their sub- fasciculate disposition, this species may be easily distinguished from C. nodulifer , Janisch. Habitat. — Oamaru deposit (Grove ! Cleve!); Yokohama(Schmidt) ; Cambridge deposit, Barbados (Kinker ! Johnson!); Bichmond, Ya. 478 Proceedings of Boyal Society of Edinburgh. [sess. (Cleve and Moller!); Balearic Islands ; Nancoori; Sta Monica deposit (Cleve and Moller!); between Aden and Bab-el-Mandeb ; Mejillones guano (Cleve !) ; Alsb-, Felsd-, Esztergaly, K6kko and Szakal deposits (Pantocsek !). Yar. curvatuloides , Grove MS. — Diam. *1 mm. Markings irregular, and smallest on a small round excentric area, elsewhere subequal, 4J in *01 mm., and in evident fasciculate rows, those in each fasciculus parallel to that at its edge ; apiculi minute, inter- fasciculate. — Cleve and Moll., Diat ., Nos. 57, 164. Through this var. C. vetustissimus is allied to C. curvatulus, var. genuina. Habitat. — Jackson’s Paddock, Oamaru deposit (Grove !) ; Bich- mond, Ya.; Sta Monica deposit (Cleve and Moller !). C. atlanticus. Cstr., Diat. Chall. Exped. , p. 158, pi. v. fig. 8. — Diam. ’046 mm. Central space and rosette absent. Markings round, granular, without order, and with hyaline interspaces from the centre to a little beyond the semiradius, thence polygonal, subequal, 10 in *01 mm., and in radial subfasciculate rows to the border. Border distinct, hyaline. Habitat. — South Atlantic, H.M.S. Challenger (Castracane). Yar. striatula , nov\ C. atlanticus , var., Cstr., ibid., p. 158, pi. iii. fig. 7. — Diam. *07 1 5 mm. Markings round, granular, and irregular from the centre to about J of radius beyond this polygonal in evident fasciculi ; those composing each fasciculus parallel to that at its centre, subequal, 6 in ‘01 mm. Border about of radius broad, striae 8 to 10 in *01 mm. Habitat. — (?) (Castracane). C. nitidus. Greg., Trans. Roy. Soc. Edin., 1857, p. 499, pi. x. fig. 45. — Diam. ’03 to *075 mm. Surface almost flat. Central space absent. Markings small, rounded, subpearly, with hyaline interspaces, largest towards the centre, decreasing slightly to the border, irregular, sometimes in inconspicuous radial subfasciculate rows around the border. Border striae, 6 in *01 mm., distinct. — Balfs in Pritch. Inf., p. 831, pi. viii. fig. 18 ; Sch., Jahresb. d. Kom. z. Untersuch. d. deutsch. Meer , Kiel, 1874, p. 94, pi. iii. fig. 32; 1888-89.] Mr John Rattray on the Genus Goscinodiscus. 479 Sch., Atl., pi. lviii. fig. 18 (excl. figs. 16, 17); Pant., Fossil Bacil. Ung ., p. 73, pi. xviii. fig. 166; Cleve and Moller’s Fiat., No. 210; Janisch, Gazelle Exjped., taf. v. figs. 12, 14-16; C. nitidus, Greg., var., Cleve. and Moll., Diat., Nos. 150, 154, 155, 208, 257, 311 ; C. foraminosus, Grev. MS. in Coll. Brit. Mus. Habitat. — Kekko, Mogyorod, Szakal and Szent Peter deposits (Pantocsek); sand washings, Cumbrae (Arnott !);* Lamlash (Greville! Gregory !) ; Ascidia, Roundstone Bay, county Galway ; Malahide, county Dublin ; Restrevor, county Down ; Kilkee, county Clare (O’Meara) ; Hvidingsoe (Schmidt) ; Manilla shells (Greville !) ; Tahiti (Kinker !) ; Numea Algae (Kinker !) ; Manilla (Firth !) ; f Tamatave (Hardman !);J Rio Janeiro (Hardman!); Oamaru deposit (Grove !); Campeachy Bay (Weissflog ! Cleve and Moller !) ; Monterey (Weissflog !) ;§ Gazelle Expedition (Weissflog !) ; Anda- man Islands (Macrae !); || coral washings, locality*? (Doeg!); shell cleanings, locality 1 (Doeg !) ; Rovigno, North Carolina, Balearic Islands, Gripp (Cleve and Moller !); Galapagos Islands, Labuan, Virgin Islands (Cleve !). Var. minor. Cleve and Moll., Diat., No. 154. — Diam. ’025 to *0325 mm. Markings angular, 4 in *01 mm. Central papillae prominent, rows evident near border ; interspaces subohsolete. Habitat. — Balearic Islands (Cleve and Moller !). Var. sjparsa. C. nitidus , Sch., All., pi. lviii. fig. 17. — Diam. *035 mm. Markings round, isolated granules, with wider interspaces, smaller near the border. Border striae more evident and longer. Habitat. — Campeachy Bank (Schmidt). Var. tenuis , nov. C. nitidus , Greg., var. Sch., Atl ., pi. lviii. fig. 19. — Diam. about *04 mm. Markings minute, with smaller hyaline interspaces, on a narrow hand adjacent to the border punctiform, and forming regular radial striae. — Sch., Atl., pi. lvii. fig. 45 (*?). Habitat. — Campeachy Bank (Schmidt). * In the collection of Dr Greville. f In the collection of Dr F. W. Griffin. X In the collection of Julien Deby. § This specimen has, on what seems to me insufficient grounds, been named on Weissflog’ s slide C. nitidus , var. by Grunow. II In the collection of Dr Greville. 480 Proceedings of Royal Society of Edinburgh. [sess. Yar. moronensis. Grun. MS. — Diana. -0875 mm. Markings with central papillae more prominent, scabrous ; secondary, irregularly, curved rows, subconcentric within the semiradius ; around the border the subradial rows short, inconspicuous. Border narrow ; striae 6 or 7 in ‘01 mm. — (PI. I. fig. 21.) Habitat. — Moron deposit (Weissflog !). C. nitidulus, Grun. Sch., Atl., pi. lviii. figs. 20, 21.- — Sometimes trilobate. Diam. *04 to *1175 mm. Central space absent. Mark- ings small, round, granular, decreasing slightly towards the border ; about 4 in -01 mm.; rows radial, beyond the semiradius subfascicu- late ; interspaces hyaline. Border distinct, narrow ; striae 6 to 8 in *01 mm. — Yan Heurck, Syn. Diat. Belg ., pi. cxxxii. fig. 2 ; Pant., Fossil. Bacil. Ung ., p. 73, pi. xxiv. fig. 214 ; Janisch, Gazelle Exped ., taf. v. fig. 13. Distinguished from C. nitidus by the smaller size of the markings, which decrease less towards the border. Habitat. — Campeachy Bay (Yan Heurck, Schmidt.) ; Szakal, Szent Peter and Dolje deposits (Pantocsek); Hong Kong (Hard- man !) ; * Cambodia (Firth !) ; Springfield deposits, Barbados (Firth!); Sta Maria deposit (Grove!); Oamaru deposit (Firth!); Bio Janeiro (Hardman !); * Macassar Straits (Grove !) ; between Aden and Bab-el-Mandeb (Cleve !). Yar. subradians, nov. — Diam. *05 mm. Markings round, granular, largest at the centre, decreasing gradually outwards, punctiform at the border; interspaces wide, smallest towards the border; rows subradial, non-fasciculate, crowded on a distinct zone at the border about i of the radius broad. Habitat. — Aegina (Schmidt). C. suspectus, Janisch. Sch., Atl., pi. lix. fig. 2. — Diam. *106 mm. Central space and rosette absent. Markings polygonal, about 7 in *01 mm., decreasing slightly towards the border ; rows radial or oblique and decussating, the former forming inconspicuous narrow fasciculi most evident beyond the semiradius, the latter straight or slightly curved outwards. Border narrow, hyaline. — Grun., Denk. Wien. Ak., 1884, p. 85. In the collection of Julien Deby. 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 481 Distinguished from C. circumdatus by the greater irregularity of the rows about the centre and the simple border. Habitat. — San Francisco, Cal. (Schmidt). C. Kiitzingii. Sch., Atl, pi. lvii. figs. 17, 18. — Diam. *0635 mm. Central space absent. Markings polygonal, about 6 in *01 mm., subequal or decreasing slightly towards the border ; rows distinctly fasciculate beyond the semiradius ; those in each fasciculus parallel to one another, secondary oblique decussating rows evident, and curved towards the border. Border sharply defined, bearing crowded oblique decussating rows of areolae. — Grun. Denk. Wien. Ak., 1884, p. 84; C. marginatus, Sch., Jahresb. d. Kom. z. Unter- such. d. deutsch. Meer , Kiel. 1874, p. 94, pi. iii. fig. 35. Distinguished from C. suspedus by the more evident fasciculi and border, and from C. subtilis in the absence of apiculi. The relationship to C. excentricus referred to by Grunow is more remote. Habitat . — Cuxhaven, Firth of Tay (Schmidt) ; Arctic and Antarctic (Grunow). Yar. glacialis. Gran., Denk. Wien. Ak ., 1884, p. 84, pi. iv. (D), fig. 18. — Diam. 045 mm. Markings 10 in 01 mm., decreasing slightly towards the border ; rows less distinctly fasciculate, a circlet of minute apiculi inserted at the border. Border striae radial, distinct, 8 to 10 in *01 mm. Habitat. — Franz Josef’s Land ; Cape Wankarema, North Siberia ; Kerguelen (Grunow). C. subglobosus. Cleve and Grun., Denk. Wien. Ak ., 1884, p. 84, pi. iv. (D), figs. 19, 20. — Diam. 025 to 04 mm. Surface some- what convex. Central space absent. Markings polygonal, decreasing slightly towards the border, 8 in 01 mm.; rows radial, on outermost portion of valve parallel and subfasciculate ; secondary oblique rows curved outwards, most evident towards the border. — Sch., Atl., pi. lviii. fig. 44 (no name); Cleve and Moller, Diat ., Nos. 114, 172, 302, 319. Distinguished from C. Kiitzingii by the more irregular markings on the central portion. VOL. XVI. 25/10/89 2 H 482 Proceedings of Royal Society of Edinburgh. [sess. Habitat. — Arctic, Davis Straits ; Franz Josefs Land, H. Siberia ; Antarctic (Grunow) ; Davis Straits (Cleve and Moller !) ; Mascara, Cape Wankarema (Cleve ! Cleve and Moller !) ; Greenland (Cleve !). C. inclusus , sp. n. Sch., Atl ., pi. lvii, fig. 47 (no name). — Diam. about *07 mm. Central space distinct, rounded, slightly excentric. Markings rounded, granular, about 5 in ’01 mm., smaller towards the border ; rows fasciculate, those in each fasciculus parallel to the central radial row, non-apiculate. Border sharply defined, striae 6 to 8 in ‘01 mm. Habitat. — Richmond deposit, Virginia (Schmidt). C. tuber culatus. Grev., Trans. Micr. Soc. Lond. 1861, p. 42, pi. iv. fig. 6. — Diam. -0375 to *0975 mm. Surface almost flat. Central space irregular, small. Markings around the central space minute, rounded, granular; beyond this polygonal 4J in ‘01 mm., subequal to the zone of the apiculi ; at the border 6 in *01 mm.; rows radial or obscurely fasciculate, straight ; apiculi distinct placed between the fasciculi. Border striae delicate, 8 to 10 in '01 mm. — Sch., Atl ., pi. lvii. fig. 42 ; Grun., Denk. Wien. Ak., 1884, p. 82. This species cannot be united with Aulacodiscus, as suggested, with some doubt, in the second edition of Habirshaw’s Catalogue, § Coscinodiscus. Small specimens in Weissflog’s collection have been distinguished as forma minor , but these are quite normal. Distinguished from C. subtilis by its larger markings and more robust apiculi. Habitat. — Barbados deposit (Greville ! Hardman ! Firth ! Cleve !) ; Chalky mound, Barbados (Weissflog ! Firth !) ; Cam- bridge deposit, Barbados (Firth ! Johnson ! Hardman !). Var. Monicas. Grun. ibid., p. 82, pi. iii. (C), fig. 29. — Diam. *0625 mm. Markings 3 to 5 in *01 mm., smaller close to the border ; a few rows in each inconspicuous fasciculus ; the apiculi nterfasciculate, more evident. — C. tuber culatus, Grev. var. ? Sch., Atl., pi. lvii. figs. 40, 41. The markings in the Barbados valves are larger than those in Sta Monica specimens. According to Schmidt, there are no pro- 1888-89.] Mr John Rattray on the Genus Coscmodiscus. 483 cesses: “die dunkeln Elecke am Rande entstehen dadurch, dass sich je 2 bis 3 Zellchen mit Schalensubstanz fiillen.” Habitat . — Sta Monica deposit (Grunow) ; Cambridge deposit Barbados (Johnson !). C. isoporus. Ehrb., Mikrog ., pi. xxxiii. 17. fig. 3. — Diam. about •055 mm. Central space and rosette absent. Markings large, 3| to 4 in *01 mm., distinct, subequal to the circumference; rows radial, subfasciculate ; secondary concentric rows evident. — Ralfs in Pritch. Inf “, p. 830. Differs from C. concavus in the absence of a distinct border, and the concentric arrangement of the markings, and from C. patina , Ehrb., by the reduction in size, in the latter, of the markings around the evident clear border and their less conspicuous con- centric arrangement. Habitat. — Rappahannock Cliff, Virginia (Ebrenberg). C. Payeri. Grun., Denk. Wien. Ak., 1884, p. 80, pi. iii. (C), figs. 12, 13. — Diam. -024 to *03 mm. Central space small, about of diam. broad, irregular. Markings around the central space rounded, elsewhere angular, often quadrilateral ; central papillae distinct, towards the central space 5 or 6, at the border 9 in *01 mm.; rows radial, in inconspicuous fasciculi. Habitat. — Franz Josefs Land (Grunow). Var. subrepleta. Grun., ibid., 1884, p. 80, pi. iii. (C), figs. 14, 15. — Diam. *034 mm. Central space subobsolete, or with small round isolated granules. Markings smaller, 4J to 5 in '01 mm., a band of small granules adjacent to the border. Habitat. — Franz Josefs Land (Grunow). C. hyalinus. Grun., Denk. Wien. Ak ., 1884, p. 108, pi. iii. (C), fig. 28. — Diam. ’025 mm. Central space minute, inconspicuous, bearing isolated puncta. Markings punctiform, subequal, 24 in -01 mm. ; rows radial to subparallel in inconspicuous fasciculi; apiculi numerous, distinct, in a single circlet. Border broad, hyaline. — Cleve and Moll., Diat ., No. 315; Odontodiscus hyalinus , Grun., Kon. Sv. Vet.-Ak. Ilandl. Stockh., 1879, p. 113. Distinguished from C. bioculatus by the absence of the two conspicuous central granules and the more numerous apiculi. 484 Proceedings of Royal Society of Edinburgh. [sess. Habitat. — From iceberg, lat. 74° 48' 4" N., long. 54° 52' 8" E., August 1872 (Grunow); Cape Wankarema (Cleve and Moller! Cleve !) ; Tindingen, Franz Josefs Land ; Kara (Cleve). C. capensis. Grun., Denk. Wien. Ak ., 1884, p. 86, pi. iv. (D), fig. 29. — Diam. '032 mm. Surface with slight circular undulation about the semiradius. Central space circular, about of diam. broad, distinct, with a few isolated granules at its centre. Mark- ings punctiform, smallest towards the border; rows straight or with slight bendings, inconspicuously fasciculate ; apiculi numerous, distinct, frequently arranged in a double row. Border sharply defined, hyaline. — Cleve and Moller, Diat ., No. 197. Distinguished from C. bipticatus, C. pellucidus , and C. bengalensis by its central space, subfasciculate rows, and apiculi. Habitat.— Brackish water, Baaken Biver, S. Africa (Grunow). C. bioculatus. Grun., Denk. Wien. Ak ., 1884, p. 107, pi. iii. (C), fig. 30; pi. iv. (D), fig. 1. — Diam. *02 to -03 mm. Surface convex, with faint concentric undulations. Central space subcircular, bearing two large conspicuous round granules, wdth evident central dot. Markings rounded, punctiform, least crowded towards the centre; rows subparallel in inconspicuous fasciculi, 18 to 22 in •01 mm. ; apiculi small, but evident, close to the border, numerous, subregular. In specimens from the Kara Sea and Cape Wankarema, N. Siberia, the markings and apiculi are more distant. Habitat. — Franz Josefs Land (Grunow). Yar. exigua. Grun., ibid., p. 108, pi. iv. (D), fig. 2. — Diam. #012 to ’015 mm. Central space minute, the large round granule single, or sometimes two of unequal sizes, the one indistinct. Markings less evident; rows 24 to 26 in *01 mm.; apiculi minute, 4 in •01 mm. Habitat. — Franz Josefs Land (Grunow). C. semipennatus. Grun., Denk. Wien. Ak., 1884, p. 83. — Diam. about '0305 mm. Central space absent or subobsolete. Markings rounded, or obtusely angular, granular; towards the centre 4, decreas- ing uniformly but considerably towards the border; rows fasciculate, 1888-89.] Mr John Rattray on the Genus Goscinodiscus. 485 slightly curved towards the same direction ; those in each fasciculus parallel to the corresponding and more conspicuous side rows ; interspaces hyaline. — Sch., Atl., pi. lvii. fig. 32 (no name). Habitat. — Springfield deposit, Barbados (Schmidt) ; Cambridge deposit, Barbadoes (Kinker !). C. Grunowii. Pant., Fossil. Bacil. Ung., p. 74, pi. ix. fig. 74. — Diam. -062 to 0*75 mm. Surface flat, slightly convex near the border. Central space small, indistinct, almost filled in by isolated round granules. Markings obtusely angular, subequal, 7 to 8, towards the border more crowded, 9 to 10 in *01 mm.; slightly smaller at the centre than on the adjacent area, the central papillae prominent; within the border from 10 to 23 small clear rounded subregular spaces; rows fasciculate, straight, and radial between the centre and the clear spaces near the border, the others parallel to these in the corresponding fasciculus. Border striae delicate, 16 in *01 mm. Habitat. — Also-, and Pelso-, Eszterg&ly deposits (Pantocsek !). Yar. minor. C. Grunowii forma minor , Pant., ibid., p. 74, pi. xiv. fig. 25. — Diam. *024 to *036 mm. Markings punctiform, 10 to 15 in -01 mm. ; the fasciculi more distinct, the clear spaces within the border more irregular, sometimes larger. Habitat. — Felso-Esztergaly deposit (Pantocsek). O. odontodiscus. Grun., Denk. Wien. Ate., 1884, p. 81, pl.iii. (C), fig. 23. — Diam. *0455 to ’1125 mm. Central space absent, but a narrow hyaline ring close to the centre. Markings hexagonal, 6 to 7 in *01 mm., decreasing slightly outwards, punctiform in a narrow zone adjacent to the border ; irregular at the centre within the clear band, elsewhere in subparallel rows ; secondary oblique decussating rows evident; fasciculi evident, the rows composing each parallel to that at one of its edges ; apiculi minute, interfasiculate. — Cleve and Moll., Diat., No. 57, 162; C. subtilis, var., Sch., Atl., pi. lvii. figs. 15, 16 ; Odontodiscus spica, Ehrb., Mon. Ber. Ale., 1845, p. 79 ; O. uranus , Ehrb., ibid., 1845, p. 79; C. odontodiscus, var. parva-tenuistriata, Cleve and Moll., Diat., No. 276. Habitat. — -Richmond, Ya. (Schmidt, Grove! Cleve and Moller !); 486 Proceedings of Royal Society of Edinburgh. [sess. Patos Island guano (Norman !);* Nancoori; California; Sta Maria deposits (Grove !) ; Balearic Islands ; Pabillan di Pico guano ; Successful Bay, Kerguelen (Cleve !). Var. subsubtilis , nov. C. subtilis , Sell., Atl., pi. lvii. fig. 14. — Diana, "125 mm. Markings sometimes increasing slightly from the centre outwards or subequal; at the centre 10, at the border 8 in ’01 mm.; rows straight, radial, fasciculate or subfasciculate towards the border. A narrow hyaline band adjacent to the border; apiculi minute, irregular, but interfasciculate, sometimes absent. Border strise distinct, 6 in *01 mm. Habited. — Peruvian guano (Schmidt); Monterey (Kinker!); Nancoori (Hardman !) ; f Lobos di Afuera guano (Grove !) ; marine deposit, Fiji Islands (Grove !); Inland Sea, Japan, H.M.S. Challenger (Grove !) ; Springfield deposit, Barbados (Doeg !). C. curvatulus, Grun. Sch., Atl., pi. lvii. fig. 33. — Diam. ’045 to *07 mm. Central space absent or indistinct, with numerous rounded granules. Markings polygonal, subequal, 6 in ’01 mm.; rows in gentle curves, fasciculate ; those composing each fasciculus parallel to that at its convex edge ; the curves on the two valves of a frustule in opposite directions, secondary oblique decussating rows obvious; apiculi absent. — Janisch, Gazelle Exped., taf. ii. fig. 7; taf. v. figs. 2, 3, 8; taf. vi. fig. 2 ; taf. xx. fig. 17; C. curvatulus , var. inermis, Grun., Denk. Wien. Ak., 1884, p. 83, pi. iv. (D), figs. 11, 12 ; Sch., Atl., pi. cxiii. fig. 6 ; Cleve and Moll., Diat., Nos. 57, 154, 162, 276, 319 ; C. curvatulus, var. densius-striata (?), Sell., Atl., pi. lvii. fig. 35; Oclontodiscus curvatulus, Cleve, Vega Exped. Vetensk. Jakttag. Stockli., Bd. iii. 1885, p. 488. In H. L. Sm., Diat. Spec. Typ., No. 99. This species is sometimes confounded with the very distinct C. symmetricus, Grev. Habitat. — Los Angelos, Cal. (Cambridge !) ; Biehmond deposit (Rae ! Cleve and Moller !) ; Caltanisetta deposit ; Barbados deposit (Johnson !) ; Peruvian guano, Franz Josefs Land (Grunow) ; Bolivian guano (Cleve !) ; Infusorial deposit, “Algeria” (Greville !); * In the collection of Dr Greville. t In the collection of Julien Deby. 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 487 Yszee (Kinker !) ; Table Bay (Weissflog !) ;* Virginia (Hardman !) ; Nancoori, Balearic Islands (Cleve and Moller !) ; Melville Bay, lat. 75° 27' N., long. 64° 34' W. (O’Meara !) ; * Faeroe Isles (Grove !) ; marine deposit, Fiji Islands (Grove !) ; Japan (H. L. Smith !) ; Monterey (Cleve, Weissflog !) ; iEgina (Schmidt) ; Cape Wankarema, California (Cleve and Moller !) ; Patagonian guano (Cleve !); Sta Monica deposit (Grove !). Var. Iatius-striata. Sch., Atl., pi. Ivii. figs. 30, 34. — Diam. *1 mm. Central space and rosette absent. Markings 3J to 4 in ‘01 mm., increasing gradually outwards to about semiradius, again decreasing similarly to the border ; fasciculi distinct, rows sometimes slightly curved; apiculi absent. Border striae, 6 in ’01 mm. Distinguished by the large size of the markings. Sometimes associated with C. snbtUis , to which it bears no affinity. Habitat. — Cambridge deposit, Barbados (Hardman !);f Bar- bados (Cleve !). Var. minor. Grun., ibid., 1884, p. 83, pi. iv. (D), figs. 8, 10. — Diam. *03 to *04 mm. Central space small and granular, or absent. Markings in more straight, less distinctly fasciculate rows ; secondary oblique rows indistinct, non-apiculate (excl. G. minor , Ehrb.). The union of C. minor , as figured in Schmidt’s Atlas , pi. Iviii. fig. 40, to the present var., as proposed by Grunow, is erroneous. Habitat. — Girgenti and Caltanisetta deposits (Grunow) ; Nancoori deposit (Hardman !).J Var. genuina. Grun., ibid., 1884, p. 83, pi. iv. (D), fig. 13. — Diam. *0325 to T25 mm. Central area small, circular, with but few granules. Markings 8 to 10 in ‘01 mm.; apiculi minute, interfasciculate. Border sharply defined ; strise delicate, 14 to 16 in ‘01 mm. — Van Heurck, Typ. Syn. Diat. Belg., No. 534 ; Sch., Atl., pi. Ivii. fig. 36 (no name) and 37. G. Szontaghii, Pant., Fossil. Bacil. Ung., p. 72, pi. xv. fig. 133. Specimens from Bolivia pass into the var. subocellata, Grun. * Procured by MfClintock. t In the collection of Julien Deby. + Ibid. 488 Proceedings of Royal Society of Edinburgh . [sess. Pantocsek’s figure does not show the fasciculation, though this is distinct in his specimens. Habitat. — From ice in lat. 74° 48' 4" N., long. 54° 52' 8" E., Aug. 2, 1872 (Grunow); San Prancisco, Cal. (Firth!); Los Angelos, Cal. (Griffin !) ;* Oran, Algeria (Van Heurck !) ; soundings off Kurile Islands, 1329 fathoms (H. L. Smith!);! Barbados (Johnson !);J Richmond deposit, Va. (Rae!); Kekko deposit (Grove!); Jack’s Ranch, Cal. (Macrae !) ; Szakal and Szent Peter deposits (Pan- tocsek !). Var. kariana. Cleve and Grun., Kong. Sv. Vet.-Ak. Handl Stockh ., 1880, p. 113, pi. vii. fig. 129. — Diam. '023 to *024 mm. Central space absent. Markings distinct, 11 to 12 rows in each fasciculus and 13 J to 14 in *01 mm.; apiculi interfasciculate, distinct. — Odontodiscus curvatulus , var. kariana , Cleve and Grun., ibid., 1880, p. 113. Habitat. — Kara Sea (Cleve and Grunow) ; Pinmark (Cleve). Yar. subocellata. Grun., ibid ., 1884, p. 83, pi. iv. (D), fig. 15, from Bolivian guano and Cape of Good Hope (Grunow) ; Ker- guelen Island and Challenger dredgings off Vancouver Island (Grove !), belongs to Actinocyclus. Yar. recta , nov. C. curvatulus , var., Cstr., Diat. Cliall. Exped ., p. 160, pi. iii. fig. 10. — Diam. *03 to *0875 mm. Markings 4 to 5 in •01 mm., arranged in straight fasciculi; apiculi distinct, inter- fasciculate, sometimes absent. Border with slight indentations opposite the apiculi. — Cleve and Moll., Diat., Nos. 57, 164, 276 ; Janisch, Gazelle Exped., taf. i. fig. 6 ; taf. iv. fig. 4. This var. approaches var. minor. Specimens have sometimes been confounded with C. barbadensis, Grev., or regarded as fasciculate forms of C. oculus-iridis. Habitat. — H.M.S. Challenger (Castracane) ; Richmond; Sta Monica deposit, California (Cleve and Moller !) ; Barbados deposit (Cleve !) ; Yokohama (Cleve !) ; Monterey stone (Cleve !) ; San Benito deposit, California (Grovel); Marstrand (Kinker !). * In the collection of Dr F. W. Griffin. + In H. L. Sm. Diat. Spec. Typ., No. 93. + In the collection of Dr Greville. 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 489 I have not seen C. curvatulu-s , var. barbadensis , Cleve MS., from Barbados, nor C. curvatidus , var. Cleve MS., from Yokohama. C. curvatidus , var. frigida , Grun. ( Vega Exped., Vetensk. Jakttag. Stockh ., Bd. iii. 1883, p. 488), remains undefined. Specimens so named were procured in Franz Josefs Land. C. crenulatus, Grun. Sch., Atl. , pi. lvii .fig. 38. — Diam. *0605 mm. Central space absent. Markings polygonal, subequal, about 5 in '01 mm.; irregular on a small area at the centre, elsewhere in fasciculate rows ; those composing the fasciculi parallel to that at their corresponding sides; apiculi distinct, inserted at inner edge of border and interfasciculate. Border sharply defined, its outer edge crenate, the indentations corresponding in position to the spines, its inner half with distinct striae 6 in *01 mm., its outer half smooth. — Grun., Denk. Wien. Ak ., 1884, p. 83, pi. iv. (D), fig. 17. Habitat. — Seychelles, Bolivian guano, ex Salpa spinosa , Southern Ocean, Balearic Islands (Grunow); Kings-Mill Islands (Schmidt). G. ceginensis. Sch., Atl., pi. cxiii. figs. 13, 14. — Diam. *061 to •085 mm. Surface flat. Central space subcircular, to of diam. broad, bearing a few isolated round granules, one much larger than the others. Markings around the central space round, subequal, else- where polygonal, increasing outwards to about the semiradius, thence decreasing gradually to the border ; towards the centre 4J, at the semiradius 3J in *01 mm.; rows radial, subfasciculate beyond the semiradius, secondary oblique rows evident. Border narrow, striae 6 in ’01 mm. Habitat. — Aegina (Schmidt). C. simbirskianus. Grun., Denk. Wien. Ak., 1884, p. 81. — Diam. ’1 to *225 mm. Central space absent, sometimes minute; rosette inconspicuous or subobsolete, sometimes distinct. Markings hexagonal, increasing slightly outwards to about the semiradius, thence decreasing gradually to the border ; towards the centre 4, at the semiradius 3 to 3J, at the border 6 in ’01 mm.; central papillae evident; rows fasciculate, those in each fasciculus subparallel to that at its centre ; oblique decussating rows evident. Border narrow ; striae 4 to 5 in '01 mm. — Sch., Atl., pi. cxiii. figs. 11, 12. 490 Proceedings of Royal Society of Edinburgh . [sess. Distinguished from C. radiatus by the fasciculate arrangement of the markings. Habitat. — Simbirsk (Grunow, Grove !); Ananino deposit (Grove ! Kinker! Rae ! Deby !); Archangelsk-Kurojedowo (Schmidt, Cleve !). C. symmetricus. Grev., Trans. Micr. Soc. Land ., 1861, p. 68, pi. viii. fig. 2. — Diam. *075 to *175 mm. Central space small, rounded, of diam. broad, sometimes absent. Markings subpearly; round or subangular, granular, 4 in *01 mm., subequal or slightly smaller at the border ; interspaces hyaline ; rows straight, fasciculate, those composing a fasciculus parallel to that at its centre ; around the border a narrow hyaline clear space, irregular on its inner side, sometimes bearing a few isolated granules. Border striae evident, 6 in -01 mm. — Excl. C. symmetricus , Sch., Atl ., pi. lvii. figs. 25-27 ; Grun., Denk. Wien. Ak ., 1884, p. 81. Narrow hyaline radial spaces sometimes, in larger valves, run outwards for some distance from the central space. Quite distinct from C. subtilis in the size and arrangement of the markings. C. ? clypeus, Ehrb. ( Mikrog ., pi. xi. fig. 6), from Bilin deposit, Bohemia, is too minute for identification. The fragments show rouud granular fasciculate markings with hyaline interspaces. Ehrenberg did not define the species, and regarded it as probably a fragment of Campylodiscus clypeus. The figure may represent a small piece of Coscinodiscus symmetricus , Grev. Habitat. — Cambridge deposit, Barbados (Johnson ! Greville ! Firth !); Manilla (Firth !); Newcastle deposit, Barbados (Firth !); “Barbados” (Kinker !). C. planiusculus, sp. n. — Diam. *0825 mm. Surface flat. Central space rounded, indistinct, about of diam. broad. Markings rounded or oval, with long axis radial; outlines faint; central papillae evident ; about 3 \ in -01 mm., subequal, on a distinct band around the border, about i to of radius broad, moniliform ; rows fasciculate, those in each fasciculus parallel to that at its middle. Border crenate, a minute dark speck at each indentation at intervals of about *005 mm. — Janisch, Gazelle Exped ., taf. vi. fig. 12. (PI. I. fig. 22.) Differs from C. subtilis in the shape of the markings, the monili- 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 491 form band adjacent to the border and the crenate appearance of the latter. Habitat. — Gazelle Expedition (Weissflog !). C. fasciculatus. O’Me., Quart. Jour. Micr. Sci ., 1867, p. 245, pi vii. fig. 1. — Diam. *0825 mm. Central space circular, distinctly defined. Markings areolate, suhequal to the border ; rows fasciculate, 9 composing each fasciculus and parallel to that at its centre, the central radial row only extending to the central space, each adjacent pair successively terminating farther and farther from it ; interspaces at origin of shorter rows hyaline ; apiculi absent. The markings are intermediate in size between those of C. sym- metricus and of C. Normani or C. subtilis. Habitat. — Dredged off Arran Islands, co. Galway (O’Meara). C. echinatus , sp. n. Sch., Atl., pi. lviii. figs. 35, 36 (no names). — - Diam. about *03 mm. Central space minute, rosette absent. Markings angular, subequal or decreasing gradually from the centre outwards, towards the centre 4J to 5, towards the border sometimes 6 in *01 mm.; rows fasciculate, those in each fasciculus parallel to the central row; fasciculi few, 3 to 5. Apiculi large, spine-like, interfasciculate, inserted some distance from border. Border sharply defined, striae 6 in *01 mm., sometimes obscure. Habitat. — Moron deposit (Schmidt). C. lentiginosus , Janisch. Sch., Atl ., pi.' lviii.’ fig. 11. — Diam. *075 mm. Central space absent. Markings round, granular, least crowded and with narrow hyaline interspaces towards the centre, angular and more crowded around the border ; towards the centre 6, at the border 8 in *01 mm.; irregular on a small indistinct central area, elsewhere in fascisculate rows, those in each fasciculus parallel to that at its middle ; a minute apiculus close to the border, readily overlooked. Border striae delicate, 16 in *01 mm. — Clove' and Moll., Diat ., No. 207 ; C. lentiginosus , var. maculatus, Grun., Cleve and Moll., Diat., No. 183. A similar spine occurs in C. leptopus and C. hryophilus. Its character is distinct from those of Eupodiscus , to which on its account it has sometimes been proposed to unite Janisch’s species. Habitat. — Patagonia, Antarctic Ocean (Cleve and Moller !) ; 492 Proceedings of Pioyal Society of Edinburgh. [sess. Kerguelen (Grove!); Antarctic ooze 1950 fms., H.M.S. Challenger (Kae ! Kinker ! Hardman ! Grunow) ; lat. 46° 46' N., long. 45° 31' E., 1375 fms. H.M.S. Challenger (Castracane !) ; Oamaru deposit (Doeg ! Marshall!) ;* Table Bay (Schmidt) ; Challenger dredgings, off Vancouver’s Island, and Globigerina ooze, off Ascension, S.S. Buccaneer (Grove). C. kryophilus. Grun., Denk. Wien. Ah., 1884, p. 81, pi. iii. (C), fig. 21. — Diam. *044 mm. Central space rounded, indistinct, about of diam. broad, bearing isolated rounded granules. Markings polygonal, minute ; rows fasciculate, those composing each fasciculus parallel to the radial row at its middle ; a single prominent spine inserted near the border, outside of this a circlet of numerous, closely placed minute apiculm — Janisch, Gazelle Exped., taf. iii. fig. 3. Distinguished from C. lentiginosus , Janisch, by the larger spine and by the apiculi, and from C. polyacanthus by the smaller hut more numerous apiculi. Habitat. — Cape Wankarema, N. Siberia (Grunow). C. symbolophorus. Grun., Denk. Wien. Ak ., 1884, p. 82, pi. iv. (D), figs. 3-6. — Diam. *085 to T75 mm. Surface convex. Colour at centre steel grey, an opaque ring towards the semiradius, whence dark radial hands proceed outwards. Central space small, and usually branching into a distinct star with 3 to 5 rays, sometimes absent, but around the central area a similar star with 3 to 5 lanceolate rays. Markings polygonal, decreasing regularly outwards ; towards the centre 6, towards the border 8 to 9 in *01 mm.; fasciculate, those composing each fasciculus parallel to the radial row at its middle ; secondary oblique decussating rows faint, often somewhat curved outwards. Border narrow, hyaline. — Symbolophora, sp. Ehrb., Grun., ibid., 1884, p. 82; S. microtrias , Ehrb., Mon. Ber. Ah, 1884, p. 205; Mikrog., pi. xxxv. a. 21. fig. 16; S. tetras, Ehrb., ibid., 1844; p. 205; Abh. Ber. Ah, 1872, pi. xii. 2. fig. 1; S. pentas, Ehrb., ibid., 1844, p. 205; Mikrog., pi. xxxv. a. 22. fig. 19; S. hexas, Ehrb., * In the collection of F. W. Griffin. Mr Grove remarks that he has never seen this species in the Oamaru deposit, and greatly doubts its presence there, it being a form which may easily adhere to tubes and beakers, 1888—89.] Mr John Rattray on the Genus Coscinodiscus. 493 1844, p. 205; S. microtetras ,* Ehrb., Mon. Ber. Ak., 1855, p. 302; S. micropentas ,* Ehrb., ibid , 1855, p. 302 ; S. microhexas * Ehrb., ibid., 1855, p. 302; not S. trinitatis, Ehrb., ibid., 1844, p. 88; Ralfs in Pritch. Inf., p. 833, pi. xi. fig. 36, as indicated in the second edition of Habirshaw’s Catalogue, §' Symbolophora. By Grunow this species is brought near C. subtilis, but it is readily distinguished from the latter by the central star and the absence of apiculi. Symbolophora acuta, Ehrb., from Hollis Cliff, Virginia, of which hut a fragment is figured (. Mikrog ., pi. xxxiii. 15. fig. 21), probably belongs to the present species. S. euprepia, Ehrb., from Licuare River, coast land Mozambique, remains a nomen nudum , but may come here (see Ehrb., Mikrog., p. 228), Small specimens of C. symbolophorus approach C. excentricus. Habitat. — Nottingham deposit, Mors deposit (Cleve ! Grunow) ; Simbirsk Polirschiefer (Ehrenberg, Grunow) ; Kekko, Szakal and Szent Peter deposits (Pantocsek) ; Eranz Josefs Land (Grunow); pancake ice, Antarctic Ice Barrier, and sounding of 190 fms., lat. 78° 10' S., long. 162° W.; sounding 270 fms., lat. 63° 40' S., long. 55° W. (Hooker) ; Ananino deposit (Grove ! Linker ! Hardman !);f Lumfiord, Jutland (Hardman !) ; Bermuda tripoli (Greville !); Oamaru deposit (Grove ! Marshall !) ; Ananino deposit (Rae ! Deby !). C. stellaris. Roper, Quart. Jour. Micr. Sci., 1858, p. 21, pi. iii. fig. 3. — Diam. ‘075 to 1 mm. Surface slightly convex. Markings, around the centre 5 or 6 large areolae, at subequal distances apart, and tapering towards both ends, elsewhere scarcely visible, 16 to 20 in *01 mm., angular, most evident towards the centre, the rows forming fasciculi, those in each fasciculus parallel to the radial row at its middle. — C. stellaris, var. Cstr., Diat., Chall. Exped., p. 155, pi. iii. fig. 2 ; C. stellaris, var. fasciculata, Cstr., Diat., Chall. Exped., 1886, p. 158, pi. v. fig. 9. The colour when dry is brownish. Castracane’s Antarctic frustule seems to differ only in having four markings at the centre. Transitional forms to C. symbolophorus occur. Habitat. — Caldy, Pembrokeshire (Roper ! Rev. J. Guillemard) ; * Nomina nuda , probably identical with S. tetras , S. pentas, S. hexas. t In the collection of Julien Deby. 494 Proceedings of Royal Society of Edinburgh. [sess. Tenby Bay (Boper !) ;* oyster shells, Dublin Bay (O’Meara) ; Mediterranean (Grunow) ; Balearic Islands (Weissflog !) ; Antarctic Ice Barrier, H.M.S. Challenger (Castracane). Var. Mejillonis. Grun., Denk. Wien. Ak., 1884, p. 82. — Diam. •23 mm. Markings 18 in *01 mm., central rosette with 8 large areolae. Habitat. — Mejillones guano (Grunow, Grove !). C. minutellus , sp. n. — Diam. *0225 mm. Surface flat. Central space and rosette absent ; a minute, somewhat excentric triangular clear area (*0025 mm. broad) with a single central dot evident. Markings polygonal, 10 in *01 mm., still smaller towards the border ; rows faintly fasciculate, subradial, the secondary oblique outwardly concave, decussating rows more distinct ; apiculi at the border prominent, at subequal intervals of *007 6 mm., or somewhat less.— Border narrow, hyaline. — (PI. II. fig. 5.) Habitat. — From Salpa spinosa, locality ? (Weissflog !). C. subtilis. Ehrb., Abli. Ber. Ak ., 1841, p. 412, pi. i. 3. fig. 18 ; pi. iii. 7. fig. 4. — Diam. *0425 to ’1125 mm. Central space and rosette absent. Markings polygonal, 6 to 10 in '01 mm., decreasing somewhat towards the border, without order on a small central area, elsewhere in fasciculate rows, about 12 forming each fasciculus at its outer extremity, and arranged parallel to the central radial row ; secondary oblique decussating rows evident ; apiculi sometimes present at the border, interfasciculate. Border striae delicate, faint, 12 to 14 in. ’01 mm. — Ehrb., Mikrog., pi. xviii. fig. 35, a.b.; pi. xxxiii. 13. fig. 4 ; pi. xxxiii. 16. fig. 7; pi. xxxiv. 7. fig. 6; pi. xxxv. 22. fig. 5 ; pi. xxxv. 23. fig. 5. Grev., Quart. Jour. Micr. Sci ., 1859, p. 81 ; Balfs in Pritch. Inf. , p. 830 ; Jan., Abh. Sch. Ges. vdter. Cult., 1862, Heft ii. p. 4, pi. i. A, fig. 2 ; Janisch, Gazelle Exped., taf. ii. fig. 8 ; iv. figs. 1, 2 ; v. figs. 5, 7 ; vi. figs. 1, 5; xx. fig. 5; Grun., Sitzungsb. naturw. Ges. Isis., Dresden, 1878, p. 124; Sch., Jaliresb. d. Kom. z. Untersuch. d. deutscli . Meer, Kiel , 1874, ii. p. 94; Sch., Atl., pi. lvii. figs. 11, 13, 28, 29 (no name); pi. lviii. fig. 37 (no name) ; Grun., Denk. Wien. Ak., 1884, p. 81, * In the collections of Dr Greville and E. Grove. 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 495 pi. ii, (C), fig. 26; Raben., Alg. Europ., Nos. 2142, 2187, 2558; Van Heurck, Syn. Diat. Belg ., p. 218, pi. cxxxi. fig. 1 ; Typ. Syn. Diat. Belg., Nos. 519, 520, 532, 533; Cleve and Moll., Diat, Nos. 57, 125, 162, 164, 207, 211, 257, 258, 319; H. L. Smith, Diat. Spec. Typ., No. 100 ; C. subtilis % Sch., Atl., pi. lvii. fig. 12 (excl. C. radiolatush= G. subtilis, Ehrb., Mikrog, pi. xxii. fig. 4). According to Ehrenberg’s original definition, there should he 12 markings in *01 mm. Fasciculi were hardly indicated in his figures, several of which approach his C. intermedins. There is no close affinity to C. punctatus, Ehrb., as he at one time believed (Mon. Ber. Ak., 1844, pp. 186, 188-191). The species approaches C. Normani, Greg., but in the latter the markings are less regular, the fasciculi at the border consist of about 6 rows instead of 12, and the lines between the fasciculi are less distinct. Janisch erroneously describes the markings as round. Schmidt, in 1874, erroneously refers to the fasciculi as branched towards the same side by a bent ray. Specimens are sometimes confounded with C. odontophorus, Grun., and G. Rothii, Grun., and are distinguished from C. symmetricus , Grev., and C. denarius, Sch., only by their smaller markings, to which transitional forms occur, Grunow having observed forms from Monterey and Australia with 5J to 7 markings in -01 mm. Habitat. — Stratford (Ehrenberg) ; Richmond, Ya. (Erhenberg, Rae ! Cleve and Mo Her !) ; Caltanisetta (Ehrenberg) ; Moron deposit (Schmidt) ; Peruvian guano (Janisch, Schmidt) ; Angamos guano (Janisch) ; Bolivian guano (Grunow) ; Patos guano (Greville!); Assistance Bay, San Francisco (Ehrenberg); lat. 71° 19' N., long. 11° 28' W., 129 fms., in yellowish-grey mud ; lat. 73° 16' N., long. 15° 48' W., 1300 fms., in dark greyish-brown mud ; lat. 63° 40' N., long. 5° 28' E., 569 fms., in light grey fine mud; lat. 74° 11' N., long. 15° 19' W., 224 fms., in fine greyish-brown mud; lat. 74° 33' N., long. 18° 39' W., 90 fms., in coarse sandy mud (Ehrenberg) ;* pancake ice, Antarctic Ice Barrier, lat. 78° 10' S., long. 162° W.; melted ice, lat. 75° S., long. 170° W.; snow and ice, near Vancouver Island, lat. 70° S., long. 165° W.; floating ice, lat. 64° S., long. 160° W.; Gulf of Erebus and Terror, lat. 63° 40' S., long. 55° W., 207 fms.; ex Salpd, lat. 66° S., long. 157° W. * Specimens procured by the second German North Polar Expedition. 496 Proceedings of Royal Society of Edinburgh. [sess. (Hooker) ; Canton (Ehrenberg) ; Yokohama and Arica (Schmidt) ; Ascidia , Hull (Greville !), rice fields, Georgia (Grove ! Bailey,* Greville !) Yszee (Kinker !) ; dredged in 28 fms., Royal Sound, Kerguelen, by H.M.S. Challenger (Rae!);f “Antarctic Ocean” (Cleve and Moller !) : rice field mud, Savannah (Cleve ! H. L. Smith !) Cambodia (Hardman ! J Firth !) ; Port Elizabeth (Hardman !) ; Humber (O’Meara !) ; Richmond Tunnel (O’Meara !) ; “India” (Macrae !); § Holstein (Van Heurck!); Maryland (O’Meara!); Japan (H. L. Smith!); Los Angelos deposit (O’Meara !) ; Cannibal Islands (Greville !) ; Gazelle Expedition (Weissflog !) ; Macassar Straits (Grove!); mud from Gliickstadt ; Elbe, above Cuxhaven (Rabenhorst and Schwarz !) ; Archangelsk (Cleve); Campeachy Bank, Gulf of Mexico (Rabenhorst and Gersten- berger !) ; Rappahannock River, Ya. (Rogers ! Greville !) ; on Dutch rushes, Hull (Norman !) ; Virginia (Greville !) ; Indian Ocean soundings, Capt. Pullen, 2200 fms. (Greville !) ; Kannaliack, Cannibal Islands (Greville!); Woodlark Island (Roberts!); Bass Straits (Greville!); Kamtschatka, 1700 fms. (Greville!); Jersey (Wallich !) ; Patos Island guano (Greville !); shell cleanings from Singapore (Hardman!); Nicobar shell cleanings (Doeg !) ; Santa Marta deposit (Doeg !) ; coral washings, Mauritius (Doeg !) ; Newcastle deposit, Barbados (Doeg!); N. Atlantic, lat. 51° 20' N., long. 52° 25' W., 232 fms. (O’Meara!); Wexford (O’Meara!); Eaeroe Isles (Grove !) ; Monkstown (O’Meara !) ; Jack’s Ranch California (Macrae !) ; Monterey (Schmidt) ; Upolu (Weissflog) ; Nancoori; Sta Monica deposit; Patagonia; Delaware; North Carolina ; Pensacola ; Cape Wankarema (Cleve ! Cleve and Moller !) ; Balearic Islands (Cleve !) ; Greenland, Yeddo, Franz Josef’s Land, Peruvian guano, Pabillan di Pico guano, Patagonian guano, Schleswig Holstein; Elbing, West Prussia; Labuan, Virgin Islands (Cleve !). Var. siberica. Grun., Kongl. Sv. Vet. -Ah. Handl. Stochh ., * Found in various localities in United States in 1850. Its presence, with other brackish or salt water species in the rice fields, has been held to ndicate the presence of salt water much further up the river formerly than at present. t In the collection of Dr F. W. Griffin. t In the collection of Julien Deby. § In the collection of Dr Greville. 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 497 1880, No. 2, p. 115.— Diam. *044 mm. Markings more delicate, 15 to 16 rows in *01 mm., irregular on a small central area; fasciculi numerous, each consisting of about 12 rows ; apiculi absent. This forms transition to C. glacialis. Habitat . — Jenissey (Cleve and Grunow). Var. lineolata , nov. — Diam. *0575 mm. Central space subcircular, from this a single narrow straight line passing to the border and at intervals of about J of the surface, two other pairs of straight rows in contact with one another. Markings in irregular inconspicuous fasciculi; apiculi at the border, inter-fasciculate; indistinct. — (PI. I. fig. 16.) Grunow has regarded this var. as an abnormal form of C. subtilis. Habitat. — Mors Island (Weissflog !). Yar. scabra, nov. — Diam. '0375 to *055 mm. Central space absent. Markings 12 to 14 in '01 mm., somewhat smaller towards the border ; fasciculi inconspicuous ; apiculi few, indistinct at wide intervals, not between each pair of fasciculi ; minute scattered hyaline specks (apiculi1?) most numerous towards the centre. — (Pl. III. fig. 6.) Habitat. — Nancoori (Weissflog !). C. whampoensis. Grove M.S. — Diam. *075 mm. Surface with a distinct narrow elevated ring about § of the radius from the centre. Central space and rosette absent. Markings hexagonal, decreasing slightly towards the border; towards the centre 8 to 10, towards the border 12 in "01 mm. ; irregular on a minute round central area, elsewhere in substraight fasciculate rows, deflected slightly near the border, those in each fasciculus parallel to that at or near its middle ; apiculi minute, inserted at the border at wide intervals of -0175 mm. or more, between or upon the fasciculi. Border narrow, hyaline. — (PI. I. fig. 24.) The elevated zone, the less regular appearance of the fasciculi at their outer ends, and the distant apiculi, distinguish this species from C. subtilis and G. Rothii. Habitat. — Canton River, Whampoa (Grove !). YOL. xvi. 25/10/89 2 I 498 Proceedings of Royal Society of Edinburgh. [sess. C. odontophorus. C. ( subtilis var. V) odontophorus, Grun., Denk. Wien. Ah., 1884, p. 82, pi, iii. (C), fig. 24. — Diam. *05 to *175 mm. Surface slightly concave at the centre, and convex towards the border. Central space and rosette absent. Markings polygonal, 6 in *01 mm.; punctiform on a distinct narrow zone extending outwards from the apiculi, thence gradually becoming more delicate and indistinct to the border ; rows irregularly fasciculate, those in each fasciculus subparallel to that at its centre, or to a row near one of its edges ; secondary oblique decussating rows distinct, most evident on the narrow zone at the apiculi ; apiculi prominent, inserted at unequal intervals upon or between the outer ends of the fasciculi, at a considerable distance from the border. Border distinct, narrow ; striae delicate, 8 in *01 mm. Distinguished from C. subtilis by the markings and apiculi. Habitat. — California deposits (Grunow) ; “ chalk beds between White Plains and Hot Spring Station,” California (Grove ! Kinker !). C. glacialis. C. ( subtilis , var.1?) glacialis , Grun., Denk. Wien. Ak., 1884, p. 82, pi. iii. (C), fig. 27.- — Diam. *0215 mm. Surface slightly convex. Central space and rosette absent. Markings poly- gonal, 15 to 16 in *01 mm., on a small rounded central area, irregular, elsewhere forming 8 broad radiating fasciculi ; apiculi at border minute, interfasciculate. Habitat. — Under side of iceberg, lat. 74° 48' 4” U., long. 54° 52' 8” E., Aug. 1872 (Grunow). C. poly acanthus. Grun., Kong. Sv. Vet.-Ak. Handl. Stockh ., 1880, Ho. 2, p. 112, pi. vii. fig. 127.-- Diam. *026 mm. Central space absent. Markings polygonal, minute, irregular on a small indis- tinct central area, elsewhere in fasciculate rows, the rows composing each fasciculus 12, parallel to that at the middle, 15 to 16 in *01 mm. ; secondary oblique decussating rows faint, apiculi distinct, numerous, about 5 in *01 mm., placed upon and between the outer ends of the fasciculi. Border narrow, hyaline. — Odontodiscus poly acanthus, Grun., ibid., 1880, p. 112. Distinguished from C. Rothii and C. odontodiscus by the more numerous and stronger apiculi. Habitat. — Jamal (Cleve and Grunow); Eranz Josefs Land 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 499 (Grunow); Baltic Sea (Grunow); Tindingen (Cleve !); San Benito deposit, California*? (Grunow!?). Yar. davisiana. Grun., Denk. Wien. Ak ., 1884, p. 81, pi. iii. (C), fig. 19. — Diam. *035 mm. Markings larger, hexagonal, 10 in •01 mm.; fasciculi less evident, secondary oblique rows more indis- tinct ; apiculi closer to the border, arranged in two concentric rows. Habitat. — Davis Straits, Franz Josefs Land (Grunow). Yar. intermedia. Grun., ibid., 1884, p. 81, pi. iii. (C), fig 25. — Diam. *06 mm. Markings in evident fasciculi, the secondary oblique rows distinct ; apiculi distinct, less numerous, at some distance from the border, forming a single row, and inserted at the middle of and between the fasciculi. Habitat. — Cape Wankarema, N. Siberia (Grunow). Var. baltica. Grun., ibid., 1880, No. 2, p. 112. — C. polyacan- tlius, Grun., Sitzungsb natnrw. Ges. Isis, Dresden, 1878, p. 125. — Diam. -03 to '1 mm. Central space minute and irregular, some- times distinct, about of diam. broad, having minute isolated granules. Markings somewhat larger, rows fasciculate, 12 to 14 in *01 mm., straight and regular; apiculi some distance from the border, of variable size, especially among larger specimens, arranged usually in two subconcentric indistinct rows. — Cleve and Moller, Diat., No. 237; C. poly acanthus, var.? baltica, Grun., ibid., 1884, p. 81, pi. iii. (C), figs. 17a, 5; C. balticus, Grun., in Cleve’s Coll. Though this var. was originally named C. poly acanthus in 1878, it remained imperfectly defined, C. poly acanthus being sub- sequently diagnosed from Jamal valves ; the original C. poly acanthus being at the same time (1880) reduced to var. baltica. Habitat. — Waxholm, Hernosand; Rathan, Baltic Sea (Cleve! Grunow); Baltic bottom clay, black clay from Roslagen, silt from Ronneby, Dalaro, Furusund, Norrteljie (Juhlin-Dannfelt). C. divisus. Grun., Sitzungsb. naturw. Ges. Isis, Dresden, 1878, p. 125. — Diam. -08 mm. Central space indistinct, rounded, with numerous isolated granules, about ^ of diam. broad. Markings round, granular about the central space, elsewhere polygonal; about 10 in -01 mm., decreasing slightly outwards, on a distinct band 500 Proceedings of Royal Society of Edinburgh. [sess. around the border, minute, 15 to 16 in *01 mm. ; rows fasciculate, the fasciculi consisting of 8 to 10 rows, their sides almost straight or more curved ; apiculi interfasciculate, minute, inserted at inner edge of marginal hand. — C. ( curvatulus , var.?) divisus , Grun., Denk. Wien. Ah ., 1884, p. 83, pi. iv. (D), fig. 16). In 1878 Grunow distinguished as var. arcuata specimens with the edges of the fasciculi somewhat bent, those of the type being straight. Habitat. — Peruvian guano, on Macrocystis and Lessonia from the coast of Peru (Grunow). G. Normani , Greg., in Grev., Quart. Jour. Micr. Sci ., 1859, p. 80, pi. vi. fig. 3. — Diam. *0625 to *1125 mm. Surface slightly con- vex. Central space and rosette absent. Markings polygonal, about 8 in '01 mm., decreasing somewhat towards the border; rows radial, fasciculate, converging slightly towards the periphery, towards the border 6 rows composing each fasciculus ; apiculi absent or minute. Border with delicate striae of closely placed punctiform markings. — Janisch, Gazelle Exped., taf. v. fig. 6 ; G. fasciculatus ; Sch.,* Jahresb. d. Kom. z. Untersuch. d. deutsch. Meter., Kiel , 1874, ii. p. 94 ; Sch., Atl., pi. lvii. figs. 9, 10; Odontodiscus subtilis , Grun., in Sch., ibid., 1874, p. 95 ; C. subtilis, var. Normanii , Van Heurck, Syn. Diat. Belg ., p. 218; C. normanicus , Van Heurck, ibid., Ex- plan. pi. cxxxi. fig. 1; in Van Heurck, Typ. Syn. Diat. Belg., Ho. 532 ; G. subtilis, Ehrb., Eul. Diat. Sp. Typ., No. 115 (fide Van Heurck); C. curvatulus, var. Normanii, Cleve, Vega Exped. Vetensk. Jakttag. Stockh., 1883, Bd. iii. p. 488. An undulation of the surface, resulting from greater prominence of the lines between the fasciculi, as referred to by Greville, is not constant. Well-preserved specimens of G. fasciculatus , Sch., show indications of minute processes. Habitat. — Ex Ascidiis, Hull (Norman) ; Boundstone Bay, co. Galway (O’Meara) ; Dutch rushes from Holland (?) ; Arran Islands, co. Galway (O’Meara); Cuxhaven (Schmidt); marshy ground, Wedel (Grunow) ; locality? (Kinker !); Richmond tunnel (O’Meara !); Hol- stein (Van Heurck !); [Richmond, Va. (Rae !) ; Californian guano (Norman !);+ Arafura Sea, H.M.S. Challenger (Doeg !). * Not C. fasciculatus, O’Me., Quart. Jour. Micr. Sci., 1867, p. 245, pi. vii. fig. 1. t In the collection of Dr Greville. 1888-89.] Mr John Rattray on the Genies Coscino discus. 501 C. margihulatus, Rattray. C. marginulatus , var. gallopagensis, Gran, ; Yan Heurck, Syn. Died. Belg., pi. xciv. fig. 30. — Diam. •0315 mm. Central space and rosette absent. Markings puncti- form ; rows straight, inconspicuous, fasciculate ; those in each fasciculus parallel to the central row; at intervals evident, radial striae not quite reaching the centre ; apiculi at the border minute, at somewhat regular intervals. Border broad, striae evident, 8 in •01 mm. Habitat. — Galapagos Islands (Van Heurck). Var. curvato -striata, Grun. Sch., Atl ., pi. lvii. fig. 5. — Diam. •045 to *05 mm. Central space -Jg- to ^ of diam. broad, hyaline. Fasciculi numerous, almost straight or curved ; apiculi minute, sometimes hardly visible. Border striae 8 to 10 in *01 mm. — Yan Heurck, Syn. Diat. Belg., pi. xciv. fig. 32. In Moron (?) valves, in Weissflog’s collection, the striae on the border are still more delicate, 12 to 14 in *01 mm. - Habitat. — Campeachy Bay (Weissflog! Yan Heurck) ; Moron (?) deposit (Weissflog?). Yar. stellulifera, Grun. Yan Heurck, ibid., pi. xciv. fig. 34. — Diam. ’02 mm. Central space as in var. curvato-striata. Fasciculi separated around the central space by short straight lines, indistinct on the outer half; apiculi less evident. Border striae more delicate. Habitat. — Campeachy Bay (Yon Heurck). Var. sparsa , Grun. Yan Heurck, ibid., pi. xciv. fig. 31. — Diam. •035 mm. Central space absent. Markings irregular; apiculi absent. Border striae distinct. The var. campechiana, Grun. (Yan Heurck, ibid., pi. xciv. fig. 33), from Campeachy Bay, differs only in having evident apiculi at the border. Habitat. — Campeachy Bay (Yan Heurck). C. angulatus, Grev., Trans. Micr. Soc. Bond., 1864, p. 9, pi. ii. fig. 11. — Diam. *075 mm. Surface flat, showing an octagonal figure at inner edge of border. Central space and rosette absent. Markings polygonal, 3J to 4 in -01 mm., slightly smaller at the border ; the rows straight parallel to those passing from the centre to the apiculi, secondary oblique rows less evident. Apiculi 502 Proceedings of Royal Society of Edinburgh. [sess. obvious, placed at the ang'les of the octagon. Border with its inner edge distinct, about of radius broad, striae evident, 4 to 5 in *01 mm. Habitat . — Cambridge deposit, Barbados (Cleve, Greville !) ; Oamaru deposit (Grove !). C. Rothii. Grun., Denk. Wien. Ah, 1884, p. 29, pi. iii. (C), figs. 20a, 20 b, 22. — Diam. -07 to -175 mm. Surface with faint undula- tions. Central space absent. Markings polygonal, 6 to 8 in *01 mm., decreasing slightly towards the border, irregular on a small central area, elsewhere in straight fasciculate rows, subparallel to that at centre of each fasciculus- or subradial; apiculi small, placed at the middle of the outer ends of each fasciculus. Border distinct, with uniform strise, 14 in *01 mm. — Cleve and Moll., Diat ., No. 57 ; C. Rothii forma minor , Grun. : Van Heurck, Typ. Syn. Diat. Belg ., No. 533; C. symmetricus, Kitton and Weissflog (not Grev.), Sell., Atl., pi. lvii. figs. 25, 26, 27 ; Heterostepliania Rothii (a) octonaria , Ehrb., Mikrog ., pi. xxxv. a, 13 b. fig. 4 ; II. Rothii (/?) denar ia, Ehrb., ibid., pi. xxv. a, 13b. fig. 5; II. Rothii, Balfs in Pritch. Inf., p. 833., pi. v. fig. 33. Ehrenberg established but did not define Heterostephania, of which the only known species was H. Rothii ; his forms octonaria and denaria founded only on the number of fasciculi may be abandoned. C. Rothii is sometimes distinguished from C. subtilis by the smaller number of rows in each fasciculus, and especially by the position of the apiculi. C. Rothii forma minor , Grun., differs only in its small size, *025 to *0375 mm. in diam. Habitat. — Elbe Tertiary mud (Ehrenberg).* Ceylon (Macrae !);f Caspian Sea (Grunow) ; locality1? (Bae!). Porto Seguro (Hardman !); Manilla (Firth !) ; Amboina shell scrapings (Kinker !) ; surface, Arafura Sea, H.M.S. Challenger (Rae !) ; Oamaru deposit (Mar- shall!) Chalky Mt., Barbados (Firth !) Para River, S. America (Hardman !);§ India (Macrae!) ;|| Antwerp (Van Heurck!); Oamaru deposit (Grove !) ; Whampoa (Grove !) ; rice fields, Georgia (Gre- ville!); IF Cambridge deposit, Barbadoes (Johnson !); || Rio Janeiro * This diluvial formation was discovered by Roth. + In the collection of Dr Greville. + In the collection of Dr F. W. Griffin. § In the collection of Julien Deby. || In the collection of Dr Greville. IT Rare, amongst abundance of C. subtilis. 1888—89.] Mr John Rattray on the Genus Coscinodiscus. 503 (Weissflog !) ; Curtis Straits (Roberts !) ; Richmond (Cleve and Moller !) ; Successful Bay, Kerguelen (Cleve !) ; Virgin Islands (Cleve !). Var. singciporensis, nov. — Diam. *085 mm. Central rosette evident, large. Markings 4 in ’01 mm. Adjacent to the border a sharply defined broad band with smaller markings, 6 in *01 mm. Apiculi large, with a median constriction and rounded extremities, inserted at inner edge of the marginal band. Border narrow, striae 6 in *01 mm. Habitat. — Singapore (Schmidt). Var. adinocycloides. C. actinocycloides, Pant., Fossil. Bacil. Ung., p. 71, pi. ix. fig, 72. — Diam. *075 to *1125 mm. Surface flat towards centre, slightly convex towards the border. Central space small, subcircular, punctate. Markings 6 towards the border, gradually diminishing to 8 in '01 mm.; rows parallel to that at centre of each fasciculus, secondary oblique decussating rows straight, the fasciculi separated by evident rows of small subulate hyaline interspaces ; apiculi distinct. Habitat — Kekko, Szakal, Felso-Esztergaly deposits (Pantocsek !) ; K6kko deposit (Grove !); Szent Peter deposit (Grove !). Var. grandiuscula , nov. Sp. n A Sch., Atl., pi. lvii. fig. 23. — Diam. -04 mm. Markings 6 in *01 mm. Apiculi prominent, placed at a considerable distance from the border. Habitat. — Rio de Janeiro (Schmidt). C. doljensis. Pant., Fossil. Bacil. Ung., p.72, pi. xii. p. 10 5. — Diam. -036 to *1 mm. Surface slightly convex towards the border. Central space minute, indefinite, with minute isolated or subangular granules. Markings delicate, 12 to 15 in ’01 mm,, somewhat less crowded towards the centre, towards the border punctiform ; rows radial and subparallel, obscurely fasciculate ; minute subulate hyaline spaces opposite origin of the shorter rows, on a distinct band adjacent to the border, the oblique decussating rows more manifest. Apiculi prominent at internals of '006 to -01 mm. Border narrow, hyaline. Habitat. — Dolje Klebschiefer (Pantocsek !). 504 Proceedings of Royal Society of Edinburgh. [sess. C. barbadensis. Grev., Trans. Micr. Soc. Land ., 1861, p. 43, pi. iv. fig. 9. — Diam. *035 mm. Surface fiat. Central space absent. Markings polygonal, 8 in *01 mm., 9 subsymmetrical prominent rows proceeding from the centre to the border, the intervening rows subradial. This species, like G. senarius , Sch. (. Atl. , pi. lvii. fig. 24), forms the transition to AulacoUiscus. In the second edition of Habirshaw’s Catalogue, the unnamed Springfield valves figured by Schmidt (. Atl ., pi. lvii. fig. 32), subsequently justly separated by Grunow as C. semipennatus ( Denk . Wien. Ak., 1884, p. 83), are erroneously associated with this species. Habitat. — Barbados deposit (Greville !). C. Gregorii. O’Me., Proc. Roy. Ir. Acad., 1875, p. 263, pi. xxvi. fig. 23. — Central space small, angular. Markings subquadrangular, smaller and more equal than in C. nitidus, Greg.; rows radial from the angles of the central space, fasciculate ; those in each fasciculus parallel to that at its centre or subradial. This is not Carnpylo discus ? an C oscinodiscus ? Greg., from Glen- shira Sand (Trans. Micr. Soc. Lond., 1857, p. 84, pi. i. fig. 50), as stated by O’Meara, Gregory’s specimens being devoid of a central space, and having large rounded sparsely placed markings in rows partly parallel and partly radial, among the rows a broad rectangular cross being faintly visible. The C. semipennatus , Grun. (Sch., Atl., pi. lvii. figs. 32, 32*), from Barbados, is not so close to C. Gregorii, O’Me., as it is to Gregory’s valves. C. Gregorii differs from C. senarius , Sch., in the presence of a central space, and in having less evident rays between the fasciculi. Habitat. — Arran Island; Ascidia, Roundstone Bay, co. Galway; Ascidia, co. Clare (O’Meara). C. denarius. Sch., Atl., pi. lvii. figs. 19, 20, 21. — Diam. ’053 to *0755 mm. Central space absent. Markings polygonal, equal, 3|- to 4 in *01 mm., rows fasciculate, those of each fasciculus parallel to the radial row at its centre, secondary straight oblique decussating rows obvious. Border strise sometimes distinct, 8 in *01 mm. — C. * In the collection of Dr Griffin, and procured in the original sample sent to Firth by Kitton. 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 505 denarius , var., Sch., Atl., pi. lvii. fig. 22 .; C. (symmetricus, var.) denarius , Sch., Cleve and Gran, in Moll., Diat ., No. 183. The fasciculi sometimes do not reach the centre, because of the extension inwards of the rows belonging to the adjacent fasciculi. Distinguished from C. subtilis by the larger more uniform markings. Habitat. — Springfield deposit, Barbados (Doeg ! Schmidt !) ; Campeachy Bank, Sansego (Schmidt); Cambridge depo.sit, Barbados (Greville !); Chalky Mount, Barbados (Firth !); * Antarctic Ocean (Cleve and Moller !). Var. variolata. C. variolatus, Cstr., Diat. Chall. Exped., p. 155, pi. ii. fig. 5. — Diam. *068 mm. Surface spotted at wide irregular intervals with small groups of more prominent granules. Border narrow, hyaline. Habitat. — Phillipine Islands, H.M.S. Challenger (Castracane). Coscinodiscus ? Ehrb., Abh. Ber. Ak ., 1871, p. 140, pi. i. (B), fig. 20. — A minute fragment showing closely placed angular markings in straight fasciculate rows, which are parallel to that at the centre of the fasciculus. Border hyaline, distinct. In the second edition of Habirshaw’s Catalogue, this is associated with a valve indicated as Coscinodiscus , sp.1 Sell., Jahresb. d. Kom. z. TJntersucli. d. deutsch. Meer, Kiel , 1874, p. 95, pi. iii. fig. 42), but it is more nearly allied to C. denarius , var., Sch. (Atl., pi. lvii. fig. 19), from Sansego. C. senarius. Sch., Atl., pi. lvii. fig. 24. — Diam. '04 mm. Central space absent. Markings polygonal, equal, 4 in *01 mm.; rows fasciculate, those composing each fasciculus parallel to that at its middle, the interfasciculate rows most prominent; secondary oblique decussating rows straight, non-apiculate. Border narrow, striae delicate, 12 to 14 in '01 mm. — Janisch, Gazelle Exped., taf. vi. fig. 5. Habitat. — Springfield deposit, Barbados (Schmidt). C. partitus. Grove and Sturt MS. — Diam. -05 mm. Central space minute, rosette absent. Markings subactinocycloid, rounded, granular, towards centre 6, towards border more crowded, sub- punctiform, 8 in '01 mm.; rows straight, fasciculate, inconspicuous, 506 Proceedings of Royal Society of Edinburgh. [sess. those in each fasciculus parallel to the central row ; interfascicnlate radial rows most evident, secondary oblique decussating rows most obvious towards border; interspaces minute, most evident towards centre ; apiculi distinct, inserted a short distance from border, inteifasciculate. Border narrow, striae delicate, 10 in *01 mm. — Cleve and Moll, Dial , Nos. 114, 162.— (PI. III. fig. 5.) Habitat. — Totara, Oamaru (Grove !) ; Mascara, Nancoori (Cleve and Moller !). C. extravagans. Sch., Atl ., pi. lviii. fig. 33. — Diam. *053 mm. Central space distinct, circular, about -Jg- of diam. broad. Markings small, granular, about 6 in *01 mm., more crowded, somewhat smaller on a distinct marginal zone about i to i of radius broad ; rows radial, fasciculate, those between the fasciculi most prominent ; those composing each fasciculus parallel to the radial row at its middle, secondary oblique rows evident on the marginal zone ; interspaces hyaline at the inner ends of the shorter rows ; apiculi large, conical, interfasciculate, inserted at inner edge of marginal zone. Border hyaline. Habitat. — Yokohama (Schmidt). C. interlineatus , sp. n. — Diam. *06 mm. Surface flat. Central space and rosette absent. Markings hexagonal, 8 to 10 in *01 mm., somewhat smaller towards the border, rows fasciculate, those in each fasciculus parallel to that at its middle; secondary oblique decussating rows straight, slightly flexuous or concave outwards, obvious ; between each fasciculus a distinct radial row, the fasciculi 7, unequal ; apiculi evident, interfasciculate, inserted at the border. Border indistinct. — (PI. I. fig. 6.) Differs from C. senarius, Sch., by the more delicate markings and apiculi. Habitat. — Newcastle deposit, Barbados (Weissflog !) ; Nancoori deposit (Cleve !). C. actinosiis , Grove MS. — Diam. -06 mm. Surface slightly convex near the border. Central space inconspicuous, round, with rounded granules at its middle. Markings actinocycloid, round, granular, faint ; towards the border angular and in contact, towards the centre 8, towards the border 10 in ‘01 mm.; interspaces small and 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 507 hyaline towards the centre, most evident opposite origin of shorter rows; rows fasciculate, straight, those in each fasciculus parallel to that at its centre, interfasciculate radial rows evident ; secondary oblique decussating rows most evident towards the border. Border distinct, striae delicate, 14 to 16 in '01 mm. — (PI. II. fig. 7.) Habitat. — Manilla Algae (Grove !). C. obnubilus , Rattray. C. umbonatus ,* Cstr., Diat., Cliall. Exped ., p. 156, pi. ii. fig. 8. — Diam. '077 mm. Surface rising steeply from the centre for about J of radius, thence descending rapidly outwards, becoming slightly concave, near the border flat. Central space subcircular, about of diameter broad, with several isolated granules at its middle. Markings punctiform, subequal, about 8 in *01 mm., most crowded towards the border ; rows fasciculate, those composing each fasciculus parallel to the radial rows at one of its sides or subradial ; interspaces small, hyaline, opposite the origin of the shorter rows ; apiculi distinct, inserted at the border, interfasciculate. Border distinct, narrow. Habitat. — Pacific Ocean, 2900 fms., H.M.S. Challenger (Castracane). § YI. Radi ati, Grun., Denlc. Wien. Ah., 1884, p. 70; Pant., Fossil. Bacil. Ung., p. 69. Pseudostephanodiscus, Grun., ibid., p. 85. Clivosi , Pant., ibid., p. 72. Eleganti, Pant., ibid., p. 73. Surface flat, rarely undulate, centre sometimes depressed. Markings rounded or areolate; rows radial, sometimes faintly fasciculate towards the border ; apiculi sometimes present. C. diversus. Grun., Denh. Wien. Ah., 1884, p. 72. — Diam. '07 to '135 mm. Central space absent. Markings rounded, pearly, with hyaline interspaces from the centre for \ to f of radius, increasing slightly outwards ; on the outer portion polygonal, 2| to 3, at the border 6 in '01 mm. ; central papillae distinct; secondary oblique rows indistinct. Border with inner edge indistinct ; striae obvious, radial or subradial, about 6 in '01 mm. — Sch., Atl., pi. lxii. figs. 13-15 (without name); C. caraibicus, Tru. and Witt., Jeremie Diat., p. 13, pi. ii. fig. 3. * Name preoccupied by Gregory for a different species. 508 Proceedings of Royal Society of Edinburgh. [sess. Grunow considers that this may he an abnormal1 form of C. radi- atus. To me it seems to be more allied to C. marginatus. Habitat. — Springfield deposit, Barbados, and Cambridge deposit, Barbados (Schmidt, Hardman!);* “Barbados deposit” (Rae ! Greville !). Yar. completa. — Diam. T125 to T4 mm. Central space small, angular, *0035 to '005 mm. broad ; the bounding areolae incon- spicuous. Markings polygonal and in contact to the central space, towards the centre 3|-, increasing outwards to 2J in ’01 mm. at about J of the radius, again decreasing to the border, punctate ; secondary oblique rows obvious. Habitat. — -Barbados deposit (Rae !). C. profundus. Ehrb., Mon. Ber. Ak ., 1854, p. 238. — Diam. h Central space and rosette absent. Markings somewhat larger at centre, about semiradius 6J to 7 in *01 mm., near border smaller and more crowded ; interspaces distinct, most evident opposite the shorter rows, at about -| of radius from centre. — Ehrb., Mikrog ., pi. xxxv. B.b. fig. 8 ; Ralfs in Pritch. Inf., p. 830. Ehrenberg’s figure shows the markings as subcircular and as decreasing gradually from the centre outwards, but more rapidly near the border. Habitat. — Atlantic Ocean, 2000 fms.; lat. 62° 6' N., long. 32° 21' W., 1540 fms.; lat. 59° 12' K, long. 50° 38' W.; lat. 58° 3' 1ST., long. 51° 50' W.;f northern and equatorial zone, 16 to over 2000 fms. (Ehrenberg). J C. antarcticus. C. ( subglobosus , var.1) antardicus, Grun., Denk. Wien. Ak ., 1884, p. 84, [pi. iv. (D), fig. 23. — Diam. ’03 mm. Central space and rosette absent. Markings irregular, polygonal, increasing from the centre to about the semiradius, again diminish- ing towards the border ; at the centre 8, at the semiradius 4, towards the border 8 in ’01 mm.; rows indistinct; on outer half of valve inconspicuous, irregular, concentric bands ; apiculi numerous, in- serted close to the border, Border with inner edge indistinct ; striae irregular, 10 to 12 in ’01 mm. * In the collection of Dr Greville. t Mon. Ber. ATc., 1861, p. 280. t Abh. Ber Ak., 1872, p. 263. 1888-89.] Mr John Rattray on the Genus Coseinodiscus. 509 The valve named C. decipiens , Gran. (Sch., Atl., pi. lix. fig. 18), from Table Bay, is distinct, though Grunow has proposed to unite them ; in the latter the markings are largest at the centre, and the apiculi are more prominent. Habitat. — Antarctic, Kerguelen (Grunow). C. lanceolatus. Cstr., Diat. Chall. Exped ., p. 164, pi. xvih fig. 19. — Elliptical to subdiamond-shaped. Major axis *0775 mm., about 2J times minor. Central space and rosette absent. Markings polygonal, largest and subequal on a small indefinite central area, thence decreasing to the border; at the centre 3, at the border 6 in *01 mm.; irregular or in faint radial rows. This species forms a transition to the untenable genus Stoschia. Habitat. — South of Australia, H.M.S. Challenger (Castracane). C. velatus. Ehrb., Mon. Ber. Ah ., 1844, p. 78. — Diam. about •055 mm. Central space and rosette absent. Markings angular, robust, pearly, about 2J in *01 mm., punctate; rows obscurely radial, subregularly concentric. — Ehrb., Mihrog ., pi. xviii. fig. 37 ; Ralfs in Pritch. Inf., p. 830. Ehrenberg regarded this species as probably belonging to Eupo- discus. With this, owing to the absence of processes, it cannot be united. It is closely allied to Steplianopyxis. It also approaches C. marginatus. Habitat. — Richmond deposit, Va. (Ehrenberg). O. marginatus. Ehrb., Abli. Ber. Ah., 1841, p. 142. — Diam. from *0375 to T5 mm. Central space absent. Markings polygonal, somewhat pearly, with large, round central papillae ; towards the centre 2 to 2J in *01 mm., decreasing gradually towards the border. Border distinct, *0025 to *0075 mm. broad, with coarse striae, 4 in *01 mm. — Ehrb., Mihrog., pi. xviii. fig. 44; pi. xxxiii. 12. fig. 13; pi. xxxviii. B. 22. fig. 8; Cleve and Moll., Diat., Ho. 114, 164, 215 ; H. L. Sm., Diat. Sp. Typ., Nos. 94, 95; Sch., Atl., pi. lxii. figs. 1, 2, 3, 4, 5, 9, 11, 12 ; pi. lix. fig. 11 (no name) ; C. fimbriatus limbatus ,* Ehrb., Mihrog., pi. xix. fig. 4 ; Sch., Atl., pi. lxv. figs. 3-6 ; pi. cxiii. fig. 2; C. limbatus, Ehrb., Mon. Ber. Ah., 1840, p. 206; * Quoted “ C. fimbriato-limbatus by Grunow, Denk. Wien. Ah., 1884, p. 72. 510 Proceedings of Royal Society of Edinburgh. [sess. Mikrog pi. xx. figs. 29a, b; Sch., Atl., pi. lxv. fig. 7;* Raben., Alg. Europ., Nos. 2484, 2485 ; 0. radicdus f. heterostida, Gran, in Pant., Fossil. Bacil. TJng ., p. 70, pi. xx. fig. 184; C. oculus-iridis, Sch. in Atl., pi. cxiii. fig. 2 ; 0. subconcavus forma major , Sch., Atl., pi. lxii. fig. 7 ;f (excl. C. limbatus, Jan. et Raben. — Raben., Beitr. Kennt. u. Verbreit. Alg., Leipz., 1863, p. 7, pi. iv. fig. 1; and C. marginatus, Kiitz., Bacil., p. 131, pi. i. fig. 7), Hardman’s original Monterey specimen, on which C. robustus, Grev., was founded, is not in the Grevillean Collection of the British Museum, but two specimens on a slide of Bermuda tripoli, labelled by Greville “ C. robustus ,” and now in this collection, belong to C. marginatus. The C. limbatus, Jan. et Raben., has a central space, markings increasing to about the semiradius, and again decreasing to the border. Forms occur in Cambridge deposit, Barbados, similar to that figured by Schmidt (Atl., pi. lxv. fig. 7). Stokes has labelled specimens of this species C. ambiguus. Schmidt is only prohibited from uniting the specimens figured on pi. lxii. figs. 11, 12, with C. velatus by the small size of the mark- ings and the absence of the fine puncta which, according to Grunow, cover the surface of that species. Habitat. — Richmond, Va.; tripoli from Columbia River Oregon, Patagonian tufa ; plastic clay, Aegina (Ehrenberg, Schmidt); Notting- ham, U.S. (Hardman oa San Diego, San Pedro (H. L. Smith !) ; Bajtha, Elesd Also-, Felso-Esztergally, Kekko, Mogyorod, Szakal, and Dolje deposits (Pantocseck !); sounding from 2950 fms., H.M.S. Challenger (Rae !) ; “California” (Deby ! Cleve!); Cambridge de- posit, Barbados (Hardman !) ;t Monterey (Stokes ! § Firth ! Greville ! Cleve !) ; Sta Barbara deposit (Kinker !) ; Moron deposit (Kinker !); Faeroe Channel (Grove !) ; Nagy-Kurtos deposit, Hungary (Rae ! Deby !) ; Nancoori (Hardman !) ; soundings off Kurile Islands, 1329 fms. (H. L. Smith!); Behring Sea, 1681 fms. (H. L. Smith!); Los Angelos (Hardman ! Cambridge !) ; || Oamaru deposit (Grove !) ; * Quoted “ 0. fimbriatus , Sch. {nee. Ehrb.)” by Grunow, ibid., p. 72. t Grunow ( DenTc . Wien. AJc., 1864, p. 72) refers to this simply as C. subconcavus , Grun., and proposes to name it C. marginatus, var. subconcava, or better, to unite it to C. robustus , Grev., the markings towards the centre being 2, towards the border 3 to 3| in ‘01 mm. The central papillae are prominent, as in C. robustus. 4 In the collection of Julien Deby. § In the collection of E. O’Meara. || In the collection of Dr Griffin. 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 511 “Barbados” (Johnson! Cleve!); Rappahannock River, Va. (Rogers !) ; * Kekko and Sta Maria deposits (Grove !) ; Szent Peter deposit (Pantocsek ! Grove !) ; San Benito deposit, California (Grove!); Kamtschatka Sea, 1700 fms. (Greville ! Bailey!); Atlantic Telegraph soundings (Roper !) ; * Piscataway deposit (Greville !) ; Santa Monica deposits (Cleve and Moller ! Firth !) ;* King’s Mill (Schmidt) ; Nicobar Islands (Schmidt) ; Mascara (Cleve and Moller !) ; Nottingham deposit, Md. (Cleve and Moller!); Briinn Tegel (Cleve !) ; Holland’s Cliff (Cleve !) ; anchor ground, Laguna Harbour, 20 miles N. of Laguna in the sea (Rahenhorst and Schwarz !) ; Marstrand (Kinker !). Yar. decussata, nov. — Diam. T15 mm. Markings 3 in '01 mm., subequal to the zone at the border, the secondary oblique decussating rows obvious ; radial rows not differentiated. Border more sharply defined. Habitat. — Roundstone Bay, Ireland (O’Meara !). Yar. latemarginata. Pant., Fossil. Baeil. Ung ., p. 70, pi. xxii. fig. 201. — Diam. *057 mm. Markings subequal, 3 in *01 mm. Border sharply defined. Striae more distant. Habitat. — Elesd deposit, Hungary (Pantocsek !). Yar. intermedia , Rattray. C. robustus , var. intermedia , Grun., Denk. Wien. Ah., 1884, p. 72. — Diam. T65 mm. Markings increasing slightly outwards, those at the centre somewhat larger than those on the adjoining area; near the centre 2, towards the border 1J, at the border 3 to 3J in ’01 mm. — G. robustus , Sch. (not Grev.), Atl., pi. lxii. fig. 6. C. robustus. Grev., Trans .Micr. Soc., Lond., 1866, p. 3, pi. i. fig. 8. — Diam. from ’0825 to *325 mm. Surface slightly convex towards centre. Central space and rosette absent. Markings pearly, 1J to 2 in ’01 mm., suhequal for about § of radius, smaller towards the border, at intervals smaller areolae sometimes distinct among the larger, the central papillae prominent ; radial rows inconspicuous, sometimes secondary, suhconcentric or short oblique decussating rows visible. Border prominent, sharply defined, from to 2V °f radius broad ; striae evident, 4 to 6 in '01 mm. — Sch., * In the collection of Dr Greville. 512 Proceedings of Royal Society of Edinburgh. [sess. Atl ., pi. lxii. figs. 16, 17 ; Grun. Denk. Wien. Ak ., 1884, p. 72; H. L. Sm., Diat. Spec. Typ ., No. 99 ; Janisch, Gazelle Exped ., taf. iv. figs. 10, 11 ; (7. marginatus , var. submarginata, Grun,, ibid ., p. 72; (7. subvelatus, Grun. — Sell., AiZ., pi. lxv. fig. 9; C. linker - ianus , Tru. and Witt, Jeremie Diat., p. 13, pi. iii. fig. 1. In a Santa Monica form *13 mm. in diam., discovered by Dr Rae, the usual striated border was surrounded by a second more sharply defined but narrower band, with a slightly convex surface, and bearing delicate striae, 8 to 10 in '01 mm.; at one place this band is interrupted and somewhat more convex on the two sides of the break. This gives it the appearance of an elastic spring envelop- ing the valve. Habitat. — Santa Monica deposit (Kinker ! Hardman ! Weissflog ! Rae ! Firth !) ; Santa Maria deposit (Rae ! Grove !) ; Santa Marta deposit (Doeg !) ; Nagy-Kurtos deposit, Hungary (Rae ! Deby !) ; Monterey (Weissflog! Hardman!); Mejillones (O’Meara! Hardman!); Jeremie deposit (Truan and Witt); Los Angelos (O’Meara !); Japan (H. L. Smith!); Sea of Kamtschatka, 1700 fms. (Bailey!); Briinn Tegel (Cleve) ; San Pedro (Grove !). Var. kittoniana , nov. — Diam. T125 to *225 mm. Markings 1J to If in '01 mm., central papillae prominent, forming transversely truncated cones, with finely but evidently and closely furrowed sides. Habitat. — Holothurians, China (Macrae !). Var. fragilis , nov. — Diam. J875 mm. Markings more minute ; at the centre 2J, at the border 3, in -01 mm.; adjacent to border a single band of markings elongated radially ; central papillae more faint, puncta distinct ; oblique decussating rows more evident. Habitat. — Santa Maria deposit (Rae !). C. implicatus , sp. n. — Diam. T5 to -25 mm. Surface somewhat convex towards the centre. Central space and rosette absent. Mark- ings hexagonal, decreasing but very slightly outwards ; at the centre 3J, towards the border 4J in ,01 mm.; rows irregular, oblique, straight or curved, forming short, inconspicuous broad fasciculi, that are interrupted by those meeting them at variable angles. 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 513 Border sharply defined, to of radius hroad ; striae distinct, 4 in *01 mm.— (PI. III. fig. 1.) This species does not strictly belong either to the Radiati or to the Fasciculati. It is placed here for convenience, since it approaches C. robustus in general appearance. Habitat.— Sta Maria deposit (Rae !) ; Sta Marta deposit (Doeg !). Yar . pidurata, nov. — Diam. *3 mm. Markings angular or subro- tund ; surface mottled with dark, mostly quadrangular spots, which are somewhat more crowded on the flattened central portion than towards the border. Border striae 6 in *01 mm. — (PI. III. fig. 11.) Habitat. — Saida Monica deposit (Thum !).* C. glaberrimus , sp. n. — Diam. *1 to *1075 mm. Surface flat from the centre to about semiradius, thence sloping somewhat steeply to the border. Central space and rosette absent. Markings polygonal, pearly, suhequal to semiradius, largest about § of radius from the centre, thence decreasing to the border, towards the centre 3J, at -J of radius 3, in *01 mm.; secondary oblique decussating rows faint. Border hroad, with inner edge indistinct; striae 4 to 5 in *01 mm. —(PI. I. fig. 19.) Distinguished from C. diversus by the presence of a central space and the polygonal outline of the markings on the central half of the valve. Habitat. — Cambridge deposit, Barbados (Rae !). C. obscurus. Sch., Atl., pi. lxi. fig. 16. — Diam. '09 to *165 mm. Central space minute, sometimes absent. Markings subpearly, with central dots evident, increasing but slightly outwards ; towards the centre 2|, at about i of radius from centre 2, at the border 4 to 5 in *01 mm.; secondary oblique rows inconspicuous, at the origin of the shorter rpws are small clear spaces, readily overlooked. Border striae coarse, 4 to 6 in *01 mm. — Grun., Denk. Wien. Ak., 1884, p. 74; Cestodiscus obscurus , Van Heurck, Syn. Diat. Belg.f pi. cxxix. fig. 4. This species is intermediate in the appearance of its markings between C. crassus, Bail., and C. radiatus, Ehrb., on the one hand, * In the collection of Jnlien Deby. VOL. XVI. 25/10/89 2 K 514 Proceedings of Royal Society of Edinburgh. [sess- and C. marginatus , Ehrb., on the other. Yan Heurck’s figure is from a photograph by Dr Woodward. Habitat. — Moron deposit (Greville ! Grunow) ; Sta Monica deposit (Rae!); Szent Peter and Dolje deposits (Pantocsek); sounding, lat. 3° 1' S., long. 33° 50' W., H.M.S. Challenger (Rae !); Yirginia (Greville !); Sta Maria deposit (Grove !). Var. minor , nov. C. obscurus, var. ? Sch., Atl., pi. lxi. figs. 17, 18. — Diam. *05 to *06 mm. Markings smaller, 3 in *01 mm. Border striae longer. — Grun., ibid., 1884, p. 74. Habitat. — Moron deposit (Greville !). C. radiatus. Ehrb., Abh. Ber. Ah., 1839, p. 148, pi. iii. figs. 1 a-c (excl. d). — Diam. *0675 to *18 mm. Central space absent. Mark- ings polygonal, 2 to 2J in *01 mm., subequal from the centre for about f of radius, thence decreasing sometimes to 6 in *01 mm., in inconspicuous radial sometimes subfasciculate rows, central dots faint. — Ehrb., Mihrog ., pi. xix. fig. 1 ; pi. xxii. fig. 3 ; pi. xxxiii. 13. figs. 2, .2*; pi. xxxiii. 16. fig. 6; pi. xxxv. a. 17. fig. 6 (excl. pi. xx. 1. fig. 27 ; pi. xxi. fig. 1); Ralfs in Pritch. Inf., p. 831, pi. xi. figs. 39, 40; Sch., Atl., pi. lx. figs. 5, 6, 9 ; pi. lxii. fig. 18; pi. lxv. fig. 8; Grun., Denk. Wien. Ah., 1884, p. 71 ; pi. iii. (c), figs. 4, 7 ; in H. L. Smith, Diat. Spec. Typ ., No. 99 ; Cleve and Moll., Diat., Nos. 57, 114, 155, 164, 207, 211, 215, 257; Raben., Alg. Europ., Nos. 2263, 2437, 2487 ; Coll., Kiitz. Diat., No. 902 ; C. caspius, Ehrb., Abh. Ber. Ah., 1872, p. 170, pi. xii. fig. 14 ; C. argus, Sch. (non Ehrb.), Atl., pi. lxi. fig. 13 (excl. C. radiatus, Weisse, Bull. Acad. Imp. St Petersb., 1868, p. 122, pi. i. fig. 25 ; and C. radiatus, Bail, Amer. Jour. Sci ., xlii. 1842, p. 95, pi. ii. fig- 14). Schumann ( Schrift . Phys. Oeh. Ges. Konigsberg, 1867, p. 61) proposed to break up this species restricting C. radiatus for those forms in which the markings are angular, and decrease from the centre outwards from about 6 to 7J in *01 mm. Other forms from the Baltic, with round markings and fine short furrows within the border, 16 J in *01 mm., he named C. vicinus ; but the definition of this last species, which is not accompanied by a figure, is inadequate, and the name may be abandoned. 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 515 His C. fallax (ibid., p. 62, pi. iii. fig. 76) with, in the dry state, round markings, between which smaller faint granules, each resolv- able under high powers into two, occur, is also C. radiatus, Ehrb. In balsam C. fallax , like C. radiatus , showed hexagonal markings, and by good illumination smaller granules at their angles. C. caspius , Ehrb., was only distinguished by having the rows inconspicuously radial and the secondary oblique curved rows evident. Oamaru specimens show transitions to C. argus , and have the rows sub fascicu- late, with sometimes a distinct central rosette. Schmidt misinterprets Ehrenberg’s C. argus , in overlooking the increase of the markings outwards ( Atl ., pi. lxi. fig. 13). Habitat. — White chalk marl, Caltanisetta; Polirschiefer, Stratford Cliff, Va.; Zante, Plattenmergel ; plastic clay, Aegina; tripoli, San Francisco, Cal. (Ehrenberg) ; Ichahoe guano (O’Meara !); dredged in 1319 fathoms, lat. 71° 19' N., long. 11° 28' W., in yellowish- grey mud; in 1300 fathoms, lat. 73° 16' N., long. 15° 48' W., in dark greyish-brown mud; and in 569 fathoms, lat. 63° 40' N., long. 5° 28' W., in clear grey fine mud (Ehrenberg) ; Caspian Sea, 14 to 422 fathoms; North Sea, at Cuxhaven ; Baltic, at Wismar (Ehrenberg) ; Mors deposit (Schmidt and Cleve !) ; Yera Cruz, Mexico (Ehrenberg) ; off Ascension Island, 1845 fathoms, S.S. Buccaneer (Grove!); Cambridge deposit, Barbados (Hardman!); sounding, lat. 3° 1' S., long. 33° 50' W., H.M.S. Challenger (Rae!); “Atlantic Ocean” (Schmidt); Mascara (Cleve and Moller !) ; “ Virginia ” (Hardman !) ; Kekko deposit (Grove !) ; Oamaru deposit (Grove! Doeg!); Hong Kong and Monte Gubbio (Grove!); marine deposit, Fiji Islands (Grove!); Japan (H. L. Smith!); “Barbados” (Johnson!); Newcastle deposit, Barbados (Grove !); marshy ground from Wedel (Schmidt); Aegina (Schmidt); San Benito deposit, California (Grove !) ; Balearic Islands, Sta Monica deposit, Patagonia, Delaware, North Carolina (Cleve and Moller !); Yedo, Mejillones guano, Bohuslan, mud from Elbing, West Prussia, Saldanha Bay guano, Patagonian guano, Schleswig- Holstein, Lahuan, Nancoori deposit, Cape Wankarema, between Aden and Bab-el-Mandeb (Cleve !); Grip, 70 fathoms; Kiel; Briinn Tegel (Cleve) ; anchor ground, Reikjavik, Iceland ; mud from Gliickstadt ; Elbe, above Cuxhaven (Rabenhorst and Schwarz !) ; coast of St Paul Island, South Sea (von Frauenfeld !) ; Oran marl 516 Proceedings of Royal Society of Edinburgh. [sess. (Ehrenberg, Kiitzing !) ; coasts of Britain (Grove ! Eattray !) ; Kirkwall and Eaeroe Islands (Grove !) ; Marstrand (Kinker !). Yar. subcequalis. Grun., ibid., p. 72, pi. iii. (C), fig. 3. — Diam. •12 mm. Markings as in the type, but subequal almost to the border, around which on a narrower zone they are larger than in the type. — C. radiatus, var. abyssalis , Cstr., Diat. Chall. Exped ., p. 165, pi. xxix. figs. 2, 11, 15; Sch., Atl ., pi. cxiii. fig. 15 (no name). Castracane’s var. abyssalis , which is not sufficiently characterised, is provisionally placed here from his note that the markings gradually diminish in size to the border. Habitat. — Oran, Monterey and Nancoori Island deposits (Grunow) ; Atlantic Ocean, H.M.S Challenger (Castracane) ; Monkstown, in tide pool (O’Meara !) ; Cambridge deposit, Barbados (Greville !) ; San Diego (Griindler). Yar. gladalis. Grun., ibid., Expl. pi. iii (C), fig. 1 ; C. borealis, Ehrb., Mon. Ber. Ah., 1861, p. 294 (not C. borealis, BaiL, Amer. Jour. Set., 1856, vol. xxii. p. 3). — Diam. *1 to *15 mm. Surface flat; central rosette absent. Markings subequal, 3 to 3J decreasing to 4 in *01 mm. at the border ; central papillse delicate. — C. radiatus, var. borealis, Grun., ibid., p. 72; Sch., Atl., pi. cxiii. fig. 8; C. radiatus, Sch., Atl., pi. cxiii. fig. 8. The varietal name gladalis given, by Grunow in the explanation of his plate, is better than borealis, as it avoids confusion with Bailey’s species. Habitat. — Lat. 62° 40' N., long. 29° W., 1000 fathoms ; lat. 62° 6' N., long. 32° 21' W., 1540 fathoms; lat. 59° 12' N., long. 50° 38' W., 1833 fathoms; lat. 58° 3' K., long. 51° 50' W., 1840 fathoms; lat. 60° 5' N., long. 50° 27' W., 2090 fathoms (Ehrenberg) ; Franz Josef’s Land (Grunow ! Cleve !) ; Aegina (Schmidt). Yar. media. Grun., ibid., p. 72, pi. iii. (C), fig. 2. — Diam. *075 to 44 mm. Markings 3J to 4 in '01 mm., gradually decreas- ing towards the border, where they are 6 to 6J in *01 mm. — Sch., Atl., pi. cxiii. fig. 21; C. radiatus, Sch., Atl., pi. lx. fig. 10 ; C. radio - 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 517 latus, Sch., Jahresb. d. Kom. z. Untersuch. d. deutseh. Meer , Kiel , 1874, p. 94. Schmidt, in 1878, stated that this form is the traditional type of C. radiatus , Ehrb., but in Ehrenherg’s original definition the mark- ings are given as about 2 in ’01 mm. Los Angelos specimens have been observed by Grunow occasionally to have on their valves groups of larger markings, forming 4- to 6-angled rosettes. Habitat. — Atlantic sounding for telegraph cable (Greville!); Oran, Nancoori and Los Angelos deposits (Grunow); King’s Bay, Spitzbergen, 160 fathoms (Cleve) ; Davis Straits (Cleve) ; Franz Josefs Land (Grunow); Baltic (Schumann); Gulf of California (H. L. Smith!); Sussex (Dickie!); Cambridge deposit, Barbados (Hard- man !);* Peruvian guano (Hardman !);* Lumfiord, Jutland (Hard- man !);* Nottingham, U.S. (Hardman !);* Compeachy Bay (Hard- man !); Rio Janeiro (Hardman !); Californian guano (Norman !);f Teignmouth (Arnott !);f Melville Bay, lat. 75° 27' N., long. 64° 34' W. (O’Meara !); Nottingham deposit (Hardman !);* Lamlash Bay (Dickie ! Gregory !);f Ascidia, Belfast (O’Meara !); Maryland (O’Meara !); Mejillones deposit (O’Meara !); rice fields, Georgia (Greville !); Gulf of Mexico (Schmidt); Algeria (Arnott !); Indian Ocean soundings, Capt. Pullen, 2200 fathoms (Greville ! Roper !);f shell cleanings from Singapore (Hardman !);f Bay of Bengal (Macrae !) Yar. minor. C. radiatus f. minor , Sch., Jahresb. d. Kom. z . Untersuch. d. deutseh. Meer , Kiel , 1874, p. 94, pi. iii. fig. 34. — Diam. *03 to *0525 mm. Markings 4 in *01 mm. at the centre, decreasing to 8 to 9 in *01 mm. at the border, the radiating rows less obvious. Border strise, 6 to 8 in *01 mm. — C. radiatus, var. parva, Grun., Sitzungsb. Naturw. Ges. Ids , Dresden , 1878, p. 124, pi. iv. fig. 16 ; C. devius, Sch., Atl., pi. lx. figs. 1-4 ; Van Heurck, Syn. Diat. Belg., pi. exxx. fig. 3 ; Cleve and Moller, Diat ., No. 150. Habitat. — Peruvian guano, Rio de Janeiro, Santos, Campeachy Bay, Japan, and Baku Harbour, Caspian Sea (Grunow); Hvidingsoe (Schmidt); Nancoori (Hardman!);* Manilla (Hardman!).* * In the collection of Jnlien Deby. t In the collection of Dr Greville. 518 Proceedings of Boyal Society of Edinburgh. [sess. Var. irregularis , Gran. Van Heurck, Sign. Diat. Belg., pi. cxxix. fig. 1. — Obtusely triangular, sometimes elliptical. Diam. about -105 mm. Markings 5^ to 6 in -01 mm., subequal almost to tbe border ; radial rows straight or curved, evident. This var. is distinguished from var. glacialis by the size of the markings and the arrangement of the rows. Transitional to the untenable genus Stoschia. Habitat. — Naparima deposit (Yan Heurck). Yar. crenidata , Rattray. C. radiatus , var., Wallich, Trans. Micr. Soc. Lond., 1860, p. 48, pi. ii. fig. 22. — Diam. about *025 mm. Markings sub-equal. Border crenate. Habitat. — From Salpce, Indian Ocean (Wallich). C. luctuosus , Grove MS. — Diam. *0875 to *125 mm. Surface rising gradually from centre to about semiradius, thence descending with a similar slope to border, convex. Central space and rosette absent. Markings at centre obtusely angular, soon becoming acutely angular and areolate, subequal, 3J in *01 mm.; rows straight. Border sharply defined, Jg- to -fa of radius broad ; striae obvious, 5 to 6 in *01 mm. — (Plate III. figs. 8, 9.) Habitat. — Troublesome Gully, Oamaru (Grove !). C. compositus , Rattray. Sp. n.? Sch., Atl., pi. lix. fig. 10. — Diam. *023 mm., central space and rosette absent. Markings angular, towards the centre about 6 in ’01 mm., decreasing slightly towards the border, central dots absent; rows inconspicuous, secondary oblique rows obscure. Border sharply defined, about i of radius broad ; striae evident, about 4 in ’01 mm. Habitat. — Nottingham (Schmidt). C. egregius , Rattray. Sp. nl Sch., Atl., pi. lvii. fig. 39. — Diam. •03 mm. Central space and rosette absent. Markings angular, increas- ing to about semiradius, thence decreasing gradually to the border ; towards the centre 4, at semiradius 3, towards border 3J in ‘01 mm ; central dots evident, radial rows inconspicuous, secondary curved rows evident ; a distinct sharply defined band with faint striae 6 in *01 mm. adjacent to border, prominent truncate, but small markings 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 519 (processes?) at intervals of about *01 mm. inserted at inner edge of marginal band. Border narrow, hyaline. Habitat. — Table Bay (Schmidt). C. pectinatus , Rattray. C. decipiens , Grun.; Sch., Atl., pi. lix. figs. 18, 19. — Diam. '024 to *0515 mm. Central space and rosette absent. Markings angular, subequal, or increasing somewhat to about semiradius, again decreasing to border; towards centre 4J to 5, about semiradius 3J, near border 4, in °01 mm. ; secondary oblique decussating rows manifest ; apiculi prominent, long acicular, inserted at inner edge of border, and reaching outwards to its outer edge. Border distinct; striae faint, 6 in '01 mm. The name decipiens cannot be adopted here, having been already applied to a distinct form. Habitat . — Table Bay (Schmidt). C. bulliens. Sch., Atl., pi. lxi. figs. 11, 12. — Diam. '05 to '1075 mm. Central space absent. Markings polygonal ; at the centre 2 \ to 3, increasing to the semiradius to 1J or 2, again gradually decreasing to the border to 6, in '01 mm.; the largest areolae forming a conspicuous zone. Border indistinctly defined ; striae short, incon- spicuous, 6 in *01 mm. — Cleve and Moll., Diat ., No 215. C. ebulliens, var. Cstr., Diat. Ghall. Exped ., p. 159, pi. v. fig. 1. Some small specimens from Cambridge deposit show only a single band of large markings. This species has sometimes been con- founded with C. heteroporus. Habitat. — Springfield deposit, Barbados (Schmidt, Grunow) ; Maryland (Kinker !) ; Szent Peter deposit (Pantocsek) ; Cambridge deposit, Barbados (Greville ! Johnson! Hardman!);* Oamaru deposit (Grove!); Barbados (Johnson!);! Nottingham deposit (Cleve and Moller !) ; Maryland (Cleve !). C. asperulus. Grun., Denk. Wien. Ak., 1884, p. 73. — Diam. '088 to ‘093 mm. Central space absent. Surface somewhat curvex, but with slight slope to the border. Markings polygonal ; towards the centre 3 to 3J, at the border 4, in '01 mm.; distinctly punctate. * In the collection of Julien Deby, t In the collection of Dr Greville. 520 Proceedings of Royal Society of Edinburgh. [sess. Distinguished from C. radiatus by the more convex surface and the evident punctation of the markings. Habitat. — Church Hill, Richmond (Grunow) ; Dolje deposit (Pantocsek !). C. subangidatus. Grun., Denk. Wien. Ak.} 1884, p. 73. — Outline irregular, obtusely angular. Diam. *06 mm. Surface very convex at the border. Markings polygonal; towards the centre 3, at the border 4 to 4J in *01 mm., distinctly punctate. Border striae evident, radial or oblique, 4 to 5 in *01 mm., its inner edge indistinct. Habitat. — Moron deposit (Grunow, Greville !). C. nodulifer , Janisch. Sch., Atl., pi. lix. figs. 21-23. — Diam. *065 to *1 mm. Central space and rosette absent, but one (rarely a few) evident nodules present. Markings angular, 3J to 4 in *01 mm., decreasing slightly around the border ; radial rows inconspicuous, the oblique decussating rows more distinct. Border sharply defined ; striae evident 4 to 6 in ’01 mm. — Cleve and Moll., Diat ., No. 57, 155 ; Janisch, Gazelle Exped ., taf. ii. figs. 4-5. Habitat. — Richmond, Va. Balearic Islands (Cleve and Moller !) Sta Monica deposit (Grove !) ; California, Gazelle Exped. (Schmidt) ; Islay, Peru (Kitton !) ; Isle of Muntok, near Sumatra (Grove !) ; Macassar Straits (Grove !) ; Indian Ocean sounding, Capt. Pullen, 2200 fathoms (Greville !) ; coral washings, locality % (Doeg !) ; Atlantic Ocean, lat. 3° 3' N., long. 15° W. (O’Meara!); off Ascension Island, S.S. Buccaneer (Rattray ! Grove !) ; Galapagos Island (Cleve!) Patagonian guano; between Aden and Bab-el-Mandeb (Cleve !). Yar. apiculata , nov. C. nodulifer (Janisch), Sch., Atl., pi. lix. fig. 20. — Diam. *0685 to *15 mm. Central nodule single, larger. Markings decreasing more around the border; apiculi numerous, subregular, placed at inner side of border. Habitat. — Campeachy Bay (Schmidt) ; trawled at lat. 12° 42' N., long. 152° r W., in 2900 fathoms, by H.M.S. Challenger (Rae !). C. radiosus, Grun. Van Heurck, Syn. Diat. Belg.} .pi. cxxxii. fig. 7 — Diam *09 to *11 mm. Surface almost flat, or somewhat convex towards the centre. Central space absent. Markings poly- 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 521 gonal; towards the centre 6 to 9, gradually decreasing towards the border to 9 or 10 in '01 mm.; secondary oblique rows evident; minute, subulate spaces at origin of the shorter rows. — Grun., Denk. Wien. Ak., 1884, p. 72; Janisch, Gazelle Exped ., taf. v. fig. 9 ; taf. vi. fig. 4. This species was formerly associated by Grunow with C. radiolatus , Ehrb., a species that cannot be determined with certainty. The specimens referred to from Macassar Straits are more convex towards the centre, and have been authenticated by Grunow. Habitat. — Monterey and Barbados deposits (Grunow) ; South Sea (Grunow) ; Los Angelos (O’Meara !) ; Macassar Straits (Grove !). Yar. kerguelensis. Grun., Denk. Wien. Ak ., 1884, p. 73. Diam. •047 mm. Markings towards the centre 6, at the border 9 in ‘01 mm.; close to the border a circlet of minute apiculi. Habitat. — Kerguelen (Grunow). C. subaidacodiscoidalis , sp. n. Sch., Atl, pi. lvii. fig. 8. — Diam. *0425 mm. Surface convex. Central space absent, rosette minute. Markings angular, 6 to 8 in *01 mm., decreasing gradually towards the border ; secondary oblique decussating rows evident ; apiculi, six large, with a slight median constriction inserted some distance from the border at subuniform intervals. Border narrow, hyaline, sharply defined. This species in its general characters approaches Aulacodiscus concinnus , Kitton, but there are no primary rays, and the processes are reduced to stout apiculi. Habitat. — Baldjik (Schmidt). C. B alley i, Rattray. Cestodiscus Bailey i, H. L. Sm., Amer. Quart. Jour. Microscopy , 1878, p. 16, pi. iii. fig. 8. — Diam. *04 to *0925 mm. Central space small, rounded, indistinct, bearing isolated granules. Markings 12 in ‘01 mm.; rows radial, straight; distinct hyaline subulate spaces opposite origin of shorter rows, secondary oblique rows evident ; apiculi distinct, at wide intervals inserted a short distance within border ; inner layer of valve with a clear central space surrounded by a zone of closely disposed costae 6 in ’01 mm., outside the latter a second broad hyaline zone 522 Proceedings of Royal Society of Edinburgh. [sess. adjacent to border. — Cestodiscus Baileyi , H. L. Sm., Diat. Spec. Typ ., No. 67. This species was first collected by Lieut. Williamson ( Explor . and Surveys for Railroad Route from Mississippi River to Pacific Ocean, vol. vi. pt. 2, “Geology,” chap. iv.). Prof. H. L. Smith, in his remarks on the species, first throws doubt on the validity of the genus Cestodiscus. Habitat. — Lost River, lower Klamath Lake, Oregon fossil (H. L. Smith !). C. fragilissimus, Grun,, in Van Heurck, Syn. Diat. Belg., pi. cxxviii. fig. 4. — Diam. *3165 mm. Central space minute, indistinct, rounded. Markings minute, 12 in *01 mm.; secondary oblique rows manifest ; apiculi distinct, scattered at wide unequal intervals, most crowded towards the border. Border narrow, hyaline. — Ethmo- discus convexus, Cstr., Diat. Chall. Exped ., 1886, p. 167, pi. iii. fig. 9. Habitat. — Arafura Sea (Yan Heurck, Castracane !). C. asteroides. Tru. and Witt, Jeremie Diat., p. 13, pi. iii. fig. 2. — Diam. T5 to 2 mm. Surface usually showing a circlet of six to twelve small shallow depressions at a distance of J to \ of the radius from the centre. Central space absent. Markings hexagonal, on a small central area 2 to 2\ in ‘01 mm., decreasing somewhat suddenly at about i of the radius to 3 or 3J, thence increasing gradually outwards to 1J in -01 mm., again becoming somewhat smaller at the border. Central papillse distinct ; secondary oblique, curved, decussating rows manifest. Border narrow. In Maryland specimens the shallow depressions are not found. Habitat. — Monte Gubbio (Grove !) ; Jeremie deposit (Truan and Witt !) ; Cove, Calvert County, Maryland (Greville !) ; South Naparima, Trinidad (Greville !) ; Nottingham deposit, Maryland (Johnson !) ; * Rappahannock, Ya (Greville !). C. lunatus. Grove MS. — Diam. ’09 to T5 mm. Central space minute ; a rosette frequently distinct, occasionally subobsolete. Surface with an evident lunate unilateral depression, its long axis at right angles and subequal to or somewhat longer than the radius, * In the collection of Dr Greville. 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 523 about twice its greatest breadth, its outer edge more distinct, convex towards the border, its inner less curved, the extremities obtuse ; the slope to the border gentle. Markings areolate, sub- equal, 3 to 3| in *01 mm. on the depression round, granular ; with hyaline interspaces, the central papillae prominent; rows radial, straight; secondary oblique decussating rows uniformly curved, manifest. Border relatively narrow, with coarse evident, subradial striae about 6 in *01 mm. Habitat — Santa Barbara County, California (Grove !). O. excavatus. Grev., Ralfs in Pritch. Inf, ’, p. 829, pi. viii. fig. 26. — Diam. from *1 to *255 mm. Surface with 1 to 3 rounded or subcuneate elevations, and alternate depressions around the centre, elsewhere subplain. Central rosette sometimes distinct. Markings hexagonal, increasing regularly outwards, but becoming somewhat smaller around the border ; near the centre 4, towards the border 1J, at the border 2, in *01 mm., the central dots faint; secondary oblique decussating rows evident. Border striae 4 in ’01 mm. — Grun., Denk. Wien. Ah, 1884, p. 73. Habitat — Piscataway deposit (Dallas ! * Rae !) ; Newcastle deposit, Barbados (Rae !); Holland’s Cliff (Cleve); “ Artesian Well ” (Febiger). Var. genuina. Grun., ibid., p. 73. — Rarely elliptical. Diam. greater than that of the other vars., from *25 to *3 mm. Surface elevations and depressions 3. — G. excavatus, Grev., Sch., Atl., pi. lxv. fig. 1. Habitat. — Piscataway deposit (Dallas !* Rae ! Grunow, Deby !) ; Naparima (Grunow) ; Newcastle deposit, Barbados (Grunow, Rae !) ; Naparima, Trinidad (Firth ! Kinker !) ; Richmond, Ya. (Kitton) ; Rappahannock (Rogers !).* Var. quadrioeellata. Grun, ibid., p. 73. — Circular or roundly elliptical. Diam. T5 to T9 mm. Surface elevations and depres- sions 2. Central rosette inconspicuous. Markings more uniform ; towards the centre 4, towards the border 2, in *01 mm. Border striae, 6 in -01 mm. — C. diophthalmus, Cstr., Diat. Chall. Exped., p. 163, pi. xvi. fig. 4. * In the collection of Dr Greville. 524 Proceedings of Royal Society of Edinburgh. [sess. Habitat. — Newcastle deposit, Barbados (Bae! Weissflog! Febiger!* Grunow) ; trawled by H.M.S. Challenger, in 2900 fathoms, lat. 12° 42' N., long. 152° V W.; “Barbados” (Firth! Febiger !). Yar. biocellata. Gran, ibid ., 1884, p. 73. — Diam. *0875 to •15 mm. Surface elevation and depression opposite, roundly elliptical, edges abrupt. Markings on the elevation 6 towards its central edge, 3 towards the peripheral, in ’01 mm., in radial, diverging rows ; on the depression more equal and larger, 2 \ in *01 mm., elsewhere as in var. quadriocellata. — C. diophthalmus , var. monophthalma , Cstr., Diat. Chad. Exped ., p. 163, pi. xvi. fig. 7. Habitat. — Newcastle deposit, Barbados (Bae! Kitton ! Weiss- flog ! Firth ! Febiger !) ; * Cambridge deposit, Barbados (Bae !) ; Hardman!)! “Barbados” (Febiger!); trawled by H.M.S. Challenger, in 2900 fathoms, lat. 12° 42' N., long. 152° 1' W. (Bae !). Yar. semilunaris. Grun., ibid., 1884, p. 73. — Diam. *1 to •1175 mm. Surface elevation semilunate, with rounded ends, sometimes short and broad, uniformly disposed with respect to the centre round which it curves, the depression slight, half inclosed by the elevation. Central rosette inconspicuous. Markings towards the centre 4, towards the border 3 in *01 mm. — C. semi- lunaris, Grun., ibid, 1884, p. 71. The vars. quadriocellata, biocellata, and semilunaris approach G. crassus , Bail., in the character of the markings ; those of var. genuina recall C. gigas, Ehrb. Habitat. — Newcastle deposit, Barbados, (Weissflog ! Grove !). Yar. deliquescens, nov. — Diam. ’0475 mm. Surface elevation and depression opposite, the former indistinct, the latter evident, but elliptical. Markings on the depression round free granules, with hyaline interspaces, elsewhere angular, 4J in ,01 mm.; towards the border smaller, rounded ; between the outer ends of the radial rows * In the collection of Herr E. Weissflog. t In the collection of Julien Deby. 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 525 narrow hyaline areas bearing a few minute markings, attenuating inwards; secondary oblique rows obvious. — In H. L. Sm., Diat. Spec. Typ., No. 99 (no name). — (PI. III. fig. 7.) Habitat. — Japan (H. L. Smith !). C. decrescens, Grun. Sch., Atl ., pi. lxi. figs. 7 to 9, 10 ('?). — Diam. *038 to *05 mm. Central space and rosette absent. Mark- ings polygonal, with evident central papillae; at the centre 3 in *01 mm., decreasing rapidly on outer of radius to the border ; rows on the outer portion sometimes subfasciculate, secondary oblique sometimes outwardly curved decussating rows distinct towards the border. — Grun., Denk. Wien. Ak., 1884, p. 80. Distinguished from C. marginatus by the rapid decrease in size of the markings on the outer third of the valve. Sometimes obtusely triangular specimens occur. Habitat. — Springfield deposit, Barbados (Grunow) ; Dolje (Pantocsek) ; Barbados (Cleve !) ; west coast “Florida” U.S. Sur- vey (Febiger !) ; Faeroe Channel (Grove !). Yar. irregularis. Grun., ibid., 1884, p. 80. — Obtusely triangular, two of the angles more evident than the third. Diam. *068 mm. Markings increasing slightly from the centre to about the semi- radius. Habitat. — Springfield deposit, Barbados (Grunow). Var. venusta. Grun., ibid., 1884, p. 80; C. heteroporus , Ehrb., forma major, Grun., Sch., Atl., pi. lxi. fig. 6. — Diam. *1135 mm. Central space minute. Markings increasing distinctly from central space for about \ to f of radius, thence decreasing rapidly to the border; towards the centre 4, increasing to 2J in *01 mm. — C. argus, Grun. (non Ehrb.) in Sch., Atl., pi. cxiii. fig. 7. The central space and increase of the markings outwards bring this var. near to C. heteroporus, but the appearance of the markings and the arrangement around the border bring it nearer to C. decrescens. The transition from specimens like that shown in Schmidt’s Atlas, pi. cxiii. fig. 7, to C. bulliens, A. S., is easy. Habitat. — Springfield, Barbados (Grunow) ; iEgina (Schmidt) ; “ Barbados earth ” (Greville !). 526 Proceedings of Royal Society of Edinburgh. [sess. Y&Y.valida. Gran., ibid., 1884, p. 80; C. decrescensl * Sch., Atl., pi. lxi. fig. 15.— Diam. *1 mm. Central space small, about of diam. broad, angular. Markings increasing but little from centre for ^ of radius, from the semiradius decreasing rapidly to tlie border ; at the central space 2i, from i to \ of radius li to 2 in *01 mm.; rows radial, irregularly concentric bands indistinct. Habitat. — Springfield deposit, Barbados (Schmidt, Grunow). Var. polaris. Grun., ibid., 1884, p. 80, pi. iii. (C), fig. 11. — Diam. f047 to ‘055 mm. Central space small, about of diam. road. Markings increasing slightly outwards to the semiradius to about 3 in *01 mm. ; rows subfasciculate, and secondary subconcen- tric rows evident. Habitat. — Franz Josefs Land (Grunow) ; Monterey deposits (Hardman !)f Barbados deposits (Greville !). Yar. repleta. Grun., ibid., 1884, p. 80, pi. iii. (C), fig. 18. — Diam. •0325 to *0825 mm. Surface convex towards the centre. Central space absent. Markings 3 to 3^, sometimes 4, in *01 mm. ; secondary concentric rows faint, oblique decussating rows undiffer- entiated near the border. — In H. L. Sm., Diat. Spec. Typ., No. 99. Habitat. — Franz Josefs Land (Grunow) ; Oamaru deposit (Grove !) ; Japan (H. L. Smith !); Macassar Straits (Grove !). C. epiphanes , sp. n. — Diam. *165 to *21 mm. Surface rising slightly from the centre for about f of the radius, here descending abruptly, and continuing thence on one plain to the border. Central space absent, rosette distinct. Markings hexagonal, increas- ing slightly from the centre to the highest zone, here decreasing suddenly, and from this subequal to the border ; towards the centre 3, at the highest zone 2 \, towards the border 3 in ’01 mm. ; central papillae faint, secondary oblique decussating rows evident. Border narrow; striae faint, 8 to 10 in *01 mm. — (PI. II. fig. 14.) Habitat. — Bichmond deposit (Deby !). J * Quoted “ C. decrescens, var.?” by Grunow ( Derik . Wien. Ah., 1884, P- 80). + In the collection of Julien Deb}T. + In a Coscinodiscus type-plate by Thum, in the collection of Julien Deby. 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 527 C. patina. Ehrb., Abh. Ber. Ak., 1839, p. 147, pi. iii. figs. 3 a-e. —Diam. -035 to -1125 mm. Surface flat. Central space and rosette absent. Markings angular, subequal, somewhat smaller near the border ; rows concentric, obvious, radial rows undifferen- tiated. Border narrow, hyaline. — Ehrb., Mikrog ., pi. xx. 1. fig. 31 ; Ralfs in Pritcli. Inf., p. 830; Janiscb, Gazelle Exped., taf. v. fig. 1; C. patina, Ehrb., pro parte , Abh. Ber. Ak, 1838, p. 129, pi. iv. figs. 10-12 d. (Excl. C. patina , Bail., Amer. Journ. Sci., 1842, vol. xlii. p. 96, pi. ii. fig. 13.) The Simbrisk valve, figured by Weisse (Bull. Acad. Imp. St Petersb., 1855, p. 276, pi. i. fig. 6) has round free markings, with a more prominent circlet at the border, and is probably distinct. Ehrenberg at first embraced in C. patina bis G. radiatus (conf. Abh. Ber. Ak., 1839, p. 148). The specimen figured by Janiscb (Gazelle Exped.) shows the concentric arrangement of the mark- ings most clearly towards the border ; the areolae are at intervals irregular. Habitat. — Zante, Caltanisetta, Oran and Grecian deposits, Cux- baven (Ehrenberg). G. argus. Ehrb., Abh. Ber. Ak., 1838, p. 129. — Diam. from *0675 to T75 mm. Central space absent, a rosette sometimes present. Markings polygonal, increasing gradually outwards; at the centre 4, near the border 2 to 3, on a narrow zone adjacent to the border 4 or 5, in *01 mm. ; secondary oblique rows indistinct or undifferentiated. — Ehrb. ibid., 1839, p. 145, Mon. Ber. Ak., 1844, p. 79; Mikrog., pi. xxi. fig. 2 (excl. pi. xxii. fig. 5, 8); Grun., Denk. Wien. Ak., 1884, p. 72; G. irradiatus, Harting, Verh. Kon. Ak. Wetensch , Amsterdam, 1864, No. ii. p. 8, pi. i. fig. 1); C. radiatus, Ehrb., Mikrog., pi. xxi. fig. 1 ; C. Woodwardii, Sch., (not Eul.) Atl., pi. lxi. fig. 2; G. heteroporus, Grun., in Sch., Atl., pi. lxi. fig. 2. Ehrenberg regarded this as probably a var. of G. radiatus ; Bright well and Grunow more correctly accept it as a distinct species. Harting points out the affinity of his G. irradiatus to G. radiatus, Ehrb., but its markings increase towards the border as in G. argus, to which it seems rather to belong. Some specimens belonging to this species were labelled by O’Meara G. sinensis. 528 Proceedings of Royal Society of Edinburgh. [sess. Cleve has named C. argus, var. subimpressa , some Oamaru specimens that differ from more typical valves only in showing a subfascicu- late arrangement of the markings, chiefly visible when the papillae are in focus. Habitat. — Oran deposit (Ehrenherg, Grunow, Greville!); Caltanisetta deposit (Ehrenherg); Bichmond, Va. (Kiitzing); Szent Peter deposit (Pantocsek) ; Cuxhaven (Ehrenherg); Carpentaria Bay (Schmidt); N. America (Grunow); Banda Sea, 1200 fms. (Hart- ing); locality ? (Deby !); Japan (Kinker !); Cambodia (Hardman !); Mejillones (O’Meara !); stomach of oysters at Howth (O’Meara!); Los Angelos deposit, Cal. (O’Meara!); Cambridge deposit “Barbados” (Johnson !* Weissflog ! Hardman !); Canton Biver, Whampoa (Grove !); Indian Ocean sounding, by Captain Pullen, 2200 fathoms (Greville !); Maryland (O’Meara !); Oamaru deposit (Cleve ! Grove !). Yar. subtraducens, nov. — Diam. *15 to *225 mm. Central space absent or minute, rosette absent or obscure. Markings hexagonal, increasing from the centre almost to the border ; towards the centre 4, near the border 3 in *01 mm.; central papillae evident ; secondary oblique curved decussating rows distinct. Border narrow ; striae, 5 to 6 in ’01 mm. — (PI. I. fig. 20.) Transitional between C. argus and O. traducens. Specimens have sometimes been erroneously associated with O. fimbriatus. C. intermedins , Ehrb. ( Mikrog ., pi. xxxiii. 13. fig. 3), may perhaps belong to this variety, its markings being figured as more delicate than those of C. argus {Mikrog., pi. xxi. fig. 2). Here may also come the valve figured by Ehrenberg as C. radiolatus ? ( Mikrog . , pi. xxxix. 2. fig. 18), but C. radiolatus , Ehrb. {Mikrog., pi. xxii. fig. 4), is distinct (see infra). Habitat. — Jackson’s Paddock, Oamaru deposit (Grove !). C. traducens, sp. n. — Diam. T mm. Surface flat. Central space and rosette absent; a small central area surrounded by a sub- circular hyaline line evident. Markings hexagonal, gradually in- creasing in size from the central area outwards ; towards the centre 8, at the border 6 in *01 mm. ; irregular on the central area, * In the collection of Dr Greville. 1888-89.] Mr John Eattray on the Genus Coscinodiscus. 529 secondary oblique curved decussating rows distinct ; a narrow hyaline band adjacent to the border. Border narrow, sharply defined, with small evident granules 6 in -01 mm. — C. nebula , Ehrb.? Abh. Ber. Ah., 1872, p. 167, pi. xii. fig. 15. Sp. n. ? Sch., Atl., pi. lviii. fig. 12. C. nebula , Ehrb., is an imperfectly defined species, approaching C. radiolatus, Ehrb., and C. intermedins , Ehrb.; its insertion here is provisional. Habitat. — Railway cutting, Oamaru (Grove !) S.E. of Bhjuts- kaja Kossa, Sea of Azof (Ehrenberg). Yar. hispida, nov. Sch., All ., pi. lviii. fig. 38 (no name). — Diam. about ’035 mm. Markings 6 to 7 in *01 mm ; apiculi prominent, numerous, at intervals of about '0075 mm., inserted some distance within border. Habitat. — Yokohama (Griindler). C. exutus , sp. n. — Diam. *0775 mm. Central space and rosette absent. Markings polygonal, increasing slightly outwards to the marginal band ; at the centre 6 to 7, about the semiradius 5 to 5 J, on the sharply defined marginal band, 10 in *01 mm., this band about J of radius broad; rows radial from the centre to the marginal band, upon the latter the oblique decussating rows more evident. Border narrow, distinct. Habitat. — Los Angelos (Hardman !).* C 1 debilis , sp. n. — Diam. *3 mm. Surface almost flat, a gentle slope near the border. Central space and rosette absent. Mark- ings hexagonal, 5 to 7 in '01 mm., slightly smaller at the centre, and towards the border submoniliform ; central papillae distinct ; secondary curved oblique decussating rows evident ; minute subu- late areas at the origin of the shorter rows. Border sharply defined, usually opaque, about A_ 0f radius broad ; its broad or inner portion closely and irregularly punctate ; the outer with evident striae, 6 or 7 in *01 mm. — (PI. I. fig. 4.) Habitat. — Jackson’s Paddock, Oamaru deposit (Grove !). C. dubiosus, Grun. MS. — Diam. *0925 to *15 mm. Central * In the collection of Julien Deby. VOL. XVI. 25/10/89 2 L 530 Proceedings of Royal Society of Edinburgh. [* space and rosette absent. Markings hexagonal, minute, subpuncti- form, smallest and most crowded towards the border ; towards the centre 8 to 10, towards the border 14 to 16 in *01 mm.; secondary rows slightly oblique or irregularly subconcentric, the latter more evident near the centre, narrow radial subulate clear lines opposite the origin of the shorter rows ; clear irregularly disposed puncta sometimes evident. Border narrow; striae, 10 to 12 in -01 mm. — Cleve and Moll., Diat ., No. 164; Janisch, Gazelle Exped ., taf. v. figs. 10, 11. This species sometimes approaches Podosira hormoides, Mont. (Van Heurck, Syn. Diat. Belg ., pi. lxxxiv. figs. 3-6), but the markings are not in fasciculate rows. In Podosira liormoides Grunow has noted that the markings are grouped in almost radial lines (Sitzungsb. naturw. Ges.Isis, Dresden , 1878, p. 33). Habitat. — Oamaru deposit (Grove! Doeg!); California (Arnott!);* Monterey deposit (Hardman ! * Cleve !); Sta Monica deposit (Cleve and Moller !); Java (Cleve !); Successful Bay, Kerguelen (Cleve !). Yar. curvans , nov. — Diam. T1 mm. Markings similar, but numerous hyaline subulate spaces towards the centre; the rows uniformly curved between centre and border, radial ; the secondary rows but slightly oblique or almost parallel to the border. Habitat. — Troublesome Gully, Oamaru (Grove !). C. plicatus , Grun. Sch., Atl ., pi. lix. fig. 1. — Diam. -0425 to *05 mm. Surface with a short transverse central plication. Central space absent. Markings polygonal, increasing from the centre outwards, again decreasing slightly at the border; towards the centre 7, towards the border 6 in *01 mm.; a circlet of evident, but small, apiculi at the border. — Grun., Denk. Wien. Ah, 1884, p. 73, pi. iii. (C), fig. 10.; Cleve and Moll., Diat., Nos. 114, 276. Habitat. — Polycistinous rock, Nancoori (Cleve ! Grunow, Schmidt, Hardman !) ; f Mascara and California (Cleve and Moller !). C. corolla. Sch., Atl., pi. lviii. fig. 32. — Diam. -047 mm. Central space and rosette absent. Markings polygonal, 8 to 10 in *01 * In the collection of Dr Greville. t In the collection of Julien Deby. 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 531 mm., decreasing slightly to the border ; irregular on a small central area, elsewhere the rows radial, sometimes subfasciculate ; secondary oblique decussating rows obvious ; apiculi numerous, forming a circlet placed some distance from the border. Border distinctly defined, about Jg- of radius broad; striae distinct, 5 or 6 in ’01 mm. Habitat. — Yokohama (Schmidt). C. denticidatus. Cstr., Diat. Chall. Ex]oed., p. 155, pi. iii. fig. 8. — Diam. *13 mm. Surface slightly convex, on outer of radius. Central space and rosette absent. Markings polygonal, subequal, 8 in *01 mm.; central dots distinct; apiculi scattered over the surface at irregular wide intervals, distinct. Border sharply defined; striae 8 to 10 in *01 mm. This species is nearly allied to C. radiosus , but differs in the more uniform markings and in the apiculi Compare also Podosira hormoides, Mont. ( = Melosira nummuloides , Ehrb.) from Lima (Van Heurck, Syn. Diat. Belg ., pi. lxxxiv. fig. 3). Habitat. — Pacific Ocean (Castracane). C. imjpressus, Grun. Van Heurck, Syn. Diat. Belg., pi. cxxxii. fig. 5. — Diam. ‘08 mm. Surface with a long depression near the centre. Central space minute, irregular, about ~ of diameter broad. Markings increasing gradually from the margin of the central space to the border ; towards the centre 8, toward the border 7, in *01 mm.; secondary oblique rows indistinct. Border striae, subregular, distinct, 8 to 10 in *01 mm. — Grun., Denk. Wien. Ah., 1884, p. 73. Habitat. — Sta Monica deposit (Grunow). C. coneinnus. W. Sm. Syn., Brit. Diat., ii. p. 85. — Diam *062 to ’35 mm. Surface somewhat convex. Central rosette of large but delicate areolae, sometimes inconspicuous, *0075 to *01 mm. broad. Markings polygonal, delicate, most evident towards the centre, where there are 7 or 8 in ’01 mm., decreasing outwards to 12 in -01 mm.; rows obscurely fasciculate ; near the border at subequal intervals short narrow radial clear areas, whence faint subbyaline lines proceed inwards ; a circlet of apiculi at the border, minute ; 2 larger apiculi unsymmetrical close to the border, a few long acicular apiculi also sometimes present on a zone within the processes. — Roper, Quart . 532 Proceedings of Royal Society of Edinburgh. [sess. Jour. Micr. Sci ., 1858, p. 20, pi. iii. figs. 12, 12a; Ralfs in Pritch. Inf, 828, pi. v. fig. 89 ; Janisch, Gazelle Exped., taf. ii. fig. 6 ; H. L. Sm., Amer. Jour. Micr., 1877, No. 8, p. 102; Sell., Ail., pi. cxiii. figs. 8, 9 ; Cleve and Moll., Diat., No. 215, 319 ; C. ? tenuis , Bail., Boston Jour. Nat. Hist., 1862, p. 333, pi. vii. fig. 9; C. centralis, Schulze, fide Grunow, Jour. Roy. Micr. Soc ., 1879, p. 688 (excl. Eupodiscus gregorianus, de Breb., Jour. Quek. Micr. CL, 1870. p. 41). Bailey has seen specimens from Para River with 3 processes, hut this species cannot he united to Eupodiscus, as suggested hy Bailey, since the character of the processes of the latter is distinct. C. concinnus, var. Tcerguelensis, Grun., differs in the markings, decreasing outwards from 5 to 7 in '01 mm., and C. concinnus, var. arafurensis , Grun., in having a small clear circular central space and the markings 9 to 12 in ’01 mm., the clear radii being very long. Eupodiscus gregorianus, de Breb., is Eup. subtilis , Greg. (Rattray, Jour. Roy. Mic. Soc. Lond ., 1888, p. 915). C. con- cinnus agrees with C. centralis, Ehrb., in its two large unsymmetrical apiculi, but differs in the degree of fineness and arrangement of the markings. Habitat. — Peruvian guano (Schmidt) ; Para River (Bailey) ; Kerguelen, 25 fathoms (O’Meara, Grunow, Rae!); Baltic (Flogel) ; stomach of Pecten, Sussex coast and Kinsale Bay (W. Smith !) ; Firth of Clyde (Hennedy) ; Cumbrae (Arnott !); Hull (Ralfs) ; Loch Fyne and Inveraray (Gregory !); Ascidia, Hull (Greville ! Gregory !); Gorleston (Roper !) ; Caldy, Pembrokeshire (Rev. J. Guillemard) ; Humber dredgings (Norman) ; seaweeds, Ballybrack, and oyster shells, Dublin Bay (O’Meara) ; San Francisco, Cal. (Firth ! Schmidt) ; Helder Algae (Kinker !); Mejillones deposit (O’Meara !) ; “Barbados” (Johnson!);* stomach of Pecten, Penzance (Mont- gomery!) ;* Nottingham deposit, Cape Wankarema (Cleve and Moller ! Grunow!); Schleswig-Holstein (Cleve!); Heligoland (Schulze !* Weissflog !) ; Firth of Forth (Grove ! Rattray !) ; Sheer ness (Grove !) ; Marstrand (Kinker !). Var .jonesiana, Rattray. Eupodiscus jonesianus, Grev., Trans. Micr. Soc. Lond., 1862, p. 22, pi. ii. fig. 3. — Rarely triangular. Diam. -21 to *45 mm. Markings coarser, and more sharply defined ; towards * In the collection of Dr Greville. 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 533 the centre 6, towards the border 8, in *01 mm. ; the unsymmetrical processes 2, larger, obtusely conical, and placed towards the same side of the valve ; the clear radial lines less distinct ; long apiculi sometimes present within the processes, as in the type. — C. concinnus, H. L. Sm., Diat. Spec. Typ., No. 92 ; Eup. jonesianus , Grev.; Cleve, Bih. Sv. Vet.-AJc. Handl. , 1873, No. 11, p. 5, H. L. Smith, Diat Spec. Typ., No. 163; Eup A commutatus, Grun., and Coscinodiscus commutatus, Grun., Denk. Wien. Ak., 1884, p. 79; Van Heurck, Typ. Syn. Diat. Belg., No. 490 ; Eup. concinnus, var. triangularis , ibid. Greville erroneously states that the processes are 3. In triangular specimens they occur at the middle of two of the sides. Habitat. — Peruvian guano (Grunow) ; Hong Kong (H. L. Smith ! Hardman! Grove! Palmer!* Greville!); Yokohama (Hardman!); Canton River, Whampoa (Grove !) ; Schleswig (Van Heurck !) ; shell cleanings, Singapore (Hardman !);* Sand Heads, Bay of Bengal, Ceylon, and edible seaweeds, India (Macrae !);* Port Elizabeth (Hardman !) ; Java Sea (Cleve, Grunow) ; surface, Arafura Sea, H.M.S. Challenger (Rae !) ; Cherbourg (H. L. Smith !) ; Cux- haven, Brazil, and China (Grunow); North Sea (Griffin !); trawled at lat. 34° 36' N., long. 140° 22' E., by H.M.S. Challenger (Rae !);f Kusu (Cleve ! O’Meara !) ; from Pecten, Penzance (Greville !); Bay of Bengal (Macrae!); Tindingen, Greenland (Cleve!); Java (O’Meara! Cleve !) ; Cape Wankarema (Cleve !) ; lat. 4° 20' S., long. 105° 22' E. (Cleve !). Var. Moseleyi, Rattray. C. Moseleyi , O’Me., Quart. Jour. Micr. Sci., 1875, p. 330. — Diam. *28 to ‘55 mm. Colour iridescent when dry. Central rosette distinct, of large unequal areolae. Markings towards the centre 5 to 6, at the border 8, in *01 mm.; rows obscurely fasciculate, unsymmetrical ; processes 2, minute ; apiculi obscure. Border narrow. — O’Meara, Jour. Lin. Soc. (Bot.), 1877, p. 57, pi. i. fig. 6 ; Cstr., Diat. Chall. Exped., p. 153 ; C. concinnus , var. kerguelensis , Grun., Denk. Wien. Ak., 1884, p. 79. Habitat. — Kerguelen, at 25 fathoms (Rae ! Hardman ! J Peal ! J O’Meara, Grunow) ; Royal Sound, Kerguelen (Rae !). * In the collection of Dr Greville. t In the collection of Dr F. W. Griffin. + In the collection of Julien Deby. 534 Proceedings of Royal Society of Edinburgh. [sess. Yar. arafurensis. Gran., Denh. Wien. Ah., 1884, p. 79. C. papuanus , Cstr., Diat. Chall. Exped., p. 154, pi. iii. fig. 3. — Diam. *152 to '475 mm. Central space minute, surrounded by a minute and inconspicuous rosette, sometimes hardly differentiated. Markings 9 to 12 in '01 mm. This form is frequent in the Arafura Sea. The central space is smaller than figured by Castracane in all the specimens I have observed, and is not of specific value, as he maintains. He has over- looked the distinct marginal processes characteristic of the species. Habitat. — Arafura Sea, H.M.S. Challenger (Rae ! Castracane) ; between Kerguelen and Heard Island, H.M.S. Challenger (Rae !). C. africanus , Janisch. Sch., Atl., pi. lix. figs. 24, 25. — Sub- circular or roundly elliptical. Diam. -035 to '088 mm. Central space and rosette absent. Markings polygonal, increasing gradually outwards, and again smaller at the border ; towards the centre 6, towards the border 4 in '01 mm.; irregular on an indistinctly defined somewhat excentric area, elsewhere the rows radial, straight, or slightly bent, sometimes indistinctly subfasciculate towards the border. Border regularly striated, sometimes double ; the inner portion about J of the breadth of the outer. Sometimes also with more evident striae ; oblique curved, more distant and more distinct lines. — Janisch, Gazelle Exped. y taf. iii. fig. 2. This species is readily distinguished by the character of its border. From C. vetustissimus, Pant., it differs by the absence of a nodule from the excentric area. Habitat. — Gazelle Expedition (Janisch); Newcastle, Barbados (Firth) ; off Ascension Island, S.S. Buccaneer (Grove !). Yar. wallichiana, Grun. Cleve and Moll., Diat., No. 183, 207. — Diam. '05 to '0575 mm. Central space irregular, excentric, with a few isolated rounded granules. Markings rounded, granular ; towards the excentric area 8, about the semiradius 5 to 5J, at the border 6 in *01 mm.; rows radial, straight; at irregular intervals hyaline, narrow radial spaces passing inwards for a short distance from border. Border distinct; striae delicate, 10 to 12 in '01 mm. — C. africanus, var. rotunda , Cstr., Diat. Chall. Exped., 1886, p. 159, pi. xxiv. fig. 3. — (PL II. fig. 4.) 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 535 Habitat. — Antarctic Ocean, Patagonia (Cleve and Moller !) ; Table Bay (Cleve !). C. mirificus. Cstr., Diat. Chall. Exped., p. 154, pi. iii. figs. 6, 6a. — Diam. ’326 mm. Central space irregularly subcircular, about j!t of diam. broad. Markings hexagonal, their sides composed of closely placed round granules, the central dots distinct, decreasing but slightly to the border ; towards the centre 6, at the border 8, in *01 mm. Border formed by a simple line. In its large central space, this species approaches C. asteromphalus var. pabellanica , Grun., but is distinguished by the absence of a distinct band around the central space, and by the greater uniformity of the markings. Habitat . — Hong Kong, in the sea, H.M.S. Challenger (Castra- cane). C. Hauckii , Grun. Van Heurck., Syn. Diat. Belg ., pi. xciv. fig. 29. — Diam. '0365 mm. Central space absent. Markings obscure, punctiform, irregular, and with hyaline interspaces on the central portion, which extends outwards for § of radius, on outer J- more minute, closely disposed in crowded radial lines, 16 in '01 mm.; apiculi sometimes evident, inserted near border about 2 in *01 mm. — Cleve and Moll., Diat., No. 210. Habitat. — Rovigno (Van Heurck, Cleve and Moller !) ; Green land (Cleve !) ; Labuan ? (Cleve !). C. liocentrum. Ehrb., Abh. Ber. Ak., 1870, p. 53, pi. ii. 2. fig. 9. — Diam. '075 mm. Central space extending almost to semiradius, smooth. Markings polygonal, delicate, increasing slightly outwards ; rows distinct. This species is at once characterised by the large size of the central space, in which it approaches C. mesoleius, Cleve. Habitat. — Humboldt Valley deposit, Cal. (Ehrenberg). C. vacuus , sp. n. Melosira ? Sch., Atl., pi. Iviii. fig. 29. — Diam. about *03 mm. Central space large, extending outwards for about j of radius. Markings minute, punctiform, forming closely dis- posed radial rows, on outer ^ of radius. Border narrow, indistinct. This is distinguished from C. Hauckii , Grun. (Van Heurck, Syn. 536 Proceedings of Boyal Society of Edinburgh . [sess. Diat. Belg ., pi. xciv. fig. 24), by the absence of punctiform granules on the central space. In Schmidt’s figure of C. vacuus , a minute round central areola (nodule?) is faintly indicated. Habitat. — Cape of Good Hope (Schmidt). C. mesoleius. Cleve, Vega Exped. Jakttag. Stockh ., Bd. iii., 1883, p. 503, pi. xxxviii. fig. 2. — Diam. -03 mm. Central portion hyaline, extending outwards to about § of the radius, sharply defined. Markings on outer portion punctiform, 28 in *01 mm., forming radial striae. The valve is very thin and transparant. It approaches C. Hauckii , Grun. (Van Heurck, Syn. Diat. Belg., pi. xciv. fig. 29) ; but in the latter the central portion is covered with irregular scattered puncta. Both approach the genus Cyclotella. Habitat. — Labuan, near Borneo, among Algae* (Cleve !). C. lutescens , sp. n. — Diam. *125 mm. Surface flat. Central space circular, minute, surrounded by a subobsolete band of large cuneate areolae. Markings hexagonal, distinct, increasing outwards for about -f- of the radius, thence decreasing to the border ; towards the centre 5, at -f- of radius 3J, in *01 mm.; central papillae faint ; rows straight, at intervals with the markings oblique; apiculi minute, inserted at the border at intervals of about *005 mm. Border narrow; striae faint, 8 to 10 in *01 mm. — (PI. II. fig. 2.) Distinguished from G. cribrosus, Tru. and Witt, by the gradual increase of the markings outwards. Habitat. — Jeremie deposit, Hayti (Bae !). C. modestus , sp. n. — Diam. *325 mm. Surface flat, becoming convex towards the border. Central space round, indistinct, about — 3 of diam. broad. Markings delicate and faint for inner § of radius, more evident on the outer J, increasing from the central space outwards, towards the centre punctiform 8, beyond the semi- radius distinctly hexagonal, with evident central papillae, 4 in •01 mm.; rows around the central space of unequal lengths, and separated by hyaline interspaces, towards the border subfasciculate, the fasciculi separated by fossil subhyaline radial lines ; apiculi 2, unsymmetrical, separated by an interval equal to about J of the Collected by Dr Kjellman. 1888-89.] Mr John Battray on the Genus Coscinodiscus. 537 circumference, small but distinct ; a circlet of more minute apiculi at the border absent. Border narrow. — (PI. I. fig. 3.) Habitat. — Eio Janeiro (Kitton !) ; * Peruvian guano (Macrae !). O. patellceformis. Grev., Trans. Micr. Soc. Lond., 1861, p. 80, pi. x. fig. 4. — Diam. *07 to *1 mm. Surface slightly convex. Central space and rosette absent. Markings polygonal, in contact and without order for about ^ of radius, thence obtusely angular, pearly, 4 in *01 mm., increasing slightly outwards and forming straight rows ; secondary suhconcentric rows also evident ; at the origin of the shorter rows are sharply defined, irregular, elongate hyaline spaces ; a narrow hyaline hand within the border. Border with a single row of granules* 4 in ‘01 mm. — C. detritus , Sch., Atl ., pi. lviii. fig. 15. In Greville’s specimen, the undulating concentric lines crossed by stronger radial costae — as shown in his figure — do not occur. Habitat. — Springfield deposit, Barbados (Hardman !).f C. oblongus. Grev., Trans. Micr. Soc. Lond., 1866, p. 4, pi. i. figs. 9, 10. — Elliptical or oval, major axis *07 to ‘145 mm., from 2-| to 3 times minor. Surface depressed at the centre. Central area roundly or elongately elliptical, with rounded granules irregular, or in rows parallel to the major axis. Markings round, granular, increasing slightly from the edge of the central area outwards, again decreasing towards the border ; at the semiradius 4, near the border 6, in ‘01 mm. ; rows straight only along the major and minor axes, between these slightly curved towards the major axis. — Sch., Atl., pi. lxvi. figs. 10, 11 ; C. oblongus forma typica, Truan and Witt, Jeremie Diat., p. 14, pi. ii. fig. 16. Distinguished from C. pundatus by the presence of a central depression, the absence of a central space, and of oblique decussating rows near the border. Habitat. — Chalky Mount., Barbados (Firth !) ; Barbados (Weiss- flog! Deby! Hardman! Kinker ! Cleve ! Greville !) ; Springfield deposits, Barbados (Johnson! Firth! | Hardman); § Pacific Ocean, * In the collection of Herr E. Weissflog. t In the collection of Dr Greville. £ In the collection of Dr F. W. Griffin. § In the collection of Julien Deby. 538 Proceedings of Royal Society of Edinburgh. [sess. H.M.S. Challenger (Castracane); Jeremie deposit, Hayti (Truan and Witt). C. ellijpticus. Grun., Reise. d. Novara Wien., 1870 (Bot. Th.), p. 104, pi. i. a. fig. 18 a , b. — Elliptical, major axis ‘04 to *075 mm., minor *02 to *035 mm. Central space absent. Markings rounded, granular ; on the central portion large, subradial or irregular, decreas- ing slightly outwards, on a well-defined band at the border minute, in radiating delicate crowded striae. Distinguished from Cestodiscus ovalis , Grev., by the absence of apiculi, and from C. oblongus , Grev. (Trans. Micr. Soc. Lond ., 1866, p. 4, pi. i. figs. 9, 10), in the relatively larger size of the central area, and the greater diminution of the markings on the marginal band. In the explanation of Cleve and Moller’s Diat ., No. 57, C. ellipticus is given as equivalent to C. lewisianus , var.; with this opinion I am unable to agree. Habitat. — Polycistinous Rock, Nancoori (Grunow, Cleve !). G. obovatus. Cstr., Diat. Chall. Exjped ., p. 160, pi. viii. fig. 4; pi. xviii. fig. 7 ; pi. xxii. fig. 9. — Roundly elliptical or oval, major axis '099 mm., about 1 \ times minor. Surface almost flat. Central space and rosette absent. Markings polygonal ; towards the centre 4, near the border decreasing to 6 or 8 in '01 mm. ; rows straight and parallel to the major axis on the central portion, which is distinctly defined, and extends to about f of the radius from the centre, beyond this in radial rows. Habitat. — Pacific Ocean (Castracane !). Var. circularis , nov. — Circular. Diam. '0625 mm. Markings similar, but the central area less distinctly defined, the rows radiating from its outer edge to the border sometimes slightly curved. Specimens of this var. have been labelled C. subtilis by O’Meara. Habitat. — Humber (O’Meara !) ; stomach of oysters, at Howth (O’Meara!); locality? (O’Meara!); Atlantic Ocean, lat. 3° S., long. 15° W. (O’Meara!); off Ascension Island, lat. 0° 1'6' S., long. 15° 56'5' W., 1845 fathoms, S.S. Buccaneer (Grove! Rattray J). C. dubius, sp. n. Sch., Atl., pL lxi. fig. 14. — Diam. '094 mm. Surface flat from centre for about § of radius, thence convex to the 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 539 border. Central space angular, about t3q of diam. broad. Markings rounded, adjacent to the central space, elsewhere polygonal, 2J to 3 in ‘01 mm., subequal for about § of radius, thence decreasing slightly to the border, smooth; secondary oblique rows inconspicuous. Border indistinctly defined ; striae somewhat irregular, 4 to 6 in *01 mm. — C. crassus , Bail. var. % Grun., Denk. Wien. Ak., 1884, p. 74. Habitat. — Springfield deposit, Barbados (Schmidt). C. cingulatus. Ehrb., Mon. Ber. Ak ., 1844, p. 200. — Diam. '049 mm. Central space small, hyaline, indistinct. Markings minute, about 13 in -01 mm. Border smooth, distinct. — Ehrb. Mikrog ., pi. xxxv. a. 21. fig. 6; Ralfs in Pritch. Inf., p. 829. In Antarctic diabomaceous gatherings I have not observed speci- mens similar to this. Is it possible that Ehrenberg was dealing with a valve of C. subtilisl Habitat. — Antarctic Ice Barrier, lat. 78° 10' S., long. 162° W. (Hooker !).* C. crassus. Bail., Amer. Jour. Sci., 1856, pi. xxii. p. 4. — Diam. •12 to *14 mm. Central space small, angular to elliptical. Mark- ings subpearly ; towards the centre 3, towards the border 2 to 2 \ in *01 mm., at the border suddenly smaller, secondary oblique rows inconspicuous. Border striae 4 in *01 mm. — Ralfs in Pritcli Inf., p. 830; Grun., Denk. Wien. Ak., 1884, p. 74. C. crassus , var., Sch., Atl., pi. Ixi. fig. 19. Grunow, in his synopsis of this genus {Denk. Wien. Ak., 1884, p. 71), states correctly in his section 15 that the markings become smaller both towards the centre and towards the border, but erroneously states in the following section (16) that only the outer- most markings are smaller, and by this means he differentiates this species. Habitat. — Soundings, Sea of Kamtschatka (Bailey) ; Monterey (Bailey); Kekko, Szakal and Dolje deposits (Pentocsek !) ; Barbados (Cleve !). Yar. morsiana. Grun., ibid., 1884, p. 74. — Diam. J74. Mark- * Fide Ehrenberg. 540 Proceedings of Royal Society of Edinburgh. [sess. ings towards the centre 3, increasing outwards to 2, at the border 8 in *01 mm., forming 3 concentric zones. Habitat. — Mors deposit (Grunow). Var. gelida. Grun., ibid., 1884, p. 74, pi. iii. (C), fig. 6. — Diam. •114 mm. Markings increasing, hut little from the centre outwards ;• towards the centre 3, towards the border 2f, at the border 7, in *01 mm. Distinguished from var. morsiana by the markings towards the border, and from C. apiculatus, var. Woodwardii, by the increase of the markings outwards. Habitat. — Franz Josefs Land (Grunow). Var. algida. Grun., ibid., 1884, p. 74, pi. iii. (C), fig. 5. — Diam. *094 mm. Central smooth area small. Markings subequal, 2 to 2i in -01 mm., at the border 3 \ in ’01 mm. This var. approaches C. marginatus in the character of the markings. Habitat. — Franz Josefs Land (Grunow). C. heteroporus. Ehrb., Mon. Ber. Ah., 1844, p. 265. — Diam. •072 to T125 mm. Central space small, or replaced by a small rosette. Markings at the centre 3^ to 4, increasing outwards to an annular, somewhat elevated area about the semiradius to 2 or 21, again decreasing to the border to 6 or 7 in ’01 mm.; rows sometimes indistinctly radial ; secondary oblique rows irregular, obscure. Border striae evident, 6 in -01 mm. — Balfs in Pritch. Inf., p. 831; Grun., Denk. Wien. Ah., 1884, p. 74. C. heteroporus, var. Grun., in Sch., Atl., pi. lxi. fig. 4. Recent Manilla valves show a transition to C. apiculatus, var. Woodwardii. It differs from C. crassus in the less robust appearance of the markings and the distinctly striated border. Habitat. — Nottingham and Monterey deposits (Grunow); Elephant Point, Bengal (Grunow); Piscataway deposit (Griffin!); Santa Monica deposit (Rae !) ; Delaware, Maryland (O’Meara!); Manilla (Grove !) ; Labuan (Cleve !) ; Maryland (Cleve). Var. moronensis. Grun., Denk. Wien. Ah., 1884, p. 75. — Diam. T mm. Central rosette distinct. Markings on the elevated ring 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 541 more prominent ; rows distinctly radial, secondary rows more obvious. Border more sharply defined. — C. heterojoorus, var. Grun., Sch. in Atl., pi. lxi. fig. 1. Habitat. — Moron deposit (Grunow). C. boliviensis. Grun., Denk. Wien. Ak., 1884, p. 76. — Diam. *15 to ’22 mm. Central space small, irregular. Markings hexagonal, non-punctate, increasing gradually outwards for about § of the radius; towards the centre 4, at about -| of radius 3 in *01 mm., thence decreasing rapidly to the border to 6 in -01 mm.; secondary oblique rows inconspicuous. — C. Woodwardii. varj Sch., Atl., ** pi. lx. fig. 8. Distinguished from C. apiculatus , var. ambigua , by the increase of the markings outwards, and from C. gigas by their arrangement in contact around the centre. The increase in the markings out- wards is greater in some specimens than in others. Habitat. — Bolivian guano, Sta Monica deposit (Grunow) ; Darien (Grunow, Schmidt); Cambridge deposit, Barbados (Hardman!* Kinker !) ; Lobos de Afuera guano (Grove !). Yar. spinulosa. Grun., ibid., 1884, p. 76; C. boliviensis, Grun.; Van Heurck, Syn. Diat. Belg., pi. cxxxii. fig. 4. — Diam. *16 to ’17 mm. Central space large, circular, about A of diam. broad. Markings towards the centre 5, towards the border 3i in •01 mm.; a circlet of numerous minute apiculi close to the border. Habitat. — Bolivian guano (Grunow). C. gigas. Ehrb., 1841, Abh. Ber. Ak., 1841, p. 412. — Diam. *159 to *31 mm., easily seen by the naked eye. Central space subcircular, about to of diam. broad. Markings obtusely angular, and least crowded towards the centre, with central dots faint, soon becoming hexagonal, and increasing gradually outwards ; towards the centre 4, towards the border If to 2 in ’01 mm., at the border again small ; secondary oblique decussating rows distinct. Border narrow ; strise radial, about 4 in ’01 mm., subregular. — Ehrb., Mikrog., pi. xviii. fig. 34 ; Ralfs in Pritch. Inf., p. 829 ; Jan., Sch. Ges. voter. Cult., 1862, Heft ii. p. 3, pi. 1a. fig. 12; Gazelle In the collection of Julien Deby. 542 Proceedings of Royal Society of Edinburgh. [sess. Exped ., taf. iii. fig. 4 ; vi. fig. 13 ; Sch., Atl ., pi. ixiv. fig. 1 ; Grun., Denk. Wien. Ak ., 1884, p. 76; Cleve and Moll., Diat ., Nos. 57, 162, 164; C. radiatus, Bail., Amer. Jour. Sci., 1842, vol. xlii. p. 95, pi. ii. fig. 14. In all the specimens there is a central space, not indicated in the earlier figures. Ralfs first noted the striated border, Janisch the more robust character of the markings towards the border, and Grunow the delicate puncta on the markings and their small size at the border. Habitat. — Richmond, Ya. (Ehrenberg, Kinker ! Cleve and Moller !) ; Nancoori (Schmidt, Cleve !) ; Sta Monica deposit (Weissflog! Grove!); locality 1 (Deby !); Hong Kong (Hardman!); Peruvian guano (Cleve ! Hardman !) ; Macabees guano (Firth !) ; Crescent City (Hardman !) ; Sea of Java (O’Meara !) ; Maryland (O’Meara !) ; Los Angelos deposit (O’Meara !) ; Cove Calvert county, Maryland (Greville !); Bay of Bengal (Macrae !); Sta Monica deposit, Patagonian guano (Cleve !); Marstrand (Kinker !). Yar. pundiformis, nov. C. gigas, var., Grun., ibid., 1884, p. 76. — Diam. *151 mm. Markings free towards the centre, from the semiradius to the border polygonal, punctiform, almost invisible markings at origin of shorter radii. — G. Woodwardii , Eul., var., Grun. in Sch., Atl., pi. lxv. fig. 2. According to Schmidt, this is intermediate between C. gigas and C. diorama, Sch. Habitat. — Aegina (Schmidt). Yar. diorama. Grun., ibid., 1884, p. 76. C. diorama, Sch., Atl., pi. Ixiv. fig. 2. — Diam. 45 to *25 mm. Surface slightly con- vex. Markings towards the central space rounded 4 in ’01 mm., increasing gradually but to a less degree outwards, subequai — 12^ in ’01 mm. — on outer half of valve, again decreasing gradually near the border. — C. gigas , var. Montereyi , Grun., ibid., 1884, p. 76. Grunow’s var. Montereyi differs only in having somewhat smaller markings, 3J in ’01 mm. Habitat. — Santa Monica deposit (Schmidt, Cleve !); Monterey deposit (Grunow, Cleve !); Isle of Muntok, Indian Archipelago (Kitton !); Pabillan de Pico guano (Cleve !); Patagonian guano (Cleve !). 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 543 Yar. duplicata. Grun., Hid., 1884, p. 76. — Diam. *0215 mm. Markings hexagonal, placed obliquely so that their lower and upper ends are united by oblique walls. — Cleve and Moll., Diat., No. 57. Habitat. — Shokoe Hill deposit, Richmond, U.S. (Grunow) ; Richmond Ya. (Greville ! Cleve and Moller !). Yar. calif ornica, nov. C. calif ornicus, O’Me. MS. — Diam. *22 mm. Markings towards the central space 6 to 8, increasing outwards to 4, again decreasing to 8, in *01 mm. at the border. Habitat. — “ Guano ” (O’Meara !); Medway (Dallas!).* Yar. guineensis, Rattray. C. guineensis , Grun., Denk. Wien. Ah., 1884, p. 76. — Diam. *05 to ‘1 mm. Central space distinct. Mark- ings hexagonal; towards the centre rounded, granular, 5J in *01 mm., not distinctly punctate ; the interspaces provided with small hexa- gonally arranged puncta. Habitat. — Brackish water, Lagos (Grunow). Yar. laxa, nov. C. guineensis, Grun., ibid., p. 76. — Diam. *13 mm. Central space round with irregularly placed puncta. Markings free. Habitat. — Monterey deposit (Grunow). C. Janiscliii. Sch., Atl., pi. lxiv. figs. 3, 4. — Diam. *16 to' '245 mm. Central space subcircular, hyaline, to of diam. broad. Markings faint, but sharply defined, on a narrow band at the border, obtusely angular, increasing but slightly outwards, to 4 in -01 mm., non-punctate, the central dots indistinct ; rows radial, secondary rows obscure. — Grun., Denk. Wien. Ak., 1884, p. 76; Janisch., Gazelle Exped., taf. iv. figs. 3, 5; C. marginatus , Janisch. ( non Ehrb.), Abh. Sch. Ges. vater. Cult., 1882, p. 3, pi. i. a. fig. 20. This species has sometimes been confounded with C. gigas. Habitat.— 11 Guanos ” (Grunow ! Macrae ! Firth !) ; Ichaboe guano (Janisch, Joshua! Greville!); Felso-Eszterg&ly, Kekko, Szent Peter deposits (Pantocsek !) ; Peruvian guano (Firth !);f soundings, Gulf of California (H. L. Smith !) ; J Richmond (O’Meara !) ; Chincha guano (Grove !) ; Saldanha Bay guano (Cleve, Greville !) ; Cambridge deposit (?) (O’Meara !) ; Java, (Cleve !). * In the .collection of Dr Greville. + In the collection of Dr F. W. Griffin. J H. L. Smith, Diat. Spec. Typ ., No. 91. 544 Proceedings of Royal Society of Edinburgh. [sess. Var. arafurensis. Grun., ibid., 1884, p. 76. — Diam. from ’29 to *425 mm. Markings increasing more from the central space outwards ; towards the centre 3^-, towards the border 2^ in ’01 mm. — O. arafurensis var., Cstr., Diat. Chall. Exped ., p. 153, pi. ii. fig. 4; C. craspedodiscus , O’ Me., Quart. Jour. Micr. Sci., 1877, p. 463. The presence of the large central space referred to by O’Meara in his Coscinodiscus craspedodiscus excludes it from Coscinodiscus craspedodiscus , Kiitz., where there is a central rosette, and identifies it with the present var. The hoop-like appearance of the valve under a low power referred to by Castracane is not shown in his figure {Diat. Chall. Exped ., p. 152-154, pi. iii. fig. 5). Habitat. — Arafura Sea, H.M.S. Challenger (Grunow, Weissflog ! Rae !) ; Gazelle Expedition (Weissflog !) ; Bay of Bengal (Macrae !) ; “ Atlantic Ocean ” (Cleve). C. entoleion. Grun. Sch., Atl., pi. cxiv. fig. 3. — Diam. *25 to •3 mm. Surface slightly convex near the border. Central space circular, to Jj- of diam. broad. Markings hexagonal, increasing from the central space outwards, again decreasing near the border ; towards the centre most delicate, 3 to 2-|, near the border 2 to 2-| in *01 mm.; oblique decussating rows evident. Border sharply defined, narrow ; striae 4 to 5 in -01 mm. Distinguished from C. gigas by the more delicate and smaller markings and the absence of narrow hyaline interspaces radiating outwards from the central space. Specimens have sometimes been confounded with C. perforatus , var. cellulosa , from which they differ by the marked increase of the areolae outwards. Habitat. — Thames mud, at Southend (Dickie !) ; Hungarian marl (Thum).* C. flexilis, sp. n. Sch., Atl., pi. cxiv. fig. 6. — Diam. T5 to T2 mm. Surface almost flat. Central space distinct, Jg- to of diam. broad, surrounded by an inconspicuous band of larger areolae. Markings polygonal, mostly hexagonal, increasing for a short distance outwards from the band surrounding the central space, again gradually decreasing towards the border ; towards the centre 4i to 5, about the semiradius 4, at the border 6, in *01 mm.; rows Fide A. Schmidt. 1888—89.] Mr John Rattray on the Genus Coscinodiscus. 545 straight, secondary oblique decussating rows evident. Border narrow, sharply defined ; strise faint, 6 in *01 mm. Distinguished from C. apiculatus by the more delicate markings that are devoid of prominent central papillae, and by the distinct central space. Habitat. — 1 (Griffin !) ; Chincba guano (Schmidt). C. conformis, sp. n. Sell., Ail., pi. cxiv. fig. 4. — Diam. ‘2 mm. Surface somewhat depressed at the centre. Central space circular, about -Jg- of diam. broad, surrounded by an inconspicuous band of areolae subobsolete on their central side. Markings 4- to 6-angled, without puncta at the angles, increasing gradually to the semiradius, again decreasing to the border ; towards the central space 4y, at the semiradius 3^, at the border 5 or 6 in *01 mm.; central papillae indistinct ; delicate puncta at the origin of the shorter rows ; secondary oblique rows short, substraight, evident. Border narrow ; striae 8 in ’01 mm. Habitat. — Arica (Schmidt). C. josejinus. Grun., Denk. Wien. Ak., 1884, p. 75, pi. iii. (C), fig. 16. — Diam. ‘08 mm. Surface convex. Central space small, subcircular. Markings smooth, decreasing but slightly near the border, 7 to 8 in ’01 mm. Border striae delicate, 14 in -01 mm., about -Jg- of radius broad, at its middle a sharp line concentric with the outer edge. Distinguished from C. radiosus, Grun., and C. fimbriatus, Ehrb., by the character of the border. Habitat. — Pranz Josef’s Land (Grunow). C. nobilis. Grun., Jour. Roy. Micr. Soc. Lond., 1879, p. 687, pi. i. fig. 1. — Diam. '375 to *54 mm. Central space distinct, hyaline. Markings minute, about 7 in ’01 mm., hexagonal towards the border, separated into obscure fasciculi by inconspicuous radial lines. — Cleve and Moll., JDiat., Nos. 145, 146, 162; C. regius , Grun., Sitzungsb. naturw. Ges. Isis., Dresden , 1878, p. 124. Sometimes mistaken for C. eoneinnus, but distinguished by its large central area and more distinct radial rows of markings. Habitat. — In Noetiluca, at Gorleston Pier, Suffolk, Harwich (Grunow !) ; Ascidia, Hull (Greville !) ; Hong Kong and Arafura Sea VOL. xvi. 29/10/89 2 M 546 Proceedings of Royal Society of Edinburgh. (Grunow ! Rattray !) ; Java Sea (Grunow ! Cleve and Moller !) ; lat. 4° 12' 7" H., long. 3° 57' 5" E, 1460 fms. (Rae !) ; Isle of Muntok, In- dian Archipelago (Kitton !) ; * Hancoori (Cleve and Moller !) ; Java (Cleve and Moller! Cleve!); lat. 4° 20' S., long. 105° 22' E. (Cleve!); surface, Gulf of Guinea, S.S. Buccaneer Exped. (Grove !). 0. Gazellce. Janisch., Jour. Roy. Micr. Soc., Lond ., 1879, p. 688. — Diam. 1*8 to 1*9 mm. Central space circular, about -0375 mm. broad, hyaline, bearing at its centre a group of irregular evident apiculi, and having at its boundary a circlet of similar apiculi at wide unequal intervals. Markings delicate, punctiform, 6 to 7 in *01 mm.; rows straight, short secondary transverse or oblique rows obvious ; adjacent to the border a distinct narrow (about '003 mm. broad) hyaline zone.t Border sharply defined, about -005 mm. broad, hyaline. — Ethmodiscus tympanum , Cstr., Diat. Chall. Exped ., 1886, p. 170, pi. xiv. fig. 3; E., sp. {frag- menta ) Cstr., ibid., p. 170, pi. xiv. figs. 4 a-c\ E. gigas , Cstr., ibid., p. 169, pi xiv. fig. 5. E. ivyvilleanus , Cstr. {ibid., p. 170, pi. xiv. fig. 6), differs only in having the angles of the valves rounded, and may be a var. of the present species. To the same var. belongs E. sphoeroidalis, Cstr., ibid., p. 170, pi. xxii. fig. 10; in the specimen figured divi- sion has recently been completed. Habitat — Gazelle sounding Ho. 125, lat. 30° 53' S., long. 177° 6' E., depth 4151 metres (Janisch!); and sounding Ho. 96, lat. 9° 57' S., long. 121° 52' E. (Weissflog!); H.M.S. Challenger, station 265, depth 2900 fms. (Grunow !) ; Hottingham deposit in fragment (Grunow). C. imperator, Janisch MS. — Diam. ? Central space and rosette 1 Markings minute, delicate, angular, 8, towards the border recog- nised with greater difficulty, 10 to 12 in *01 mm.; rows straight ; the oblique decussating rows faint ; hyaline band adjacent to the border absent. Border narrow, hyaline. — (PI. I. fig. 5.) Habitat. — Gazelle Expedition, sounding Ho. 96, lat. 9° 57' S., long. 121° 52' E. (Weissflog!). G. praetor, Grove MS. — Diam.1? Surface flat. Central space + This also occurs in C. rex. In the collection of E. Grove. 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 547 absent, a distinct central area *04 mm. broad, bounded by a narrow ring of closely disposed minute apiculi. Markings angular recognised with difficulty ; within the central area 6, at the borde 8 to 9 in *01 mm.; secondary oblique decussating rows short, obvious; no hyaline band at the border. The connecting zone bears straight parallel rows of short striae, 10 in '01 mm., the rows separated by narrow hyaline lines. — (PI. III. figs. 2 and 3.) Habitat. — “Barbados” (Grove !). C. punctatus. Ehrb., Mon. Ber. Ak ., 1884, p. 78. — Elliptical or subdiamond-sbaped, major axis -0625 to T2 mm., If to If times minor. Central space small, circular sometimes bearing a few isolated round granules. Markings rounded, granular, 6 to 8 in *01 mm.; interspaces hyaline, largest towards the centre, towards the border more crowded ; rows radial, straight ; oblique decus- sating; rows evident near the border. Border indistinct on inner side ; striae delicate, 8 to 10 in *01 mm. — Ehrb., Mikrog ., pi. xviii. figs. 40, 41 ; Ralfs in Pritch. Inf., p. 830 ; H. L. Sm., Diat. Spec. Typ ., No. 97 ; Cleve and Moll., Diat., No. 57. This species is sometimes mistaken for C. lewisianus, but it differs in the arrangement of the markings. It differs from Cestodiscus ovalis, Grev., by the absence of obtuse submarginal processes. In the original definition Ebrenberg gives the markings as 12 to 13 in *01 mm. Kitton has found frustules with very dissimilar valves in the Richmond deposit, Virginia. In one case one valve was normal, whilst the second showed a large rounded central area, with round isolated granules disposed without order, the zone adjacent to the border only appearing normal. Grove inclines to associate this form with Adinocyclus, as bis forms in H. L. Smith’s series show pseudonodules. Habitat. — Richmond, Va. (Greville ! Bailey ! * Grove ! Kitton ! Cleve and Moller!); Nancoori (Hardman!); Crescent City, Cal. (Weissflog !); Paita deposit, Peru (H. L. Smith !). Var. rhombica, nov. C. rhombicus, Cstr., Diat. Chall. Exped., p. 164, pi. xxii. fig. 11. — Major axis T19 mm., about 2f times * In the collection of Dr Greville. 548 Proceedings of Royal Society of Edinburgh. [sess. minor. Central space absent. Markings without order, but around the border smaller, and forming short radial lines. Habitat. — Sea of Japan, H.M.S. Challenger (Castracane). C. reniformis. Cstr., Diat. Chall. Exped ., p. 160, pi. xii. fig. 12. — Reniform, somewhat broader at one end than at the other. Major axis *1725 mm., about 2|- times the greatest breadth. Central space and rosette absent. Markings polygonal, gradually increasing from the centre outwards ; towards the centre 8, at the border 6, in *01 mm.; rows straight. Grunow correctly notes (Bot. Centralb ., Bd. xxxiv. p. 40) that Janisch had named this form Stoschia admirabilis in his still un- published manuscript of his report on the Diatoms of the Gazelle Expedition. Habitat. — ? (Castracane). C. sarmaticus. Pant., Fossil. Bacil. Ung., p. 74, pi. viii. fig. 62. — Elliptical, major axis ’016 to ‘025 mm., about \\ times minor. Central space and’ rosette absent. Markings delicate, punctiform, least crowded and most evident towards the centre; radial rows obscure ; apiculi 2, minute, inserted at the extremities of the minor axis and close to the border, sometimes absent. Border narrow but distinct, smooth. Habitat. — Dolje deposit (Pantocsek !). G. biangulatus. Sch.,^.,pl. lxiii. fig. 1 3. — Diam. *125 to *175 mm. Surface slightly convex. Central space and rosette absent. Mark- ings decreasing gradually from the border outwards ; towards the centre 2 J to 3, towards the border 4, in *01 mm ; central papillae distinct ; rows straight, secondary oblique curved decussating rows evident. Border sharply defined, from to TTg of radius broad, its inner edge with 2 unsymmetrical deep constrictions at an interval from each other about equal to the radius ; striae coarse, moniliform, 4 or 5 in *01 mm. — C. asteromjohalus , var. biangulata , Cleve and Moll., Diat., No. 215. Habitat. — Nottingham deposit, Md. (Schmidt, Cleve and Moller !); Bermuda tripoli (Greville !) ; Calvert, county Md. (Cleve); Nagy- Kurtos deposit, Hungary (Rae ! Deby !) ; Moron deposit (Grove). 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 549 C. asteromphalus. Ehrb., Mon. Ber. Ak., 1844, p. 77. — Diam. *085 to '3 mm. Surface slightly depressed at the centre, and convex towards the border. Central space small, obtusely angular, surrounded by a distinct rosette. Markings polygonal, robust, punctate, 3J to 4 in *01 mm., increasing slightly towards the border to 2J or 3 in ’01 mm.; central dots distinct. Border distinct; striae obvious, coarse, about 4 in *01 mm. — Ehrb., Mikrog ., pi. xviii. fig. 45 ; pi. xxxiii. 15. fig. 7 ; Ralfs in Pritch. Inf., p. 828; Grun., Denk. Wien. Ak ., 1884, p. 78; Pant., Fossil. Bacil. Ung ., p. 71; Sell., Atl ., pi. cxiii. fig. 23; Van Heurck, Typ. Syn. Diat. Belg., No. 508; Cleve and Moll., Diat., Nos., 57, 164 ; Janiscb, Gazelle Exped., taf. iv. fig. 9 ; C. asteromphalus, var. genuina , Grun., Denk. Wien. Ak., 1884, p. 78 ; C. asteromphalus, var. conspicua, Grun.; Van Heurck, Syn. Diat. Belg., pi. exxx. figs. 1, 2, 5, 6; Grun., Denk. Wien. Ak., 1884, p. 78 ; sp. n. 1 Sell., Atl., pi. lxiii. fig. 5. This species is distinguished by the evident puncta on the markings. Grunow has observed in Richmond specimens a finely punctate layer detached. The places corresponding to the angles of the markings have coarse, mostly triangular dots, the round central dots are only found in perfect valves. Habitat. — Richmond (Ehrenberg, Cleve and Moller !) ; and Holies Cliff deposit, Va. (Ehrenberg) ; guano (Grunow) ; Sta Monica deposit (Weissflog! Rae !) ; Fernando Noronha guano (Rattray !) ; New York, in the sea (Grunow) ; Arica (Schmidt). Var. eximia, Grun., ibid., 1884, p. 78. — Diam. '475 mm. Central space irregular, small. Markings increasing slightly outwards from central rosette to about f of radius, 2 to 2J in '01 mm. — C. aster- omphalus, Ehrb; Sch., Atl., pi. lxiii. fig. 12. Habitat. — Santa Monica deposit (Schmidt, Rae ! Deby !). With type (Grunow). Var. omphalantha. Grun, ibid., 1884, p. 78; C. omphalanthus, Ehrb., Mon. Ber. Ak., 1844, p. 266. — Diam. *255 to '45 mm. Central space sometimes absent, rosette conspicuous or less obvious. Markings subequal to about of radius, 2J in '01 mm., thence decreasing gradually to the border, sometimes slightly smaller towards the central rosette. Border somewhat sharply constricted 550 Proceedings of Royal Society of Edinburgh. [sess. at two somewhat distant points. — C. omphalanthus , Ehrb.; Ealfs in Pritch. Inf. , p. 828 ; Cleve and Moll., Diat., Nos. 57, 215 ; Ehrb., Diat. Spec. Typ ., No. 6 (excl. Sch., pi. lxiii. fig. 2). The constriction at the border is similar to that of C. biangidatus (Sch., Atl ., pi. lxiii. fig. 13), but the latter is devoid of a rosette, and the dark band at the border is relatively much wider. This var. is frequently mistaken for C. ocidus-iridis, and sometimes for C. borealis. Habitat. — “Bermuda” (Eulenstein ! * Greville !) ; Nottingham deposit (G. M. Brown ! Firth ! Cleve and Moller ! O’Meara ! Cleve! Hardman !);f Bichmond, Ya. (O’Meara! Cleve and Moller!); Maryland (O’Meara! Hardman!); “Virginia” (Hardman!); Calvert, county Maryland (Kinker ! Weissflog!); Kekko deposit (Grove l); Rappahannock (Greville !) ; Piscataway deposit (Cambridge ! f Greville !) ; Delaware, U.S. (Cambridge !) ; f Patuxent earth (Rae !) ;f Holland’s Cliff (Cleve !) ; San Benito deposit, California (Grove !). Yar. brightwellioides. Grun., ibid ., 1884, p. 78. — Diam. T125 to *26 mm. Surface rising from the centre to the semiradius, thence descending to the border. Markings polygonal, gradually increasing outwards to the semiradius, where there is a distinct circlet of large areolae, 2J in *01 mm. at right angles to the radius, thence decreas- ing to the border; at the centre 3J, at the semiradius 3, at the border 3 in *01 mm. — Pant., Fossil. Bacil. Ung ., p. 71, pi. xvii. fig. 155. The circlet of larger markings forms a transition to C. bidliens , Sch., and to the genus Brightivellia. Habitat. — Santa Monica deposit (Grunow, Kinker !) ; Szakal and Szent Peter deposits (Pantocsek). Yar .pulclira. Grun., ibid., 1884, p. 78. — Diam. -36 mm. Central space smooth, *01 mm. broad, surrounded by a distinct band of larger markings. Markings 2 to 2J in *01 mm., decreasing quite close to the border. * As to this locality, Eulenstein notes — “Locus dubius, verisimiliter ad pagum hujus nominis Americse borealis non ad insulas referendus. ” t In the collection of Dr Griffin. 1888-89.] . Mr John Rattray on the Genus Coscinodiscus. 551 This var. approaches var. princeps in the character of the central space, but is distinguished by its larger markings. Habitat. — Santa Monica deposit (Grunow). Var. macrantha. Grun., ibid., 1884, p. 78. — Diana. *45 mm. Central space still larger, ’026 mm. broad, round, smooth, or punctate. Markings subequal, 4 in ‘01 mm. Habitat. — Found on a mass of diatoms floating on the Elbe by Moller (fide, Grunow). Var. princeps. Grun., ibid., 1884, p. 78.- — Diam. '5 mm. Central space subcircular, *025 mm. broad. Markings decreasing outwards to about semiradius thence increasing slightly towards the border, 5 to 6 in ‘01 mm. — Van Heurck, Syn. Diat. Belg., pi. cxxviii. figs. 1-3. Separated from the preceding by its smaller markings. Habitat. — With var. macrantha (Grunow, Van Heurck). Var. pabellanica. Grun., ibid., 1884, p. 79. — Diam. T45 to *16 mm. Central space "0065 mm. broad, the surrounding band of markings evident, but less prominent than in vars. princeps and macrantha. Markings 5 in *01 mm., somewhat smaller at the border, faintly punctate. — C. asteromphatus, var. pabellana, Grun. in Van Heurck, Syn. Diat. Belg., pi. cxxviii. fig. 5. Habitat.— Pabellan de Pico guano (Grunow) ; Peruvian guano (Hardman !).* Var. hybrida. Grun., ibid., 1881, p. 79, pi. iii. (C) fig. 9. — Diam. •175 to '35 mm. Surface convex toward the border. Central rosette evident, rarely indistinct. Markings 4 in *01 mm., somewhat smaller close to the rosette than about the semiradius thence gradually decreasing towards the border to G in *01 mm., sometimes indis- tinctly punctate, a circlet of minute apiculi sometimes present at the border. — Pant., Fossil. Bacil. TJng ., p. 71 ; Sch., Atl., pi. cxiii. fig. 22 ; C. centralis, Sch., Atl., pi. Ixiii. fig. 1. Habitat. — Felso-, Esztergaly, Kekko and Szent Peter deposits (Pantocsek) Hvidingsoe, N. Sea, Cuxhaven, Marahon mouth, Davis In the collection of Julien Debv. 552 Proceedings of Boyal Society of Edinburgh. [sess. Straits, Franz Josefs Land (Grunow); “ Yszee” (Kinker !); Melville Bay (Griffin !) ; Los Angelos deposit (O’Meara !). C. bisinuatus. Sch., Atl., pi. lxiii. figs. 14, 15. — Diam. from *106 to *145 mm. Surface showing 2 unsymmetrical depressions, convex inwards, close to the border. Central space absent ; rosette distinct, composed of few (4 to 6) areolae. Markings polygonal, 4 in *01 mm., decreasing for a short distance from the rosette, thence subequal, and again decreasing near the border ; close to the border 7 in *01 mm. Approaching C. oculus-iridis and C. centralis. Habitat. — North Celebes (Grunow, Schmidt). C. Weyprerhtii. Grun., Denk. Wien. Ah., 1884, p. 78, pi. iii. (C), tig. 8. — Diam. *094 mm. Surface convex. Central space minute, rosette small. Markings gradually decreasing outwards ; towards the centre 5J, towards the border 7, in *01 mm., each with a distinct central dot and minute puncta, 20 in .01 mm. Border broad, of two portions, the inner with delicate striae 15 in '01 mm., the outer hyaline. This species is readily distinguished by its border. Habitat. — Franz Josefs Land (Grunow). C. undulans, Rattray, C. undulatusf Cstr., Diat. Chall. Exped ., p. 159, pi. viii. fig. 3. — Diam. *390 mm. Surface with two con- centric elevations — one near the centre, the second close to the border. Central space irregular, about of diam. broad. Markings polygonal, largest on the elevations, increasing gradually from the central space to the inner edge of the inner elevation, decreasing suddenly at its outer edge, and again increasing to the inner edge of the outer elevation, smallest towards the border, about the semi- radius If to 2 in *01 mm. ; rows indistinct upon the elevations, secondary oblique decussating rows evident. Habitat. — Pacific Ocean, H.M.S. Challenger (Castracane). C. convexus. Sch., Atl ., pi. lx. fig. 15. — Diam. T mm. Central space absent. Markings hexagonal, distinctly punctate, central * Name preoccupied by Cleve ( Kongl . Sv. Vet.- Ale. Handl. StocTch. , 1881, No. 5, p. 20). 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 553 dots minute ; towards the centre 3, towards the border 5, and at the border 8, in *01 mm.; secondary oblique decussating rows evident. Habitat. — Springfield deposit, Barbados (Schmidt, Grunow, Hardman !). Var. bengalensis. Grim., Denk. Wien. Ah., 1884, p. 73. — Diam. •08 to T‘4 mm. Markings smaller ; towards the centre 4, at the border 10, in *01 mm. Habitat. — Brackish water, coast of Bengal (Grunow). Yar. diminuta. Grun., ibid., 1884, p. 73. — Markings still more minute ; towards the centre 6 in *01 mm., less distinctly punctate. This var. approaches G. radiosus, Grun. Habitat. — Brackish water, coast of Bengal (Grunow). C. fimbriatus. Ehrb., Mon. Ber. Ah., 1844, p. 78. — Diam. -1 to -1125 mm. Central space and rosette absent. Markings poly- gonal ; towards the centre 4, decreasing gradually outwards to 6 or 7, in *01 mm.; radial rows most evident towards the border. — Ehrb., Mihrog., pi. xxii. fig. 2 ; Ralfs in Pritch. Inf., p. 829 ; Grun., Denh. Wien. Ah., 1884, p. 74. C. radiolatus, Ehrb., Sch. Ati, pi. lx. fig. 11. Distinguished from C. radiosus, Grun., by its larger markings, which decrease less rapidly outwards in Grunow’s species. Habitat. — Caltanisetta (Ehrenberg) ; Sicilian marls (Grunow) ; Los Angelos deposit, Cal. (O’Meara !). Var. subradiata, nov, C. fimbriatus, var., Yan Heurck, Syn. Diat. Belg., p. cxxxi. fig. 2. — Diam. -063 mm. Markings 7, increasing gradually outwards to 6, in ’01 mm., about f of radius from the centre, again diminishing more rapidly to the border. Grunow, followed by Pantocsek, have united this to the species. The affinities to C. radiatus are striking. Habitat. — Oran deposit (Yan Heurck). Yar. californica. Grun., ibid., 1884, p. 74. — Diam. ’07 to -09 mm. Centre, often with small irregular puncta, one frequently larger than the others, and like the rudiment of a bristle. Markings 554 Proceedings of Royal Society of Edinburgh. [sess. towards the centre 5 to 6 in *01 mm., often smaller than nearer the border, at the border 8 to 9 in -01 mm. Habitat. — California deposits, San Diego (Grunow). C. obversus , sp. n. Sch., Atl., pi. lx. fig. 14 (no name). — Diam. *075 to ‘105 mm. Central space absent or subobsolete ; rosette distinct. Markings polygonal, subequal or decreasing slightly for about -§■ of radius, thence decreasing more rapidly to the border ; towards the centre 2 to 2J, at the border 8, in '01 mm.; central papillae distinct ; rows sometimes obscurely fasciculate ; on the outer J of the radius the rows are separated by clear radial lines. Border narrow, distinct. From C. fimbriatus this is distinguished by the larger more robust markings and their more rapid decrease from the centre outwards. Habitat. — Hong Kong (Hardman!); marsh ground, Wedel (Schmidt) ; Cambodia (Hardman !). Var. tenuior , nov. — Diam. ’0625 mm. Central space and rosette absent. Markings subequal to about |- of radius, thence decreasing to the border ; at the centre 6, near the border 10, in '01 mm. Habitat. — Kio Janeiro (Hardman !). C. grandineus , sp. n. Sp. aff. C. concinno , Sch., Atl., pi. lx. fig. 16. — Diam. "075 to *19 mm. Surface moderately convex, flat on a narrow zone adjacent to the border. Central space absent; rosette distinct, sometimes inconspicuous. Markings polygonal, decreasing gradually from the centre outwards; towards the centre 4, towards the border 8, in '01 mm.; rows straight; secondary oblique decussating rows manifest; minute apiculi at intervals of about *006 mm., sometimes present at the border. Border with inner edge indistinct ; strise 4 to 6 in '01 mm. This species differs from C. concinnus, var. jonesiana, by the absence of apiculi and processes, and from C. aster omphalus, var. hybrida, by the absence of a zone of somewhat larger markings about the semiradius than nearer the centre. Habitat. — Dredged off Heard Island, in 75 fathoms, by H.M.S. Challenger (Bae !) ; dredged in Boyal Sound, Kerguelen, 28 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 555 fathoms, by H.M.S. Challenger (Rae !) ; Bally brack (O’Meara !) ; Nottingham, Maryland, (O’Meara !) ; Cambridge deposit, Barbados (Johnson !).* Yar. dentata , nov. — Diam. ’1075 mm. Central rosette small ; markings towards the centre 4, at the border 6, in *01 mm.; apiculi large, at subequal intervals of about -006 mm. Habitat. — A (Greville !) ; Barbados deposit (Johnson !).* C. centralis , emend.; G.? centralis. Ehrb., A bh. Ber. Ak ., 1838, p. 129. — Diam. -12 to *255 mm. Central space absent or minute; a rosette obvious. Markings variable in size, subequal — 4 to 4J in ‘01 mm. for about of the radius, thence decreasing gradually outwards on the outer sometimes 5 in ‘01 mm., and decreasing gently from the semiradius ; rows straight, sometimes subfasciculate towards the border, the fasciculi separated by narrow clear areas proceeding inwards from the apiculi ; apiculi delicate, inserted at the border at intervals of about ‘006 mm., 2 larger unsymmetrical, at an interval from each other somewhat greater than the semiradius. Border narrow; striae 6 in ‘01 mm. — Ehrb., Mon. Ber. Ak. 1844, p. 78 ; Mikrog ., pi. xviii. fig. 39; pi. xxii. fig. 1; Greg., Trans. Roy. Soc. Edin ., 1857, p. 501, pi. xi. fig. 49 ; Ralfs in Pritch. Inf., p. 828 ; Grun., Sitzungsb. naturw. Ges. Isis , Dresden , 1878, p. 123 ; Van Heurck, Syn. Diat. Belg., pi. ciii. fig. B ; H. L. Smith, Diat. Spec. Typ., Nos. 91, 92 ; Cleve and Moll., Diat., Nos. 57, 164, 207, 215 ; C. aster omphalus, var. centralis, Grun., Denk. Wien. Ak., 1884, p. 79; C. centralis, var. micraster, Grun.; Cleve and Moll., Diat., No. 172 ; G. centralis , var., Cstr., Diat . Chall. Exped. , p. 155, pi. ii. fig. 3 ; G. centralis forma minor, Van Heurck., Typ. Syn. Diat. Belg., No. 531 (excl. G. centralis ,f Weisse, Bull. Acad. Imp St Petersb., 1867, p. 122, pi. i. fig. 18 ; G. centralist O’Me., Proc. Roy. Irish Ac. 1875, p. 260, pi. xxvi. fig. 19; G. centralis, § Sch., Atl., pi. lxiii. fig. 1 ; C. centralis, Ehrb., Mikrog., pi. xxi. fig. 3). This species is sometimes confounded with C. oculus-iridis, G. * In the collection of Dr Greville. t This is C. radiatus, Ehrb. X This is probablly G. subtilis, as stated by Grunow. § This is C. . aster omphalus, var. hybrida, Grun . 556 Proceedings of Royal Society of Edinburgh. [sess* radiatus , and C. condnnus. In balsam preparations the apiculi are hardly visible. It has been employed along with Rhizosolenice and Denticellce by Max Schultze in 1859 in studying the movements of the granules in the interior of the frustule. In Ehrenberg’s de- finition, the markings are given as about 6 in ,01 mm., as in some forms of C. radiosus, Grun. Castracane erects a var. solely on the presence of a striated border — a character clearly shown in Ehren- berg’s figure (MiJcrog., pi. xviii. fig. 39). Some specimens named by Van Heurck Coscuiodiscus centralis forma minor (Typ. Syn. Diat. Belg ., No. 531), from the Baltic, may belong rather to C. oculus-iridis , owing to the increase in the size of the markings out- wards ; C. centralis is readily distinguished from C. aster omphalus by the presence of the 2 unsymmetrical apiculi, and cannot be united with it in the same species as proposed by Grunow. The two processes were first observed by Mr E. Grove, F.R.M.S. Habitat. — Oran and Caltanisetta deposits (Ehrenberg); Glenshira sand (Gregory !) ; Cherbourg (H. L. Smith !) ; grey mud dredged in 569 fathoms, lat. 63° 40' N., long. 5° 28' E. (Ehrenberg) ; Richmond (Ehrenberg, Cleve and Moller !) ; some guanos (Grunow); South Sea, Lagos, North Sea, Baltic (Grunow) ; Heligoland (Schultze!);* Ascidia , Hull (Gregory!* Greville !) ; Loch Fyne (Gregory ! Greville !) ; Lamlash (Gregory ! Greville ! Dickie !) ; Firth of Forth at Granton (Rattray!) ; Melville Bay (Deby ! Barnett !) ; f soundings, Gulf of California (H. L. Smith !) ; Bally brack (O’Meara !) ; from Laminaria saccharina , Dalkey (O’Meara !) ; Ascidia , Dublin Bay (O’Meara !) ; Bermuda tripoli (Greville!); Euur deposit, Jutland (O. N. Witt!); North Atlantic, lat. 51° 20' N., long. 52° 25' W., 232 fathoms (O’Meara!); Richmond Tunnel (O’Meara !) ; Davis Straits (O’Meara !) ; Liim- fiord, Jutland (Hardman!); Patos Island guano (Hardman !) ;J Virginia (Hardman !);J Oran deposit (Deby!); Malahide (O’Meara!); Behring Sea, 1681 fathoms (H. L. Smith!); surface, Hong Kong Harbour, H.M.S. Challenger (O’Meara !) ; Monks- town, in tide pools (O’Meara !) ; Ascidia , Belfast (O’Meara !) ; Thames, at Sheerness (Grove !) ; Spezzia (Kinker !) ; Los Angelos * In the collection of Dr Greville. t In the collection of Dr Griffin. + In the collection of Julien Deby. 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 557 deposit (O’Meara !) ; Baltic Sea (Van Heurck !) ; Teignmouth Grove !) ; Peruvian guano (Cleve ! Macrae !) ; Sta Monica deposit (Cleve and Moller !) Sta Maria deposit (Grove !) ; Davis Straits, Patagonia ; Nottingham deposit, Md. (Cleve and Moller !) ; Moller’s Elbe material (Cleve!); Patagonia, 1375 fathoms, H.M.S. Challenger (Cleve !) ; Faroe Channel (Grove !) ; west coast Florida, U.S. Survey (Febiger !) ; Yeddo, Bohuslan, Kusu, Pabillan de Pico guano ; Greenland ; Successful Bay, Kerguelen (Cleve !). C. Jloridulus. Sch., MfZ., pi. cxiii. fig. 16, a , b, c. j — Diam. •1135 mm. Surface flat. Central space irregular, small, about 2^- of diam. broad. Markings polygonal, largest round the central space, thence decreasing slightly outwards ; towards the centre 4, towards the border 6, in *01 mm.; minute puncta at the origin of the shorter rows ; rows substraight ; secondary oblique decussating rows evident ; minute apiculi at angles of areolae (seen in oblique aspects). Border narrow. Distinguished from C. obscurus by the size of the markings and character of the border. Habitat. — Sta Monica (Schmidt). C. incequisculptus, sp. n. Roundly elliptical; major axis *175 mm., about l-Jy times minor. Surface slightly convex towards the centre. Central space absent ; rosette distinct. Markings hexagonal, and increasing slightly outwards to about the semi- radius, 4 to 4J in ‘01 mm. in radial rows ; beyond the semiradius large, unequal, and without order ; central papillae of the smaller faint, in the larger a faint elevated central ridge. Border indistinct ; striae 6 in *01 mm. — (PI. I. fig. 17.) Habitat. — Moron deposit (Greville !). C. megacentrum. Grove, MS. — Diam. -12 to -125 mm. Surface somewhat convex. Central space and rosette absent. Markings angular, 1J to 3 on the central area, which extends to J or J of the radius, beyond this 2J in *01 mm., and increasing gradually outwards to about § of radius, thence decreasing to the border ; the central papillae prominent; irregular on the central area, beyond this in straight rows, with secondary oblique decussating 558 Proceedings of Royal Society of Edinburgh. [sess. rows manifest. Border narrow ; striae evident, 6 to 8 in *01 mm. —(PI. II. fig. 13.) Habitat. — Oamarn deposit (Grove !). C. secernendus. Sch;, Atl., pi. cxiv. fig. 1. — Diam. -272 mm. Surface flat, towards the border convex. Central space absent; rosette distinct. Markings hexagonal, towards the border obtusely angular with long axes radial, pearly, decreasing gradually outwards; towards the centre 1J to 2, towards the border 2 to 2J, in •01 mm. Central papillae prominent, those on the markings near the border placed towards their central side ; secondary oblique decussating rows distinct, towards the border coarsely submoniliform, and separated by more distinct clear lines. Border narrow; striae 8 to 10 in *01 mm. The appearance of the markings recall those of Aulacodiscus margaritaceus , var. Kinkeri. Habitat. — Maryland (Thum). C. moravicus, Grun. Sch., Atl., pi. cxiv. fig. 2. — Diam. *24 mm. Surface convex. Central space small ; rosette distinct. Markings hexagonal, obscurely punctate, towards the border submoniliform, increasing slightly outwards for some distance from the central space, again decreasing towards the border ; towards the centre 2 J, at the border 5, in. *01 mm.; central papillae faint, rows towards the border separated by evident clear lines ; secondary oblique decussating rows evident. Border narrow, hyaline. Distinguished from C. aster omphalus by the size and appearance of the markings. Habitat. — Hungarian marl (Thum). C. borealis. Bail., Amer. Jour. Sci., 1856, p. 3. — Diam. T5 to *25 mm. Surface slightly depressed at the centre, somewhat convex towards the border. Central rosette usually distinct, of 6 to 8 large areolae, sometimes subobsolete. Markings increasing regularly from the central rosette outwards ; towards the rosette 3 to 3 J, near the border 2, in. -01 mm.; central papillae distinct, the division lines composed of rounded granules; secondary oblique rows distinct. Border distinct, dark, with close irregular coarse striae. — Balfs in 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 559 Pritch’Inf.j p. 828; Sell., Atl., pi. lxiii. fig. 11; H. L. Sm., Diat. Spec. Typ., Kos. 90, 93, 95; C. oculus-iridis, var. borealis, Cleve, Vega Exjped., Vetensk. Jakttag. Stockh., Bd. iii. 1883, p. 488; Grim. Denlc. Wien. Ak., 1884, p. 77 (excl. C. borealis, Ehrb., Mon. Ber. Ak., 1861, p. 294). Distinguished from C. oculus-iridis by the coarser and more robust markings. Kitton’s Holies cliff specimens sometimes show under low powers 4 minute elevations at the centre, indicating a transition to C. excavatus and C. asteroides. Habitat.' — Kamstchatka Sea, 1700 fathoms (Bailey !); Kurile Islands, 1329 fathoms (H. L. Smith !); Nottingham deposit (Kitton!); Sta Monica deposit (Weissflog, Type plate); California (Firth,* Grunow); Hong Kong (Weissflog !) Japan (Deby!) ; Napar- ima, Trinidad (Johnson !); Sta Barbara deposit (Kinker !); Cambodia (Hardman !); f Behring Sea, 1681 fathoms (H. L. Smith !); Holies cliff (Kitton !); Barbados (Johnson !); Cape Wankarema (Cleve). C. inegajporus. Ehrb., Mon. Ber. Ak., 1861, pp. 280, 294. — Diam. about '23 mm. Surface slightly convex. Central rosette absent. Markings subequal, 2|- to 3 in *01 mm.; near the border smaller, central papillae small. Border broad. Probably, as stated by Ehrenberg, a var. of C. borealis. Habitat. — Lat. 60° 40' N. long. 29° W. 1000 fathoms ; lat. 62° 6' N., long. 32° 21' W., 1540 fathoms; lat. 59° 12' N., long. 50° 38' W., 1833 fathoms (Ehrenberg). C. oculis-iridis. Ehrb., Abh. Ber. Ak., 1839, p. 147. Diam. T35 to *3 mm. Central rosette distinct, sometimes small. Markings polygonal, n on-punctate ; towards the rosette usually smaller, from 3 to 4, thence increasing gradually outwards to 2J in •01 mm., towards the border again decreasing to 5 or 6 in ’01 mm.; central papillae sometimes prominent ; secondary decussating rows well marked. Border narrow ; striae 6 in ’01 mm. — Ehrb., Mikrog., pi. xviii. fig. 42; pi. xix. fig. 2; Jan., Abh. Schl. Ges. vater. Cult., 1862, Heft ii. p. 3, pi. i. B. lig. 6 ; pi. ii. A. fig. 4; Gazelle Exjped., taf. ii. fig. 2 ; Sch., Atl., pi. lxiii. figs. 6, 7, 9 ; pi. cxiii. figs. 1, 3-5, * In the collection of Dr F. W. Griffin, f In the collection of Julien Deby. 560 Proceedings of Royal Society of Edinburgh. [sess. 20 ; Raben., Alg. Europ ., Nos. 2487, 2558 ; Cleve and Moller, Diat ., Nos. 3, 57, 162, 215, 258, 259, 276, 319 ; C. ocidus-iridis , var. genuina , Grun., Denk. Wien. Ah, 1884, p. 77 ; C. centralis, Ehrb., Mikrog., pi. xxi. fig. 3 ; H. L. Sm., Diat. Spec. Typ., No. 92 ; C. omphalanthus, Gran., in Sell., Atl., pi. lxiii. fig. 2. Grunow believes that C. oculus iridis, var. ? pacifica, may belong to C. aster omphalus, the pnnetation of the markings having escaped Schmidt’s observation. The specimen figured by Schmidt (Atl., pi. cxiii. fig. 1) shows a less obvious increase of the markings outwards, and a more distinctly marked zone adjacent to the border ; but gradations occur to such forms, as shown in fig. 3 of same plate. Habitat. — Franz Josef’s Land (Grunow !); Cherbourg (H. L. Smith!); Liimfiord, Jutland (Hardman!); mud from Gluckstadt, Elbe above Cuxhaven (Rabenhorst and Schwarz !); Aegina (Schmidt); Bohuslan, and mud from Elbing, West Prussia (Cleve !); Java (Schmidt, Cleve and Moller ! O’Meara !); Japan (Deby !) Inland Sea, -Japan, H.M.S. Challenger (Bae !) ; Arafura Sea, H.M.S. Challenger ! (Doeg !); Japan oysters (Doeg !); Hong Kong (Deby ! Hardman ! Greville ! Grove ! Firth !); surface, Hong Kong Harbour, H.MkS. Challenger (O’Meara !) ; Isle of Muntok, Indian Archipelago (Grove !); Sand Heads, Bay of Bengal (Macrae !) ; shell cleanings, Singapore (Hardman !); Ceylon (Macrae !) ; edible sea- weeds, India (Macrae !); East Indies (Macrae !); Spitzbergen, Santos (Clevej; Cape WTankarema (Cleve and Moller!); Greenland (Cleve !) ; Penang Plarbour (Bae !) ; Kusu (Cleve ! O’Meara !) ; H.M.S., Challenger, lat. 32° 31' N., long. 135° 39' E., 1675 fathoms (Bae !) ; Atlantic Telegraph soundings (Boper !) ; North Atlantic, lat. 51° 20' N., long. 52° 25' W., 232 fathoms (O’Meara!); “Atlantic sounding” (Weissflog!); Sea of Kamstchatka, 1700 fathoms (Bailey!); Nancoori, Pensacola, Trinidad, California (Cleve and Moller !) ; Cambodia (Hardman !) ; marine deposit, Fiji Islands (Grove !) ; Patagonian and Ichaboe guanos (Janisch) ; Lobos di Afuera guano (Grove !) ; Peruvian guano (Hardman ! Cleve ! Janisch) ; Bolivian guano (Cleve ! Greville !) ; “guano ” (O’Meara !) ; Patos Island guano (Greville !) ; Californian guano (Greville !) ; Baytha, Elesd, Also-, Felso-, Esztergaly, Kekko, Mogyorod, Szakal, Szent-Peter and Dolje deposits (Pantocsek !) ; Moron deposit (Greville !) ; Brlinn Tegel (Cleve !) ; Mors (Cleve) ; Monterey (Stokes !) ; Santa Maria 1888-89.] Mr John Rattray on the Genus Ooscinodiscus. 561 deposit (Kinker !) ; Mejillones (Cleve ! O’Meara !) ; Santa Monica (Schmidt, Grove ! Kinker !) ; Nottingham deposit, Md. (Greville ; Cleve and Moller !) ; “Maryland” (O’Meara!); Richmond tunnel (O’Meara!); Richmond, Ya. (Cleveand Moller!); Bermuda tripoli (Greville!); “Barbados” (Johnson!); Cambridge deposit, Bar- bados (Johnson!) ; Holland’s Cliff (Cleve !) ; Marstrand (Kinker !). Yar. morsiana. Grun., ibid., 1884, p. 77. — Diam. *23 mm. Surface slightly depressed towards the centre, the highest zone about | of radius from centre. Central rosette distinct. Markings robust, increasing from the centre for § of radius, thence diminish- ing to the border ; towards centre 3, on highest zone 2 to 2 \, in ‘01 mm.« — Sch., Atl., pi. lxiii. fig. 9 (no name) ; C. asteromphalus, Ehrb., ( fide Grun., ibid., p. 77); Sell., Atl., pi. lx. fig. 7 ; Cestodiscus radiatus, Ehrb.; Van Heurck, Syn. Diat. Belg., pi. cxxix. fig. 5. Distinguished by the wider zone at the border, upon which the markings decrease. Habitat. — Mors deposit (Schmidt, Cleve !). Yar. subspinosa. Grun., ibid., 1884, p. 67. — Diam. *1665 mm. Central rosette small, often with a small central space. Markings more delicate, 4 to 5, at the border 7 to 8 in -01 mm. Apiculi minute, numerous, at subregular intervals, inserted close to border. — Sch., Atl., pi. lxiii. fig. 4 (no name). Habitat. — Mejillones guano (Grunow, Grove, O’Meara!) ; Ichaboe guano (Schmidt). Yar. tenui-striata. Grun., ibid., 1884, p. 77. — Diam. *14 to *15 mm. Surface very convex on outer portion. Markings at the centre 5, towards the border 6 or 7, in ’01 mm., non-apiculate. A smaller form with smaller markings, 7 to 10 in *01 mm., occurs in the Caspian. Habitat. — Campeachy Bay (Grunow). Yar. stelliger. Sch., Atl., pi. lxiii. fig. 8 (no name). — Diam. 2 mm. Central rosette distinct. Markings subequal to, but more irregular than those of the type ; towards the border larger, and forming a circlet of distinct rosettes, placed at regular intervals about i of radius from the border. Habitat. — Java (Schmidt). vol. xvi. 29/10/89 2 n 562 Proceedings of Royal Society of Edinburgh. [sess. Yar. loculifera , nov. — Diam.. *17 mm. Central space distinct, from *0025 to -005 mm. broad, subcircular, usually surrounded by a distinct band of large areolae. Markings sometimes most evident on a narrow band around the border, the central papillae faint.- — (PI. I. fig. 2.) This var. is readily distinguished by the central space. Oamaru specimens have the band around the central space less evident, and the markings towards the centre 5, near the border 3J in *01 mm. Habitat. — Manilla Algae (Grove!); Jackson’s Paddock, Oamaru deposit (Grove !) ; Springfield deposit, Barbados (Doeg !) ; Mejillones (Grove !). C. annulatus. Grun., Denk. Wien. Ah. , 1884, p. 74, pi. v. (E), fig. 57. — Diam. *08 to *165 mm. Surface with a distinct annular depression, about of radius broad, its inner edge about -J- of radius from the centre, beyond this slightly convex to the border. Central space minute, subcircular ; surrounded by a distinct rosette or band of large areolae. Markings pent- or hex-agonal, gradually increasing outwards ; near the centre 4, towards the border 3, in ’01 mm.; on the annular depression the areolae replaced by isolated round granules with hyaline interspaces. Border narrow; striae delicate, 8 to 10 in *01 mm. This species is readily distinguished by the annular depression. Habitat. — Mors deposit (Grunow, Deby !). C. groveanus* sp. n. — Roundly elliptical, major axis ’21 mm. about times minor. Surface flat from the centre for about J- of the radius, here rising abruptly to form a distinct elevated ring about *01 mm. broad, thence descending more gradually with a slight outward concavity to the border. Central rosette distinct. Markings hexagonal, subequal, 3J in *01 mm., but near the border enlarging to about 2 \ ; central papillae faint ; rows radial, straight ; secondary oblique decussating rows most distinct near the border. Border striae coarse, indistinct. — (PL I. fig. 11.) Distinguished from all others by the prominent elevated ring. The transition to G. excavatus , var. semilunaris , Grun., is not abrupt. Habitat. — Newcastle desposit, Barbados (Grove !). * Dedicated to Edmund Grove, Esq., F.R.M.S., the well-known investi- gator of the Oamaru deposit, New Zealand. 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 563 C. suboculatus, sp. n. Sell., Atl.} pi. lxi. fig. 5 (no name) — ■ Diam. *1 1 35 mm. Surface convex towards the border. Central space minute, rosette distinct. Markings polygonal, increasing from the rosette to the semiradius, thence gradually decreasing to the border ; towards the centre 4, at semiradius 3, in *01 mm. Central dots distinct. Border sharply defined, about TL of radius broad ; striae coarse, moniliform, 4 in *01 mm. According to Grunow, this is allied to C. oculus-iridis. It is, however, readily distinguished by its more robust markings and its broader border. Habitat. — Springfield, Barbados (Schmidt). C. joacificus , Rattray. C. oculus-iridis , var.? pacifica, Grun., Benk. Wien. Ak ., 1884, p. 77. — Diam. *25 mm. Central space absent or small ; rosette distinct. Markings polygonal, decreasing from the centre to the border ; towards the centre 4, towards the border 6, in -01 mm. ; rows radial, straight, or subparallel; secondary oblique decussating rows evident. Border indistinct. — Sch., Atl., pi. lx. fig. 13 (no name); C. asteromphalus, Ehrb.? fide Grun., ibid,, 1884, p. 77. Habitat. — Monterey (Schmidt). C. intermixtus , sp. n. — Diam. *224 mm. Surface almost flat. Central space absent ; rosette distinct. Markings hexagonal, increasing subregularly from the rosette outwards; towards the centre 4, near the border 2J, in -01 mm.; at J- of the radius from the centre a distinct zone about ’01 mm. broad, of much larger, unequal and irregularly arranged areolae for the most part in 3 rows, those at the centre being largest ; the central papillae evident, rows straight. Border narrow, sharply defined; striae 4 to 6 in *01 mm., well marked. — (PI. I. fig. 13.) Habitat. — Santa Monica deposit (Weissflog !). C. Monicoe , Rattray. O. Janischii , var.? Monicce , Grun., Denk. Wien. Ak ., 1884, p. 76. — Diam. -2275 mm. Central^space rounded, about, of diam. broad. Markings around the central space large, free, circular, or elliptical, forming a distinct single hand, beyond this obtusely angular, subpearly, with faint central dots, gradually increasing from the central space for about ^ of the radius, again 564 Proceedings of Royal Society of Edinburgh. [sess. diminishing slightly towards the border ; towards the centre 5, towards the border 3J, in *01 mm. — Sch., Atl., pi. lxiii. fig. 10 (no name). Habitat. — Santa Monica deposit (Grunow). C. Kurzii, Grim. Sch., Atl ., pi. cxiii. fig. 17. — Diam. *1 1 35 mm. Central space small, surrounded by an inconspicuous band of areolae. Markings sometimes rounded and free on one side, hexa- gonal and in contact on the opposite, increasing from the central space for about -J of the radius, thence decreasing rapidly to the border ; towards the centre 4, at f of the radius 2, at the border 6, in ’01 mm.; central papillae evident; minute puncta at the origin of the shorter rows; secondary oblique decussating rows distinct. Border narrow. Habitat. — Elephant Point (Grunow). C. spinuligerus, sp. n. Sch., Atl. , pi. lxiii. fig. 3 (no name). — Diam. 44 to #24 mm. Central space subcircular, ’0065 mm. broad, surrounding band of large areolae distinct. Markings polygonal, 3J in *01 mm., distinctly punctate, decreasing on outermost i of radius to the border; small apiculi at the border obvious, at intervals of about *0075 mm. — Sch., Atl., pi. cxiii. fig. 19 (no name); C. aster omphalus, var. spinuligera, Grun., Denh. Wien. Ah., 1884, p. 79. Sometimes united to C. oculus iridis. The apiculi resemble those of C. concinnus, W. Sm. Habitat. — Monterey deposit (Schmidt, Grunow) ; Ichahoe guano (Schmidt ; Sta Monica deposit (Firth !); Arica (Griindler). C. oamaruensis. Grove and Sturt, Jour. Quek. Micr. Cl., 1887, p. 68. — Diam. *185 to *25 mm. Surface convex towards the centre. Central space obtusely angular, about of diam. broad, surrounded by a distinct hand of markings somewhat larger than those adjacent to them. Markings 4- to 6-angled, gradually decreas- ing from the centre outwards; towards the centre 4 to 4|, towards the border 6 to 7, in •01 mm.; central papillae small, indistinct; rows straight, or showing wide gentle curves ; apiculi near the border minute, at unequal intervals of -0075 to ’01 mm.; at the origin of the shorter rows minute puncta sometimes present. Border narrow, 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 565 hyaline. — Grun., Bot. Centralbl ., Bd. xxxiv. Nos. 2, 3, p. 35. — (PI. i. fig. i.) The central space and surrounding band are similar to those of C. asteromphalus, var. pabellanica , Grun., but the markings of the latter are quite distinct. I agree with Mr Grove that this species cannot be assimilated to C. perforatus , var. cellulosa, as stated by Grunow. Habitat. — Oamaru deposit (Grove !) ; Cambridge deposit Bar- bados (Johnson !).* C. umbonatus. Greg., Trans. Roy. Soc. Edin ., 1857, p. 500, pi. x. fig. 48. — Diam. *13 mm. Surface with central portion convex for about f of radius, sharply defined, thence almost flat to the border. Central space absent; rosette distinct, *0075 mm. broad. Mark areolate, subequal, 4 in. *01 mm.; close to the border submoniliform rows, distinct; the secondary oblique rows less obvious. — Jan. Abh. Schl. Ges. vdier. Cult., 1861, pi. ii. fig. 5, 1862, p. 5. The radiating lines between the markings noted by Janisch are not found in the type. Habitat. — Lamlash (Gregory!);* Peruvian guano (Janisch). C. Weissjlogii. Sch., Atl., specimen plate (Einladung zur Subscription), 1874, fig. 5. — Diam. '0375 mm. Surface with 8 submarginal bullations having the peripheral portion most prominent. Central space circular, obscure, a distinct rosette of 3 areolae meeting at its middle. Markings minute, 12 to 14 in *01 mm., rows straight; no primary rays corresponding to the bullations, surface scabrous. —(PL I. fig. 25.) This species is, in the opinion of Kitton, a Melosira. In the 8 submarginal bullations it shows affinity to the Injlati section of the genus Aulacodiscus. Habitat. — New Zealand deposits (Kitton!); Samoa (Schmidt). C. theskelos, f sp. n. — Diam. *13 mm. Surface bearing a number of subradial curved or flexuous distinct lines, simple, or once dicho- tomous, and uniting around the centre to form a sharply curved * In the collection of Dr Grevillo. + 0e ' j border, not robust. Border about f of radius, broad, with distant evident striae, . L Markings otherwise, f Markings areolate to about semiradius, on outer 36. -J portion distinct radial flexuous lines, ( No such lines, f Markings towards centre 3, decreasing rapidly j outwards, at border 6, in *01 mm. Border o-k j striae coarse, ....... '* j Markings 3^ in *01 mm.; apiculi long, delicate, j inserted at inner edge of border, and extending l beyond circumference, 31. Markings rounded, granular ; towards centre 5, towards border 8, in ’01 mm. ; at intervals near border delicate, clear, elevated lines, Markings largest and irregular beyond semi- radius, smaller towards centre and border ; rows radial. A rosette, .... Appearance otherwise, . gg / At centre a large prominent nodule, . ‘ \ No such nodule, .... r Markings in conspicuous, regular, concentric circles, . . . . . . Markings robust, 2| in *01 mm., punctate, sub- regularly, but more obscurely concentric ; og ! radial rows obscure, * j Markings small, round, granular, subequal ; I radial rows inconspicuous, secondary irregu- I larly concentric bands evident. A minute | central rosette, ...... I No such concentric arrangement, f Rows inconspicuously radial, on outer portion obscurely fasciculate secondary oblique out- j wardly curved decussating rows distinct ' j towards border, ...... | Rows radial, ....... I Rows oblique and decussating, .... ' Surface rising for about f of radius, here descend- ing abruptly, thence flat to border ; markings areolate 3, increasing outwards to 2| in ’01 mm. on highest zone, here decreasing suddenly, and from this to border subequal. A rosette, Surface with central portion convex and sharply ^ defined. Markings areolate. A rosette, ' j Surface with central portion funnel-shaped. Markings granular, decreasing from edge of . central portion to border, .... Surface regularly convex between centre and border. Markings subequal, areolate, 3| in •01 mm. Border striae 5 to 6 in ’01 mm., ^No such surfaces, ...... ( Markings somewhat excentric, 4 to 4 \ in 43. -| ’01 mm. ; rows curved, ..... [No such excen tricity or curvature, megacoccus. irregularis. 36. subareolatus. 37. turgidus. nottmghamensis. armatus. inoequisculptus. 38. nodulifer. 39. patina. velatus. radiopundatus. 40. decrescens. 41. 42. epiphanes. umbonatus. bathyomphalus. luduosus. 43. elegantulus. 44. 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 613 44. -{ 45.- 'Elongated subulate spaces at origin of shorter rows. Markings polygonal, without order to g of radius, thence obtusely angular, 4 in •01 mm., and in radial rows, .... Subulate spaces opposite shorter rows. Markings 12 in ’01 mm. Apiculi some distance from border. Inner layer with a zone of costae round central space, Subulate spaces most evident towards centre. Surface undulate beyond semiradius. Mark- ings 7| to 8, at border 10, in *01 mm., at wide intervals minute clear spaces close to border, . L No such appearance, Near centre a sharply defined band of 2 rows of large areolae. Markings towards centre 4, at border 2J, in *01 mm. A distinct rosette, No such band, ....... ^ f Markings resolved with difficulty, ' \ Markings larger, easily resolved, 47. f Markings 12 in '01 mm.; apiculi scattered at J wide intervals over surface, but most crowded J towards border, ...... L Appearance otherwise, 49, f Markings more delicate, rows radial. J with scattered clear dots, | A distinct central depression. \ evident at its edge, Surface patelloeformis. Baileyi. Kochii. 45. intermixtus. 46. 47. 48. fragilissimus. 49. pundulatus. depressus. ,o / Markings areolate, .... ’ \ Markings round, granular or punctiform, gQ / Apiculate, ° \ Non-apiculate, . .... 50. 51. 52. 53. Apiculi 7 to 14, prominent. Markings at centre 8, increasing outwards to 6, in ‘01 mm., then subequal, ....... Apiculi 6, large, with slight median constriction. Markings 6 to 8 in *01 mm., decreasing near border. Rosette minute, .... Apiculi 2, asymmetrical at border. Markings 4 to 5 in -01 mm., decreasing gradually out- wards. A rosette, Apiculi prominent, long, acicular, inserted at inner edge of border, and reaching its outer edge. Markings towards centre 4J to 5, at 52. semiradius 3 J, near border 4, in ’01 mm. , Apiculi prominent, truncate. Markings towards centre 4, at semiradius 3, towards border 3|, in *01 mm., radial rows inconspicuous, . Apiculi scattered irregularly over surface. Mark- ings towards centre 8 to 10, towards border 14 to 16, in *01 mm. Subulate clear lines opposite shorter rows, ....... Apiculi scattered over surface at irregular inter- vals. Markings 8 in ’01 mm., subequal, cen- tral dots distinct, ...... Apiculi minute, in a circlet at border. Markings l towards centre 4, towards border 8, in ’01 mm., apiculiferus. subaulacodiseoidalis. centralis. pectinatus. egregius. dubiosus. denticulatus , grandineus. 614 Proceedings of Eoyal Society of Edinburgh. [sess. 53. 54. 55. { { { 56.- Two asymmetrical distant curved depressions at border. A rosette, ..... No such depressions, Border broad, its inner edge with 2 asymmetrical constrictions. No rosette, .... No such constrictions, Largest markings forming a conspicuous, not sharply defined zone towards semiradius, No such zone, Markings towards centre 6 in *01 mm., decreasing slightly outwards, secondary rows obscure. Border sharp, \ of radius broad, striae evident, 4 in *01 mm., Markings 3i, towards border 2^, in *01 mm. Surface with an elevated ring about \ of radius from centre, ....... Markings 1| to 2, at border 2 to 2|, in *01 mm., pearly ; central papillae prominent, directed centrally towards border, .... Markings 5 to 7 in *01 mm., slightly smaller towards centre, papillae evident, minute sub- ulate areas at origin of shorter rows. Border opaque, . Markings increasing gradually outwards, at centre 8, at border 6, in *01 mm. ; irregular on a small central area, secondary oblique rows distinct, ....... Markings 2| to 3 in *01 mm., subequal, central papillae small. Border broad, Markings 6^ to 7 in *01 mm., decreasing rapidly outwards, interspaces evident opposite shorter rows, . . Markings towards centre 4, towards border 6, in *01 mm., decrease outwards gradual. A rosette, Markings towards centre 6 to 9, near border 9 to 10, in *01 mm. ; minute subulate lines at origin of shorter rows, ...... Markings towards centre 4, decreasing outwards to 6 or 7, in *01 mm. ; radial rows most evident towards border, fimbriate, .... Markings towards centre 2 to 2|, at border 8, in *01 mm., decrease outwards rapid; radial rows separated by distinct lines around border, central papillae distinct, .... Markings towards centre 2 to 2J, near border 6, in *01 mm. Border narrow, .... Markings towards centre 2 to 2\, robust. Border broad, sharply defined, striae coarse, Markings towards centre 1^ to 2J, central papillae prominent. Border conspicuous, raised, striae coarse, ... .... Markings towards centre 3 to 4, increasing to 2\ or 3, at border 5 or 6, in *01 mm. Central rosette large, Markings towards centre 8, at semiradius 4, towards border 8, in *01 mm., on outer half of valve in inconspicuous, subcon centric bands, . Markings at centre large, unequal, suddenly de- creasing to 4 or 4| in *01 mm. , again increas- ing to about semiradius, thence decreasing to border. Border distinct, smooth, . bisinuatus. 54. biangulatus. 55. bulliens. 56. compositus. groveanus. secernendus. debilis. ■ traducens. megaporus. profundus. pacificus. radiosus. fimbriatus. obversus. radiatus. marginatus. robustus. oculus-iridis. antarcticus. biharensis. 1888-89.] Mr John Rattray on the Genus Coscinodiscus. Markings at centre 6 to 7, at semiradius 5 to 5-|, on a sharply defined marginal band (g of radius broad) 10 in ’01 mm. ; rows on marginal band oblique, decussating, Markings towards centre 4, increasing outwards to 2 or 3, at border 4 or 5, in "01 mm. , secondary oblique rows indistinct, ..... Markings 3^, subequal to semiradius, at § radius 3 in ‘01 mm., thence decreasing to border. Border broad, striae 4 to 5 in *01 mm , . Markings towards centre 3, decreasing outwards to 5, at border 8, in '01 mm. ; secondary oblique rows evident, ...... Markings towards centre 3 to 3J, towards border 4, in ’01 mm. Surface convex, / Apiculate, ^ . ' \ Non-apiculate, ....... 'Apiculi robust, spine-like, with blunt free ends inserted at middle of small round hyaline spaces. Markings round, granular, 6 in *01 mm., subpunctiform towards border. Central space small, ..... Apiculi robust, 6, symmetrical, free ends blunt. Markings round, granular, 4 in *01 mm., subequal. Central space absent, Apiculi distinct. Valves dissimilar — the one with markings round, granular, 3 in *01 mm., on a submarginal band 8 in ’01 mm., and in oblique decussating rows, interspaces distinct — the other with markings angular, without interspaces, ....... Apiculi evident, about *015 mm. apart. Mark- ings round, granular, about 5 in *01 mm., toward border smaller interspaces hyaline, largest towards centre, ..... Apiculi conspicuous in a circlet some distance from border. Markings granular, secondary rows irregularly concentric, interspaces hyaline, distinct, ....... Apiculi distinct, a circlet close to border, within this a few scattered apiculi forming an irregular inner circlet. Markings 6 to 10 in ’01 mm., . Apiculi in two circlets, those of the inner promi- nent, distant, and at opposite sides of valve replaced by a narrow curved hyaline band, those of the outer minute, close. Markings (_ 16 in *01 mm., ...... f Markings in radial ■ rows separated by wide cuneate interspaces. Border sharp, striae obvious, ....... Markings small, on a zone adjacent to border punctiform, interspaces wide, smaller towards border, rows obscurely radial towards centre, obviously radial near border, .... Markings decreasing gradually outwards, towards centre 4 to 4^ in ’01 mm., at border puncti- form ; rows radial, those reaching centre few, on a narrow band at border many, . Markings round for \ to f of radius, on outer portion polygonal 2J to 3, at border 6, in l ’01 mm., 57. 58. exutus. argus. glaberrimus. convexus. asperulus. 57. 58. johnsonianus. stokesianus. superbus. evadens. plicatulus . lacustris. ludovicianus. duriusculus. agapetos. subdivicus. diversus. 616 Proceedings of Royal Society of Edinburgh. [sess. f A distinct elevated ring at § of radius from 12.-! centre. Apiculi minute, . . . ( No such ring, f Around centre three or more large lanceolate dark I distant spaces, sometimes meeting at centre to form a stellette. Marking towards centre 6, 59-1 border 8 to 9 in *01 mm., . I Around centre, five or six large areolse. Mark- ings 16 to 20 in '01 mm., .... LNo such markings, ...... ( An apiculus at middle of outer ends of each 60. -! fasciculus ; no octagonal figure, (Appearance otherwise, f An apiculus at middle of outer ends of each j fasciculus. Surface with an octagonal figure. 61. -{ Apiculi placed at the angles of the octagon, . | Apiculi not confined to middle of outer ends of L fasciculi, . . . f Apiculi many upon and between the fasciculi. Markings 15 to 16 in *01 mm., | Apiculi absent or minute, the rows in the 62. «{ fasciculi converging towards border, about 6 in each fasciculus at border. Markings 8 in •01 mm., decreasing a little outwards, . ..Apiculi interfasciculate, ..... f Apiculi 3 to 5, spine-like, inserted some distance from border. Markings towards centre 4| to 5, towards border 6 ; rows in each fasciculus parallel to central row, ..... Apiculi distinct. Markings cestodiscoid, angular, 8 in '01 mm., on a broad band at border 63. ■{ punctiform 10 in "01 mm. ; rows fasciculate, those in each fasciculus parallel to central row, ........ Apiculi minute. A narrow hyaline band around central area. Markings 6 to 7 in *01 mm., rows sub parallel, ...... Appearance otherwise, . . . . . f Fasciculi evident only beyond semiradius, j Markings 6 in '01 mm. Border with crowded J oblique decussating rows, non-apiculate, ‘ j At border a circlet of small round clear spaces, j non-apiculate, LNo such markings and spaces, .... f Rows in each fasciculus parallel to central row, . | Rows in each fasciculus parallel to corresponding 65. -j side rows, | Nine asymmetrical prominent rows from centre L to border ; intervening rows subradial, . (A single minute apiculus close to border. Mark- ings towards centre 6, at border 8, in ’01 mm., No such apiculus, ..... gg / Prominent interfasciculate radial rows, ' L No such rows present, . whampoensis. 59. symbolophorus . stellaris. 60. Rothii. 61. angulatus . 62. poly acanthus. Normani. 63. echinaius. pusillus. odontodiscus. 64. Kutzingii . Grunowii. 65. 66. 67. barbadensis. lentiginosus. 68. 69. 70. 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 71. f A distinct marginal band. Markings granular, decreasing slightly outwards, secondary oblique 69. ■{ rows on marginal band. Apiculi large, inter- fasciculate, ....... I No such marginal band, ..... f Markings actinocycloid, towards centre 8, towards border 10, in '01 mm., decussating rows most evident towards border. Non-apiculate, Markings sub-actinocycloid, towards centre 6, towards border subpunctiform 8, in ‘01 mm. Apiculi interfasciculate, distinct, some dis- tance from border, ..... Markings equal, 4 in *01 mm., secondary oblique rows straight. Non-apiculate, Markings equal, 8 to 10 in '01 mm., secondary oblique rows straight. Apiculi interfasci- culate, ........ Markings 3^ to 4 in ‘01 mm. ; secondary rows straight, obvious. Non-apiculate, _ Markings robust, increasing to semiradius, 70. thence decreasing, at centre 4, at semiradius 3 to 3|, at border 6, in ’01 mm. Central papillae evident, f Markings punctiform, ..... f Markings 15 to 16 in '01 mm., irregular on a small round central area, elsewhere in 8 broad 72. proteus. 71. actinosus. partitus. senarius. interlineatus. denarius. simbirsJcianus. 72. fasciculi. Apiculi interfasciculate, glad alis. Markings punctiform on a distinct band at border, elsewhere 6 in *01 mm. Apiculi pro- minent, interfasciculate, Markings arranged in striae, at intervals not reaching centre. Apiculi distinct in a circlet near border. Border broad, .... odontophorus. marginvlatus. Border crenulate, ...... crenulatus. Border uniformly curved, 73. Fasciculi and rows curved, .... 74. Fasciculi and rows straight, .... 75. Markings areolate, 6 in '01 mm., sometimes non- . apiculate, . . . . curvatulus. Markings rounded, decreasing rapidly outwards. Non-apiculate, ...... semipennatus. Markings 6 to 10 in *01 mm. Apiculi inter- fasciculate, subtilis. Markings 10 in '01 mm. Apiculi prominent. Valve minute, ...... minutellus. 73, 74. 75. fTwo asymmetrical distant processes. Minut | apiculi at outer ends of radial clear lines 13. -J Markings 7 to 8, decreasing outwards to 12, in j ‘01 mm., I No such processes and apiculi f Markings excentric, . ’ \ Markings non-excentric, ( Markings irregular on a small excentric area 77 J Border prominent, ...... | Markings similar. A distinct small central k nodule. Border simple, concmnus. 76. 77. 78. africanus. vetustissimus. 618 Proceedings of Royal Society of Edinburgh. [sess. Markings polygonal, 7 in ’01 mm., decreasing outwards ; rows fasciculate beyond semi- radius, oblique rows concave outwards, . Markings 8 in ’01 mm., decreasing slightly 78. towards border ; rows radial, on outermost portion of valve parallel, subfasciculate secondary rows obscure, slightly concave out- wards, ........ (.Markings otherwise, ( A distinct irregularly broad band adjacent to mq J border. Markings 12 in '01 mm., irregular * 1 on a small central area, I No such irregular marginal band, f Markings round, granular, and without order, with hyaline interspaces to about semiradius, 80. ■{ thence more crowded in rows and subequal, 10 in *01 mm. ; rows subfasciculate to border, L Markings not so disposed, .... ( Markings round, granular, subequal, 7 to 8 in *01 mm., irregular on a small indistinctly defined central area, elsewhere in radial sub- parallel and irregularly subfasciculate rows ; 81. ■{ secondary concentric bands faint, . Markings 3^ to 4 in *01 mm., angular; radial rows faint, secondary concentric bands evident, ....... No concentric arrangement visible, . "Markings punctiform, 12 to 15 in '01 mm. Apiculi large, distant. A distinct band at border, with evident oblique decussating rows, Markings round, granular about 4 in '01 mm., 82. -J beyond semiradius subfasciculate, interspaces wide, Markings polygonal, 8 to 10 in '01 mm. A circlet of numerous apiculi a short distance within border. Border distinct, striae evident, f Markings consisting of a few wavy lines diverg- 10 J ing outwards from central space, and confined ' j to central half of valve, . . I No such markings, ...... f Markings round, granular, 6 in '01 mm., smaller towards border, radial rows obscure, short transverse somewhat flexuous rows more evident, non-decussating except near border, . Markings small, round, granular, interspaces | unequal, radial rows obscure, radial hyaline lines at regular intervals extending from centre, a distinct apiculus at outer end of each, . . . . . . . Markings round, granular, without order to semiradius, beyond this punctiform in radial rows ; radial clear spaces at subregular intervals, extending outwards from semi- radius, Markings obtusely angular or oval, 5J to 6, at border 8, in ‘01 mm. ; rows radial, towards border uniformly curved towards same direction, with hyaline curved lines extending inwards a short distance at subregular intervals, ....... suspedus. subglobosus. 79. subsalsus. 80. atlanticus. 81. subnotabilis. isoporus. 82. doljensis. nitidulus. corolla. * venulosus. 83. densus. splendidulus. apages. obliquus. 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 619 Markings punctiform, on inner half of valve in faint radial rows on outer half forming striae 14 in '01 mm., at subregular intervals straight hyaline radial lines, most prominent at their outer ends, ....... Markings faint, minute, granular, most loosely disposed towards centre, at semiradius 6 in *01 mm., towards border punctiform in radial strise, 8 to 10 in ’01 mm., between the striae at subregular intervals indistinct short hyaline spaces, ........ I Appearance otherwise, ..... 'Outline hexagonal. Markings round, granular, about 15 in *01 mm. , rows radial or somewhat oblique, Outline polygonal, ...... Outline diamond-shaped. Markings rounded and granular towards middle of large central 84. -J area ; on a distinct zone adjacent to border subpunctiform, 10 in ’01 mm. ; rows radial, apiculi distinct, ...... Outline elliptical or diamond-shaped. Markings rounded. 6 to 8 in *01 mm. ; rows radial, oblique | decussating rows at border ; interspaces hyaline, l Outline circular, ...... f Central space excentric. Markings punctiform j in distant rows, on a band around central space | larger, .... .... 85. 4 Central space slightly excentric. Markings round, granular, 5 in ‘01 mm. , rows fasciculate, those in each fasciculus parallel to central row, L Central space non-excentric, .... r Central space extending to § radius, hyaline. Markings punctiform ; rows close, radial, Central space extending almost to semiradius. Markings polygonal, delicate, increasing slightly outwards ; rows radial, Central space extending to -J radius, sharply defined. Markings punctiform, 28 in ’01 mm., 86. ■{ forming radial striae, ..... Central space large, not smooth. Markings 15 in ‘01 mm., . . . . Central space large (‘0375 mm. broad), at its centre a group of irregular apiculi, at its edge a circlet of similar apiculi at wide intervals. Markings 6 to 7 in ‘01 mm., .... L Central space otherwise, ..... ( Radial plications from centre to border, produc- 87. -J ing a wheel-like aspect, ..... [ No such plications, ( Markings non -fasciculate, 88. -J Markings obscurely fasciculate, .... ( Markings obviously fasciculate, .... ( Surface with a distinct central portion extending I to about f- radius, and with its outer edge ] crenulate, sharp, ...... I No such crenulate portion, .... perikompsos. tenuis. 84. flaqrans. polygonus. rhombicus. punctatus. 85. rotula. inclusus. 86. vacuus. liocentrum. mesoleius. disciger. Gazelles. 87. trochiscos. 88. 89. 90. 91. Trinitatis. 92. 620 Proceedings of Royal Society of Edinburgh. 92 I •^e^ca^e puncta at origin of shorter rows, . ’ \ No such puncta, ...... fPuncta at origin of shorter rows sometimes visible. A distinct band of large markings qo j round central space. Markings towards ' j centre 4 to 4|, towards border 6 to 7, in *01 mm. j Apiculi near border in a circlet, minute, l No such band nor apiculi, . . qn f Distinct apiculi at angles of areolse, . ‘ \ No such apiculi, f An evident rosette. Markings towards centre 2|, increasing outwards for a short distance, gg J at border 5 in -01 mm., submoniliform, ’ j obscurely punctate ; central papillae faint. | Rows radial, ....... I No rosette, 97. f A distinct band of large round markings around central space. Markings towards centre 5, J increasing slightly for ■£■ of radius, towards j border 3^ in -01 mm., subpearly. Central papillae faint, I No such distinct band, . 98. f Markings rounded or angular, 3| to 4 in’Ol mm. Central dots distinct ; central space small, Markings robust, rounded on one side of valve, hexagonal on the other ; towards the centre 4, at f of radius 2, at border 6, in ’01 mm. ; decrease to border rapid. Central papillae pro- minent ; rows radial, ..... Markings towards centre 4|, at semiradius 3J, at border 5 or 6, in ’01 mm. Central space rounded ; no distinct central papillae or dots, . Markings towards centre 2J, at -£ of radius 2, at border 4 to 5, in ’01 mm. , subpearly. Central papillae faint. Central space minute or sub- ^ obsolete, ....... f Markings unequal and irregular. Central space 94. -J irregular, . ...... V. No such irregular markings, . . . . f An elevated band at semiradius ; markings inside this and towards border moniliform 6, on the elevated band 4, in *01 mm. , Highest zone reaching inwards to about £ of 99. -{ radius, most abrupt on inner edge. Markings towards centre 8, on highest zone 4, beyond this 6, in -01 mm. Subequal subulate spaces opposite shorter rows, ..... I No such sharp elevated band, .... f Markings of 2 kinds — the larger distant, rounded, | in inconspicuous radial rows — the smaller Iqq j closer, punctiform, in evident radial rows ; * j hyaline spaces opposite shorter rows, I Markings not of such distinct kinds — rounded, . L Markings not of such distinct kinds— angular, . 93. 94. oamaruensis. 95. fioridulus. 96. moravicus. 97. Monicas. 98. perforatus. Kurzii. conformis. obscurus. anastomosans. 99. grayianus. spiniferus. 100. diplostictus. 101. 102. 1888- 101. < 103. - 104. 106. 107. 108. 105. 109. 110. -89.] Mr John Rattray on the Genus Coscinodiscus. r Markings 6 in *01 mm. ; rows radial, separated by wide hyaline intervals ; adjacent to border a distinct band, with markings similar to those on rest of valve, ...... lunce. Markings minute, granular, faint beyond semi- radius, radial rows distinct, .... stelliger. Marking minute, granular, most distinct towards centre, beyond semiradius puncti- form ; rows radial. Border sharply defined, . exiguus. Markings punctiform ; rows radial, interspaces widest opposite shorter rows. Adjacent to border a band of large faint areolse. A few scattered prominent round granules on a circlet some distance from border and one at centre, . perminutus. L Markings otherwise, ...... 103. Adjacent to border a distinct band, with smaller markings than on rest of valve, No such band, ....... 104. 105. f The marginal band narrow, .... 106. I The marginal band broad, .... 107. { Central space elongately elliptical. An undula- tion at semiradius, with inner edge less abrupt l than outer, ....... undulatus. f Markings rounded, granular, 4 to 4| in *01 mm. | Interspaces hyaline, large opposite shorter | rows, elegans. j Markings 6 in '01 mm. A marginal hyaline l band. Apiculi distinct, .... tabularis. (Rows radial, less crowded, and prominent within a distinct marginal band, .... actinochilus. No such appearance, 108. f Markings towards centre 3 J, at semiradius 3, at border 6, in ’01 mm. Central space irregular, 4 small ; rows radial, secondary rows indistinct, cribrosus. j Markings obtusely angular ; central space sub- L quadrate. Border hoop-like, strise evident, . aethes. C Apiculate, 109.' \ Non-apiculate, 110. f Apiculi large, distant, inserted some distance within border. Markings rounded or obtusely angular; towards centre 6, towards border 8 to 9, in ‘01 mm., and punctiform; secondary rows most evident near border, . . . hungaricus. Apiculi minute. Markings 6 in ‘01 mm., de- creasing slightly near border ; rows radial, l_ interspaces narrow, ..... griseus. ( Markings rounded, granular, about 5 in *01 mm. , conspicuous, decreasing rapidly from centre out- I wards, towards border punctiform, more faint ; j rows radial, ....... ] Markings rounded, granular, near border puncti- form ; without order, ..... | Markings rounded, granular; interspaces hyaline, l unequal; rows radial, ..... gemmifer. confusus. 111. 622 Proceedings of Royal Society of Edinburgh. ill. 'Markings about 4 in ‘01 mm., least crowded to- wards centre. Border narrow, Markings about 4 in -01 mm. Border broad, sharply defined, striated, .... - Markings towards centre 3^, towards border 4|, in *01 mm. ; smaller round granules at intervals among the larger ; rows inconspicuous, monili- form at border, interspaces narrow, L Markings otherwise, ...... galapagensis. gemvnatulus. pauper. 112. f Markings granular; about semiradius 6, at border )8 to 9, in ‘01 mm.; rows radial, secondary rows distinct only beyond semiradius. Central space circular. No rosette, . . . . decussatus. I No such secondary rows, . . . . . 113. ( Surface with an elongated somewhat curved de- j pression on each side of valve, about § of 1 radius from centre. Rows radial, . . . biplicatus. I No such depressions, 114. f Markings obtusely angular, robust, subpearly, 4 in ’01 mm. ; rows radial, alternately longer I and shorter, spaces at origin of shorter rows 114. ^ hyaline, ....... | Markings delicate, granular, rounded, 5 in j '01 mm.; rows radial, interspaces wide, l No such wide interspaces, .... 115 / Markings obviously punctate, . * \ Markings not obviously punctate, biradiatus. apollinis. 115. • 116. 117. { Border broad, of two parts, inner striated, outer | hyaline, Weyprechtii. 116.4 Border simple. Markings robust, 3| to 4, in- creasing outwards to 2\ or 3, in '01 mm.; I central dots prominent. A rosette, . . asterornphalus. . , m / Markings in contact and angular throughout, . ' \ Markings not in contact and angular throughout, 118. 4 'Markings towards centre 3, increasing outwards to 2 at border, 5 in *01 mm. ; secondary oblique rows not conspicuous, Markings more delicate, towards centre 3J, in- creasing outwards to 2| on an angular some- what elevated zone, at border 6 or 7 in ‘01 mm. ; radial rows indistinct, ..... Markings towards centre 4, at § radius 3 in *01 mm., thence rapidly decreasing to border ; radial rows distinct, secondary rows incon- _ spicuous, ....... f Markings rounded, sometimes angular, towards | centre 6, increasing slightly outwards, at border | 6 to 8 in "01 mm. ; central papillse prominent, 119.4 rows radial, ....... I Markings towards central space rounded 4, soon becoming hexagonal and gradually increasing l outwards to If or 2 in ’01 mm. near border, . f Adjacent to border a distinct band, with mark- ings well-defined, elsewhere outlines of mark- 102. -{ ings faint. Markings 3^ to 4 in -01 mm., increasing but little outwards, l No such band adjacent to border, 118. 119. crassus. heteroporus. boliviensis. apiculatus. gigas. Janischii. 120. 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 623 f Markings minute, delicate 8, towards border re- cognised with difficulty, 10 to 12 in *01 mm. ; rows radial, oblique rows faint ; no hyaline band adjacent to border, Border narrow, 120.-! hyaline, Markings punctiform, 15 in ’01 mm.; rows radial. Apiculi distinct in a circlet near border, .... I Appearance otherwise, 121 -f-^0 rosette> ' \ A rosette distinct, f A distinct band of large areolae around large j central space. Markings punctate 3J in *01 122. ] mm. , decreasing on outer j of radius. Apiculi j at border obvious, .... LNo such band, ..... 124 -f Markings punctiform, ’ \ Markings larger, .... ( Markings 13 in ’01 mm. ; rows radial, 125 J R°ws ra(tial, only a few reaching large central ' J space, the others mostly ending about £ of l radius from border ; interspaces large, hyaline, ' Markings rounded, granular, 6 in ’01 mm. ; rows radial; interspaces wide towards centre, Markings obtusely quadrangular, pearly, in- creasing to } of radius to 2 or 2J, thence de- creasing to border to 3 in ’01 mm. ; irregular 126.] around centre; rows radial, .... Markings obtusely angular, subpearly; towards centre 5, at semiradius 4^, at border 8, in ’01 mm. ; radial rows separated by narrow clear lines. Secondary subconcentric rows evident, l Markings hexagonal, ..... f Markings increasing outwards to f °f radius, j towards centre 5, at f of radius 3J in ’01 mm. ; ] rows radial, at intervals the markings in ' | isolated oblique rows. Apiculi minute, form- ing a circlet at border, ..... L Markings not oblique at intervals, f Markings towards centre 8, beyond semiradius 4, in ’01 mm. ; rows towards border subfascicu- 128. ] late. Apiculi 2, at an interval equal to about [ ^ circumference, ...... LNon-apiculate, 129. ( Markings 2| to 3 in ’01 mm., subpearly, with central papillae obscure, .... Markings towards centre 3 to 3|, increasing out- wards almost to border to 2 or 2| in ’01 mm. ; rows radial, with secondary rows evident, - Markings towards centre 4|, at semiradius 4, in ’01 mm., smaller at border, . . . . Markings 7 to 8 in *01 mm. , decreasing but little towards border. Border striae delicate, Markings towards centre 6, towards border 8, in l ’01 mm. Central space broad diam. ) imperator. Martonfii. 121. 122. 123. spinuligerus. 124. 125. 126. cingulatus. comptus. Thumii. mossianus. notabilis. 127. lutescens. 128. modestus. 129. dubius. entoleion. flexilis. josefinus. mirificus. 624 Proceedings of Royal Society of Edinburgh. 123. A distinct band of smaller markings adjacent to border. Apiculi numerous, minute, inserted at inner edge of border, .... I No such band nor apiculi, .... f Markings towards centre 3 to 3|, near border 2, .j j in *01 mm. ; robust, central papillae prominent, ’ 1 Markings towards centre 4, at semiradius 3, in L ‘01 mm. Border radius broad, striae coarse, ( Central space bearing 2 large conspicuous round 90. -j granules, . . . . . (No such granules, ...... {Apiculi numerous, forming a double circlet; undulation about semiradius, slight, No such arrangement of apiculi, f Apiculi many in one circlet. Markings 24 in *01 mm. Border broad, hyaline, . jon I Apiculi large. Markings towards centre rounded * j 4|, outwards polygonal 6, in ‘01 mm. Border striae delicate, vNo such apiculi and markings, .... C Distinct subulate spaces at origin of shorter rows, j Markings 7 in '01 mm., the fasciculi separated j by inconspicuous radial lines. Yalve large, . I No such distinct subulate spaces, [ Markings towards centre 4|, at semiradius 3|, in | ‘01 mm., thence decreasing outwards. Centre 1 ^4. j with one large irregular marking, . ' ‘ Markings towards centre 5 or 6, at border 9, in ‘01 mm. ; around central space rounded, else- . where often quadrilateral, .... f Fasciculi unequally curved, rows parallel to side j row of each. Markings rounded, granular 10, q, j in an indistinct band at border 15 to 16, in ‘ j ‘01 mm. Apiculi interfasciculate, | Fasciculi not so curved. Rows parallel to central L row in each fasciculus, apiculus near circlet of minute 133. 135. -{ f A single prominent spine-like ' border, outside of this a apiculi L No such apiculus, f Markings oval, with long axis radial or a little 136. -J oblique. Border broad, strise coarse, ^ No such markings, ...... , f Distinct radial interfasciculate rows, ' * \ No radial interfasciculate rows, ( Apiculi interfasciculate, robust ; a distinct zone ! adjacent to border, j Markings sometimes subradial, decreasing out- L wards, . fEach adjacent pair of rows in each fasciculus | originating farther and farther from central 139. \ row ; interspaces hyaline. Non-apiculate, j Markings subpearly, 4 in ‘01 mm. ; around border l a narrow clear space, ..... s / Central space absent or minute, ‘ \ Central space present larger, .... 138. blandus. 130. borealis. suboculatus. bioculatus. 131. capensis. 132. hyalinus. tuberculatus. 133. nobilis. 134. ceginensis. Payeri. divisus. 135. Jcryophilus. 136. planiusculus. 137. 138. 139. extravagans. Gregorii. fasciculatus. symmetricus. 140. 141. 1888-89.] Mr John Rattray on the Genus Coscmodiscus. 625 nn i Non-apiculate, * | Apiculate, f Surface with several concentric zones of different j and brilliant hues. A broad hyaline band } within border. Central space indistinct, INo such zones, f Near border a circlet of radially elongate clear 144 -! sPaces- Surface with two concentric undula- * | tions. Central space absent, l No such clear spaces, . "An annular depression about g of radius from centre. Central space minute. Rosette distinct, ....... A wide undulation extending from a short dis- tance within the semiradius almost to border. Markings towards centre 6|, angular, soon be- 145 4 coming rounded, and 5| in *01 mm. No hyaline ’ | band at border. Central space absent. Minute hyaline spaces at subregular wide intervals near border, ........ Undulation faint near semiradius. Markings punctiform, delicate, least crowded towards centre ; rows substraight, radial, . .Appearance otherwise, ..... f Alternate or opposite rounded or cuneate eleva- tions and depressions around centre. Central space absent. A rosette, .... I An elongated unilateral depression near centre, 146. Central space minute, . Surface with 6 to 12 shallow depressions near centre. Markings at centre 2 to 2|, at 4 radius I 3 to 3|, thence increasing to 1^ in ‘01 mm.; I smaller at border, ...... ( A short transverse central plication. Central 143 -1 sPace ahsent, ...... ’ j Two shallow concentric undulations. Central l space indistinct, granular, .... f Surface hat-shaped, ...... ' \ Surface with faint undulations, "Centre much depressed. Markings punctiform, fasciculate. Apiculi prominent, inserted at border, . ' 147. -J Centre less depressed. Markings punctiform, j faintly fasciculate, the rows at unequal inter- j spaces stopping short of border, and leaving l small hyaline areas, ..... f Undulation single, about § of radius from centre, j Central space indistinct. Markings 7 to 8 in 148-1 ’^1 mm' Apicub small, .... ’ j Two undulations, one near centre, the other close | to border. Markings increasing outwards between these, largest on the elevations, YOL. XVI. 7/11/89 142. 143. clivosus. 144. neogradensis. 145. annulatus. intumescens. pellucidus. 146. excavatus. impressus. asteroides. plicatus. undatus. 147. 148. obnubilus. patera. bengalensis. undiclans. 2 R 626 Proceedings of Royal Society of Edinburgh. [sess. r Adjacent to border a distinct zone of large areolse, with inner edge wanting, and outer convex outwards, ....... 42. ■{ Adjacent to border a single zone of still larger areolse, perfect, with inner and outer edges convex, ........ I No such zone, r Outer portion of valve scarcely siliceous from 3 to \ of radius broad, with distant subuniform costse, Four subgelatinous, cuneate, symmetrical pro- tuberances. A distinct band adjacent to border ; within this markings in oblique rows 2|, upon it 6, in '01 111m. , and in radial rows, 149. -{ Markings round, robust, pearly, smooth, decreas- ing rapidly at border; at centre 2, at border 4, in '01 mm. Border sharp, of 1 to 2 bands of round granules, Markings angular, 10 in '01 mm. A narrow hyaline band adjacent to border. Apiculi j many, outer ends obtuse, . I No such outer portion or markings, . f Inner § of border hyaline between distant radial lines, outer ^ with 8 to 10 strise in '01 inm., . Border prominent, hoop-like, strise 1| in '01 mm. Markings 1| in '01 mm., a few more minute 150. -J at wide intervals ; papillae obscure, Border subopaque, prominent, with coarse striae. Markings 2 to 2b in '01 mm., radial rows inconspicuous, ...... (.Border otherwise, ...... 151. A single large apiculus near border, outside it a circlet of more minute ones. Markings on a narrow zone adjacent to border, granular, No such isolated large apiculus, { Surface convex towards centre. Markings towards centre 4 to 4J, towards border 3, in I '01 mm., ....... 152. \ Surface convex; markings at centre 2, at bor- ‘ der 3, in '01 mm. ; at border a single band of quadrate, equal areolse, . ’ l Surface almost or quite flat, .... ( Markings hexagonal, 2 in '01 mm. ; at intervals I more minute, more distinct areolse. A distinct | band of larger areolse adjacent to border, \^No interspersed minute areolse, .... ( Markings hexagonal, 2^ to 3 in '01 mm. Apiculi 154.4 clavate at inner edge of hyaline border, . No such apiculi, ( Markings 1 J, decreasing outwards to 2 in '01 mm. ; , - _ I a ring of very large areolse adjacent to border. ‘ 1 Non-apiculate, V No such distinct ring, . zonulatus. heteromorphus. 149. sol. bipartitus. vigilans. cristatus. 150. circumdatus. aphrastos. concavus. 151. leptopus. 152. tumzdus. Molleri. 153. pulchelbjis. 154. macraeanus. 155. splendidus . 156. 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 627 f Markings to 4 in *01 mm., oblique rows straight. Apiculi minute or absent, I Markings 3J to 4 in '01 mm., oblique rows sub- j straight. Border f of radius broad, with dis- I tinct striae ; inner and outer parts separated by . J a narrow line, ...... ' | Markings 6 in '01 mm. ; rows straight. Apiculi minute. Border narrow, .... I Markings 9 in ’01 mm., decreasing but slightly outwards, on a narrow band at border 14 to 16 I in "01mm., oblique rows substraight, . I Oblique rows more curved, .... 'Markings at centre 5 to 6, towards border 10, in -4)1 mm. ; robust, decrease outwards rapid. Apiculi many, at unequal intervals, , -h ! Markings towards centre 2, at border 4 to 6, in im j *01 mm.; not robust, decrease outwards rapid. Apiculi many, prominent in a circle at border, I Decrease in size of markings outwards more l gradual, ....... f Markings obtusely angular, subpearly, towards centre 2J, at border 3, in ‘01 mm. Non- apiculate, ....... Markings at centre 4, towards border 8, in *01 -mm.; oblique rows concave outwards. Apicu- late, ........ Markings in ’01 mm. Border narrow. 158. -{ (Diam. ‘02), Markings towards centre 5, at border 6, in ’01 mm. Apiculi numerous, distinct. Border broad, hyaline, ...... Markings at centre 6, at border 9 to 10, in ‘01 mm. ; oblique rows curved outwards. Apiculi j minute. Border narrow, .... I Markings otherwise, ...... /"Markings towards centre 4, at border 6, in *01 J mm., secondary larger hexagonal areolse j distinct, ....... l^No such secondary areolse, .... f Markings 8 to 9 in '01 mm. , decreasing somewhat | outwards. Centre occupied by .a single, evident „„ j circular areola. Apiculi in a circlet at border, ’ j Markings towards centre 3, towards border 4, in '01 mm. No such central areola. Border L strise 3 to 4 in *01 mm., .... llneatus. marginato • lineat us. anguste-lineatus. sublineatus. 157. decipiens. minuens. 158. antimimos. excentricus. subconcavus. peruanus. minor. 159. labyrinthus. 160. pseudo-lineatus. antiquus. ACTIROGONIUM. Ehrb. emend., Mon. Ber. Ah ., 1847, p. 54. — Circular or ob- tusely angular. Surface with a central area, sometimes distinct. Colour pale grey, the rays more pearly. Markings on central area areolate, obtusely angular or rounded; rays broad, distinct, straight, curved or flexuous, sometimes subclavate, regular or irregular, confined to a zone about the semiradius, or extending nearer to the border, radiating to the angles of the valve, more 628 Proceedings of Royal Society of Edinburgh. [sess. rarely at unequal intervals opposite its sides ; on the interradial areas round or angular irregularly disposed unequal granules; interspaces unequal, hyaline. Border hyaline, sometimes broad and angular. Van Heurck has recorded his belief that Actinogonia are hut the “ valves interieures ” of Asterolamprce. Of this I have observed no evidence, hut the possibility of their being so is to be borne in mind. A. multiradiatum , sp. n. — Diam. *05 mm. Surface with central area, extending to about -§ of radius from centre. Colour pale grey, somewhat darker towards border. Markings on central area sub- angular, unequal areolae about 4 in *01 mm., most crowded at the centre ; rays short, occupying middle third of valve, broad, subclavate, or with lateral irregular lobes ; the interradial areas hyaline, on outer third bearing rounded or irregularly angular pearly granules, without order, and decreasing outwards, 4 to 4J in -01 mm.; adjacent to the border a hyaline band of unequal breadth. Border narrow, hyaline. — (PI. III. fig. 15.) Habitat. — Barbados deposit (Hardman !).* A. septenarium. Ehrb., Mon. Ber. Ah., 1847, p. 54. — Obtusely angular. Diam. *055 to ’07 mm. Surface with central area indistinctly defined, about *01 mm. broad. Markings on central area small, round or obtusely angular subequal granules, with hyaline unequal interspaces; rays distinct, extending between central area and angles of valve ; short similar rays sometimes intercalated towards outer portion of interradial areas, rarely’ two rays proceeding side by side towards the same angle, straight or gently curved, rarely flexuous; on the interradial areas round unequal irregularly disposed granules, with hyaline unequal inter- spaces, few close to the outer sides of those areas. Border angular, about -Jg- of radius broad, widest opposite the middle of the inter- radial areas. — Ehrb.; Mikrog ., pi. xxxvi. fig. 39 ; Ralfs in Pritch. Inf., p. 813, pi. v. fig. 55; Ehrb., Abh, Ber. Ah., 1875, p. 38, pi. i. fig. 4; Van Heurck, Syn. Diat. Belg., pi. cxxvii. fig. 8; A. quinarium, Habirsh., Cat. Diat. § Adinogonium. Habitat. — Cambridge deposit, Barbados (Van Heurck); Spring- field deposit, Barbados (Hardman !). * In the collection of Mr Julien Deby. 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 629 Artificial Key. f Circular. Rays confined to a narrow area about I semiradius. Central area sharply circum- ^ 1 scribed. Border narrow, .... multiradiatum. ' j Angular. Rays passing between central area and angles of valve. Central area not sharply cir- l cumscribed. Border angular, broad, . , septenarium. BRIGHTWELLIA. Ralfs, Pritch. Inf., p. 940. — Circular. Surface flat from the centre outwards to the circle of large areolae, beyond this slightly convex, and sloping downwards to the border. Colour pale smoky grey, when dry sometimes purplish or brown. Central space circular or obtusely angular, distinct, hyaline ; more rarely minute or absent ; a rosette rarely differentiated. Markings areolate, rarely subcircular towards the centre; a distinct circlet formed by a single row of large areolae, and situated between J and § of radius from centre ; within this circlet the rows evident, substraight, or oblique, curved, and decussating, beyond the circle radial ; evident costae and primary rays sometimes radiating at regular intervals between the circlet of large areolae and the border ; minute apiculi sometimes present. Border narrow, hyaline, with delicate radial striae. — Craspedodiscus, pro parte , Brightw., Quart. Jour. Micr. Sci ., 1860, p. 95 ; Heterodictyon, Grev., Trans. Mic. Soc. Lond., 1863, p. 67. This genus approaches Coscinodiscus through C. bidliens. § 1. AcOSTATiE. Non-costate. B. splendida, Rattray. — Heterodictyon splendidum , Grev., Trans. Micr. Soc. Lond., 1863, p. 67, pi. iv. fig. 7. — Diam. -045 mm. Central space small, indistinct. Markings toward the centre subcir- cular, soon becoming obtusely angular ; the circlet of large areolae at about J- of radius from centre, within this circlet the markings about 4 in '01 mm., in obscure slightly curved radial rows; the secondary decussating rows undifferentiated ; the hyaline interspaces largest towards the centre ; beyond the circlet the rows moniliform, with markings decreasing gradually outwards from 5 to 8 in *01 mm., 630 Proceedings of Royal Society of Edinburgh. [sess. and separated around the border by narrow hyaline lines ; non- apiculate. Border hyaline. Habitat. — Cambridge deposit, Barbados (Johnson !). B. excellens , sp. n. — Diam. *07 mm. Central space obtusely triangular, about tl of diameter broad, not sharply defined. Mark ings towards the central space subcircular, soon becoming areolate ; the circlet of large areolae about f of radius from centre ; within this circlet the markings subequal, 4 in -01 mm., in evident curved oblique decussating rows ; beyond the circlet decreasing gradually outwards from 5 or 5-|- to 8 in *01 mm.; minute apiculi at intervals of ’0075 mm., inserted close to the border. Border striae delicate, 14 in -01 mm.— (PI. III. fig. 16.) This species was labelled by Greville B. splendida , but his specific name cannot be adopted, since the distinct Heterodidyon splendidum now becomes B. splendida. Habitat. — Barbados deposit (Greville !). B. hyperborea, Grun. Van Henrck, Syn. Diat. Belg ., pi. cxxviii. fig. 8. — Diam. *065 mm. Central space and rosette absent. Mark- ings areolate; the circlet of large areolae about J of radius from centre, within this circlet the markings subequal, 3 to 3^ in '01 mm., in substraight decussating rows, beyond the circle subequal, or decreasing but slightly near the border, 5 in *01 mm. Border hyaline. — Grun., Denk. Wien. Ah, 1884, p. 70, pi. ii. (B), fig. 64. Habitat. — Dredged by U.S.S. Gettysburg, lat. 35° 25' N\, long. 69° 42' E., in 2924 fathoms; marine deposit, Franz Josefs Land (Grunow). B. elaborata. Grev., Trans. Micr. Soc. Lond., 1861, p. 73, pi. ix. fig. 1. — Diam. *08 mm. Central space absent; an inconspicuous rosette. Markings areolate ; the circlet of large areolae about f of radius from the centre, within this circlet the markings increasing slightly outwards from the rosette 3 to 3J in *01 mm., in inconspicu- ous radial, and secondary oblique curved decussating rows ; beyond the circlet increasing in breadth uniformly, but somewhat rapidly outwards, but subequal radially, 4 in *01 mm. Border hyaline. Habitat. — Barbados (Johnson !). B. coronata. Balfs in Pritch. Inf., p. 940 ; Craspedodiscus 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 631 coronatus , Brightw., Quart. Jour. Micr. Sci ., 1860, p. 95, pi. v. fig. 6. — Diam. J2 to ’185 mm. Central space subcircular, y1^ to 2I3 of diam. broad, sometimes surrounded by a band of evident areolae. Markings areolate ; the circlet of large areolae from § to yy of radius from centre, within this circlet the markings subequal, 4 in ■01 mm., in regular, oblique, greatly curved, decussating rows, beyond the circlet decreasing and slightly curved outwards from 5 to 8 in •01 mm.; minute apiculi at intervals of about ‘0075 mm. sometimes visible, inserted close to the border at the outer ends of faint sub- hyaline lines. Border striae 12 to 14 in ’01 mm. — B. pulchra, Grun,; Van Heurck, Syn. Diat. Belg ., pi. cxxviii. fig. 9; Bot. Centralbl., Bd. xxxiv. 1888, Nos. 2, 3, p. 35 ; Grove and Sturt, Jour. QueJc. Micr. Cl., 1887, p. 67 ; B. Murray i, Cstr., Diat. Chall. Exped.,- 1886, p. 138, pi. x. fig, 2. Habitat. — Cambridge deposit, Barbados (Greville! Johnson!); Oama'ru deposit (Grove!); Bridgewater deposit, Barbados (Johnson!); “Barbados” (Greville! Johnson!). Var. radians, nov. — Diam. -15 mm. Central space obtusely triangular, with outwardly convex sides. Markings areolate ; the circlet of large areolae about J of radius from centre, beyond this circlet the markings decreasing more distinctly outwards from 4 to 8 or 9 in ‘01 mm.; primary rays evident at intervals of "0075 mm., non-costate. — (PI. III. fig. 14.) Habitat. — Barbados deposit (Greville!). § 2. Costat^e. Costae distinct. B. Johnsonii. Ralfs, Trans. Micr. Soc. Bond., 1866, p. 4, pi. i. fig. 11. — Diam. *07 to *1075 mm. Central space minute, round or angular, a rosette sometimes evident. Markings areolate, some- times subcircular towards the centre, with narrow hyaline inter- spaces, the circlet of large areolae from -f- to f- of radius from centre, within this circlet the markings subequal or somewhat smaller towards the centre, 3 in -01 mm., in obscure radial and more evident, sometimes irregularly curved or flexuous decussating secondary rows, beyond the circlet decreasing gradually outwards from 4 to 6 or 6J in ,01 mm. at border, with evident primary rays at 632 Proceedings of Royal Society of Edinburgh. [sess. subregular intervals, and narrow costate ridges towards the outer ends of these rays. Border with delicate striae, 16 to 18 in '01 mm. — Walker and Chase, New and Rare Diat ., ii. p. 2, pi. v. fig. 10. Habitat. — Springfield deposit, Barbados (Hardman !); “Barbados” (Johnson! Greville !); Cambridge deposit, Barbados (O’Meara!). Artificial Key. /'Primary rays obvious outside of circlet of large 1 I areolse, narrow prominent costse at their outer M ends, .......... Johnsonii. l^No such costse, 2. Central space subcircular, surrounded by a band of large markings. Markings within the circlet of large areolse in uniform, curved, decussating rows, beyond the circlet decreasing gradually outwards. Apiculi faint, coronata. Central space obtusely triangular, without a dis- 2 I dinct limiting band of markings. Markings sub- ‘ ' circular towards central space. Apiculi minute, but more evident, ...... excellens. Central space minute, indistinct. Markings within the circlet of large areolse rounded, in obscure curved radial rows, beyond the circlet the rows moniliform. Non-apiculate, . . . sylendida. LNo central space, ....... 3. (An inconspicuous rosette. Markings outside circlet of large areolse increasing gradually in breadth towards border, ..... elabomta. 3. No rosette. Markings within the circlet of large areolse in substraight decussating rows without order, beyond the circlet subequal or slightly l smaller near border, hyberborect. STELLADICUS, gen. n. Circular. Central space and rosette absent. Markings : rays clavate, their inner ends rounded, meeting at centre, attenuating towards border, at middle of the interradial areas similar, but narrow faint rays extending to middle of inner ends of compartments, and continued thence as straight narrow lines to the border; the com- partments each consisting of three subequal parts separated by narrow straight lines reaching the border, the inner end of each part convex towards the centre; areolate, the areolse forming oblique, straight, decussating rows. Border narrow, hyaline. — Asterolampra , pro jparte Norman, Trans. Micr. Soc. Lond ., 1861, p. 6. S. stella , Battray. Asterolampra stella. Norman, Trans. Micr . Soc. Lond., 1861, p. 6, pi. ii. fig. 1. — Diam. ’09 mm. Markings: 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 633 clavate rays 6, about *005 mm. broad at their widest part, the faint interradial rays about '0025 mm. broad, the compartments reaching about of radius from circumference, the areohe subequal, 8 (?) in *01 mm. Norman provisionally united this species to Asterolamjpra. The remarkable appearance of the compartments and rays are sufficient to justify its separation as the type of a new genus. Habitat. — Sierra Leone (Norman).* ASTEROLAMPRA. Ehrb. emend., Mon. Ber. Ah., 1844, p. 73. — Circular, rarely obtusely and regularly angular. Surface subplain, a central or subcentral portion hyaline with distinct rays, the outer portion with evident compartments separated by distinct, rarely sub- obsolete, hyaline intervals. Colour subhyaline to subpearly or pale grey. Central space sometimes distinct, hyaline, granular or subareolate ; a rosette rarely differentiated, a central areolate area frequent. Markings on central area areolate, rarely arranged in concentric zones ; rays diverging from centre or from outer edge of areolate area, rarely from an excentric point, straight, arcuate of sharply geniculate at or near their middle, frequently dichotomous, narrow, more rarely broad or subobsolete, sometimes confined to a relatively narrow, submarginal zone ; interradial areas hyaline, or with faint diffuse lines passing to centro-lateral angles of compart- ments ; compartments with inner ends convex or concave towards the centre, sometimes transversely or obliquely truncate, rarely asymmetrical with respect to the rays; granular, with irregular, wide, hyaline interspaces or areolate, rarely subhyaline or minutely punctate ; the areolae in oblique, decussating, substraight rows parallel to the edges of the inter-compartmental intervals ; fre- quently a single row of larger areolae fringing the compartments ; those bounding its inner edge most evident, rarely large and con- spicuous ; at each centro-lateral angle a single areola, frequently larger and sometimes protuberant into the interradial spaces ; single or double granules rarely present at outer ends of intervals. Border narrow, hyaline, rarely broad, obtusely angular and subpearly. — * Communicated by Mr Frederick Kitton. 634 Proceedings of Royal Society of Edinburgh. [sess. Asterolampra , Grev., pro parte , Trans. Micr. Soc. Lond ., 1860, p. 162; Or aspedodiscus, pro parte, Brightw., Quart. Jour. Micr. Sci., 1860, p. 95. § 1. Margin at®. A large central areolate area. Rays many, short, confined to a submarginal zone. Compartments minute. A. marginata. Grev., Trans. Micr. Soc. Lond., 1862, p. 50, pi. viii. fig. 30. — Diam. *0625 to *1125 mm. Central area extending outwards for § to ~ of radius, its outer edge sinuate, subregular ; a rosette distinct. Markings on central area areolate, the areolae decreasing slightly from the rosette outwards from 4 to 5 in -01 mm.; rows radial, straight, non-fasciculate ; secondary oblique decussating rows faintly marked ; rays short, straight, radiating from the distal angles of the sinuations about 4 of radius long; the compartments restricted to a narrow (about *0025 mm. broad) zone adjacent to the border, their inner ends convex towards the centre ; the intervals represented by shallow indentations. — Eulenst., Diat. Spec. Typ., No. 16; Craspedodiscus marginatus, Brightw., Quart. Jour. Micr. Sci., 1860, p. 95, pi. v. fig. 7 ; Ralfs in Pritch. Inf, p. 832 ; A. marginata, var. minor, Walker and Chase, New and Rare Diat., ii. p. 7, pi. v. fig. 8. This species forms the connecting link between Coscinodiscus , Brightwellia, and Asterolampra. In the first named genus it ap- proaches C. bultiens, the markings on its central area also recall those of C. concinnus, var. jonesiana. The interspaces between the rays are larger, more peripherally placed, and less sharply defined than in any Brightivellia, whilst its regularly areolate central portion is homologous to the irregular inconstant areolate areas of Astero- lampra decora , A. affnis, A vulgaris, & c. Habitat. — Barbados deposit (Johnson ! Greville ! Grove ! Eulen- stein !) ; Springfield deposit, Barbados (Walker and Chase) ; Cambridge deposit, Barbados (Deby !) ; Chalky Cliff, Barbados (Deby !). § 2. Ductiles. A central areolate area. Compartments conical in outline with distant subradial branching, but evident lines. 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 635 A. ralfsiana. Grey., Trans. Micr. Soc. Lond., 1862, p. 50, pi. viii. fig. 31. — Diam. *07 to ‘08 mm. Central space sub-circular, •0025 to '003 mm. broad, sometimes slightly excentric ; an evident subcentral areolate area reaching between \ and i of radius from centre. Markings on subcentral area unequal, 2^ to 4 in *01 mm., the smaller sometimes confined to a band contiguous to the central space ; rays many, straight, those on one side of valve sometimes somewhat longer than those on opposite side ; the compartments reaching from -j to Afr of radius inwards, outline obtusely conical, sides subuniformly convex, non-areolate but with distinct subradial lines at intervals of ‘0025 to *003 mm., frequently dichotomising towards the border ; intervals wide, reaching close to the border ; adjacent to the border a circlet of delicate subregular striae, 4 to 4J in. *01 mm. — Eul., Diat. Spec. Typ., No. 16. Habitat.-— -Barbados deposit (Greville ! Grove, Eulenstein !); Cambridge deposit, Barbados (Deby !). § 3. SUBMARGARITACE/E. Aspect subpearly. A central space usually distinct. Rays broad or undifferentiated, the inner ends of the compartments extending to edge of central space. A. ambicfua. Grev., Trans. Alter. Soc. Lond., 1862, p. 54, pi. viii. figs. 42-45. — Diam. '0175 to *0525 mm. Central space angular, distinct, hyaline, up to about *01 mm. broad. Markings : rays straight, sometimes slightly constricted at the middle or expanding regularly outwards and merging into the compartments ; inter- radial spaces expanding outwards, at outer end trilobate, the small central lobe homologous to the interval between the com- partments ; two straight or slightly curved lines (concave towards each other) passing from the inner end of this lobe, and meeting at or near the inner ends of the interradial spaces ; the compart- ments reaching about of radius inwards, sometimes indistinctly defined, subhyaline ; small isolated round granules sometimes present close to the sides of the interradial spaces. Rarely the outline of the valve is obtusely angular. Habitat. — Cambridge deposit, Barbados (Johnson !) ; Barbados (Greville !) ; locality 1 (Deby !). 636 Proceedings of Boy at Society of Edinburgh. [sess. A. dabia. Grey., Trans. Micr. Soc. Lond ., 1862, p. 54, pi. vii. fig. 41. — Diam. *0375 mm. Central space quadrate, with sides slightly concave, hyaline. Markings : rays hardly differentiated, interradial spaces 4, cruciform ovate, their inner ends rounded, the outer subacute, homologous to the intervals between the com- partments; two distinct subparallel radial lines extending from their inner ends to the sides of the protuberant outer extremities, an evident round clear granule adjacent to the border and opposite the outer ends of each interradial space; the compartments extend- ing to edges of central space, subhyaline. Border distinct, its inner edge round. Habitat. — Barbados deposit (Greville !). A. aliena. Grev., Trans. Micr. Soc. Lond ., 1862, p. 55, pi. viii. fig. 46. — Diam. -0525 mm. Central space circular, about *01 mm. broad, not sharply defined. Markings : a minute clear round granule at middle of central area ; rays broad ; interradial spaces expanding uniformly outwards, the sides straight, their outer ends placed obliquely to the direction of the intervals, two faint sub- parallel lines extending between their inner ends and the edges of the intervals; the compartments reaching almost to the semiradius, subhyaline; the intervals narrow, of uniform breadth, a small round granule at their outer extremities. Border about f of radius broad, subhyaline, its inner edge slightly angular, the angles corresponding in position to the outer ends of the intervals. Habitat. — Barbados deposit (Greville !). § 4. Traducentes. Sometimes regularly polygonal. Central space distinct, granular or subareolate, rarely hyaline. Kays narrow. Compartments sub- hyaline, or their inner edge with a distinct band of larger areolae, their outer portion subareolate or granular. A. stellulata. Grev., Trans. Micr. Soc. Loncl., 1862, p. 54, pi. vii. fig. 40. — Sometimes regularly and obtusely polygonal. Diam. •045 mm. Central space irregularly concavo-convex, about *0075 mm. long by *005 mm. broad. Markings: rays 7 to 9, straight, regular; the compartments reaching about •§ of radius inwards, their inner ends sigmoid on each side of, and protuberant 1888-89.] Mr John Rattray on the Genus Coscinocliscus. 637 towards the centre opposite the rays ; the puncta obscure, isolated towards their inner, more crowded around their outer extremities, 10 to 12 in *01 mm.; the intervals narrow, of uniform width, their outer ends convex outwards not reaching the border, a small round clear granule intervening between them and the border. Border narrow, indistinct. Habitat. — Barbados deposit (Greville !). A. hittoniana. Grey., Trans. Micr. Soc. Lond ., 1862, p. 53, pi. viii. fig. 39. — Diam. '04 to *05 mm. Central space heptangular, about i of diam. broad, its sides deeply concave outwards. Mark- ings : central space punctato-areolate ; rays 7, straight ; the com- partments reaching from to J of radius inwards, their inner ends concave inwards on each side of, hut protuberant opposite, the rays, sometimes bluntly conical, their centro -lateral angles rounded, the markings obscure ; the intervals alternating rapidly outwards, not reaching the border, their outer ends acute, with a minute rounded granule at each side sometimes visible. Habitat. — Barbados deposit (Kitton, Dehy !). A. traducens , sp. n. — Diam. *0875 mm. Central area circular, '025 mm. broad. Markings on central area isolated, rounded or irregular and minute granules; rays 16, narrow; at middle of interradial spaces a narrow, sharply defined, suhlinear area ex- tending from their inner ends to the intervals; the compartments with inner ends faintly defined, an outwardly concave row of minute puncta stretching between the extremities of the adjacent intervals, those contiguous to the intervals somewhat more promi- nent, elsewhere the puncta more minute and obscure; intervals tapering rapidly outwards, their outer ends acute, not reaching the border. Border obtusely and subregularly polygonal. — (PI. III. fig. 22.) Habitat. — Barbados deposit (Deby !). A. pulclira. Grev., Trans. Micr. Soc. Lond., 1862, p. 53, pi. viii. figs. 37, 38. — Obtusely and regularly octagonal. Diam. '0625 mm. Central space distinct, ^ to J of diam. broad, its sides deeply concave outwards between the radii. Markings : on central space a few irregular, obtusely angular or subareolate granules ; 638 Proceedings of Royal Society of Edinburgh. [sess. rays 8, straight ; at middle of interradial spaces 2, somewhat faint subparallel lines radiating to the inner edges of the intervals ; the compartments reaching about J of radius inwards, the inner ends concave towards the centre on each side of, but somewhat pro- tuberant opposite, the rays, subhyaline or obscurely and minutely punctate, sometimes with a band of more prominent oblong mark- ings at their inner ends ; the intervals alternating gradually out- wards, their outer ends convex, and not reaching the border with a distinct oval, obliquely placed granule at one or both sides, those belonging to one interval sometimes meeting each other at the extremity of the interval. Habitat. — Barbados deposit (Greville). A. scutula. Grev., Trans. Micr. Soc. Lond., 1862, p. 52, pi. viii. fig. 47. — Diam. *04 mm. Central space regularly hexagonal, about *005 mm. broad, its sides convex, towards the centre hyaline. Markings : rays 6, straight ; the compartments reaching about § of radius inwards, their inner ends flat or somewhat concave, pro- tuberant at outer ends of rays ; a single large areola at their centro- lateral angles, tapering slightly outwards, its inner end convex but hardly protuberant, elsewhere the puncta obscure or unresolved; the intervals evident, expanding somewhat rapidly outwards, their outer ends rounded, reaching the border. Border narrow, hyaline. Habitat. — Barbados deposit (Johnson !). A. simulans. Grev., Trans. Micr. Soc. Lond., 1862, p. 52, pi. viii. fig. 36. — Circular or regularly and obtusely polygonal. Diam. •0375 to ’06 mm. Central space polygonal, *01 mm. broad, its sides convex towards the centre, hyaline. Markings : rays 9, straight ; at middle of interradial spaces somewhat broader, more faint lines radiating to the edges of the intervals ; the compart- ments reaching about i of radius inwards, their inner ends concave, concentric, with circumference sometimes slightly protuberant at ends of rays ; a distinct band of narrow radially elongate areolae fringing their inner ends, those at the centro-lateral angles most distinct and largest ; contiguous to the outer ends of the areolae a narrow, hyaline, indistinct band concave towards the border, beyond this the puncta obscure, about 5 in *01 mm., sometimes 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 639* subareolate ; intervals narrow, unequal, sometimes expanding slightly at their outer ends. Border narrow, its inner edge with minute irregularities, hyaline. This species is sometimes confounded with A. punctata, from which it differs in the appearance of its central space and the sculpturing of its compartments. Habitat. — Barbados deposit (Greville !) ; Springfield deposit, Barbados (Hardman !).* A. cemulans. Grev., Trans. Micr. Soc. Lond ., 1862, p. 52, pi. viii. figs. 34, 35. — Diam. '05 to *075 mm. Colour pale brownish-grey at centre and along the rays, elsewhere pale smoky grey. Central portion distinct, extending outwards for J to T%- of radius. Markings on central portion areolate, the areolse 3J to 4 in ‘01 mm.; rays straight or but slightly arcuate; the compart- ments reaching from f to of radius inwards, their inner ends convex, with short distinct radial lines about 4 in '01 mm., and extending outwards for i to J of their length, their outer portion with delicate areolae decreasing rapidly outwards from 5 or 5J to 8 in -01 mm.; the intervals of equal breadth, their outer ends rounded, not reaching the border. Border distinct, hyaline. Both of the typical specimens in Greville’s collection show a dis- tinctly areolate central portion, and there is no indication of its ever being solid, as stated by Greville. In neither are the markings on the compartments isolated puncta. Habitat. — Barbados deposit (Greville !). § 5. Eximi^. Central areolate area sometimes present. Rays rarely unequal, and meeting at an excentric point or absent. Compartments areolate, their inner ends convex, concave, or obliquely truncate towards the centre, sometimes bounded by a prominent areolate fringe; intervals rarely obsolete or occluded at their inner ends, reaching the border. A. nicobarica , Grun. Van Heurck, Syn. Diat. Belg., pi. cxxvii. fig. 7. — Diam. *055 mm. Central space angular, about ’01 mm. In the collection of Mr Julien Dely. 640 Proceedings of Royal Society of Edinburgh. [sess. broad. Markings : a single rounded granule at middle of central space; rays between central space and middle of inner ends of compartments absent, but well-defined, subpearly, subclavate areas, tapering gradually outwards, sometimes slightly flexed, and each with a narrow delicate flexuous line close to their edges, extending between central space and outer ends of intervals, the lines from the sides of the adjacent areas meeting one another at their inner ends ; the compartments reaching about i of radius inwards, their inner edge convex towards the centre, the areolae distinct, 6 to 8 in *01 mm.; at irregular wide intervals a few more prominent rounded dots ; the intervals not reaching the border, tapering outwards, their outer ends convex. Habitat. — Nancoori deposit (Van Heurck). A. punctata. Grev., Trans. Micr. Soc. Lond ., 1862, p. 51, pi. viii. fig. 32. — Diam. ‘0625 mm. Central areolate area subelliptical to round, about *0075 mm. broad. Markings on central area unequal, 4 to 5 in *01 mm.; rays 6 to 7, straight; the compartments reach- ing about \ of radius inwards, their inner ends transversely truncate or slightly concave inwards opposite the radii ; the areolae few, granular towards their inner ends, towards the border 6 to 7 in *01 mm., with hyaline interspaces; the intervals wide, sometimes expanding slightly outwards, their outer ends transversely truncate, sometimes reaching the border. Border, narrow, indistinct, sub- hyaline. The smaller areolae on the central area may be so disposed as to form an arc round a somewhat larger unilateral areola. Habitat. — Barbados deposit (Greville !). A. balearica. Cleve, Kongl. Sv. Vet.-Akad. Handl. Stockh ., 1881, Bt. xviii. No. 5, p. 20, pi. v. fig. 59. — Diam. *0715 mm. Central space circular, hyaline, about of diam. broad. Mark- ings : rays straight, symmetrical ; the compartments reaching about of radius inwards, their inner ends uniformly and con- siderably convex, areolate; the areolae subequal or decreasing but slightly outwards, 9 in ’01 mm.; the intervals of uniform breadth, their outer ends convex, reaching about |- of radius from centre. From A. Grevillei this differs not only in the coarser character 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 641 of the areolae, but also in the appearance of the rays and length of the intervals between the compartments. In Asterom. centraster , Johnson, the intervals between the compartments are continued to the centre, and their outer ends are swollen and knob-like. From A. brebissoniana it is readily distinguished by the straight instead of geniculate rays. Habitat. — Balearic Islands (F. Soderlund). A. Icevis. Grev., Trans. Micr. Soe. Lond., 1862, p. 51, pi. viii. fig. 33. — Diam. *0275 mm. Central rosette and areolate area absent. Markings : rays 6, straight ; the compartments reaching to about f of radius inwards, their inner ends transversely truncate, sometimes slightly flexuous or concave towards . the centre ; the areolae obscure, least evident towards the border; the intervals extending to about \ of distance between their inner ends and border, tapering gradually outwards with the inner ends convex. Border narrow, hyaline, indistinct. Habitat. — Barbados deposit (Greville !). A. marylandica. Ehrb., Mon. Ber. Ah., 1844, p. 76, fig. 10 (June 1844). — Diam. '0375 to J5 mm. Central areolate area absent. Markings : the rays 4 to 12, straight, or but slightly arcuate, diverging from the centre, sometimes dichotomous or dividing into three equal rami ; the compartments reaching from § to J of radius inwards, their inner ends uniformly convex towards centre, some- times truncate or slightly emarginate, rarely unsymmetrical ; the areolae distinct, subequal on inner § of compartments, or decreasing more regularly outwards from 6 to 10 in *01 mm.; a band of more prominent submuriform areolae around their edges; the intervals expanding gradually outwards, their outer ends reaching close to the border. — Muller, Abh. Ber. Ah., 1841, p, 232, pi. vi. fig. 4 (no name); Bail., Amer. Jour. Sci., 1845, vol. xlviii. pi. iv. fig. B; Brightw., Quart. Jour. Micr. Sci., 1860, p. 94, pi. v. fig. 3; Wallich, Trans. Micr. Soc. Lond., 1860, p. 47, pi. ii. figs. 14, 15 ; Grev., Trans. Micr. Soc. Lond., 1860, p. 108, pi. iii. figs. 1-4 ; Ralfs in Pritch. Inf., p. 836, pi. xi. fig. 33 ; Grev., Trans. Micr. Soc. Lond., 1862, p. 44, pi. vii. figs. 1-3 ; A. septenaria, Johnson, Amer. Jour. Sci., 1852, ser. 2, vol. xiii. p. 33; A. impar, Shadb., Trans. Micr. Soc. vol. xvi. 7/11/89 2 s 642 Proceedings of Royal Society of Edinburgh. [sess. Lond ., 1854, p. 17, pi. i. fig. 14; A. pelagica, Ehrb., Mon. Ber. Ah., 1854, p. 238; A. hexactis, Ehrb., Abh. Ber. AJc ., 1872, pi. ix. figs. 1, 2 ; A. marylandica , var. ausonia , Cstr., Atti Accad. pontif. nuov. Lincei , Roma , 1875, p. 393, pi. vi. fig. 4. Dr Wallich distinguished as var. ft (Wallich, ibid., 1860, p. 48, pi. ii. fig. 14), forms having compartments with rounded inner ends and simple rays, and as var. y (ibid., pi. ii. fig. 15), those having the inner ends of the compartments truncate, and the rays con- spicuously ramose near their central ends, the branching of the rays being symmetrical with respect to a diameter of the valve. Peragallo (Diat. Baie d. Villefranche, 1888, p. 74) 'still adheres to Wallich’s .var. f3, naming it A. marylandica, var. major, but it appears to me hardly sufficiently distinct from the type to merit varietal recognition. Wallich’s var. y may be named A. mary- landica, var. ramosa. Forms named by Thum A. adriatica , Grun., and now in the collection of Mr Julien Deby, do not differ from A. marylandica. One of the specimens named A. adriatica has a delicate round granule (probably an apiculus) at the outer extremity of each interval, thus recalling A. marylandica, var. ausonia , Cstr. Mr Edmund Grove has proposed to establish a var. approjpinquans, for specimens in which one of the compartments is smaller than the others, two of the intercompartmental intervals being in conse- quence unusually approximated. Habitat. — Maryland (Ehrenberg) ; Barbados deposit (Greville !) ; Springfield deposit, Barbados (Hardman !) ; Cambridge deposit, Barbados (Greville ! Hardman !) ; Newcastle deposit, Barbados (Grove!); Piscataway deposit (Johnson, Greville! Roper!);* Monterey stone (Greville ! Walker- Arnott) ; Richmond deposit, Ya. (Johnson); Rappahannock (Greville); Bermuda tripoli (Dallas); Oamaru deposit (Grove!); Indian Ocean soundings, lat. 5° 37' S., long. 61° 33' E., 2200 fathoms, Captain Pullen (Greville !); Indian Ocean, off Zanzibar (Ehrenberg) ; from Salpoe, Bay of Bengal (Wallich) ; Alexandria (Hardman !); Port Natal (Shadbolt) ; from Comatulce, Mediterranean Sea (Muller) ; Holothuria, China (Thum); South Naparima, Trinidad (Greville) ; Rembang Bay (Deby !) ; f Tegel von Mahren, Austria (Deby !). * In the collection of Dr R. K. Greville. t In a type slide of material from this locality prepared by Herr Thum. 1888-89.] Mr John Rattray on the Genas Coscinodiscus. 643 A. rotula. Grev., Trans. Micr. Soc. Lond., 1860, p. Ill, pi. iii. fig. 5. — Diam. *11 mm. Central areolate area absent. Markings: rays diverging from a central point, simple, or rarely dichotomous, straight or slightly arcuate ; on the interradial spaces faint, indefinite, narrow dark hands extending for some distance inwards from the centro-lateral angles of the compartments ; the compartments reach- ing about § of radius inwards, their inner edges somewhat obliquely truncate on each side of the rays; areolae faint; the intervals of. uniform breadth, their outer ends convex close to the border. — Cstr., Atti Accad. pontif. d. nuov. Lincei , 1875, p. 393, pi. vi. fig. 3a; A. Grevillii Wallich , var. adriatica , Grun. in Yan Heurck, Syn. Diat. Pelg., pi. cxxvii. fig. 12. This species is distinguished from those forms .of Asteromphalus variabilis having a faintly differentiated subobsolete ray by the more straight rays and the less obliquely truncate inner ends of the compartments. Peragallo (Diat. \Baie d. Villefranche , Paris , 1888, p. 74) has already pointed out the identity of Grunow’s A. Grevillii ) var. adriatica with A. rotula. Habitat. — Monterey stone ( Walk er- Arno tt); Adriatic Sea, Balearic Islands (Yan Heurck). A . dallasiana. Grev., Trans. Micr. Soc. Lond., 1860, p. 115, pi. iv. fig. 10. — Diam. ‘075 mm. Central areolate area absent. Markings : rays 7, slightly and uniformly arcuate towards the same direction ; the compartments reaching to § of radius inwards, their inner ends transversely truncate, areolate; the areolae sub- equal 10, near border 12, in *01 mm.; the intervals attenuating slightly outwards, and again somewhat swollen at the extremities, reaching close to the border. — Asteromphalus dallasianus , Ralfs in Pritch. Inf., p. 836. Although a centro-lateral area is present there is no such subobsolete interval as would be found in Asteromphalus. Compare also in this respect some forms of Asteromphalus variabilis. Habitat. — Nottingham deposit, U.S. (Greville !). A. brebissoniana. Grev., Trans. Micr. Soc. Lond.) 1860, p. 114, pi. iii. fig. 9. — Diam. *075 mm. Central areolate area absent. Markings : rays sharply bigeniculate about f of their length from the centre, two or three adjacent rays meeting 644 Proceedings of Royal Society of Edinburgh. [sess. one another a short distance from centre whither they are con- tinued as a simple line ; the compartments reaching from -| to £ of radius inwards, their inner ends transversely truncate or hut slightly convex, areolate ; the intervals attenuating gradually outwards, their outer ends slightly swollen and knob-like. Greville has pointed out that the geniculate flexure of the rays found also in Aster omphalus imbricatus , Aster om. Darwinii , Aster om. elegans , Aster om. Brookei , and in a less degree in Aster om. sliadboltianus , goes to establish the validity of the union of Aster omphalus and Asterolamyra into a single genus, but this ignores the apparently more constant characters in the latter con- nected with the subobsolete interval between two of the compart- ments. Habitat. — Monterey stone ( Walker- Arnott). A. Grevillei. Grev., Trans . Micr. Soc. Lond ., 1860, p. 113, pi. iv. fig. 21. — Diam. *075 to -085 mm. Central areolate area absent, sometimes represented by two plano-convex areolae. Mark- ings : rays straight, or gently arcuate, diverging from an angular arched central line rarely from a central point ; the compartments reaching about J of radius inwards, their inner ends curved or flattened on opposite sides of rays convex towards the centre, their margin formed by a narrow subhyaline band, elsewhere minute sub- punctiform granules, least evident towards the border, sometimes unresolved; the intervals of uniform breadth; their outer ends trans- versely truncate, sometimes terminating a considerable distance from the border. Border narrow, indistinct. — Aster omphalus Grevillei , Wallich, Trans. Micr. Soc. Lond., 1860, p. 47, pi. ii. fig. 15. The radii vary in number from 7 to 17, and exhibit considerable variation in their mode of origin, an adjacent pair frequently uniting and being connected by a common short central stalk with the central portion. Habitat. — Moron deposit (Greville ! Hardman !) ; Indian Ocean 2200 fathoms, Captain Pullen (Wallich); Eappahannock deposit, U.S. (E. W. Dallas) ; Monterey stone (Walker- Arnott). A. princeps * nov. A. Grevillei , var. eximia. Cstr., Diat. Chall. * The name eximia “cannot be adopted, as it has been preoccupied by Greville {Trans. Micr. Soc. Lond., 1865, p. 99) for a distinct form (see infra). 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 645 Exped ., 1886, p. 136, pi. v. fig. 5. — Diam. -18 mm. Central areolate area absent. Markings : rays regularly arcuate, rarely substraight, 2 or 3 short lines diverging from the centre and branching somewhat irregularly 2 to 4 times before reaching half the distance to the inner ends of the compartments ; the compartments reaching about § of radius inwards, their inner ends transversely truncate or slightly concave towards the centre, the areolae forming a distinct hand along their outer edge and 6, elsewhere in straight oblique decussating rows and about 8, in *01 mm., decreasing hut slightly towards the border ; the intervals attenuating regularly outwards and reaching the border. Border distinct, hyaline. Habitat. — Equatorial Atlantic, H.M.S. Challenger (Castracane). A. brightwelliana. Grev., Trans. Micr. Soc. Lond ., 1862, p. 48, pi. viii. figs. 26, 27. — Diam. *075 to *095 mm. Surface markedly convex. Central areolate area absent. Markings : rays straight or suhuniformly curved, springing from a somewhat excentric point, the compartments reaching about J of radius inwards, their inner ends concave towards the centre, unequal, areolate ; a single hand of areolae adjacent to their inner ends large, 3 to 3J in *01 mm., the others much smaller 6, decreasing gradually outwards to 8 or 9 in *01 mm., the intervals of equal breadth (about *0035 to *004 mm.), their outer ends reaching the border. Habitat. — Springfield deposit, Barbados (Hardman !) ; “ Bar- bados” deposit (Greville !); Cambridge deposit, Barbados (Dehy !). A crenata. Grev., Trans. Micr. Soc. Lond., 1862, p. 47, pi. viii. figs. 4-16. — Diam. *05 to *075 mm. Central areolate area absent. Markings : rays straight ; the compartments reaching about J of radius inwards, their inner ends concave towards the centre and concentric with the border, areolate, a single band of large unequal areolae at their inner ends 4 in *01 mm., and from 3 to 3 J times as long as broad, beyond this hand 6, decreasing rapidly outwards to 10 in *01 mm.; the intervals expanding slightly outwards and of uniform breadth, reaching the border. This species is distinguished from A. concinna by the shape of the inner ends of the compartments and from A. vulgaris by their regularity. Habitat. — Barbados deposit (Greville! Hardman!). 646 Proceedings of Royal Society of Edinburgh. [sess. A. eximia. Grev., Trans . Micr. Soc. Lond ., 1865, p. 99, pi. viii. fig. 10. — Diam. ’0875 to *15 mm. Central rosette distinct, a central areolate area about *036 mm. broad. Markings : on central area unequal areolse 2 to 2J in *01 mm.; rays 22, straight ; the compartments reaching about f of radius inwards, their inner ends convex, bounded by a single band of large areolae about 3 in •01 mm., and gradually decreasing in length from the radii out- wards, elsewhere the areolae quadrate, decreasing gradually from 3 to 4 in *01 mm.; intervals of equal breadth throughout, their outer ends convex, reaching close to the border. Habitat. — Cambridge deposit, Barbados (Hardman) ; Chalky Cliff, Barbados (Deby !). A. concinna. Grev., Trans. Micr. Soc. Lond., 1862, p. 46, pi. vii. figs. 10-12. — Diam. *075 to *1 mm. Central areolate area absent. Markings : rays straight, springing from centre ; the compartments reaching from -J- to \ of radius inwards, their inner ends transversely truncate or slightly convex, bounded by a single band of large areolse 4 in ‘01 mm. at their inner ends, sometimes a single large areola at each of the centro-lateral angles of the compartments ; elsewhere the markings 6, decreasing gradually out- wards to 8 in *01 mm.; the intervals expanding slightly outwards, their outer ends convex, reaching close to the border. Habitat. — Barbados deposit (Greville !). A. vulgaris. Grev., Trans. Micr. Soc. Lond., 1862, p. 47, pi. vii. figs. 17-20. — Diam. ‘0375 to ‘055 mm. A central areolate area sometimes present. Markings on central area unequal, about 4 in *01 mm., rarely a single round areola only present; rays straight; the compartments reaching from ^ to J of radius inwards, their inner ends concave at the middle, bounded by a distinct band of large areolse about 4 in *01 mm., those at the centro-lateral angles largest, and protruding into the interradial spaces with their central ends rounded, sometimes those of the adjacent compartments expanding towards one another at their inner ends so as almost to exclude the intervals; elsewhere the areolse 8 to 10 in ’01 mm.; the intervals expanding for some distance outwards, their outer ends sometimes reaching close to the border, convex outwards. — 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 647 A. vulgaris , var. a, Grev., ibid, 1862, p. 47 ; A. vulgaris, var. b, Grev., ibid., p. 47, pi. vii. fig. 21 ; Eul., Diat. Spec. Typ., No. 16. Rarely in valves from Cambridge deposit, Barbados, the areolae on the central area are unequal and much larger, from 1J to 1-J in *01 mm., and are disposed subconcentrically towards its outer margin. Habitat. — Barbados deposit (Greville ! Hardman ! Eulenstein !) locality ? (Deby !) ; Springfield deposit, Barbados (Hardman !) ; Cambridge deposit, Barbados (Hardman !) ; Oamaru deposit (Grove ! Deby !). Yar . planior. A. vulgaris, var. c. Grey., ibid., 1862, p. 47, pi. vii. fig. 22. — Diam. *07 mm. Central area distinct, subcircular, from to -J of diam. broad. Markings on central area 2J to 3 in •01 mm., unequal; the compartments reaching about T2T of radius inwards, the largest centro-lateral areolae but slightly protuberant, sometimes bulging laterally towards their inner ends ; the intervals clavate, their outer ends convex, close to the border. Habitat.— -Barbados deposit (Greville ! Grove !). Yar. cellulosa. A. vulgaris, var. d. Grev., ibid, 1862, p. 47, pi. vii. fig.- 23 ; pi. viii. fig. 24. — Diam. -055 to *1 mm., central area subcircular, from J to f- of diam. broad, a central rosette sometimes distinct. Markings on central area subequal, 4 in '01 mm., without order or in obscure oblique decussating rows ; the compartments reaching from ^ to f of radius inwards, the band of areolae at their inner ends extending outwards almost to border, about 4 in *01 mm. across ; the marginal areolae protruding slightly at their central inwardly convex ends. Habitat. — Barbados deposits (Greville ! Deby ! Grove !). A. decorata. Grev., Trans. Micr. Soc. Lond ., 1862, p. 46, pi. vii. fig. 13. — Diam. *08 mm. Central areolate area absent. Mark- ings : rays straight, the compartments reaching about J of radius inwards, their inner ends transversely truncate, bounded by a single band of oblong areolae 2 to 2J in *01 mm. broad and about *01 mm. long, those opposite the rays with long axis radial, the others be- coming more and more oblique, those at the angles of the compartments *0175 mm. long, with outer ends attenuated ; the 648 Proceedings of Royal Society of Edinburgh. [sess. remaining areolae decreasing gradually outwards from 6 to 9 in *01 mm.; the intervals expanding gradually outwards, their outer ends swollen, knob-like, reaching close to the border. Habitat. — Barbados deposit (Greville !). A. splendida. Grey., Trans. Micr. Soe. Lond ., 1862, p. 48, pi. viii. fig. 25. — Diam. *09 mm. Central areolate area distinct. Markings on central area unequal, about 2J in ’01 mm.; rays straight, or but slightly arcuate, many ; the compartments reaching about of radius inwards, their inner ends concave, bounded by a band of large areolae 3J in -01 mm. broad, and extending outwards almost to the border, those at the edges of the compartments with inner ends protuberant and rounded ; close to the border the areolae obscure, subpunctiform, 12 in *01 mm.; the intervals of uniform (•0025 mm.) width, their outer ends close to the border, convex outwards. — A. vulgaris , var. e, Grev., ibid, 1862, p. 47. The great length of the areolae at the inner ends of the compart- ments at once distinguishes this species from A. vulgaris , to which Greville united it with some hesitation. Habitat. — Barbados deposit (Greville !). A. ur aster. Grove and Sturt, Jour. Quek. Micr. Cl., 1887, p. 143, pi. xiii. fig. 42. — Diam. *06 mm. Central areolate area distinct. Markings on central area large, 2 to 2J- in ‘01 mm.; rays straight, two passing to the apex of one of the compartments; the compartments reaching from § to f of radius inwards, conical, their inner ends obtusely angular, the areolae evident, decreasing but slightly outwards, from 5J- to 6 in ’01 mm.; the intervals broad, attenuating somewhat outwTards, not reaching the border ; a minute round granule (apiculus V) at the outer end of each. The apex of the compartment that receives the two radii is somewhat obliquely truncate, and slightly concave towards the centre. Habitat. — Oamaru deposit (Gray). A. rylandsiana. Grev., Trans. Micr. Soc. Lond., 1862, p. 49, pi. viii. figs. 28, 29. — Diam. *04 to *05 mm. Central areolate area about ’01 mm. broad. Markings on central area subequal, about 3 in *01 mm.; rays 7 to 12, straight, or but slightly flexed ; the 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 649 compartments reaching about of radius inwards, their inner ends transversely truncate, or slightly convex towards the centre, their adjacent sides formed hy a large cuneate areola, distinctly protrud- ing by an inwardly convex inner end into the interradial space and attenuating towards the border, elsewhere the areolae evident, decreasing rapidly outwards from 4 to 8 in *01 mm.; the intervals obsolete. Border narrow, indistinct. Habitat. — Barbados deposit (Greville ! T. G. Rylands, de Brebisson, Grove !) ; Springfield deposit, Barbados (Hardman !). A. tenerrima ,* sp. n. — Diam.? Central areolate area absent. Markings : rays 4 to 7, straight ; the compartments reaching from § to \ of radius inwards, their inner ends concave towards centre or transversely truncate ; a single hand of large areolae bounding the inner ends, outside of this a single large lanceolate areola extending close to the border hounding the interval; elsewhere the areolae small, evident, in distinct radial and less manifest suhregular concentric zones ; intervals extending to border of uniform width. — (PL III. figs. 18, 20.) Habitat. — ? A. affinis. Grev., Trans. Micr. Soc. Lorid., 1862, p. 45, pi. vii. figs. 7-9. — Diam. ’0675 to ’1175 mm. Colour pale grey, the rays more opaque. Central areolate area from *0125 to *0175 mm. broad, sometimes absent ; a small central space rare. Markings on central portion 2 in *01 mm., hyaline; rays straight or slightly flexuous towards the inner ends ; the compartments reaching from * This species is established on two specimens occurring in a photograph now in the possession of Mr Julien Deby, the history of which is given in a letter addressed to him by Herr E. Weissflog dated 27th July 1878. Herr Weissflog says : — “ I have received a letter from Mr F. Habirshaw of New York, in which he says — ‘The late John E. Gavit . . . engraved the fine plate in Bailey’s “New Species,” &c. (Smith’s Contrib.). He also some years ago made a plate which suddenly disappeared — neither plate nor impressions could be found. In overhauling the effects of Judge Johnson (of Asterodiscus in Silliman's J our. ) two impressions were found, and it is believed that they are the only two extant. A few days since we photographed the one sent us, and we hope that you will be pleased with the result. If there are more wanted, I would like MM. Deby and Delogne to have copies.’ Herewith you will find two proofs, and you will oblige much by remitting one to M. Delogne.” Nothing further is known of the specimens. They seem, however, to come from the Barbados deposit. 650 Proceedings of Royal Society of Edinburgh. [sess. \ to -J- of radius inwards, their inner ends slightly angular towards centre or transversely truncate, the areolae decreasing but slightly towards the border, the outermost row more conspicuous; the intervals of equal breadth throughout, their outer ends reaching the border. Border narrow, hyaline. Habitat. — Barbados deposit (Johnson ! Deby !) ; Oamaru deposit (Grove !) ; Newcastle estate, Barbados (Grove !). A. decora. Grev., Trans. Micr. Soc. Lond., 1862, p. 45, pi. vii. figs. 4-6. — Diam. *04 to *1375 mm. Central space subtriangular or absent, central areolate area distinct, up to *02 mm. broad. Markings on the central area unequal, 2 to 3 in *01 mm., those around its edges largest; rays 5 to 19, straight, sometimes meeting at the centre ; the compartments reaching about £ of radius inwards, their inner ends slightly concave or convex towards the centre, the areolse at their inner ends largest, 4 in *01 mm., the others decreas- ing rapidly outwards from 6 to 10 in *01 mm.; the intervals of uniform breadth, reaching close to the border. — A. decora , var. Cstr., Diat. Chall. Exjped ., 1886, p. 136, pi. xvi. fig. 9. In the variation in size of the areolate central area this species approaches the forms of A. vulgaris and A. rylandsiana. The size of the markings on the compartments is less, according as the specimens decrease in size. Habitat. — Cambridge deposit, Barbados (Greville ! Hardman !); “ Barbados ” deposit (Greville !) ; Springfield deposit, Barbados (Hardman !); Oamaru deposit (Deby ! Grove ! Hardman !). Yar. concentrica , nov. — Diam. *095 to *125 mm. Central areolate area from *0375 to *065 mm. broad, a rosette some- times evident. Markings on central area 1-J to 2 in *01 mm., with central dots large, and in evident concentric, less evident radial rows ; rays many ; at middle of interradial space a distinct hyaline area, with inner end close to border of central area; the compartments reaching from ^ to of radius, areolse obvious, 4 to 8 in *01 mm.; the intervals gradually increasing in breadth out- wards, their outer ends transversely truncate. Habitat. — Oamaru deposit (Grove ! Hardman ! Deby !); Cam- bridge deposit, Barbados (Johnson !). 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 651 A. Weissflogii. Van Heurck, Syn. Diat. Belg ., pi. cxxvii. fig. 9. — Diam. *06 mm. Central space minute, irregular. Markings : rays 6, straight, robust ; at middle of interradial spaces faint subhyaline areas about '0025 mm. broad, extending between central space and intervals ; the compartments distinct, extending from -J to of radius inwards, their inner ends concave inwards on each side of, but protuberant opposite the rays ; delicate, radial, subindefinite, faint striae, about *0025 jnm. long fringing their inner edge ; a somewhat larger more prominent areola at the centro-lateral angles ; elsewhere the markings punctiform, obscure ; the intervals sharply defined, tapering slightly towards, but not reaching the border with outer ends convex outwards ; a delicate, irregularly sinuate, continuous zone of unequal breadth contiguous to the inner ends of the com- partments. Border about to of radius broad, its inner edge sharply defined, hyaline.- — Pelletan, Les Diat. Nat. Hist ., tom. i. p. 206, fig. 112 ; tom. ii. fasc. i. p. 170, fig. 427 ; A. ( pulchra , var.?) Weissflogii , Grun. ; Van Heurck, loc. cit. In some specimens the outer ends of the intervals between the compartments are not swollen, as shown in Professor Van Heurck’s figure. Habitat. — Barbados deposit (Deby !); Cambridge deposit, Bar- bados (Van Heurck). Artificial Key . fRays straight or subvmiformly curved, springing 1. -J from a somewhat excentric point, .... ^No such excen tricky, f A large central areolate area ; rays short, numerous, 2. J compartments convex inwards ; intervals sub- j obsolete, ....... I No such structure, ....... f Compartments obtusely conical, ornamented with 3. -J distinct subradial dichotomising lines, . t No such markings on the compartments, . f Rays obsolete or subobsolete or broad ; interradial j spaces small, j Rays narrow, sublinear ; valve subpearly, 4. -{ Rays linear ; valve more hyaline, .... | Rays broad, subclavate, passing between central j Space and intervals between compartments, with L inner ends convex to centre, .... ( Interradial spaces tapering towards centre, g j Interradial spaces tapering towards border ; a distinct ' j round granule adjacent to border opposite outer l end of each, brightwclliana. 2. marginata. 3. ralfsiana. 4. 5. 6. 7. nicobarica. 8. dubia. 652 Proceedings of Royal Society of Edinburgh. [sess. f Interradial spaces with outer ends trilobate ; the median lobes homologous to subobsolete intervals between the compartments; a central hyaline area 8. -{ sometimes present ; radii sometimes broad, . . ambigua. | Interradial spaces with inner ends acute ; a small j round central granule and one at outer end of each l interval. Border broad, inner edge angular, . aliena. „ f Central area granular or subareolate, . . . 9. ' \ Central area hyaline, without markings, . . . 10. f Inner ends of compartments with short distinct I radial lines, 11. j No such radial lines. Markings on compartments j obscure, at each side of the outer ends of the 9. intervals a minute granule, ..... kittoniana. j No such radial lines. Compartments with inner ends j faint, an outwardly concave row of minute puncta j stretching between outer extremities of adjacent L intervals, ........ traducens. f One portion of compartments areolate ; no granules at j outer ends of intervals, ..... aemulans. 11. -j Outer portion of compartments subhyaline or I obscurely punctate, a distinct oval granule at one l or both sides of outer ends of intervals, . . pulchra. [ Markings on compartments obscure, a small round j granule opposite the outer end of each interval, . stellulata. 10. -{ A distinct areola at centro-lateral angle of each | compartment, hardly protuberant into interradial L space, ......... scutula. « j A band of large areolse at inner ends of compartments, 12. * | No such areolate band, 13. fThe band of large areolae prominent, 2 to 2^ in *01 mm., the lateral areolae of each compartment not protuberant ; inner ends of compartments convex inwards ; the intervals expanding gradually out- j wards. No central areolate area, .... decor ata. 12. \ The areolae still longer and narrower (3^ in ‘01 mm. across), the lateral areolae protuberant ; inner ends of compartments concave towards centre, intervals not expanding outwards. A central areolate area, splendida. The limiting band of areolae less prominent but _ distinct, 14. 14. ( Inner ends of compartments transversely truncate or convex towards centre, ...... Inner ends of compartments distinctly concave towards centre, Inner ends of compartments transversely truncate or concave towards centre. A single large lanceo- late areola bounding each side of the intervals be- tween compartments. No central areolate area, . Inner ends of compartments concave inwards on each side of rays; a delicate irregularly sinuate zone contiguous to inner ends of compartments, a minute irregular central space, 15. 16. temrrvma. Weissflogii. 1888-89.] Mr John Rattray on the Genus Goscinodiscus. 653 ( Lateral areolse larger, non -protuberant ; intervals I expanding slightly outwards. No areolate central j area, ......... concinna. j Lateral areolse not larger ; intervals of uniform j breadth ; a central areolate area, .... decora. , ~ j Lateral areolse not larger, but larger areolse at inner * } ends of compartments, gradually decreasing away j from the rays; a central rosette; compartments ( with inner ends convex; intervals of equal breadth, cximia. | Lateral areolse not larger. Compartments with inner | ends mostly convex. No areolate central area. L Intervals expanding somewhat outwards, . . marylandica. ( Lateral areolse protuberant, 16.-! Lateral areolse non-protuberant; intervals expanding ^ slightly outwards. No central areolate area, 17. crenata. f Intervals narrow, frequently almost occluded at the j inner ends. A central areolate area, . . . vulgaris. 17. ■{ Intervals narrow, sometimes expanding outwards ; j contiguous to the large areolse, a narrow' hyaline L band, concave towards border, .... simulans. f Intervals between compartments obsolete, a large j subcuneate areola at sides of each compartment, j and contiguous to that of adjacent compartments, rylandsiana . I Intervals present. No such subcuneate areolse, . 18. ( Markings on compartments subgranular, few towards 18.-! their inner ends, ....... punctata. V. Markings on compartments areolate throughout, . 19. ,q f No central areolate area, 20. * \ A central areolate area present, .... 21. f A subasteromphaloid centro-lateral area differentiated 1 inner ends of compartments transversely truncate * ] intervals attenuating outwards, l No such subasteromphaloid area, dallasiana. 22. (Rays sharply bigeniculate beyond their middle; | compartments with inner ends slightly swollen, 22. -{ faint lines passing inwards a short distance from j their inner ends, ....... bribissoniana. LRays straight, 23. 23. ' A circular hyaline central space ; compartments with inner ends convex centrally, the intervals of uniform, breadth, reaching f- of radius from centre, Compartments with inner ends transversely truncate to slightly concave inwards ; intervals short, ex- tending to half length of compartments, - Compartments with inner ends somewhat obliquely I truncate ; intervals of equal breadth, outer ends close to border. Rays sometimes dichotomous, Compartments with inner ends transversely truncate, to concave ; rays straight or curved, frequently branching 2 to 4 times before reaching half distance _ to compartments, balearica. Icevis. rotula. princeps. fTwo rays passing to apex of one of the compart- 21.-! ments, an apiculus at outer end of each interval, . uraster. No such radii, 24. 654 Proceedings of Royal Society of Edinburgh. [sess. ("Central areolate area inconspicuous — represented by two areolse ; rays sometimes dichotomous ; com- I partments with inner ends curved or flattened on 24 J opposite sides of rays, the areolae obscure or unre- ' | solved, Grevillei. Central areolate area sometimes large ; compartments I with inner ends concave to truncate, areolae l obvious, affinis. i ASTEROMPHALUS. Ehrb. Emend., Mon. Bex. Ah., 1844, p. 198. — Circular, more rarely flabelliform, oval, or suboblong. Colour pale grey or sub- hyaline, inconspicuous. A centro-lateral area distinct, extending to or slightly beyond the middle of the clear area, rarely quite across the latter — ovate, clavate, or with sides somewhat deeply constricted. Markings : rays distinct, simple or dichotomous, extending from the apex or also from the sides of the centro- lateral area, straight, arcuate, flexuous or sharply bigeniculate, sometimes with short lateral rami passing obliquely outwards from the geniculations; interradial spaces hyaline, or with a subdistinct median area, continuous with the intervals between the compart- ments; the compartments equal or unequal, their inner ends convex, obliquely to transversely truncate or concave ; areolae distinct or inconspicuous, the outermost row most manifest ; intervals between the compartments tapering slightly outwards, or of uniform width, their outer ends rarely expanded, sometimes not reaching border; a subobsolete interval distinct, straight, rarely distinctly arcuate; sometimes an obscure granule (apiculus?) at outer ends of intervals. Border narrow, hyaline. — Spatangidium, de Breb., Bull. Soc. Linn. Normand ., 1857, p. 296 ; Asterolampra , Grey ., pro parte, Trans. Micr. Soc. Bond., 1860, p. 102; Mesasterias, Ehrb., Abli. Ber. Ah., 1872, p. 392; Actinogramma , Ehrb., Abh. Bex. Ah., 1872, p. 254. § 1. Obscuri. The outer ends of the rays penetrating a short distance into the apices of the compartments ; the intercompartmental intervals pro- longed as definitely marked areas to the centre. Astreom. centraster. Johnston, Quart. Jour. Micr. Sci 1860, p. 12, pi. i. fig. 10. — Diam. about *072 mm. Colour of compart- 1888-89.] Mr John Rattray on tlic Genus Coscinodiscus. 655 ments pale buff. Markings : rays straight, narrow, meeting near centre the inner ends of the intercompartmental areas, their outer ends knob-like, and reaching about f- of radius from centre; the compartments extending to about semiradius, their inner ends convex, inwards ; the intervals attenuating slightly outwards, their outer ends swollen knob -like, reaching close to the border. — Grev., Trans. Micr. Soc. Lond ., 1860, p. 124; Ralfs in Pritch . Inf., p. 838. One of the intercompartmental areas is more developed than the others at its inner end, but the outer portion of the same area is not subobsolete,“as is usual in Aster omphalus. Habitat. — Elide guano (Johnson). § 2. Centrales. Clear, median portion of valve not markedly excentric. Rays straight, arcuate or geniculate ; compartments with inner ends concave or convex towards centre, sometimes transversely or obliquely truncate ; intervals rarely markedly expanded towards their outer ends. Asterom. wallichianus. Ralfs in Pritch. Inf., p. 837. — Rarely roundly elliptical. Diam. *0375 to *055 mm. Centro-lateral area Y-shaped. Markings: rays 5 or 6, straight, diverging" from centre; the compartments reaching to about semiradius, their inner ends transversely truncate or slightly concave towards centre ; the con- cavity subconcentric with the circumference ; the areolae distinct ; the intervals of uniform width, extending close to border; the subobsolete interval attenuating rapidly outwards. — Cleve, Bih. k. Sv. Vet.-Ak. Handl. Stockh.,\ 1873, Bd. 1, No. 11, p. 5, pi. i. fig. 1 ; Asterolampra wallichiana , Grev., Trans. Micr. Soc. Lond., 1860, p. 115, pi. iv. fig. 11. Eaint lines pass towards the centre on the interradial spaces from the inner angles of the compartments. Habitat. — “Bermuda tripoli ” (E. W. Dallas); Nottingham deposit, Maryland (Greville!); Santa Monica deposit (Hardman!).* Astreom. variabilis, nov. Asterolampra variabilis. Grev., Trans. Micr. Soc. Lond., 1860, p. Ill, pi. iii. figs. 6-8. — Diam. *07 to *125 In the collection of Mr Julien Deby. 656 Proceedings of Royal Society of Edinburgh. [sess. mm. Centro-lateral area Y-shaped, with inner end obtuse or sub- acute, sometimes reaching the centre. Markings : rays substraight or somewhat curved, well-defined, frequently dichotomous, sometimes meeting in a small semicircular line curving round the centre ; the •compartments reaching from f to f of radius inwards, their inner ends obliquely truncate, on each side of rays straight or slightly concave towards the centre ; areolae distinct, decreasing gradually outwards from 5 to 8 in *01 mm.; the intervals straight or slightly arcuate, their outer ends convex outwards, sometimes hardly reaching the border. Habitat. — Monterey stone (Greville ! Arnott ! Kitton!);* Santa Monica deposit (Grove !). Asterom. Hookerii. Ehrb., Mon. Ber. AJc ., 1844, p. 200, pi. (June) fig. 3. — Diam. *064 mm. Centro-lateral area with sides straight, parallel or slightly concave outwards, sometimes suddenly contracting near the subobsolete interval, its inner end conical. Markings : rays straight ; the compartments reaching from J to ts- of radius inwards, their inner ends rounded ; areolae delicate ; the intervals attenuating slightly outwards, reaching the border. — Ehrb., Mikrog., pi. xxxv. a. 21. fig. 2 ; Ralfs in Pritch. Inf., p. 836, pi. xi. fig. 34; A. Buchii , Ehrb., ibid., 1844, p. 200, pi. (June) fig. 4, — 7 rays ; A. Cuvierii , Ehrb., ibid., 1844, p. 200, pi. (June) fig. 7, — 9 rays; Mikrog., pi. xxxv. a. 21. fig. 1 ; Janisch, Abh. Schl. Ges. voter. Cult., 1861, p. 160 ; A. Humboldtii, Ehrb., ibid., 1844, p. 200, pi. (June) fig. 6, — 8 rays; Mikrog., pi. xxxv. a. 21, fig. 3; Janisch and Rabenh., Beitr., p. 4, pi. iii. fig. 11 ; Sch., Atl., pi. xxxviii. figs. 18-20 ; Asterolampra Hookerii, Grev., Trans. Micr. Soc. Bond., 1860, p. 114. Habitat. — Pancake ice, Antarctic Ice Barrier, lat. 78° 10' S., long. 162° W.; lat. 75° S., long 170° W.; lat. 64° S., long. 160° W. (J. D. Hooker); Peruvian guano (Janisch); H.M.S. Challenger, lat. 53° 55' S., long. 108° 35' E., 1950 fathoms (Grove ! Rae !). Asterom. shadboltianus. Ralfs in Pritch. Inf., p. 838. — Diam. *0775 mm. Centro-lateral area ovate, attenuating rapidly towards its outer end, angular at the centre. Markings : rays 5, straight, or with a small geniculation near their middle ; the compartments reaching about J- of radius inwards, their inner ends transversely * In the collection of Dr R. K. Greville. 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 657 truncate, those bordering the subobsolete interval more oblique and concave inwards at the middle; areolae delicate, decreasing outwards from 9 to 14 in ’01 mm.; the intervals attenuating slightly outwards, their outer ends expanded and knob-like, not reaching the border. — Asterolampra shadboltiana, Grev., Trans. Mier. Soc. Lond ., 1860 p. 121, pi. iv. fig. 19. Distinguished from A. Brookei by the outline of the centro- lateral space, the less marked geniculation of the radii, and the relatively shorter intervals between the compartments. Habitat. — Indian Ocean soundings, 2200 fathoms, Captain Pullen (Greville !); Mejillones (Grove !). Asterom. roperianus. Ralfs in Pritch. Inf., p. 838. — Diam. *07 to *165 mm. Centro-lateral area with inner end angularly rounded, the sides sharply constricted and expanding thence to their outer ends. Markings : rays bigeniculate at their middle ; the flexions more pronounced on the rays proceeding from the sides than on those passing from the extremity of the centro-lateral area; the compartments reaching to § of radius inwards, their inner ends transversely truncate, those adjacent to the subobsolete interval more oblique; areolae 12 to 16 in '01 mm., obscure; the intervals broad, their edges parallel, the outer ends sometimes slightly expanded, not reaching the border.— Sch., Atl., pi. xxxviii. fig. 15 ; Asterolampra roperiana , Grev., Trans. Mier . Soc. Lond., 1860, p. 120, pi. iv. fig. 14; Mesasterias abyssi, Ehrb., Abh. Ber. Ak., 1872, p. 392, pi. ix. fig. 7. Habitat. — Indian Ocean soundings, 2200 fathoms, Captain Pullen (Roper ! * Greville!); Mejillones guano (Deby! Grove!). Asterom. Brookei. f Bail., Amer. Jour. Sci., vol. xxii. ser. 2, 1856, p. 2, pi. i. fig. 1. — Diam. *0725 to '075 mm. Centro-lateral area with a median constriction, its central end sometimes trans- versely truncate or subrotund. Markings : rays straight or slightly flexuous, sometimes sharply bigeniculate towards their outer extremities, with short lateral rami proceeding from the angles; the compartments reaching from f- to I of radius inwards, their inner ends tranversely truncate or slightly concave inwards, some- * In the collection of Dr R. K. Greville. t Named in honour of Lieut. Brooke of the U.S. Navy. VOL. XVI. 7/11/89 2 T 658 Proceedings of Royal Society of Edinburgh. [sess. times obtusely rounded; those bordering the subobsolete interval with the inner ends nioreVblique and slightly concave inwards ; the areolse evident, decreasing gradually outwards, from 8 to 10 in -01 mm.; the intervals tapering gradually outwards, their outer ends convex, reaching close to the border. — Ralfs in Priich. Inf. , p. 837, pi. v. fig. 79 ; Cleve, Bih. h. Sv. Vet- Ah. Handl. Stochh ., 1873, Bd. i. No. 13, p. 10; Sch., Atl., pi. xxxviii. figs. 21-23; Astero- lampra Broohei, Grev., Trans. Micr. Soc. Bond., 1860, p. 119, pi. iv. fig. 18; Actinogramma Broohei, Ehrb., Abk. Ber. Ah., 1872, p. 254. Habitat. — Sea of Kamtschatka, 1700 fathoms (Bailey!); Behring Sea, 1158 fathoms (H. L. Smith!); Atlantic soundings (Roper) ; Santa Monica deposit ; (Deby !) loc.? (Grove !). Yar. robusta, nov. A. robustus, Cstr., Atti Accad. Pontif. nuov. Lincei, 1875, p. 393, pi. vi. fig. 5. — Rotundato-obovate. Markings : rays sharply bigeniculate at their middle ; the compartments reach- ing from J- to ^ of radius inwards, the inner ends of those adjacent to subobsolete interval transversely truncate, of the others some- what concave inwards ; intervals broad, with sides parallel reaching the border. — A. ( Broohei , var.?) robustus, Peragallo, Diat. Baie Villefranche, p. 75, pi. ii. fig. 15. Habitat. — Mediterranean Sea (Castracane). Asterom. Beaumontii , Ehrb., Mon. Ber. Ah., 1844, p. 200, fig. 5. — Diam. *04 mm. Centro-lateral area with sides parallel, and inner ends conical. Markings : rays sharply bigeniculate at their middle, the compartments reaching about -J of radius from circum- ference, conical, with inner ends obtusely angular ; areolse distinct, from 6 to 8 (?) in -01 mm.; intervals between compartments attenuating gradually outwards, reaching the border. From this species I exclude Spatangidium heptactis, de Breb. (Bull. Soc. I/inn. Normand., 1857, p. 292, pi. iii. fig. 2) ; S. ralfsianum, Norman (not Grev.), Quart. Jour. Micr. Sci., 1859, p. 161, pi. vii. figs. 7, 8) ; and Asterolampra heptactis, Grev. (Trans. Micr. Soc. Bond., 1860, p. 122), which Janisch (Abh. Schl. Ges. voter. Cult, 1861, p. 160) has proposed to unite with it. Janisch also proposes to unite to Asterom. Beaumontii the forms figured by 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 659 Schmidt ( Atl ., pi. xxxviii. figs. G, 7), which belong rather to Asterom. heptactis. Habitat. — Pancake ice (Pfankuchen Eise), Ice Barrier, Antarctic Ocean, lat. 78° 10' S., long. 162° W. (J. D. Hooker); H.M.S. Challenger (Deby !). Asterom. moronensis, Rattray. Asterolampra moronensis, Grey., Quart. Jour. Micr. Sci., 1863, p. 230, pi. ix. fig. 8. — Diam. *06 to •075 mm. Centro-lateral area with sides at first almost parallel, and then converging rapidly so as to meet about half way between centre and apices of compartments, beyond the point of union a simple line passing to the centre. Markings : rays sharply geniculate at or slightly beyond their middle, . short lateral rami passing obliquely outwards from the geniculations ; the compart- ments reaching from to •§- of radius inwards, their inner ends obliquely truncate, straight or slightly concave towards the centre, those adjacent to the subobsolete interval with one side much longer than the other; the areolae obvious, decreasing outwards from 6 to 10 in *01 mm., the oblique decussating rows straight or slightly curved at their inner ends ; the intervals narrow, expanding gradually towards their outer ends, which reach close to the border, at the middle of the expanded portion a distinct radial dark subconical area. — Sch., Atl., pi. xxxviii. fig. 24. Habitat. — Moron deposits near Seville (Greville ! Hardman ! Herman !) ;* Santa Monica deposit (Hardman ! Deby !). § 3. Excentrici. Sometimes elliptical, rarely suboblong. Clear median portion of valve sometimes markedly excentric ; the centro-lateral area extend- ing beyond the centre often subclavate, sometimes malleiform. The subobsolete ray rarely arcuate, the others straight or curved; a lunate ridge sometimes visible at outer ends of rays. Asterom. wyville-thomsonianus. O’Me., Jour. Lin. Soc. (Botany), vol. xv. p. 57, pi. i. fig. 5. — Diam. *06 mm. Central areolate area absent. Markings : rays 6, straight ; the compartments 6, five equal smaller, reaching about f, the sixth larger reaching about f of radius inwards, their inner ends uniformly convex towards the * In the collection of Dr R. K. Greville. 660 Proceedings of Royal Society of Edinburgh. [sess. centre; areolae distinct; intervals attenuating gradually outwards, reaching tlie border. Habitat. — Kerguelen Island, H.M.S. Challenger (O’Meara). Asterom. stellatus. Ralfs in Pritch. Inf., p. 838. — Diam. ‘045 to ‘07 mm. Centro-lateral area elongate, faintly subclavate with slight median constriction. Markings : rays straight or slightly curved, springing from apex and side of centro-lateral area, some- times dichotomous ; the compartments reaching about § of radius inwards, their inner ends conical, with sides slightly convex; areolae obscure, decreasing outwards from 14 to 20 in *01 mm., most evident near the apices of the compartments ; the intervals rapidly attenuating outwards, reaching close to the border. — Astero- lampra stellata, Grev., Trans. Micr. Soc. Lond ., 1860, p. 124, pi. iv. fig. 20. This species approaches A. hiltonianus, hut is distinguished by the more straight radii, which are never geniculate ; in the appearance of the centro-lateral area it comes near to A. elegans. Habitat. — Indian Ocean, soundings 2200 fathoms, Captain Pullen (Greville !) ; Holothvria , China (Deby !). Asterom. elegans. Grev., Quart. Jour. Micr. Sci., 1859, p. 161, pi. vii. fig. 6. — Diam. ’075 to T4mm. Centro-lateral area elongate, its inner end rounded, straight or somewhat bent near the subsolete ray. Markings: rays simple, dichotomous, rarely branching, more frequently with sharp, rarely obtuse geniculations near their middle or somewhat closer to the central area ; the compartments conical, their inner ends subacute ; the areolae most evident at the inner ends, elsewhere obscure, decreasing outwards from 12 to 16 in *01 mm.; intervals narrow, attenuating outwards, their outer ends close to the border. — Ralfs in Pritch. Inf. , p. 837, pi. v. fig. 87.; Sch., Atl., pi. xxxviii. figs. 1, 2 ; Asterolampra elgans , Grev., Trans. Micr. Soc. Lond., 1860, p. 118, pi. iv. fig. 16; Actino- gramma Jupiter , Ehrb., Abh. Ber. Ah., 1872, p. 392, pi. ix. fig. 3 ; Ac. Venus, Ehrb., ibid., pi. ix. fig. 4 ; Ac. Saturnus, Ehrb., ibid., pi. ix. fig. 5; Ac. Sol, ibid., pi. ix. fig. 6. Habitat. — Indian Ocean soundings, 2200 fathoms, Captain Pullen (Greville !) ; Californian guano (Norman !) ;* dredged by * In the collection of Dr R. K. Greville. 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 661 H.M.S. Challenger, lat. 5° 54' K, long. 147° 2' W., 2550 fathoms (Hardman!);* Indian Ocean (Ralfs); Gazelle Expedition (Janisch); S.S. Buccaneer, off Ascension Island (Grove !). Aster om. imbricatus. Wallich, Trans. Micr. Soc. Lond., 1860, p. 46, pi. ii. fig. 9. — Roundly elliptical to subcircular. Diam. *06 to *085 mm. Centro-lateral area clavate, widest at its exremity or somewhat nearer to the narrow end, frequently extending across and beyond the centre of the valve. Markings: rays sharply bigeniculate at their middle, the geniculations regular, forming a distinct roundly elliptical figure around the centro-lateral area ; the compartments reaching about ^ of radius inwards, their inner ends conical, with sides convex, those adjacent to the suhohsolete ray with ends obliquely truncate ; areolae obscure, subpunctiform ; intervals narrow, at first attenuating, then subequal in breadth, their outer ends convex outwards, reaching close to border. — Ralfs in Pritch. Inf., p. 837 ; Asterolamyra imbricata, Grev., Trans. Micr. Soc. Lond., I860, p. 119, pi. iv. fig. 17. Wallich distinguishes as var. /3, forms with “ plani sutures,” i.e., non-geniculate rays, and as var. y, forms with “ the capitate extremity of the basal ray,” i.e., of the centro-lateral area emarginate. To his former var. the name rectiradiata may be given ; the latter is unimportant. Habitat. — Indian Ocean soundings, 2200 fathoms, Captain Pullen (Greville !) ; Bay of Bengal (Wallich) ; Natal (Roper). Asterom. hiltonianus. Ralfs in Pritch. Inf., p. 837. — Diam. •075 to ‘135 mm. Centro-lateral area slightly constricted at its outer sometimes attenuating gradually towards its outer end. Markings : rays springing from apex and sides of centro-lateral area, straight or subuniformly arcuate, sometimes geniculate about their middle, and concave towards the subobsolete interval, rarely dichotomous opposite central end of centro-lateral area ; the com- partments reaching about f of radius inwards, their inner ends sharply conical, those adjacent to the subobsolete interval somewhat more obtuse, with sides somewhat convex ; areolse obscure, 20 to 24 in -01 mm., towards border resolved with difficulty ; the In the collection of Mr Julien Deby. 662 Proceedings of Royal Society of Edinburgh. [sess. intervals attenuating rapidly outwards, reaching close to the border. — Asterolampra liiltoniana , Grev., Trans. Micr. Soc. Lond., 1860, p. 117, pi. iv. fig. 15; H. L. Smith, Diat. Spec. Typ ., No. 49. Habitat. — Indian Ocean soundings, 2200 fathoms, Captain Pullen (Greville ! Roper !) ; Algoa Bay guano (Greville !) ; South Pacific, 2900 fathoms (H. L. Smith !). Aster om. Jlabellatus. Grev., Quart. Jour. Micr. Set., 1859, p. 160, pi. vii. figs. 4, 5.- — Flabelliform, subtriangular or sub- circular. Diam. *0425 to ’06 mm., the minor axis from *0375 to *05 mm. Centro-lateral area subclavate, the sides more rarely almost parallel towards the central end, inner end rounded. Mark- ings : rays straight or slightly curved ; the compartments longer towards the subobsolete interval, reaching from f to § of radius inwards, their inner ends conical, sometimes transversely truncate ; areolae obscure ; the intervals tapering slighty outwards, extending to border. — Janisch, Abh. Schl. Ges. cater. Cult., 1861, p. 160 ; Ralfs in Pritch. Inf. , p. 837; Sch., Atl ., pi. xxxviii. figs. 10, 12; A. flabellatus, var. tergestina , Grun., Yan. Heurck, Syn. Diat. Belg ., pi. cxxvii. figs. 5, 6; Asterolampra ftabellata, Grev., Trans. Micr. Soc. Lond., I860, p. 116 ; Spatangidium flabellatum, de Breb., Bull. Soc. Linn, Normand, 1857, p. 297, pi. iii. fig. 3 ; S. peltatum, de Breb., ibid., p. 298, pi. iii. fig. 4. Habitat. — Rembang Bay (Deby !); Peruvian guano (de Brebisson, Janisch) ; Campeachy Bay, Yokohama and Hong Kong (Schmidt) ; California guano (Greville !) ; Corsican algae (de Brebisson) ; Teignmouth Ascidia (Grove !). Asterom. cleveanus , Grun. Sch., Atl., pi. xxxviii. figs. 13, 14. — Roundly elliptical to oval. Major axis -045 to ’075 mm.; minor *04 to B625 mm. Centro-lateral area tapering towards outer ends, the inner end angular, sometimes with sides slightly concave outwards. Markings : rays springing from apex and sides of centro-lateral area, straight or concave towards subobsolete ray, sometimes dichotomous ; the compartments longest towards ex- tremities of major axis, shortest towards the minor, reaching from J to § of radius inwards ; their inner ends rounded or somewhat obliquely truncate; areolae delicate, 12 to 14 in *01 mm., the 1888-89.] Mr John Rattray on the Genus Goscinodiscus. 663 intervals straight or slightly arcuate, of uniform breadth, their outer ends rounded close to the border, sometimes prolonged inwards as subdistinct areas on the interradial spaces. — Aster om. wallicliianus , Cleve (not Grev.), Bill. k. Sv. Vet.-Akad. Hand-1. Stockh., 1873, Bd. i. No. 11, p. 5, fig. 1 ; Cleve and Moller, Diat ., Nos. 145, 146. Habitat. — Surface of Java Sea (Cleve, Schmidt); Manilla mud (Grove !) ; Muntok, East Indian Archipelago (Grove !). Asterom. reticulatus. Cleve, Bill. k. Sv. Vet.-Akad. Handl. Stockli., 1873, Bd. 1, No. 11, p. 5, pi. i. fig. 2. — Diam. #051 mm. Centro- lateral area with sides uniformly concave and inner end rounded, a sharp angular bend at outer extremity of one of its sides. Mark- ings : rays arcuate, flexuous or sharply bigeniculate at their middle ; the compartments reaching about of radius inwards, their inner ends transversely truncate, that on one side of subobsolete interval convex towards centre ; areolae distinct, 7 in '01 mm.; the intervals broad, expanding gradually outwards to their middle, and again contracting uniformly towards their outer subacute ends, not reaching the border, the subobsolete interval arcuate, concave towards that compartment, having the inner end convex. Habitat. — Surface of Java Sea (Cleve). Asterom. Darwinii. Ehrb., Mon. Ber. Ak., 1844, p. 200, pi. (June), fig. 1. — Diam. *0625 to *0875 mm. Centro-lateral area short and broad, sometimes subconical, or with sides almost parallel and converging suddenly to the centre. Markings : rays sharply geniculate about their middle, with short lateral rami proceeding from the angles ; the compartments few, 5, reaching from \ to f of radius inwards, of unequal length, their inner ends transversely truncate, those bordering the subobsolete interval with the inner ends more oblique ; the areolae decreasing gradually outwards from 8 to 12 in *01 mm.; the intervals tapering ontwards, their outer ends rounded, reaching close to the border. — Balfs in Pritcli. Inf., p. 837, pi. v. fig. 86 ; Sch., Atl, ,, pi. xxxviii. fig. 16 ; A. Rossii , Ehrb., Mon. Ber. Ak ., 1844, p. 200, pi. (June), fig. 2; Mikrog ., pi. xxxv. a. 21. fig. 4; A. Brookei , Grun. (not Bail.), Sch., Atl., pi. xxxviii. fig. 9; Asterolampra Darwinii, Grev., Trans. Micr. Soc. Bond., 1860, p. 116, pi. iv. figs. 12, 13. 664 Proceedings of Royal Society of Edinburgh. [sess. Habitat. — Monterey stone (Arnott !* Kitton !) ;* Antarctic Ocean, lat. 78° 10' S., long. 162° W. (Ehrenberg, Ralfs) ; Santa Monica deposit (Hardman !)t. Aster om. rarus, Rattray. A. elegans , Grev. var. Wallich., Trans. Micr. Soc. Lond ., 1860, p. 46, pi. ii. fig. 10. — Diam. *0525 mm. Centro-lateral area distinct, extending over f of disc, its extremity conical, the sides parallel, with a wide deep lateral conical indentation. Markings : rays of two kinds — one straight opposite the centro- lateral area, the others sharply bigeniculate at their middle ; the compartments symmetrical with respect to the diameter correspond- ing to the subobsolete interval, of unequal length, their inner ends obtusely rounded, that opposite the subobsolete interval most obtuse; areolae distinct, 6 in *01 mm.; the intervals tapering outwards, reaching the border. Habitat. — Salpce, Indian Ocean (Wallich). Asterom. hejptadis. Ralfs. in Pritch. Inf, p. 838, pi. viii. fig. 21. — Diam. *0425 to *175 mm. Centro-lateral area suhclavate, the sides slightly sinuate, sometimes almost parallel, the inner end conical. Markings : rays sharply, 1- or 2- geniculate at or slightly beyond their middle ; delicate lines traceable from the geniculations to the angles of the compartments ; the compartments sometimes of unequal lengths, but symmetrical with respect to the diameter corresponding to centro-lateral area, reaching from i to § of radius inwards, their inner ends transversely truncate or slightly concave towards centre; areolae delicate, 6 in *01 mm.; the rows hounding the compartments obvious; intervals broad, a distinct lunate ridge at their outer ends .—Spatangidium heptadis, de Breb., Bidl. Soc. Linn. Normand ., 1857, p. 296, pi. iii. fig. 2; S. ralfsianum, Norman, Quart. Jour. Micr. Sci., 1859, p. 161, pi. vii. figs. 7, 8 ; Asterolampra heptadis, Grev., Trans. Micr. Soc. Lond., 1860, p. 122 ; Asterom. ralfsianus, Grun., Sch., Atl., pi. xxxviii. figs. 5-8 (excl. Asterom. Beaumontii, Ehrb., Mon. Ber. Ak., 1844, p. 200). Habitat. — Californian guano (Greville !) ; Peruvian guano (Grove ! de Br4bisson, Schmidt, Greville !) ; Ichaboe guano (J. * In the collection of Dr ft. K. Greville. t In the collection of Mr Julien Deby. 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 665 T. Korman !) ; Atlantic soundings (Ralfs) ; Gazelle Expedition, Yokohama (Schmidt) ; Pabellan di Pico guano (Deby !) ; Holo- thuria , China (Deby!); Faeroe Islands, H.M.S. Knight Errant (Grove !) ; loc.? (Grove !). Aster om. arachne. Spatangidium arachne , de Br4b., Bull. Soc. Linn. Normand ., 18D7, p. 296, pi. iii. tig. 1. — Broadly ovate to subcircular. Diam. '045 to *06 mm. Markings : rays 5, the central subobsolete longest, its proximal end expanded, malleiform, reaching between proximal ends of lateral rays, excentric ; the lateral rays in two nnsymmetrical pairs, the lower pair substraight, or slightly convex towards the central ray, the upper pair more curved in their proximal portions and more convex towards the lower ; their inner ends expanded hut more rounded than that of central ray ; their outer ends sometimes slightly swollen and not reaching the border; compartments of unequal length, their inner ends convex, that opposite the subobsolete ray concave inwards, the areolae, decreasing but slightly outwards, 6 to 7 in *01 mm.; rows evident, those adjacent to the rays somewhat more prominent. Border narrow, hyaline. — Ralfs in Priteh. Inf., p. 837; Sch., Atl., pi. xxxviii. figs. 3, 4 ; Aster om. malleus, Wallich, Trans. Micr. Soc. Bond., 1860, p. 47, pi. ii. fig. 11 ; Asterom. malleiformis, Wallich, ibid., Explan. pi. ii. fig. 11; Asterolampra arachne, Grev., Trans. Micr. Soc. Lond., 1860, p. 123; Excentron cancroides , Ralfs, ibid., p. 837. Habitat. — Peruvian guano (Greville ! Grove !) ; Ichahoe guano (Korman !) ; Indian Ocean soundings, 2200 fathoms, by Capt. Pullen (Greville !) ; locality ? (Dickie !) ; from Salpoe, Indian Ocean (Wallich) ; Guanape guano (Deby !) ; Arica, and Gazelle Expedition (Schmidt) ; S.S. Buccaneer, off Ascension Island (Grove !). Asterom. nanicoorensis. Grun., Reise d. Novara, 1870, p. 104, pi. i. a. fig. 22. — Oval to subcircular. Length, *065 to ’075 mm.; breadth, -0625 to * 065 mm. Centro-lateral area with sides concave outwards, the inner end conical. Markings excentrically disposed ; rays — one more robust, arcuate, proceeding from apex of centro- lateral area, a few others more delicate ; the compartments of unequal lengths, their inner ends concave towards centre ; areolae 666 Proceedings of Royal Society of Edinburgh. [sess. delicate; intervals expanding gradually outwards, and reaching border ; their inner ends uniformly curved away from one another, closed, and continued almost to the angles of the centro-lateral area ; opposite this area and at middle of largest compartments 5 short rays, the two lateral longest, attenuating outwards, and with the inner ends slightly swollen and knob-like, the subobsolete interval nearer one of the larger intervals than the other, not reaching the border. Allied to A. arachne. Habitat. — Nancoori deposit (Grunow). Asterom. sarcophagus. Wallich, Trans. Alter. Soc. Lond., 1860, p. 47, pi. ii. fig. 12. — Subregularly oblong, the sides slightly concave about their middle, the concavities of those adjacent to the subobsolete interval greater than the others. Length "045, breadth *0225 mm. Centro-lateral area expanding gradually outwards. Markings : rays 6, suhuniformly arcuate, the three from each side uniting in two excentric points, which are connected by a short transverse line at right angles to the major axis ; the compartments reaching from jr to •§- of radius inwards, unequal, but symmetrical with respect to the major axis, their inner ends convex; areolae distinct, 4-J (V) in *01 mm., in obscure radial rows; the intervals attenuating outwards, their outer ends convex outwards, close to the border. — Asterolampra sarcophagus , Grev., Trans. Micr. Soc. Lord ., 1860, p. 124. Distinguished from A. arachne , its nearest ally, by its outline, and the different character of its rays and compartments. Habitat. — Indian Ocean (Wallich). The following species from Peruvian guano have been founded on the number of the rays on the central portion of the valve, and cannot he retained : — A. denarius. Janisch. ( Abh . Schl. Ges. voter. Cult., 1860, p. 160, pi. ii. b. fig. 22 — unpublished). Oval. Diam. -045 mm. Rays 10, straight. Areolae small. A. Brebisonii. Janisch (ibid., 1861, p. 160, pi. ii. b. fig. 28 — unpublished). Rays 12, straight. Areolae small, on compartments adjacent to subobsolete interval. A. Pringsheimii. Janisch {ibid., 1861, p. 160, pi. ii. b. fig. 25 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 667 — unpublished). Diam. ’07 mm. Rays 1 4, zig-zag. Areolae small. A. Cohnii. Janisch {ibid., 1860, p. 160, pi. ii. b. fig. 26- unpublished). Diam. *0805 mm. Rays 15, zig-zag. Areolae small. A. cleveanus , Janisch, is erroneously mentioned in Habir- shaw’s Cat. Diat ., § Aster omphalus, as found in Abh. Schl. Ges. vater. Cult, 1860, p. 160, pi. ii. b. fig. 26. The name given in the paper quoted being A. Cohnii. A. Ehrenbergii. Janisch. {ibid., 1861, p. 161, pi. ii. b. fig. 27 — unpublished). Almost circular. Diam. *095 mm. Rays 16, zig- zag-shaped. Areolae small. A. Braunii. Janisch. {ibid., 1861, p. 161, pi. ii. b. fig. 28 — unpublished). Diam. T05 mm. Rays 17, zig-zag. Areolae small. Artificial Key. f Outline flabelliform to subtriangular, oval or elliptical, ........ 2. 1. -{ Outline circular or subcircular, .... 3. | Outline subregularly oblong, with sides slightly con- L concave at their middle, ..... sarcophagus. I f Compartments longest adjacent to snbobsolete interval, decreasing away from this ; rays simple ; areolse obscure, l Compartments shortest at ends of minor axis ; rays sometimes dichotomous ; the areolse delicate, 12 to 14 in '01 mm., ....... flabellatus. cleveanus. 3. f Rays simple, \ Rays sharply bigeniculate, 4. 5. f Intervals between compartments prolonged to centre ; ^ I outer ends of rays knob-like, penetrating a short * ] distance into compartments, ..... USTo such structure, . . . centraster. 6. f Inner ends of compartments concave towards centre ; centro-lateral area acutely Y-shaped; subobsolete interval rapidly attenuating outwards, . . . wallichianus. Inner ends of compartments obliquely truncate on each side of the rays ; centro-lateral area sometimes ! with inner ends obtuse, ..... variabilis. ’ } Inner ends of compartments rounded, ... 7. I Inner ends of compartments more conical, . . 8. | Inner ends of compartments concave towards centre ; centro-lateral ared with sides concave outwards ; | one ray opposite this area more robust than the l others ; intervals expanding outwards, . . . nankoorensis. 7. f Structure markedly excentric ; central area opposite I subobsolete interval, maleiform, extending across ] the hyaline portion of valve, I Structure not markedly excentric, .... arachne. 9. 668 Proceedings of Royal Society of Edinburgh. [sess. {Compartments 6, 5 equal smaller reaching to f- of \wyville- radius, the sixth larger, reaching to f of radius, / thomsonianus. Compartments otherwise, ..... 10. f Centro-lateral area ovate ; rays straight or slightly j flexed ; intervals not reaching border, . . . shadboltianus. 10. -{ Centro-lateral area with sides more parallel, slightly concave outwards ; intervals attenuating slightly L reaching border, ....... Hookerii. [■Compartments reaching about f of radius inwards, longest at sides of subobsolete intervals ; rays 8. -j straight ; intervals rapidly attenuating outwards, . stellatus. j Compartments reaching -f of radius inwards ; rays l towards subobsolete interval simply flexed, . . hiltonianus. C Centro-lateral area sharply constricted at middle, j thence expanding markedly outwards, compart- 5. -{ ments reaching to § of radius inwards ; the inner | ends transversely truncate, . . . . . roperianus. I Centro-lateral area otherwise, ..... 11. Outer ends of intervals expanded, a distinct dark 11. -j area at middle of wider portion, .... moronensis. k ISTo such intervals, ....... 12. ^2 f Structure not markedly excentric, . . . . 13. ' \ Structure markedly excentric, ..... 14. f Compartments with inner ends conical, ... 15. I Compartments with inner ends obtusely angular ; J intervals attenuating outwards, .... Beaumontii. ' j Compartments with inner ends obtusely rounded or truncate ; rays sometimes simple, sometimes wavy ; L geniculations not regular, ..... Brookei. ( Rays often dichotomous, bigeniculate at or within | their middle ; centrodateral area elongate, . . elegans. 15. Rays simple ; the geniculations regular, forming an j elliptical figure round the centro-lateral area. L Centro-lateral area clavate, ..... imbricatus. 14 / Subobsolete ray distinctly curved, .... reticulatus. ‘ \ Subobsolete ray straight, ...... 16. ( Centro-lateral area with sides parallel, and showing a 16.4 deep regular median indentation, .... rams. t No such indentation, ...... 17. f Centro-lateral area 'short, broad, or subconical ; com- partments few, large; rays geniculate about their I middle. No lunate ridge, ..... Darwinii. 17. \ Centro-lateral area subclavate or with sides slightly sinuate to almost parallel ; rays geniculate at or I beyond their middle ; a lunate ridge frequently L present at outer ends of intervals, .... heptactis. LIRADISCUS. Grev., Trans. Micr. Soc. Lond ., 1865, p. 4.— Circular, subcircular, or elliptical. Surface slightly convex or dome-shaped, flatter towards the border. Colour pale grey. Central space absent. Markings consisting of evident or more delicate lines, anastomosing irregularly or forming subregular or unequal areolae, on a more or less irregular 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 669 band adjacent to the border subradial, with few anastomoses or lateral rami, and sometimes dichotomous ; apiculi at the angles of the meshes sometimes distinct. Border narrow, hyaline, more rarely broad, with evident striae. § 1. Circulares. Outline circular. L. fur catus. Grove, MS.' — Circular. Diam. *0875 to -095 mm. Surface slightly convex. Markings prominent, areolae at centre few, large, unequal, sometimes triramose, and reaching -06 mm. in length, mostly 1 to 1J in ’01 mm.; adjacent to border the lines straight or curved, radial, or subradial, frequently dichotomous, hut without anastomoses ; the areolae hyaline, or with minute rounded granules at their centre. Border delicate, hut distinct, about •0025 mm. broad.— (PL III fig. 23.) Habitat. — Marine deposit, Fiji Islands (Grove !). L. capensis. Cleve, Kongl. Sv. Vet.-Akad. Handl. StocJch., 1881, Bt. xviii. FTo. 5, p. 22, pi. v. fig. 61. — Circular. Diam. •04 mm. Markings irregularly radiating or oblique, sometimes ramose, but not anastomosing lines, with large hyaline interspaces; at intervals a few rounded, elongate or irregular dots distinct. Border sharply defined; striae obvious, 15 in -01 mm. Cleve places this species with some hesitation in the present genus, believing that it might be better to range it in Cyclotella. The relationships, however, which he points out with Cy. striata, Kiitz. (Van Heurck, Syn. Diat. Belg., pi. xcii. figs. 6-8), and Gy. dallasiana, W. Sm., are remote, whilst the general aspect of the lines on the surface is liradiscoid. Habitat. — Cape of Good Hope (F. Hauck). L. barbadensis. Grev., Trans. Micr. Soc. Lond., 1865, p. 5, pi. i. fig. 14.— Diam. -05 to *105 mm. Surface slightly convex for \ to £ of radius, beyond this almost flat to the border. Markings evident areolae, from 1 to 3 in "01 mm., sometimes obtusely angular, the band adjacent to the border subregular. Habitat. — Cambridge deposit, Barbados (Johnson !); “Barbados!” (Greville ! Johnson !). 670 Proceedings of Royal Society of Edinburgh. [sess. § 2. Elliptici. Outline roundly or elongately elliptical. (a) Elongately elliptical. L. ellipticus. Grev., Trans. Micr. Soc. Lond ., 1865, p. 99, pi. viii. fig. 6. — Major axis ’0775 to *105 mm., from 2 to 2f times minor, the extremities of the major axis acute. Surface but slightly convex. Markings delicate ; areolae 2 to 3^ in ‘01 mm.; the band adjacent to the border narrow indistinct, with the subradial lines 4 to 4J in -01 mm. Habitat.- — Cambridge deposit, Barbados (Johnson !) ; “ Barbados ” (Greville ! Johnson !). (p) Roundly elliptical. L. oblongus , Grun. Cleve and Moller, Diat., No. 276. — Major axis *04 to "05 mm., about twice the minor ; the extremities obtuse. Surface slightly convex. Markings delicate; areolae 4 in '01 mm.; subequal or slightly smaller adjacent to the border, without order. Border narrow, sharply defined. Habitat. — California (Cleve and Moller !). L. ovalis. Grev., Trans. Micr. Soc. Lond., 1865, p. 5, pi. i. figs. 15, 16. — Major axis *04 to '06 mm., from 1-f- to 1^ times minor. Surface markedly convex. Markings prominent; the areolae towards the centre sometimes imperfect; the band adjacent to the border irregular, narrow ; apiculi irregular, inserted at the angles of the areolae. Greville represents the girdle as a narrow hyaline band extending for a short distance beyond the convex portion of the valve. Habitat. — Cambridge deposit, Barbados (Johnson !) ; Oamaru deposit (Grove !). L. marginatus. Grove MS. — Major axis *0475 mm., about 1J times minor; extremities obtuse. Surface slightly convex. Markings robust, areolate ; the areolae irregular and unequal, 2 to 3 in '01 mm., largest at the centre bearing a few faint rounded granules ; marginal band distinct, its outer edge crenate. Border narrow, hyaline. — (PI. III. fig. 13.) Habitat. — Oamaru deposit (Grove !). 7 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 671 L. minutus. Grev. (Trans. Micr. Soc. Lond ., 1865, p. 47, pi. v. fig. 6), belongs to Cresswellia. The surface of the type is dome- shaped, and the markings on the central portion regular, and areolae 3J to 4 in '01 mm. Artificial Key. , f Outline circular, ....... ’ \ Outline elliptical, ( Lines not anastomosing. Border broad, striae 15 in 2.4 '01mm., f Lines anastomosing, ...... f Areolae unequal, 1 to 3 in '01 mm. Band adjacent to , J border subregular, . . . ' 1 Areolae unequal, few ; around border the lines radial or l subradial, often dichotomous, .... f Extremities of major axis acute. Surface slightly „ I convex. Markings delicate, areolae 2 to 3J in ] '01mm. Band adjacent to border narrow, . I Extremities of major axis not acute, .... f Surface slightly convex, ...... _ J Surface markedly convex. Markings prominent, ‘ 1 areolae often imperfect at centre ; apiculi acicular, l irregular, inserted at angles of areolae, . f Markings delicate, 4 in '01 mm., subequal; non- apiculate. Border narrow, sharply defined, . 6. Markings evident ; marginal band of areolae distinct, its outer edge crenate ; areolae unequal, irregular, f 2 to 3 in '01 mm., largest at the centre, PORODISCUS. Grev., Trans. Micr. Soc. Lond., 1863, p. 63. — Valves elliptical, circular or rhombic : sometimes the opposite valves of a frustule of unequal sizes. Surface slightly convex, dome-shaped or conical, with transversely truncated ends. Colour pale smoky grey. Central space circular to roundly elliptical, faintly punctate or hyaline, its outline smooth or finely crenulate. Markings small, round, granular, papilliform, or areolate ; rows radial, more rarely inconspicuous or undifferentiated, secondary oblique rows some- times evident ; fasciculi frequently distinct ; interspaces largest near the central space, sometimes absent ; spines long, acicular or hour-glass-shaped, frequent ; a sharply defined marginal band rare. Border inconspicuous. — Craspedodiscus , pro parte, Grun.; Sch., Atl., pi. lxvi. figs. 7-9 ; Craspedoporus, pro parte, Grove and Sturt, Jour. Quelc. Micr. Cl., 1887, p. 67. 2. 3. capensis. 4. barbadensis. furcatus. ellipticus. 5. 6. ovalis. oblongus. marginatus. P. splendidus. Grev., Trans. Micr. Soc. Lond., 1865, p. 46, 672 Proceedings of Royal Society of Edinburgh. [sess. pi. v. fig. 5. — Circular, sometimes roundly elliptical. Diam. *075 mm. Surface convex, forming a low dome. Central space circular, about ‘015 mm. broad, sharply defined, hyaline. Markings large, areolate, increasing slightly to about semiradius, thence decreasing similarly to the border; around central space 4|, at semiradius 3 to 3| in *01 mm.; rows radial, straight; secondary oblique rows inconspicuous. Border inconspicuous. Habitat. — Springfield deposit, Barbados (Hardman). Yar. marginata , nov. Crasjpedodiscus ovalis, Grun., in Sch., Atl., pi. lxvi. fig. 6. — Roundly elliptical. Major axis #065 mm., about 1J times minor. Central space with small round, free granules. Markings areolate and subequal, 4 in -01 mm. for about -J of radius, on a distinct band adjacent to the border, round, granular, 8 in •01 mm.; interspaces between radial rows evident only on a band adjacent to border ; secondary oblique decussating rows more evident. — Porodiscus splendidus, var .1 Sch., ibid. Habitat. — Sprinfield deposit (Schmidt). P. nitidus. Grey., Trans. Micr. Soc. Lond ., 1863, p. 65, pi. iv. fig. 4. — Circular or subcircular. Diam. -05 to *07 mm. Surface uniformly and moderately convex. Central space circular or roundly elliptical, hyaline, *0075 mm. broad. Markings areolate, rarely obtusely angular towards the central space, increasing for a short distance outwards from this space, thence decreasing gradually to the border ; towards the central space 4 J, near the border 8, in *01 mm.; rows radial, straight, non-fasciculate. Border narrow. The markings being areolate, there are no such hyaline inter- spaces as are shown in Greville’s figure. Sometimes faint fasciculi are observed on one valve of a frustule, the opposite valve being non-fasciculate. Habitat. — Cambridge deposit, Barbados (Johnson !). Yar. armata , nov. — Diam. *0525 to *095 mm. Central space circular, *0075 mm. broad. Markings sometimes forming coarse moniliform strise towards the border ; spine3 acicular, about '01 mm. long, sometimes shorter, inserted about j of radius from centre; interfasciculate rarely a few at irregular intervals nearer the border. Girdle *0125 mm. broad in a valve *0525 mm. in diam., the 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 673 hyaline striae at right angles to its edge undifferentiated. — (PL III. % 17.) A specimen occurs in Dr Greville’s collection in the British Museum, labelled P. conicus , and another labelled P. major. From both of these the present var. is quite distinct. Habitat. — “Barbados” (Johnson! Greville!); Cambridge deposit, Barbados (Johnson !). P. major. Grey., Trans. Micr. Soc. Lond ., 1863, p. 64, pi. iv. fig. 2. — Fragmentary. Diam. ? Surface slightly convex. Central space subcircular, '0175 mm. broad, hearing almost invisible minute puncta, and a large round more distinct slightly excentric granule, its outer edge minutely crenate. Markings small, round, granular ; towards the central space 8, nearer the border 10, in *01 mm.; rows radial, straight, in faint fasciculi, most originating at about *015 mm. from the central space, the others proceeding from the edge of this space ; interspaces at origin of shorter rows hyaline. Habitat. — Cambridge deposit, Barbados (Greville !). Yar. densa , nov. P. major , Grev., ibid., 1865, p. 46. — Diam. 1 Central space circular to oval, '0075 to '01 mm. broad, its outer edge smooth. Markings subareolate, near the central space 51 to 6 in '01 mm.; interspaces around the central space more minute. —(PI. III. fig. 21.) The central space and markings at once distinguish this var. Habitat. — Cambridge deposit, Barbados (Johnson !); “ Barbados ” (Greville !). P. elegans. Grev., Trans. Micr. Soc. Lond., 1863, p. 65, pi. iv. fig. 1. — Circular. Diam. '0625 to '095 mm. Surface rounded and dome-shaped. Central space circular, *0075 mm. broad, sharply defined, hyaline. Markings obtusely angular or sub- areolate, decreasing gradually from the central space outwards, around the central space 6, near the border 10 to 12, in '01 mm.; rows radial, straight; fasciculi distinct; interspaces' minute, largest around the central space. Girdle cylindrical, *03 mm. broad, in a frustule, '06 mm. in diam.; a narrow hyaline hand at each extremity ; vol. xvi. 22/11/89 2 u 674 Proceedings of Royal Society of Edinburgh. [sess. the interval minutely punctate ; at subregular intervals narrow hyaline straight lines at right angles to the edges of the valve. The fasciculi are hounded by two adjacent radial rows, some- what more conspicuous than the intervening rows. In one of the valves in Greville’s collection it is possible to trace downwards from the central space a cylindrical siliceous tube which is of sufficient length to have passed to a plane corresponding in position to the edges of the valve. Habitat. — Cambridge deposit, Barbados (Greville ! Johnson !); “Barbados” (Greville ! Johnson !). P. spiniferus, sp. n.: — Circular. Diam. *0875 mm. Surface dome-shaped. Central space circular, ’0075 mm. broad. Markings areolate, subequal, 7 in *01 mm.; rows radial, straight; fasciculi evident; bounded by two rows of more prominent submuriform areoke; spines robust, conical, about *03 mm. long, interfasciculate, forming a circlet at about” J of distance between central space and edge of valve. Girdle cylindrical, *0375 mm. broad; a narrow hyaline band at each extremity ; the clear substraight lines at right angles to its edges distinct. — (PI. III. fig. 19.) Habitat. — Cambridge deposit, Barbados (Johnson !). P. oblongus. Grev., Trans. Micr. Soc. Lond ., 1863, p. 63, pi. iv. fig. 5. — Subacufcely elliptical. Major axis -05 mm. long, about times minor. Surface sloping gradually downwards from edge of central space. Central space roundly elliptical, with major axis corresponding in direction to minor axis of valve. Markings angular, decreasing regularly and somewhat rapidly from central space to border; around the central space 4J-, at border 10, in •01 mm. ; rows radial, substraight. Border narrow, hyaline. — P. ovalis, Grev., ibid. ; Explan. pi. iv. fig. 5 ; Craspedodiscus oblongus , Grun.; Sch., Atl., pi. lxvi. figs. 7-9. This species approaches in appearance Coscinodiscus oblongus , Grev. {Trans. Micr. Soc. Lond., 1866, p. 4, pi. i. figs. 9, 10). Habitat. — Barbados deposit (Johnson!). P. Stolterfothii. Cstr., Diat. Ghall. Exped ., 1886, p. 139, pi. xii. fig. 8. — Bhombic, with angles obtuse. Major axis ’077 mm. long, about l/^- times minor. Surface slightly convex towards 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 6 75 centre, towards the border subplain. Central space roundly elliptical, with major axis about ’01 mm. long, and corresponding in direction to minor axis of valve, delicately punctate. Markings areolate, gradually increasing from central space outwards ; towards the central space 6 or 6|, towards the border 4 to 4J, in '01 mm.; rows radial, straight ; secondary oblique decussating rows indistinct. Border narrow, hyaline. Habitat. — Pacific Ocean, from a sounding made at the equator by H.M.S. Challenger (Castracane). P. conicus. Grev., Trans. Micr. Soc. Lond.. 1863, p. 65, pi. iv. fig. 3. — Diam. '025 to '0525 mm. Major axis of frustule from '0625 to '0875 mm. Surface a more or less elongate regular cone, transversely truncate at the extremities, the opposite valves of a frustule of unequal height. Central space ? Markings obtusely angular or subareolate, 6 in '01 mm., subequal; rows radial, straight, non-fasciculate ; secondary oblique decussating rows evident, from the truncated ends of the cone a few short tapering clear lines, distinct. Girdle cylindrical, from '025 to '0325 mm. broad ; a narrow band at each extremity, hyaline, the intervening portion clouded with diffuse parallel lines. Habitat. — Cambridge deposit, Barbados (Johnson !) ; Bar- bados (Johnson ! Greville !) ; Bridgewater deposit, Barbados (Johnson[!). P. hirsutus. Grove and Sturt, Jour. Quek. Micr. Cl., 1887, p. 143, pi. xiv. fig. 54. — Circular. Diam. '075 to '0875 mm. Surface flat from central space to the sharply-defined marginal band, the latter sloping gently to the border. Central space circular, sharply defined, \ to -J- of diam. broad, surrounded by a narrow, more hyaline, sometimes interrupted band, with irregular outer edge. Markings rounded, prominent papillae, with hyaline interspaces and without order ; processes hour-glass-shaped, at sub- regular intervals, inserted on inner edge of marginal band, between these processes delicate radial striae, 6 to 8 in '01 mm., extending outwards to about middle of band; adjacent to border a circlet of evident papillae at intervals of '0075 to '01 mm. Border narrow, bearing minute granules, 6 in '01 mm. 676 Proceedings of Royal Society of Edinburgh . [sess. This species approaches Melosira sulcata forma coronata, Grun. (Van Heurck, Syn. Diat. Belg ., ph xci. fig. 24). Habitat. — Oamaru deposit (Grove and Sturt !). P. interruptus. Grove and Sturt {Jour. QueJc. Micr. Cl. , 1887, p. 67, pi. v. fig. 8) has been found by Mr H. Morland {Jour. QueJc. Micr. Cl., 1887, p. 167) to be the opposite valve of Cras- pedoporus elegans , Grove and Sturt {ibid., 1887, p. 64, pi. v. fig. 6; Rattray, Jour. Roy. Micr. Soc., 1888, p. 919). Artificial Key. ' Yalves irregularly conical, with transversely truncated extremities, the opposite valves of a frustule of unequal heights, Yalves elliptical. Markings decreasing rapidly out- , j wards, around central space 4^, at border 10, in •01 mm., ........ Yalves rhombic. Markings increasing gradually out- wards, towards central space 6 to 6|, towards border 4 to 4 J in ’01 mm., . ... . .Yalves circular or subcircular, ..... 2. f Surface slightly convex ; interspaces large around | central space, ....... -! Surface almost flat to marginal band. Markings papilliform. Processes hour-glass-shaped, L Surface dome-shaped, ...... o f Markings distinctly fasciculate, .... ' \ Markings non-fasciculate, 'Spines robust, conical, interfasciculate, forming a circlet at about g of distance between central space and edge of valve. Markings areolate, 7 in 4. - -01 mm., . Spines absent. Markings obtusely angular or sub- areolate ; around central space 6, near border 10 to l 12, in ’01 mm., f Markings areolate, increasing slightly to about semi- I radius, thence decreasing similarly to border; - J around central space 4J, at semiradius 3 to 3£, in ' j ’01 mm. Spines absent, j Markings towards central space 4|, near border 8, in l '01 mm., sometimes with long spines, . conicus. oblongus. Stolterfothii. 2. major. hirsutus. 3. 4. 5. spiniferus. elegans. splendidus. nitidus. THAUMATONEMA. Grev., Trans. Micr. Soc. Lond., 1863, p. 76. — Concatenate, discoid. Surface flat, or rising but slightly from centre for \ to f of radius, thence sloping steeply downwards to edge of girdle and slightly concave at middle of outer portion. Colour 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 677 pale grey, the processes subhyaline. Markings punctiform or areolate, forming evident radial rows or striae; radial costae at subresular intervals, sometimes distinct. Process single, springing from centre of valve, proximal portion nodular or elongated and columnar, distal portion biramose, the rami equal, diverging sym- metrically, their outer ends swollen and knob-like, rounded or elliptical. This genus forms the transition between the circular forms of the Diatomaceae and the armed Chadocerotidce. Apart from the process, the valves approach Coscinodiscus , and the nodular proximal portion of that of Thaumatonema costatum is but a greater development of, and so homologous with, the nodule of Coscinodiscus nodulifer , this development being still more marked in T. barbadense. T. barbadense. Grev., Trans. Micr. Soc. Lond., 1863, p. 76, pi. v. fig. 26. — Circular. Diam. *03 to *04 mm.; height of central portion of valve above edge of girdle ’075 to T mm. Surface flat to about § of radius. Markings punctiform, closely arranged in evident striae, the striae 6 to 6| in ’01 mm. Process ’0185 to •0315 mm. long, with proximal portion columnar, the rami of the upper portion stout, distance between the extremities of the rami •0135 to ’0185 mm. Habitat. — Cambridge deposit, Barbados (Johnson !); “ Barbados ” (Johnson !). T. costatum. Grev., Trans . Micr. Soc. Lond., 1865, p. 97, pi. viii. fig. 3. — Fragmentary. Roundly elliptical. Major axis ’055 mm., about 1^ times minor. Surface flat to about semiradius. Markings areolate, increasing slightly to about semiradius, thence decreasing gradually and becoming more faint to border ; at semi- radius 5, at border 8 to 9, in ’01 mm., at subregular intervals of about ’0125 mm., radiating evident costae. Process evident, the proximal median portion nodular, the diverging rami more delicate, straight, their outer ends elliptical, knob-like ; length of rami including terminal knob ’0175 mm., major axis of knob ’01 mm., about 2J times minor. Habitat. — Cambridge deposit, Barbados (Johnson !). 678 Proceedings of Royal Society of Edinburgh. [sess. Artificial Key. Markings punctiform. No costae. Proximal portion of process columuar, barbadense. Markings areolate. Evident radial costae. Proximal portion of process nodular, costatum. PEPONIA. Grey., Trans. Micr. Soc. Lond ., 1863, p. 75. — Central portion roundly elliptical, rarely subquadrate, with obtuse angles and convex sides, opposite the extremities of the minor axis a small regular cone, with free end rounded. Surface subplain. Colour pale grey. Central space absent. Markings areolate, sometimes increasing slightly from centre to semiradius, and again decreasing to border; between the lateral cones and the central portion a narrow hyaline band, at the extremities of the cones a small round, hyaline area. Border narrow, hyaline. P. barbadensis. Gr'ev., Trans. Micr. Soc. Lond., 1863, p. 76, pi. v. fig. 25. — Central portion with major axis from *0375 to *075 mm.; distance between apices of lateral cones ’0475 to •0925 mm. Markings towards the centre 4, at the semiradius 3|, at the border 8, in ’01 mm., without order or in inconspicuous radial and short oblique rows ; on the lateral cones decreasing towards their apices, sometimes absent. Habitat. — Bridgewater deposit, Barbados (Johnson !) ; Cambridge deposit, Barbados (Johnson! Greville) ; “Barbados” (Johnson! Greville !). 1888-89.] Mr John Rattray on the Genus Coscinodiscus. EXPLANATION OF PLATES. Plate I. Pig. 1. Coscinodiscus oamaruensis, Grove and Sturt, x 660. Fig. 2. 9 9 oculus-iridis, var. loculifera , nov. x 660. Fig. 3. ,, modestus, sp. n. x 660. Fig. 4. 99 debilis, sp. n. x 460. Fig. 5. „ imperator, Janisch. x 250. Fig. 6. „ interlineatus, sp. n. x 660. Pig. 7. 9 9 decussatus, Grove and Sturt MS. x 660. Fig. 8. 9 9 subnotabilis, sp. n. x 660. Fig. 9. 9 9 gracilentus, sp. n. x 660. Fig. 10. 9 9 subareolatus, sp. n. x 660 (worn specimen). Fig. 11. 99 groveanus, sp. n. x 460. Fig. 12. 99 antediluvianus, sp. n. x 660. Fig. 13. 99 inter mixtus, sp. n. x 660. Fig. 14. 9 9 obliquus, Rattray, x 660. Fig. 15. 9 9 sphceroidalis , sp. n. x 660. Fig. 16. 99 subtilis, var. lineolata, nov. x 660. Fig. 17. 99 incequisculptus, sp. n. x 460. Fig. 18. 9 9 luxuriosus, sp. n. x 660. Fig. 19. 9 9 glaberrimus, sp. n. x 660. Fig. 20. 9 9 argus, var. subtraducens, nov. x 460. Fig. 21. 9 9 nitidus, var. moronensis, Grun. MS. x 660. Fig. 22. 9 9 planiusculus , sp. n. x 660. Fig. 23. 9 9 granulosus, Grun. x 460. Fig. 24. ,, whampoensis, Grove MS. x 660. Fig. 25. 99 Weissjlogii, Sell, x 660. Plate II. Fig. 1. Coscinodiscus pulcherrimus , sp. n. x 660. Fig. 2. 9 9 lutescens, sp. n. x 660. Fig. 3. 9 9 leptopits, var. discrepans, nov. x 660. Fig. 4. africanus, var. wallichiana, Grun. x 660. Fig. 5. ,, minutellus, sp. n. x 660. Fig. 6. 9 9 notabilis , sp. n. x 660. Pig. 7. 99 actinosus, Grove MS. x 660. Fig. 8. ,, aethes, sp. n. x 660. Fig. 9. 99 densus, Grove and Sturt MS. x 660. Fig. 10. 9 9 pusillus, Grove MS. x 660. Fig. 11. 9 9 antimimos, sp. n. x 660. Fig. 12. 9 9 grayianus, sp. n. x 660. Fig. 13. 99 megacentrum, Grove MS. x 660. Fig. 14. 99 epiphanes, sp. n. x 660. Fig. 15. 9 9 superbus, var. nova-zealandica, Grove MS. x Fig. 16. 99 moronensis, Johnson, x 660. Pig- 17. 99 vetustissimus, Pant, x 660. Fig. 18. 9 9 subnotabilis, sp. n. x 660. Fig. 19. 9 9 theskelos, sp. n. x 660. Fig. 20. 99 galapagensis, Rattray, x 660. 680 Proceedings of Royal Society of Edinburgh. Fig. 1. Figs. 2. Fig. 3. Fig. 4. Fig. 5. Fig. 6. Fig. 7. Figs. 8. Fig. 9. Fig. 10. Fig. 11. Fig. 12. Fig. 13. Fig. 14. Fig. 15. Fig. 16. Fig. 17. Fig. 18. Fig. 19. Fig. 20. Fig. 21. Fig. 22. Fig. 23. Plate III. Coscinodiscus implicatus , sp. n. x 330. ,, prcetor, Grove MS. x 660 (central portion) ,, prcetor, Grove MS. x 660 (periphery). „ polurrhaptos, sp. n. x 660. ,, partitus, Grove and Sturt MS. x 660. ,, subtilis, var. scabra, nov. x 660. „ excavatus, var. deliquescens, nov. x 660. „ luctuosus, Grove MS. x 660. ,, luctuosus, Grove MS. x 660 (zonal aspect). ,, lewisianus, var. similis, nov. x 660. ,, implicatus, var. picturata, nov. x 460. ,, periTcompsos, var. curta, nov. x 660. Liradiscus marginatus, Grove MS. x 660. Brightwellia coronata , var. radians, nov. x 460. Actinogonium multiradiatum, sp. n. x 660. Brightwellia excellens, sp. n. x 660. Porodiscus nitidus, var. armata, nov. x 660. Asterolampra tenerrima, nov. x ? Porodiscus spiniferus , sp. n. x 330. Asterolampra tenerrima, nov. (small form), x ? Porodiscus major, var. densa, nov. x 660. Asterolampra traducens, sp. n. x 660. Liradiscus furcatus , Grove MS. x 660. 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 681 INDEX. Synonyms are printed in Italics. PAGE Acostat.®, nov., .... 629 Actinocyclus, 460, 488, 547, 584, 587, 595, 600 alienus Grun ., . . . 602 {alienus var. ?), undatus, Cleve, 577 Barklyi, Grun., . . 600, 60] Ehrenbergii, Ralfs , . .595 ellipticus, Grun., . . . 460 incertus, Grun., . . . 602 ovalis, Grun., .... 602 quindenarius, Ehrb., . . 601 Ralfsii, 601 Roperii, 602 subtilis, Ralfs, . . . 602 Actinogonium, Ehrb., . . 627, 628 multiradiatum, sp. n., . 628, 629 quinarium, Habir., . . 628 septenarium, Ehrb., . 628, 629 Actinoqramma, Ehrb., . . .654 RrooJcei, Ehrb., . . . 658 Jupiter, Ehrb., . . . 660 Saturnus, Ehrb., . . .660 sol, 660 Venus, Ehrb., . . . 660 Actinoptychus, BarTclyi, . .601 Arachnoidiscus ornatus, Ehrb., . 566 Asterodiscus, 649 Asterolampra, Ehrb., Grev. , pro parte, adriatica, Grun., semulans, Grev., affinis, Grev., . alien a, Grev. . amibgua, Grev., arachne, Grey., balearica, Cl., . brebissoniana, Grev, brightwelliana, Grev. Brookei, Grev., concinna, Grev., crenata, Grev., dallasiana, Grev., Darwinii, Grey., 628, 632, 633, 634, 644 634, 654 . 642 639, 652 649, 654 636, 652 635, 652 . 665 640, 653 643, 653 645, 651 . 658 646, 653 645, 653 643, 653 . 663 634 641 645 PAGE Asterolampra — decora, Grev., t . 634, 650, 653 decora, Grev. var., Cstr., . 650 decora, var. concentrica nov., 650 decorata, Grev., . . 647, 652 dubia, Grev., . . . 636, 651 elegans, Grev., . . .660 eximia, Grev., . . . 646, 653 Jlabellata, Grev., . . .662 Grevillei, Grev., . 640, 644, 654 Grevillei, var. adriatica, Grun., 643 Grevillei , var .eximia, Cstr., 644 heptactis, Grev., . . 658, 664 hexactis, Ehrb , . . .642 hiltoniana, Grev. . . .662 Hookerii, Grev., . . .656 imbricata, Grev., . . .661 impar, Shadb. , . . . 641 kittoniana, Grev., . . 637, 652 Isevis, Grev., . . . 641, 653 marginata, Grev., . . 634, 651 marginata, var. minor, Walker and Chase, .... 634 marylandica, Ehrb., 641, 642, 653 marylandica, var. ausonia, Cstr 642 marylandica, var. appropin- quans, Grove, . . .642 marylandica, var. maior B era- gal., 642 marylandica, var. ramosa, nov., 642 marylandica, var. /3, Wallich, 642 marylandica, var. y, Wallich, 642 moronensis, Grev., . . . 659 nicobarica, Grun., . . 639, 651 pelagica, Ehrb., , . . 642 princeps, Rattray, . . 644, 653 pulchra, Grev., . . 637, 652 Xpulchra, var. ?) W eissflogii, Gran., . punctata, Grev., ralfsiana, Grev., roperiana, Grev., Rotula, Grev. , . 651 639, 640, 653 . 635, 651 . 657 . 643, 653 682 Proceedings of Eoyal Society of Edinburgh. PAGE 648, 650, 653 . 666 638, 652 . 641 . 657 638, 653 648, 652 . 632 . 660 636, 652 649, 652 637, 652 653 647 647 647 647 647 647 648 655 Asterolampra — rylandsiana, Grev., sarcophagus , Grev., scutula, Grev., septenaria, Johnson shadbottiana, Grev., simulans, Grev., splendida, Grev., stella, Norman, stellata, Grev., stellulata, Grev., tenerrima, sp. n., tradncens, sp. n., uraster, Grove and Sturt, 648, 653 variabilis, Grev., . . . 655 vulgaris, Grev., 634, 645, 646 648, 650 vulgaris, var. cellulosa, nov. vulgaris, var. planior, nov. . vulgaris, var. a. Grev., vulgaris, var. b. Grev., vulgaris, var. c. Grev., vulgaris, var. d. Grev. , vulgaris, var. e. Grev., wallichiana, Grev. , Weissflogii, Van Heurck, 651, 652 Asterornphalus, 566, 643, 644, 654, 655 arachne . . . 665, 666, 667 Beaumontii, Ehrb., 658, 664, 668 Braunii, Janisch, . . . 667 Brebissonii, Janisch, . . 666 Brookei, Bail., . 644, 657, 668 Brookei, Grun., . . . 663 Brookei, var. robusta nov., . ( Brookei , var. robustus), Pera- gal, Buchii, Ehrb., centraster, Johnston cleveanus, Grun., cleveanus, Janisch, Cohnii, Janisch, Cuvierii, Ehrb., dallasianus, Ralfs, Darwinii, Ehrb., denarius, Janisch, Ehrenbergii, Janisch, elegans, Grev., . 644, 660, 668 elegans, Grev., var. Wallich, 664 flabellatus, Grev., . . 662, 667 flabellatus, var. tergestina, Grun., 662 Grevillei, Wallich, . . 644 lieptactis, Ralfs, . 659, 664, 668 hiltonianus, Ralfs, 660, 661, 668 Hookerii, Ehrb., . . 656, 668 Humboldtii, Ehrb., . . 656 imbricatus, Wallich, 644, 661, 668 imbricatus, var. /3, Wallich, 661 641 644 658 658 . 656 654, 667 662, 667 . 667 . 667 . 656 . 643 668 666 667 , 663, imbricatus, var. y, Wallich, 661 PAGE Asterornphalus — imbricatus, var., rectiradiata, nov., 661 malleiformis, Wallich, . . 665 malleus, Wallich, . . . 665 moronensis, Rattray, . 659, 668 nankoorensis, Grun., . 665, 667 Pringsheimii, Janisch, . . 666 ralfsianus, Grun., . . . 664 rarus, Rattray, . . 664, 668 reticulatus, Cleve, . . 663, 668 robustus, Cstr. , . . . 658 roperianus, Ralfs, . . 657, 668 Rossii, Ehrb., .... 663 sarcophagus, Wallich, . 666, 667 shadboltianus, Ralfs, 644, 656, 668 stellatus, Ralfs, . . 660, 668 variabilis, nov. , . 643, 655, 667 wallichianus, Ralfs, 655, 663, 667 wallichianus , Cl., . . . 663 wyville-thomsonianus, O' Me., 659, 668 Aulacodiscus, Ehrb., 458, 482, 504, 565, 604, 605 acutus, Rattray, . . 568, 609 apedicellatus, Rattray, . 604, 606 concinnus, Kitton, . . .521 excavatus, .... 605 formosus, .... 605 imperfectus, Grun., . . 604 Kittoni, 605 margaritaceus, var. Kinkeri, 558 suspectus, Rattray, . .604 Auliscus, 600 Biddulphia Johnsoni, Ralfs, . 575 Brightwellia, Ralfs, . 550, 629, 634 coronata, Ralfs, . . 630, 632 coronata, var. radians nov. , 631 elaborata, Giev., . .630, 632 excellens, sp. n., . 630, 632 hyperborea, Grun., . 630, 632 Johnsonii, Ralfs, . . 631, 632 Murrayi, Cstr., . . .631 pulchra, Grun., . . . 631 splendida, Rattray, 629, 630, 632 Campylodiscus, .... 504 clypeus, 490 Centrales, ..... 655 Cestodisci, Pant., . . . 457, 591 Cestodiscoidales, . . . 457 Cestodiscus, Grev., 450, 469, 522, 575 Baileyi, H. L. Sm., . 521, 522 johnsonianus, Grev., . . 458 moronensis, Grev., . . .458 obscurus , Yan Heurck, . . 513 ovalis, Grev., . . 460, 538, 547 ovalis, var. ? Witt, . . . 460* proteus, Hardman, . . 457 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 683 PAGE Cestodisens — pulchellus, Grey., 458, 459, 469, 586 pulchellus, Habir. , . . 454, 593 ( pulchellus , var. ?) hirtulus, Gran., 454 ( pulchellus , var. ?), Trinitatis , Gran., 593 . 561 . 466 . 457 . 677 . 669 . 507 . 599 radiatus, Ehrb., sol, Wallich, . stokesianus, Grev Chsetocerotidse, Ciculahes, Clivosi, Pant. , COCCONEIFORMES, . Coscinodiscus, Ehrb., 449, 452, 469, 470, 504, 505, 587, 600, 601, 602, 606, 607, 629, 677 actinochilus, Ehrb., 567, 573, 574, 621 adinocycloides, Pant., . . 503 actinocyclus, Ehrb., . . 606 actinosus, Grove MS., . 506, 617 seginensis, Sch., . . 489, 624 sethes, sp. n., . . . 569, 621 africanus, Janisch , 462, 534, 617 africanus ?, Sch., . africanus, var. rotunda,, Cstr., 534 africanus, var. wallichiana, Grun., . . . . 477, 534 agapetos, sp. n., . . 578, 615 ambiguus, Stokes, . . .510 amplius, Ehrb., . . . 606 anastomosans, Grun., . 455, 620 angulatus, Grev., . . 501, 616 anguste-lineatus, Sch., . 474, 627 annulatus, Grun., . . 562, 625 antarcticus, Cstr., . . .453 antarcticus, Grun., 455, 508, 614 antediluvianus, sp. n., . 456, 611 antimimos, sp. n., . . 461, 627 antiquus, Grun., . 461, 470, 627 apages, sp. n., aphrastos, sp. n., apiculatus, Ehrb, 575, 618 . 469, 626 463, 545, 570, 622 . 570 ambigua, 541, 571, 572 maxima, . 571 apiculatus ?, Sch., apiculatus, var. Grun., . apiculatus, var. Grun., apiculatus, var. "Woodwardii, 540, 571 apiculatus, var. Sch., . . 570 apiculiferus, Rattray, . 591, 613 apollinis, Ehrb., 578, 580, 586, 605, 622 apollinis, var. compacta, nov., 579 arafuraensis, . . . .462 arafurmsis , var. Cstr., . .544 PAGE Coscinodiscus — arafuscenis, O’Me., . 601, 602 argus, Ehrb., 515, 527, 528, 605, 615 argus, Grun., . . . . 525 argus, Sch., . . . .514 argus, var. subimpressa, Cleve, 528 argus, var. subtraducens, , nov.. . ... . . 528 armatus, Grev., . 575, 591, 612 armatus, Pant., . . .591 asperulus, GrUn., . . 519, 615 asteroides, Tru. and Witt, 522, 559, 625 asteromphalus, Ehrb., . 549, 556, 558, 560, 622 asteromphalus (Ehrb.), Sch., 549, 561, 563 asteromphalus, var. biangulata, Cl. and Moll., . . .548 asteromphalus, var. bright- wellioides, Grun., . . 550 asteromphalus, var. centralis, Grun., 555 asteromphalus, var. conspicua, Grun., 549 asteromphalus, var. eximia, Grun., 549 asteromphalus, var. genuina, Grun., 549 asteromphalus, var. hybrida, Grun., . . . 551, 554, 555 asteromphalus, var. macran- tha, Grun., .... 551 asteromphalus, var. omphal- antha, Grun., . . . 549 asteromphalus, var. pabellana, Grun., 551 asteromphalus, var. pabell- anica, Grun., . 535, 551, 565 asteromphalus, var. princeps, Grun., 551 asteromphalus, var. pulchra, Grun., 550 asteromphalus, var. spinu- ligera, Grun., . . . 564 asymmetricus, Grun., . . . 606 atlanticus, Cstr. , . . 478, 618 atlanticus, Gran., . . . 462 atlanticus, var. striatula nov., 478 atlanticus, var., Cstr., . .478 Auliscus, Kiitz., . . . 600 Baileyi, Rattray, . . 521, 613 balticus, Grun., . . . 499 barbadensis, Grev., 488, 504, 616 Barklyi, Coates, . . . 600 bathyomphalus, Cleve, . 587, 612 bathyomphalus, var. wan- karemensis, nov., . . 588 bengalensis, Grun., 484, 580, 625 684 Proceedings of Royal Society of Edinburgh. [sess. PAGE Coscino discus — biangulatus, Sch., 548, 550, 614 ?bifrons, Cstr. , . . . 600 biharensis, Pant., . . 590, 614 bioculatus, Grun,, . 483, 484, 624 bioculatus, var. exigua, Grun. , 484 bipartitus, sp. n., . .472, 626 biplicatus, Grun., . 484, 580, 622 biradiatus, Grev., . . 569, 622 bisculptus, sp. n., . . 471, 609 bisinuatus, Sch., . . 552, 614 blandus, Sch., . . 472, 624 boliviensis, Grun., . 541, 622 boliviensis, Grun., pro parte, 541 boliviensis, var. spinulosa, Grun., 541 borealis, Bail. , 516, 550, 558, 559, 624 borealis, Ehrb., . . 516, 559 bullatus, Janisch, . . . 462 bulliens, Sch., 519, 525, 550, 614, 629, 634 calif ornicus, O’ Me., . .543 capensis, Grun., .484, 581, 624 caraibicus, Tru. and Witt, . 507 carconensis, .... 606 caspius, Ehrb., . . 514, 515 ? centralis, Ehrb., . . .555 centralis, Ehrb. , pro parte, 555, 560 centralis emend., . 532, 552, 555, 556, 572, 613 centralis, O’Me., . . . 555 centralis, Sch., . . 551, 555 centralis, Schulze, . . . 532 centralis, Weisse, . . .555 centralis, var. micraster, Grun., 555 centralis forma minor , Van Heurck, . . . 555, 556 centralis, var. Cstr., . . 555 centranthus, Ehrb., . .606 cervinus, Ralfs, . . 593, 611 Challenged, Janisch, . .608 cinctus, Kutz. , 452, 453, 454, 611 cingulatus, Ehrb., . . 539, 623 circumdatus, Sch., 466, 481, 626 clivosus, Pant., . . 595, 625 clivosus, var. latefasciata, Grun., 595 ?clypeus, Ehrb., . . . 490 cocconeiformis, Sch., . 599, 609 cocconeiformis, var. brevior, nov., 599 cocconeiformis, var. latior, nov. , 599 cocconeiformis, var. tenuior, nov., . 600 cocconeiformis, var. Sell., . 600 commutatus, Grun., . . 533 PAGE Coscinodiscus — complexus, Stodder, . . 608 compositus, sp. n., . 518, 614 comptus, Cstr., . . 583, 623 concavus, Ehrb., . . 461, 470 concavus, Greg., . 469, 483, 626 concavus, Greg., var. Sch., . 461 concavus African, Elirb., . 470 concavus, var. africanus, Kutz. 470 concavus, var. Sch., . .470 concinnus, W. Sm., 531, 532, 545, 554, 555, 564, 617 concinnus, var. arafurensis, Grun., . . . 532, 534 concinnus, var. jonesiana, Rattray, . . 532, 554, 634 concinnus, var. lcerguelensis, Grun. , .... 532, 533 concinnus, var. Moseley i, Rattray, . . . .533 confertus, sp. n., . 584, 611 conformis, sp. n., . . 545, 620 confusus, sp. n., . . 451, 621 convexus, Sch., . . 552, 615 convexus, var. bengalensis, Grun., 553 convexus, var. diminuta, Grun., 553 corolla, Sch. , . . . 530, 618 craspedodiscus, O’Me., . 544, 600, 601, 602 craspedodiscus, Kutz., . 544, 600 crassus, Bail., 513, 524, 539, 540, 622 crassus, var. algida, Grun., . 540 crassus, var. gelida, Grun., . 540 crassus, var. morsiana, Grun. , 539, 540 crassus, Bail., var.? Grun., . 539 crassus, var. Sch., . . . 539 crenulatus, Grun., . 489, 617 cribrosus, Tru. & Witt, 536, 587, 621 cristatus, sp. n., . .475, 626 cristatus, var. distans? Sch., 475 cruciatus, Kiitz., . . . 601 curvatulus, Grun. , 477, 486, 617 curvatulus, var. barbadensis, CleveMS., . . . .489 curvatulus, var. densius- striata? Sch., . . .486 ( curvatulus , var. ?) divisus, Grun., 500 curvatulus, var. frigida, Grun., .' . . . 489 curvatulus, var. genuina, Grun., .... 487 curvatulus, var. inermis, Grun., 486 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 685 PAGE Coscinodiscus — curvatulus, var. kariana, Cleve and Grun., .... 488 curvatulus, var. latius-striata, Sch 487 curvatulus, var. minor, Gran., . . . .487, 488 curvatulus , var. Normanii , Cleve, 500 curvatulus, var. recta, nov., . 488 curvatulus, var. subocellata, Grun., .... 487, 488 curvatulus, var. Cleve MS., . 489 curvatulus, var. Cstr., . . 488 cycloteres, Cstr., . . 567, 573. debilis, sp. n., . . 529, 614 decipiens , Grun., . . 509, 519 decipiens, Grun., . 464, 465, 627 decrescens, Cstr., . . .460 decresens, Grun., . . 525, 612 decrescens, Sch 526 denarius, Sch., . 495, 504, 617 denarius, var. variolata, , 505 denarius, var. Sch. , . . 505 densus, Grove & SturtMS. , 592, 618 denticulatus, Cstr. , . 531, 613 depressus, Greg. MS., . 595, 613 detritus, Sch., . . . 537 devius, Sch., . . . .517 dimorphus, Cstr., 450, 583, 611 diophthalmus, Cstr., . . 523 diophthalmus, var. mono- phthalma, Cstr., . . 524 diorama, Sch., . . 542, 603 diplostictus, Grun., . 579, 620 disciger, Ehrb., . . 593, 619 discoplea, Ehrb., . . . 606 diversus, Grun., . 507, 513, 615 diversus, var. completa, . 508 divisus, Grun., . . 499, 624 divisus, var. arcuata, Grun., 500 doljensis, Pant., . . 503, 618 dubiosus, Grun. MS., 529, 589, 594, 597, 613 dubiosus, var. curvans, nov., 530 dubius, sp. n., . . 538, 623 PAGE Coscinodiscus — duriusculus, sp. n ., . 566, 615 ebulliens, var. Sch., . . 519 echinatus, sp. n., . . 491, 616 egregius, sp. n., . . 518, 613 Ehrenbergii, O’Me., . . 473 elegans, Grev., 573, 585, 586, 621 elegans, Grev., var. parvi- punctata, Tru. & Witt, . 585 elegans, var. spinifera, Grove and Sturt, .... 586 elegantulus, Grev., . 569, 612 ellipticus, Grun., . . 538, 610 elongatus, Grun., . . 584, 610 entoleion, Grun., . . 544, 623 epiphanes, sp. n., , . 526, 612 evadens, sp. n., . . 577, 615 evadens, var. parvula, Sch., 577 exasperans, sp. n., . . 450, 611 excavatus, Grev., 523, 559, decrescens, var. irregularis, excavatus, var. biocellata, Grun., . 525 Grun., . ._ . . 524 decrescens, var. polaris, excavatus, var. deliquescens, Grun., . 526 nov. , . 524 decrescens, var. repleta, excavatus, var. genuina, Grun., . 526 Grun., . . 523, 524, 598 decrescens, var. valida, excavatus, var. quadriocel- Grun., . 526 lata, Grun., . 523, 524, 598 decrescens, var. venusta, excavatus, var. semilunaris, Grun., . 525 Grun., . . 524, 562 decrescens, var-., Grun., 526 excentricus , Ehrb., 461, 464, decussatus, Grove and Sturt 465, 471, 474, 481, 493, 627 MS., . . 579, 622 (i excentricus , var. ?) antiquus, delawarensis, Gh'un., 606 Grun., . . 461 excentncus, var. decipiens, Grun., 464 excentricus, var. hyalina, nov. 464 excentricus, var. micropora, Grun., 463 extricus, var. perpusilla, Grun., 463 excentricus, var. punctifera, Grun., 464 (i excentricus , var.?) subline- atus, Grun., . . . 474 excentricus, var. zebuensis nov., 464 exiguus, sp. n., . . 578, 621 exiguus, var. sequalis, nov., . 578 extravagans, Sch., exutus, sp. n., fallax, Schum., fasciatus, Ehrb., fasciculatus, O' Me., fasciculatus, Sch., . Febigerii, H.L. Sm. fenestratus, Ehrb., 506, 624 529, 615 . 515 . 607 491, 500, 624 . 500 . 608 . 607 fimbriatus, Ehrb., 528, 545, 553, 554, 572, 614 686 Proceedings of Royal Society of Edinburgh. SESS. PAGE Coscinodiscus — fimbriatus, Sch., . . . 510 fimbriatus, var, californica, Grun., .... 553 fimbriatus, var. subradiata, nov., 553 fimbriatus , var. Van Heurck, 553 fimbriato-limbatus, Grun., . 509 fimbriatus-limbatus, Ehrb., . 509 flagrans, sp. n., . . 573, 619 flavicans, Ehrb., . . . 601 fiavicans, Weisse., . . . 601 flexilis, sp. n., . . 544, 623 florescens, Grun., . . . 596 floridulus, Sch. , . . 557, 620 foraminosus, Grev., . .479 fragilissimus, Grun., . 522, 613 fuscus, Norman, . . . 601 galapagensis, Rattray, . 574, 622 Gaze life, Janisch, . 546, 608, 619 gemmatulus, Cstr., . 574, 622 gemmifer, Ehrb., . 573, 585, 621 gemmifer, var. campechiana nov. , 573 gemmifer, var. Grun., . .573 gigas, Ehrb., 524, 541, 542, 543, 544, 603, 622 gigas, var. californica, nov. . 543 gigas, var. diorama, Grun., . 542 gigas, var. duplicata, Grun., 543 gigas, var. guineensis, Rattray, 543 gigas, var. laxa, nov., . . 543 gigas, var. Montereyi, Grun., 542 gigas, var. punctiformis nov., 542 gigas, var. Grun., . 462, 542 glaberrimus, sp. n., . 513, 615 glacialis, Grun., . 497, 498, 617 gracilentus, sp. n., . 599, 610 grcecus, Kiitz., . . .601 grandineus, sp. n., . 554, 613 grandineus, var. dentata, nov., 555 granulatus, Ehrb., . . 594, 611 granulosus, Grun., 453, 454, 611 granulosus, var. conspicua, nov., 454 granulosus, var. distincta nov. , 454 granulatus, Ehrb., . . . 594 grayianus, sp. n.,. . 588, 620 Gregorii, O' Me., . . 504, 624 griseus, Grev., . . 586, 621 griseus, Sch., . . . .574 griseus, var. apiculata, nov., 586 griseus, var. gallopagensis, Grun., ..... 574 groveanus, sp. n., . . 562, 614 Grunowii, Pant., . . 485, 616 Grunowii forma minor, Pant., 485 Grunowii, var. minor, Rat- tray, 485 PAGE Coscinodiscus — guineensis, Grnn., . . .543 gyratus, Janisch , . . . 462 Hauckii, Grun., . 535, 536, 611 heteromorphus, sp. n., . 468, 626 heteroporus, Ehrb., 519, 525, 540, 622 heteroporus, Grun., . . 527 heteroporus, Sch., . heteroporus, Ehrb. , forma major, Grun., . . . 525 heteroporus, var. moronensis, Grun. , 540 heteroporus, var. Grun., 540, 541 heterostigma, Ehrb., . . 604 hirtulus, sp. n., . . 454, 611 humilis, sp. n., . . 452, 610 hungaricus, Pant., . 591, 621 liungaricus, var. Szaboi, Rattray, . . . .591 hyalinus, Grun., . . 483, 624 imperator, Janisch MS. 546, 623 implicatus, sp. n., . . 512, 609 implicatus, var. picturata, nov., 513 impolitus, sp. n., . . 453, 611 impressus, Grun., . . 531, 625 incequalis, Grove and Sturt, . 477 insequisculptus, sp. n., . 557, 612 inclusus, sp. n., . . 482, 619 indicus, Ehrb. , . . . 607 inexpectatus, sp. n., . 452, 610 insutus, sp. n., . . 453, 611 interlineatus, sp. n., . 506, 617 intermedins, Ehrb., 495, 528, 529, 605 intermixtus, sp. n., . 563, 613 intumescens, Pant., 578, 590, 625 irradiatus , Harting., . . 527 irregularis, sp. n., . . 455, 612 isoporus, Ehrb., . . 483, 618 Janischii, Sch., . 543, 571, 622 Janischii, var. arafurensis, Grun., .... 544, 602 Janischii, var. Monicas, Grun., 563 ? Janus, Cstr., . . . . 600 japonicus, Cleve, . . .608 japonicus, Ehrb., . . . 607 javanensis, Grun., . . . 607 johnsonianus, Rattray, . 458, 615 josefinus, Grun., . . 545, 623 Kinkerianus, Tru. and Witt, 512 Kochii, Pant., . . 589, 613 kryophilus, Grun., 491, 492, 624 Kurzii, Grun., . . 564, 620 Kiitzingii, Sch., . . 481, 616 Kiitzingii, var. gracilis, Grun., 481 labyrinthns, Roper, . 471, 627 labyrinthus, Roper, var. ? Sch. , 471 lacunosus, Grove, . . . 608 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 687 PAGE Coscinodiscus — lacustris, Grun., . . 581, 615 lacustris, var. australiensis, . 582 ( lacustris , var. ?) australiensis , Grun., 582 lacustris, var. hyperborea, 581, 582 ( lacustris , var.?) hyperboreus, Grun., 581 lacustris, var. marina, Grun., 582 lacustris, var.- septentrionalis, 581, 582 ( lacustris , var. ?) septentrion- alis, Grun., . . . 581 lanceolatus, Cstr., . . 509, 610 lentiginosus, Janisch, 462,491, 492, 616 lentiginosus, var. maculatus, Grun., 491 leptopus, Grun., . 476, 491, 626 leptopus, var. discrepans, nov. , 47 6 lewisianus, Grev., . 547, 597, 598, 610 lewisianus, var. moronensis, nov., 598 lewisianus, var. similis, nov., 598 lewisianus, var., . . . 538 limbatus, Ehrb., . . . 509 limbatus, J xnisch and Raben. , 510 lineatus, Elirb., . 472, 474, 627 lineatus, Sell., . . .476 lineatus, Weisse., . lineatus, var. tenera, Tru. and Witt, . lineatus, var.? Sch., lineatus, var. ? Habir. lineolatus, Ehrb., . liocentrum, Ehrb., . longispinus, Grun., luctuosus, Grove MS., ludovieianus, sp. n., lunse, Ehrb., . lunatus, Grove MS. , lutescens, sp. n ., luxuriosus, sp. n., . maerseanus, Grev., .468 macraeanus, Grun., margaritaceus, Cstr., 473 . 474 . 473 . 462 . 607 535, 619 . 607 518, 612 596, 615 567, 621 522, 609 536, 623 455, 611 469, 626 . 476 . 585 marginatus, Ehrb., 455, 456, 509, 510, 514, 525, 540, 614 marginatus, Janisch, . . 543 marginatus, Sch., . . . 481 marginatus, var. decussata, nov., 511 marginatus, var. intermedia, Rattray, . . . .511 marginatus, var. latemargin- ata, Pant., .... 511 marginatus, var. subcon- cava, 510 PAGE Coscinodiscus — marginatus, var. submargin- ata, Grun., .... 512 marginulatus, Rattray, 450, 453, 454, 501, 617 marginulatus, var. campea- chiana, Grun., . . . 501 marginulatus, var. gallop- agensis, Grun., . . . 501 marginulatus, var. curvato- striata, Grun., . . . 501 marginulatus, var. sparsa, Grun., .... 501, 584 marginulatus, var. stelluli- fera, Grun., . . . 501 Martonfii, Pant., . . 591, 623 marylandicus, Grun., . . 608 megacentrum, Grove MS., 557, 611 megacoccus, Cstr., . . 456, 612 megaporus, Ehrb., . . 559, 614 mesacmaius, Ehrb., . . 607 mesodictyon, Ehrb., . . 607 mesoleius, Cleve, . 535, 536, 619 microcentrum, Ehrb., . . 607 minimus, Schum., . . . 600 minor, Anglor., . . 464, 465 minor, Ehrb., 465, 487, 607, 627 minor , Sell., .... 462 minor, Weisse., . . . 600 minor, W. Sm., . 464, 465, 603 minuens, Rattray, . . 460, 627 minutellus, sp. n., . . 494, 617 minutus, Kiitz., . . . 603 minutus, Schum., . . . 603 mirificus, Cstr. , . 535, 603, 623 modestus, sp. n., . . 536, 623 Molleri, A.S., . . 467, 626 Molleri, var. macroporus, Grun., 467 Monicse, Rattray, . . 563, 620 moravicus, Grun. / . . 558, 620 moronensis, Rattray, . 458, 609 Moseleyi, O’Me., . . . 533 mossianus, Grev., . . 573, 623 ? naviculoides, Tru. and Witt, 597, 610 nebula, Ehrb., . . . 529 neogradensis, Pant., . 590, 625 nitidulus, Grun., . 480, 578, 618 nitidulus, var. subradians, nov., 480 nitidulus, Grun., var.? Sch., 578 nitidus, Greg., 451, 478, 480, 504, 574, 611 nitidus, Sch., . . . .479 nitidus, var. minor, . .479 nitidus, var. moronensis, Grun. MS., ... 480 nitidus, var. sparsa, . .479 nitidus, var. tenuis, nov., . 479 688 Proceedings of Royal Society of Edinburgh. PAGE Coscinodiscus — nitidus, Greg., var. Cl. and Moll., 479 nobilis, Grun ., 462, 545, 602, 624 nodulifer, Janisch, 477, 520, 612, 677 nodulifer, Sch. , . . 462, 520 nodulifer, var. apiculata, nov., 520 notabilis, sp. n., . 588, 589, 623 Normani, Greg., . 491, 495, 500, 616 mormanianus, Grey., . .576 normanicus, Van Heurck, . 500 nottinghamensis, Grun., 456, 612 oaraaruensis, Grove and Sturt, 564, 620 obliquus, Rattray, . 575, 607, 618 oblongus, Grev. , 537, 538, 610, 674 oblongus forma typica, Tru. and Witt, . . . .537 obnubilus, Rattray, . 507, .625 obovatus, Cstr., . . 538, 610 obovatus, var. circularis, nov., 538 obscurus, Sch., . 513, 557, 620 obscurus, var. minor nov., . 514 obscurus, var.? Sch., . .514 ob versus, sp. n., . . 554, 614 obversus, var. tenuior, nov., 554 oceanicus, Kiitz., . . .470 oculus-iridis, Ehrb., 455, 488, 550, 552, 555, 556, 559, 563, 564, 614 oculus-iridis , Sch., . . 510 oculus-iridis, var. borealis, Cleve, 559 oculus-iridis, var. genuiua, Grun., 560 oculus-iridis, var. loculifera, nov., 562 oculus-iridis, var. morsiana, Grun., 561 oculus-iridis, var.? pacifica, Grun., .... 560, 563 oculus-iridis, var. stelliger, Sch., 561 oculus-iridis, var. subspinosa, Grun., . . . . . 561 oculus-iridis, var. tenuistriata, Grun. , 561 odontodiscus, Grun., 485, 498, 616 odontodiscus, var. parva-tenui- striata, Cl. and Moll., . 485 odontodiscus, var. subsubtilis, nov., . . . . . 486 odontophorus, Grun., . 495, 498, 617 omphalanthus, Ehrb., . 549, 550 omphalanthus, Grun., . . 560 ovalis, Rattray, . 460, 602, 610 ovalis, Roper, . . . .602 Coscinodiscus — pacificus, Cstr., . . .455 pacificus, Rattray, . . 563, 614 paleacens, Rattray, . 597, 610 papuanus, Cstr., . . . 534 parma, Bail., . . . 601 partitus, Grove and Sturt MS. , 505, 617 patelkeformis, Grev., 537, 569, 613 patera, Cstr., . . . 592, 625 patina, Bail., . . . 453, 527 patina, Ehrb., . 483, 527, 612 pauper, Tru. and Witt, 584, 622 Payeri, Grun., . . 483, 624 Payeri, var. subrepleta, Grun., 483 pectinatus, Rattray, . 519, 613 pellucidus, Grun., 484, 580, 625 perforatus, Ehrb., 571, 572, 593, 605, 620 perforatus, Cl. and Moll., . 571 perforatus, var. cellulosa, Grun., . . 544, 565, 572 perforatus, var. delicatula, uov., 572 perikompsos, sp. n., . 576, 619 perikompsos, var. curta, nov., 576 perminutus, sp. n., . 567, 621 peruanus, Grun., . . 474, 627 planiusculus, sp. n., . 490, 624 plicatulus, Grun., . 581, 582, 615 plicatus, Grun., . . 530, 625 polurrhaptos, sp. n., . 596, 609 poly acanthus, Grun., . .499 polyacanthus, Grun., 492, 49-8, 616 poly acanthus, var. ? baltica, Grun., 499 polyacanthus, var. davisiana, Grun., 499 polyacanthus, var. inter- media, Grun., . . .499 polycora, Ehrb., . . . 600 polygonus, Cstr., . . 584, 619 polyradiatus, Cstr., . .476 ? polystigma, Ehrb. , . . 600 praetextus, Janisch, . . 462 preetor, Grove MS., . 546, 610 profundus, Ehrb., . 508, 594, 614 proteus, Rattray, . . 457, 617 pseudo-lineatus, Pant., . 475, 627 pulchellus, Grev., . 459, 469, 626 pulchellus, Grun., . . . 469 • pulchellus, var. hirtulus, . 454, 469 moravica. Grun., pulchellus, Grun., . . . pulchellus, var. Trinitatis Grun., pulcherrimus, sp. n., pumilo, Ehrb., pumilus, Grun. , 459 . 469 582, 611 607 608 1888-89.] Mr John Rattray on the Genus Coscinodiscus. 689 PAGE Coscinodiscus — punctatus, Ehrb ., 495, 537, 547, 581, 619 punctatus, var. rhombica, nov., ..... 547 punctulatus, Greg., 594, 604, 613 pusillus, Grove MS., . 459, 616 Pyxidicula, Kiitz., . .601 pyxis, Ehrb., .... 601 quindenarius, Habir., . . 601 radiatus, Bail., . . 514, 542 radiatus, Ehrb., 490, 508, 513, 514, 515, 517, 520, 527, 553, 555, 570, 605, 606, 614 radiatus, Ehrb. , pro parte, . 527 radiatus , Sch., . . .516 radiatus, Weisse., . . . 514 radiaius, var. abyssalis, Cstr. , 516 radiatus, var. borealis, Grun., 516 radiatus, var. crenulata, Ratt- ray, 518 radiatus, var. glacialis, Grun. , 516, 518 radiatus forma heterosticta, Grun., 510 radiatus, var. irregularis, Grun., 518 radiatus, var. media, Grun., 516 radiatus, var. minor, nov. , . 517 radiaius forma minor, Sch. . 517 radiatus, var. parva, Grun., 517 radiatus, var. subsequalis, Grun., . . . .516, 605 radiatus, var., Wallich, . 518 radiolatus, Ehrb., 495, 521, 528, 529, 604, 605, 606 radiolatus, Ehrb. , pro parte. , 528 radiolatus, Sch., . . .516 radiolatus, Weisse., . . 606 radiopunctatus, Hooting, 594, 612 radiosusj Grun., 520, 531, 545, 553, 556, 594, 605, 606, 614 radiosus, var. kerguelensis, Grun. 520 regius , Grun., . . 545, 568 reniformis, Cstr., . . 548, 609 rex, Wallich, . . 546, 568, 610 rhombicus, Cstr., . . .547 rhombicus, Grun., . . 568, 619 robustus, Grev., 510, 511, 513, 614 robustus, Sch., . . .511 robustus, var. fragilis, mu?. , . 512 robustus, var. , intermedia, Grun., 511 robustus, var. kittoniana, nov., 512 Rothii, Grun., 495, 497, 498, 502, 616 PAGE Coscinodiscus — Rothii, var. actinocycloides, 503 Rothii, var. grandiuscula, nov. , 503 Rothii forma minor, Grun., . 502 Rothii, var. singaporensis, nov. , 503 rotula, Grun., . 566, 567, 619 rudis, Cstr., .... 600 sarmaticus, Rant., . . 548, 510 scintillans, Grev. , . . .579 secernandus, Sch., . . 558, 614 semilunaris, Grun., . . 524 semipennatus, Grun. , 484,504, 617 senarius, Sch., 504, 505, 506, 617 simbirskianus, Grun. , 489, 607, 617 sinensis , O’Me., . . 527, 603 Smithii, O’Me., . . . 603 sol, Wallich, 462, 465, 466, 471, 626 sphseroidalis, sp. n., . 451, 609 sphseroidalis, var. cincta, . 451 spiniferus, Grove and Sturt, 585, 620 spinuligerus, sp. n., . 564, 623 spinulosus, Ehrb., . . 456, 610 splendidulus, Rattray, . 576, 618 splendidus, Grev., . 468, 469, 626 sp., Cstr., .... 452 sp., Sch., .... 505 ? sp. , Tru. and Witt, . .602 stellaris, Roper, . 493, 594, 616 stellaris , var. fasciculata, Cstr., 493 stellaris, var. Mejillonis, Grun., 494 stellaris, var., Cstr., . . 493 stelliger,. Grun., . . 566, 621 stokesianus, Grun., . 457, 615 stokesianus forma baldjikiana, Grun., 458 stokesianus forma minor. , Grun. , 457 striatus, Ehrb., . . . 603 striatus, Kiitz., . . 603, 604 subangulatus, Grun., . 520, 609 subareolatus, sp. n., . 454, 612 subaulacodiscoidalis, sp. n. , 521, 613 subconcavus, Grun., 466, 510, 627 subconcavus forma major, Sch., . . . . . 510 subconcavus, var. tenuior, nov., 467 subconcavus, Grun. , var. ? Sch., 467 subdivicus, Tru. and Witt. 587, 615 subglobosus, Cl. and Grun., 481, 618 2 x VOL. XVI. 22/11/89 690 Proceedings of Royal Society of Edinburgh. [sess. Coscinodiscus — (subglobosus, var. ?) antarcticus, Gran., 508 snblineatus, Gr.un. ,. . 474, 627 subnitidus, sp. n., . . 451, 611 subnotabilis, sp. n., . 589, 618 subnotabilis, var. marina, nov., 589 suboculatus, sp. n., . 563, 624 subsalsus, Juh.-Dannf., 593, 618 subtilis, Ehrb., 481, 482, 487, 490, 491, 493, 194, 497, 498, 500, 502, 505, 539, 555, 604, 617 . 538 . 486, 495 glacialis , . 498 subtilis O’ Me. MS, subtilis, Sch., . ( subtilis , . var. ?), Grun., .... subtilis, var. lineolata nov., 497 subtilis, var. Normanii, Van Heurck, 500 (subtilis, var. ?) odontophorus, Gran., 498 subtilis, var. scabra, nov., . 497 subtilis, var. siberica, Grun., 496 subtilis, var. Sch., . . . 485 subtilissimus, Ehrb., . . 607 subvelatus, Gran., . . 467, 512 superbus, Hardman MS., 458, 459, 615 superbus, var. moravica, Grun., 459 superbus, var. nova-zealand- ica, Grove MS., . . .459 suspectus, Janisch, 480, 481, 618 symbolophorus, Grun., 492, 493, 616 symmetricus, Grev., 486, 490, 491, 495, 606, 624 symmetricus, Sch;, . .490 symmetricus, Ivitton & Weiss- flog, 502 (symmetriscus, var.) denarius, Sch., 505 Szabbi, Pant., . . .591 Szontaghii, Pant., . . . 487 tabularis, Grun., . . 583, 621 tenellus, Ehrb., . . . 604 tenerrimus, Ehrb., . . 608 tenuis, Bail., . . . .532 tenuis, Rattray, . . 577, 619 tenuisculptus, sp. n., . 452, 610 theskelos, sp. n., . 565, 566, 609 Thuinii, Cleve, . 583, 584, 623 traducens, sp. n., . . 528, 614 traducens, var. hispida, nov., 529 Trinitatis, Rattray, . 593, 619 Trocliiscos, Tru. and Witt, 567, 619 tuberculatus, Grev. , 482, 624 PAGE Coscinodiscus — tuberculatus forma minor, . 482 tuberculatus, var. Monicas, Grun., 482 tuberculatus, Grev. , var. b Sch., . . . . .482 tumidus, Janisch, 462, 465, 475, 626 tumidus, var. fasciculata, nov. 476 turgidus, sp. n., . . 455, 612 unibonatus, Cstr., . . . 592 umbonatus, Greg., . 507, 565, 612 undatus, Grun., . . 577, 625 undulans, Rattray, . 552, 625 undulatus, Cstr., . . .552 undulatus, Cleve., . . 587, 621 vacuus, sp. n., . 535, 536, 619 variolatus , Cstr., . . . 505 varius, Schum., . . . 604 velatus, Ehrb., . 509, 510, 612 velatus, Sch., .... 455 venulosus, Cstr. , . . 457, 618 vetustissimus, Pant., 477, 478, 534, 617 vetustissimus, var. curvatu- loides, Grove MS., . .478 vetustissimus, var. wallichi- ana, Grun., . . . 534 vicinus, Schum., . . . 514 vigilans, A.S., . . 467, 626 vulgaris, Schum., . . . 605 wallichianus, Grun., . . 608 Weissltogii, Sch., . . 565, 609 Weyprechtii, Grun., . 552, 622 whampoensis, Grove MS., 497, 616 Woodwardii, Eul. , var. Grun., 542 Woodwardii , Eul., . 571, 572 Woodwardii, Sch., . . 527 Woodwardii, Eul., var. Grun, 542 Woodwardii, var. ? Sch., . 541 zebuensis, Grun., M.S., . . 464 zonulatus, sp. n., . . 469, 626 Cosmiodiscus, Grev., . . 450, 605 armatus, Sell., . . .575 barbadensis, .... 590 carconenis, . . . .606 elegans, Grev., . . .576 imperfectus, Grun., . . 604 normanianus, Grev., . .575 normaniamus, Grove and Sturt (not Grev)., . .576 obliquus, Grev. MS., . .575 tenuis , Grun., . . . .577 COSTAT-E, 531 Craspedodiscus, . 601, 629, 634, 671 actinochilus, Ehrb., . .574 coronatus, Brightw., . . 630 Cosinodiscus, Ehrb., . . 600 elegans, Ehrb., . . 600, 601 1888-89.] Mr John Rattray on the Genus Ooscinodiscus. 691 PAGE Craspedodiscus — rnargincttus, Bright w., . .634 oblongus, Grun., . . .674 ovalis, Grun., . . . .672 Craspedoporus, . . . .671 elegans, Grove and Sturt , . 676 CyXTella^ 536, 600, 6oi, 603, 607, 669 Castracanei, Eul. MS., . . 596 dallasiana, W. Sm., . 603, 669 pumila, Cl., . . . 450 punctata, W. Sm., . . 581 salina, Grun., . . . 603 striata, Kiltz., . . . 603, 669 striata, var. baltica, Grun., . 600 striata, var. intermedia, Grun., 603 Denticella, 556 Dietyopyxis subtilis, Ehrb., . 473 Discoplcea graeca, Ehrb., . . 601 Ductiles. 634 Elaborati, . . . . .597 Eleganti, Pant., .... 507 Elliptici, 670 Endidya, Ehrb., . . 450, 455, 470 minor, Sch., . . . .470 oceanica, Ehrb., . . .470 Ethmodiscus, Cstr., . . .450 convexus, Cstr., . . . 522 gigas, Cstr, . . . .546 sp., Cstr., .... 546 spcehroidalis, Cstr., . . 546 tympanum, Cstr., . . .546 wyvilleanus, Cstr., . . 546 Eupodiscus, , . . 491, 509, 532 commutatus, Grun., . . 533 concinnus, var. triangularis, 533 exeentricus, O’Me., . . 464 gregorianus, de Breb., . .532 jonesianus, Grev., . . 532, 533 Roperii, de Breb., . . . 602 subtilis, Greg., . . . 532 Excentrici, Pant., . . 460, 659 Excentron cancroides, Ralfs. . 665 Eximaj, 639 Easciculati, Grun., . 477, 513, 592 Gallionella, 604 Heterodictyon, Grev., . . . 629 splendidum, Grev., . 629, 630 Heterostephania, Elirb., . 450, 502 Rothii, Ralfs., . . .502 Rothii (ft) denaria, . . 502 Rothii (a) odonaria, . . 502 Hyalodiscus cervinus, Brightw., . 593 Inflati, .... PAGE . 755 Inordinati, . . 450 Janischia, Grun., . . 450 antiqua, Grun. , . 596 Lineati, Pant., . 466 Liradiscus, Grev., . . 468, 668 barbadensis, Grev., . 669, 671 capensis, Cleve, ellipticus, Grev. , . 669, 671 . 670, 671 furcatus, Grove MS., . 669, 671 marginatus, Grove MS., 670, 671 minutus, Grev. , . 671 oblongus, Grun. , . . 670, 671 ovalis, Grev., . . 670, 671 Marginal, . . 634 Melosira, . . 565, 594, Melosira ? Sch. , 601, 604, 607 . 535, 567 angulata, Greg., . 464 Borreri, Grev. , . 453 cribrosa, de Breb., . 470 distans, Kiltz. , . 603 lineata, Ag. , . . 453 moniliformis, . . 453 nivalis, W. Sm. , . 465, 603 hummuloides, Ehrb., , . .531 oceanica, Habir. , . . 470 sulcata forma coronata, Grun., 676 Mesasterias, Ehrb., . 654 abyssi, Ehrb., . .657 Micropodiscus, Grun., . . 450 Nomina nuda, . 606 Obscuri, .... . 654 Odontodiscus, Ehrb., . 450 curvatulus, Cleve, . . 486 curvatulus, var. kariana, Cl. and Grun., . . 488 exeentricus, Ehrb., . . 462 granulosus, Grun., . ' 453 hyalinus, Grun. , . ‘ 483 pellucidus, Grun., . . 580 polyacanthus, Grun. , . . 498 spica, Ehrb., . . 485 subtilis, Grun., . 500 uranus, Ehrb. , . 485 Orthosira, . 594 angulata, Greg. , . 464 oceanica, Brightw., . 470 Peponia, Grev., . 678 barbadensis, Grev., . 678 Podosira oliverana, Grun. , . . 476 hormoides, Mont., . . 530, 531 Porodiscus, Grev., . . 671 conicus, Grev. , elegans, Grev. , 673, 675, 676 . 673, 676 692 Proceedings of Boyal Society of Edinburgh, PAGE Porodiscus — liirsutus, Grove, and Sturt, 675, 676 hormoides, Mont., . . 530, 531 interruptus, Grove and Sturt, 676 major, Grev., . . . 673, 676 major, var. densa, nov., .673 nitidus, Grev., . . 672, 676 nitidus, var. armata, nov., . 672 oblongus, Grev., . . 674, 676 oliveriana, Grun., . ovalis, Grev., .... 674 spiniferus, sp. n., . . 674, 67 6 splendidus, Grev., . . 671, 676 splendidus, var. marginata, nov., 672 splendidus, var. 1 Sch. , . . 672 Stolterfothii, Ostr. , . 674, 676 Pseudostephanodiscus, Grun., . 507 Pyxidicula, . . .472, 600, 601 Coscinodiscus, Ehrb., . . 601 gemmifera, Ehrb., . . 570, 573 Weyprechtii, Grun., . . 602 Radiati, Grun., . 507, 513, 591, 592 Rhizasolenia, 556 Spatangidium, de Breb., . .654 arachne, de Breb., . . .665 jiabellatum, de Breb., . .662 heptactis, de Breb., . 658, 664 pellatum, de Breb., . . 662 ralfsianum, Norman, . 658, 664 Species EXCLUSiE, . . . 600 Stelladiscus, gen. n., . . 632 stella, Rattray, . . .632 Stephanodiscus — carconensis, Grun., . . 606 punetatus, Grun., . . . 581 PAGE Stepbanopyxis, . 456, 468, 509, 600 superba, Grun., . . . 468 turris, 602 turris, var. arctica forma macropora, Grun., . . 602 turris, var. cylindrus forma inermis, Grun., . . . 602 Stictodiscus californicus, Grev., . 601 Crozierii, Kitton, . . . 566 Stoschia, Janisch, . . 450, 509, 518 admirabilis, Janisch, . 462, 548 1 paleacea, Grun., . . . 597 punctata, Grove and Sturt, . 452 SUBMARGARITACEiE, . . . 639 Symbolophora, Ehrb., . . 450, 493 acuta, Ehrb., .... 493 euprepia, Ehrb., . . . 493 hexas, Ehrb. , . . . 492^ 493 microhexas, Ehrb. , . 493 micropentas, Ehrb., . . 493 microtetrus, Ehrb., . . 493 microtrias, Ehrb., . . . 492 pentas, Ehrb., . . 492, 493 sp., Ehrb., .... 492 tetras, Ehrb., . . . 492, 493 Trinitatis, Ehrb., . . . 493 Thaumatonema, Grev., . . .676 barbadense, Grev., . . 677, 678 costatum, Grev., . . 677, 678 Traducentes, . . . .636 Triceratium, 600 coscinoides, Grove and Sturt, 600 parma, Bail., .... 601 Willemoesia, Cstr. sp. . Cstr., 450, 452 . 452 J. Rattray del. 7. Hnth, LitTiT Edml Proc.Roy. Soc. Edinr RATTRAY ON C 0 S C I N 0 D I S C U S Yol. XVI O OO rj '?0<*g0 • :— - #|o° 02d°°o° fco o °§°o O ° °qO®Q °3>fv ^'°VCo.V^ foOOC i§5g Vo aoago°o° o ° ogcSpt [too ° o0Ooon0 o° ° 0 o°°oYotl S°°o0o oOn QO °0°g°o °°o O ol feo0° 2O00°ooo° <&d°i Vffc, o o 0 o° 0000 o OO OO ' 0° Of/ \C o o o o o o oo°oo££f0»i %ooo ooooooooL OX On°oJ n o o n a ^ O ^ w-.y.- w/ x. 660 |ooOO 0 0000 00^ °O°0C •\%>o o Q o O O o o O o° oi o Y; 11 . YW o o o O o g o n°o Aoy IXXV La y, o' O o0°0; : > P 0 ^8|0o^o°o°^oo0. Y°o9.Sgo88f« ■;Xo,po°o0o p 0 °oc>0%°oVo s °Oc§o°o 0 0°°° > O nr°0 o o _0o0 o o 0 O c O °0 o wo _o 3 0 oO?0 °Q °0° df O 0 o p.00 0 o o o O 300 o O poo0 F.Huth., Lith* Editil * Vol. XVI. j^;"0o o “ 0 S l°o°o° ° o _0°oo o go n' i< |~ og°/ g SSoo%|o o|° 0%| Ioo00° ooo°o^oo OO0oJ \r^0o0ooZ0a°?0^0%±i |SI» iSilllll? «o RATTRAY ON CO S C I N 0 D I 5 C U S ALLIED GENERA. p Yol. XVI. F. Huth, LitH E Sin? 19. X 330 . 16. X 660 21. X 660 15. X 660 PI. III. Proc.Roy. Soc. EdinT Vol. XVI. RATTRAY ON C 0 S C I N 0 D I S C .U S AND ALLIED GENERA. 1888-89.] Sir W. Thomson on Constitution of Matter. 693 Molecular Constitution of Matter. By Sir William Thomson, (Read July 1, 1889.) § 1 . The scientific world is practically unanimous in believing that all tangible or palpable matter, molar matter as we may call it, con- sists of groups of mutually interacting atoms or molecules. This molecular constitution of matter is essentially a deviation from homogeneousness of substance, and apparent homogeneousness of molar matter can only he homogeneousness in the aggregate. “ A body is called homogeneous when any two equal and similar parts of it, with corresponding lines parallel and turned towards the same parts, are undistinguishable from one another by any difference in quality.”* I now add that unless the “ part” of the body referred to consists of an enormously great number of molecules, this statement is essentially the definition of crystalline structure. It is, indeed, very difficult to imagine equilibrium, static or kinetic, in an irregular random crowd of molecules. Such a crowd might he a liquid, — I can scarcely see how it could be a solid. It seems, therefore, that a homogeneous isotropic solid is hut an isotropically macled crystal ; that is to say, a solid composed of crystalline portions having their crystalline axes or lines of symmetry distributed with random equality in all directions. The proved highly perfect optical isotropy of the glass of object-glasses of great refracting telescopes, and of good glass prisms, seems to demonstrate that the ultimate molecular structure is fine-grained enough to let there he homo- geneous crystalline portions, which contain very large numbers of molecules while their extent throughout space is very small in com- parison with the wave-length of light. § 2. An ideal skeleton or framework for a homogeneous assemblage of bodies, or of material systems of any kind, or of qualities or pro- perties of any kind, distributed periodically throughout space, is defined and explained in § 45 ( a ) to (j) below, substantially * Thomson and Tait’s Treatise on Natural Philosophy, new edition, vol. i. part ii. §§ 675-678 ; or Elements of Natural Philosophy , §§ 646-649. VOL. XVI. 23/11/89 2 Y 694 Proceedings of Royal Society of Edinburgh. [sess. taken from Bravais’ doctrine of homogeneous assemblages, which we may look upon as the grammar of molecular construction. Space-Periodic Partitioning (§§ 3-13). § 3. Given a homogeneous assemblage of points : let it be required to partition all space accordingly. The thing to be done is concisely defined in the second sentence of § 6 below. §4. The problem is clearly indeterminate. Here is a solution which has obvious relation to Brewster’s kaleidoscope and the corresponding doctrine of electric images, and which may be import- ant in respect to Yortex Theory for a crystal or ether. From P, a point of the given assemblage, draw a line, PH, of any length in any direction, provided only that H is not a point of the assemblage of P’s. Do the same relatively to every other of the P-assemblage. We thus have a homogeneous assemblage of double points, PH. Let Q be any point in space, and let S denote summation for all the PH’s. Let <£( D) be a function which decreases as D increases from 0 to oo . The equation 2ft(QP)-«KQN)] = 0, expresses a locus for Q which partitions space periodically, and divides each periodic portion into two cells containing respectively an H and a P. Every cell containing an H is a parallel pervert (footnote on § 45a below) of every cell containing a P. That this is true we see by drawing any straight line to equal distances in opposite directions through the point midway between H and P. Its ends are similarly related, one of them to all the H’s ; the other to all the P’s. § 5. Here is a perfectly general solution. Around any one of the points P describe a closed surface S, of which the greatest distance from P is less than that of P’s nearest neighbour. Describe an equal, homochirally similar, and same- ways oriented surface around every other point P. Hone of these surfaces cuts or touches any other. Expand all of them simultaneously, equally, and without altering shape or orientation, till one of them touches another. All corresponding pairs of the surfaces touch simultaneously at corre- sponding points. Continue the expansion, annulling in each case the mutually enclosed portions of the expanding surfaces, and sub- 1888-89.] Sir W. Thomson on Constitution of Matter. 695 stituting the portion of fixed surface traced, or left behind, by the expanding line of mutual intersection. This portion of surface we shall call (after my brother, Professor James Thomson) an interface. Follow the same rule when another, another, and another contact takes place. When the borders of two of the growing interfaces thus traced meet and begin to intersect, annul their projecting por- tions, so that the intersection and wdiat is left of the expansion of its previous border now constitute the boundary of the interface. Continue the process until fresh growing intersections of interfaces are formed, and the ends of these growing intersections meet, and at last nothing is left of the expanded original surfaces, and therefore nothing of space is left unenclosed by the cells — polyhedrons of interfaces — thus constructed. § 6. The interfaces formed in § 5 are generally curved, but, as we shall see (§ 7), may be plane, and are so in particular cases of special interest. In every case each cell contains one, and only one, of the P’s ; there is no interstitial space between them ; they are all equal, homochirally similar, and con-orientational. § 7. If the initiating surface, S, of § 5 is a polyhedron of plane facets, the periodic partition to which it leads is in polyhedrons of plane facets. So it is also if the initiating surface is any ellipsoid with P for centre. § 8. Let S be a sphere. The partitional polyhedron, to which it leads, is the dodekahedron obtained by drawing planes through the middle points of the lines between P and its twelve next-neighbours, perpendicular to these lines. § 9. If S is an ellipsoid similar to and con-orientational with that determined in § 47 below, the partitional polyhedron to which it leads is the rhomboidal dodekahedron to which the rhombic dodeka- hedron of § 21 below is converted by the homogeneous strain of § 46. In this case the whole number of contacts of the expanding surfaces (§ 5) is twelve, and they all take place simultaneously. § 10. If the assemblage becomes equilateral, the partitional dode- kahedrons of §§ 8, 9 become, each of them, the rhombic dodeka- hedron of § 21. § 11. If S is an ellipsoid, having conjugate diameters along lines from P to other three points of the assemblage, and of magnitudes proportional to the distances from P to the nearest points in these 696 Proceedings of Royal Society of Edinburgh. [sess. lines, the partitional polyhedron to which it leads is a parallele- piped. § 12. If the three points chosen are nearest neighbours of P (§ 45 i below), we are led to the best conditioned (or least oblique) of all the infinity of parallelepipedal partitions possible. This is the most obvious and the best known of the periodic partitions of space. §13. Taking the parallelepipedal partitioning of §11, let P' he the farthest corner from P, so that PP' is the longest diagonal of the parallelepiped. Let PA, PB, PC he conterminous edges and A'P', B'P', C'P' their opposites conterminous in P'. Draw the planes ABC, A'B'C'. We thus divide the parallelepiped into three parts — an octohedron ABCA'B'C' ; and two tetrahedrons, PABC, P' A'B'C', which are parallel mutual perverts (footnote on § 45a below). This grouping of eight points of a homogeneous assemblage is, as we shall see later, important in the dynamics of molecular structure, or at all events in Boscovich’s theory.* On Boscovich’s Theory (§§ 14-44 and §§ 62-71). §14. Without accepting Boscovich’s fundamental doctrine that the ultimate atoms of matter are points endowed each with inertia and with mutual attractions or repulsions dependent on mutual distances, and that all the properties of matter are due to equilibrium of these forces, and to motions, or changes of motion, produced by them when they are not balanced; we can learn something towards an understanding of the real molecular structure of matter, and of some of its thermodynamic properties, by consideration of the static and kinetic problems which it suggests. Hooke’s exhibition of the forms of crystals by piles of globes, Navier’s and Poisson’s theory of the elasticity of solids, Maxwell’s and Clausius’ work in the kinetic theory of gases, and Tait’s more recent work on the same subject — all developments of Boscovich’s theory pure and simple — amply justify this statement. § 15. Boscovich made it an essential in his theory that at the * Theoria Philosophise Naturalis redacta ad unicam legem virium in natura existentium, auctore P. Rogerio Josepho Boscovich, Societatis Jesu, nunc ab ipso perpolita, et aucta, ac a plurimis prseeendentium editionum mendis ex- purgata. Editio Yeneta prima ipso auctore prsesentm, etcorrigente. Yenetiis, mdcclxiii. Ex Typographia Remondiniana superiorum permissu, ac privi- legio. 1888-89.] Sir W. Thomson on Constitution of Matter. 697 smallest distances there is repulsion, and at greater distances attrac- tion ; ending with infinite repulsion at infinitely small distance, and with attraction according to Newtonian law for all distances for which this law has been proved. He suggested numerous transi- tions from attraction to repulsion, which he illustrated graphically by a curve, — the celebrated Boscovich curve, — to explain cohesion, mutual pressure between bodies in contact, chemical affinity, and all possible properties of matter — except heat, which he regarded as a sulphureous essence or virtue. It seems now wonderful that, after so clearly stating his fundamental postulate which included inertia, he did not see inter-molecular motion as a necessary consequence of it, and so discover the kinetic theory of heat for solids, liquids, and gases ; and that he only used his inertia of the atoms to explain the known phenomena of the inertia of palpable masses, or assemblages of very large numbers of atoms. §16. It is also wonderful how much towards explaining the crystallography and elasticity of solids, and the thermo-elastic pro- perties of solids, liquids, and gases, we find without assuming more than one transition from attraction to repulsion. Suppose, for in- stance, the mutual force between two atoms to he zero for all dis- tances exceeding a certain distance, I, which we shall call the radius of the sphere of influence ; to be repulsive when the distance between them is <£; zero when it is =•-£; and attractive when it is >£ : and consider the equilibrium of groups of atoms under these con- ditions. A group of two would he in equilibrium at distance £ ; and only at this distance. This equilibrium is stable. A group of three would be in stable equilibrium at the corners of an equilateral triangle of sides £ ; and only in this configuration. There is no other configuration of equilibrium except with the three in one line. There is one, and there may he more than one, con- figuration of unstable equilibrium, of the three atoms in one line. § 17. The only configuration of stable equilibrium of four atoms is at the corners of an equilateral tetrahedron of edges £. There is one, and there may be more than one, configuration of unstable equi- librium of each of the following descriptions : — (1) Three atoms at the corners of an equilateral triangle, and one at its centre. 698 Proceedings of Royal Society of Edinburgh. [sess. (2) The four atoms at the corners of a square. (3) The four atoms in one line. There is no other configuration of equilibrium of four atoms, sub- ject to the conditions stated above as to mutual force. Important questions as to the equilibrium of groups of five, six, or greater finite numbers, of atoms occur, but must be deferred. The Boscovichian foundation for the elasticity of solids with no inter-molecular vibrations is the subject of §§ 62-71 below. A few preliminary remarks here may be useful. § 18. Every infinite homogeneous assemblage * of Boscovich atoms is in equilibrium. So, therefore, is every finite homogeneous assem- blage, provided that extraneous forces be applied to all within in- fluential distance of the frontier, equal to the forces which a homogeneous continuation of the assemblage through influential dis- tance beyond the frontier, would exert on them. The investigation of these extraneous forces for any given homogeneous assemblage of single atoms — or of groups of atoms as explained below — con- stitutes the Boscovich equilibrium-theory of elastic solids. § 19. To investigate the equilibrium of a homogeneous assemblage of two or more atoms, imagine, in a homogeneous assemblage of groups of i atoms, all the atoms except one held fixed. This one experiences zero resultant force from all the points corresponding to itself in the whole assemblage, since it and they constitute a homo- geneous assemblage of single points. Hence it must experience zero resultant force also from all the other i- 1 assemblages of single points. This condition, fulfilled for each one of the atoms of the compound molecule, clearly suffices for the equilibrium of the assemblage, whether the constituent atoms of the compound molecule are similar or dissimilar. § 20. When all the atoms are similar — that is to say, when the mutual force is the same for the same distance between every pair — it might be supposed that a homogeneous assemblage, to be in equilibrium, must be of single points ; but this is not true, as we see syntheti- cally, without reference to the question of stability, by the following * “ Homogeneous assemblage of points, or of groups of points, or of bodies, or of systems of bodies ,” is an expression which needs no definition, because it speaks for itself unambiguously. The geometrical subject of homogeneous assem- blages is treated with perfect simplicity and generality by Bravais, in the Journal de Vficole Poly technique, cahier xxxiii. pp. 1-128 (Paris, 1850). 1888-89.] Sir W. Thomson on Constitution of Matter. 699 examples of homogeneous assemblages of symmetrical groups of points, with the condition of equilibrium for each when the mutual forces act. §21. Preliminary. — Consider an equilateral* homogeneous assem- blage of single points, 0, O', &c. Bisect every line between nearest neighbours by a plane perpendicular to it. These planes divide space into rhombic dodekahedrons. Let AjOA^ A2OA6, A3OAY, A4OA8, be the diagonals through the eight trihedral angles of the dodekahedron inclosing O, and let 2 a be the length of each. Place atoms Qls Q5, Q2, Q6, Q3, Q7, Q4, Q8, on these lines, at equal dis- tances, r, from 0 ; and do likewise for every other point, O', O", &c., of the infinite homogeneous assemblage. We thus have, around each point A, four atoms, Q, Q', Q", Q'", contributed by the four dodekahedrons of which trihedral angles are contiguous in A, and fill the space around A. The distance of each of these atoms from A is a - r. § 22. Suppose, now, r to be very small. Mutual repulsions of the atoms of the groups of eight around the points O will preponderate. But suppose a - r to be very small ; mutual repulsions of the atoms of the groups of four around the points A will preponderate. Hence for some value of r between zero and a, there will be equilibrium. There may, according to the law of force, be more than one value of r between zero and a giving equilibrium ; but whatever be the law of force, there is one value of r giving stable equilibrium, supposing the atoms to be constrained to the lines 0 A, and the distances r to be constrainedly equal. It is clear from the symmetries around O and around A, that neither of these constraints is necessary for mere equilibrium ; but without them the equilibrium might be unstable. Thus we have found a homogeneous equilateral distribution of 8-atom groups in equilibrium. Similarly, by placing atoms on the three diagonals, BxOB4, B2OB5, B3OB6, through the six tetrahedral angles of the dodekahedron around O, we find a homogeneous equi- lateral distribution of 6-atom groups in equilibrium. § 23. Place, now, an atom at each point O. The equilibrium will be disturbed in each case, but there will be equilibrium with a * This means such an assemblage as that of the centres of equal globes piled homogeneously, as in the ordinary triangular-based, or square-based, or oblong- rectangle-based, pyramids of round shot, or of billiard-balls. 700 Proceedings of Royal Society of Edinburgh. [sess. different value of r (still between zero and a). Thus we have 9-atom groups and 7-atom groups. § 24. Thus, in all, we have found homogeneous distributions of 6-atom, of 7-atom, of 8-atom, and of 9-atom groups, each in equilibrium. Without stopping to look for more complex groups, or for 5-atom or 4-atom groups, we find a homogeneous distribution of 3-atom* groups in equilibrium by placing an atom at every point 0, and at each of the eight points A1? A5, A2, A6, A3, A7, A4, A8. There are four obvious ways of seeing this, found by choosing one or other of the four diagonals through trihedral angles referred to in § 21. Take, for example, A1OA5, and its congeners for all the dodeka- hedrons. These triplets include all the A’s. (Compare § 25 below.) § 25. Lastly, choosing A2, A3, A4, so that the angles AjOA^ AjOAg, A1OA4, are each obtuse, f we make a homogeneous assem- blage of 2-atom | groups in equilibrium by placing atoms at O, A1} A2, A3, A4. There are four obvious ways (compare § 24 above) of seeing this as an assemblage of di-atomic groups, one of which is as follows : — Choose A4 and O as one pair. Through A2, A3, A4, draw lines same-wards parallel to A40, and each equal to A40. Their ends lie at the centres of neighbouring dodekahedrons, which pair with A2, A3, A4 respectively. § 26. For the Boscovich theory of the elasticity of solids, the consideration of this homogeneous assemblage of double atoms is very important. Remark that every 0 is at the centre of an equi- lateral tetrahedron of four A’s ; and every A is at the centre of an equal and similar, but contrary-ways oriented, tetrahedron of O’s. The corners of each of these tetrahedrons are respectively A, and three of its twelve nearest A-neighbours ; and 0 and three of its twelve nearest O-neighbours. By aid of an illustrative model showing four of the one set of tetrahedrons with their corner atoms painted blue, and one tetrahedron of atoms in their centres painted red, the mathematical theory which had been communicated to the Royal Society of Edinburgh, was illustrated to Section A of the British Association at its recent meeting in Newcastle. * This is the assemblage described in the footnote on § 71 below. t This also makes A2OA3, A2OA4, and A3OA4 each obtuse. Each of these six obtuse angles is equal to 180° - cos-1(l/3). + This is the assemblage described in § 69 below, and used in §§ 67, 68, 70. j 1888-89.] Sir W. Thomson on Constitution of Matter. 701 § 27. In this theory* it is shown that in an elastic solid constituted by a single homogeneous assemblage of Boscovich atoms, there are in general two different rigidities, n , w15 and one hulk-modulus, k ; between which there is essentially the relation 3k = ?>n + 2 nv whatever he the law of force. Here nY denotes what are called the diagonal rigidities, and n the facial rigidities relative to the primi- tive cube of § 53 below. By facial and diagonal rigidities relative to any given cube I mean rigidities defined in the usual manner,! one of them according to shearing parallel to any face of the cube, the other according to shearing in planes parallel to any plane-diagonal of the cube. § 28. A remarkable result of my mathematical investigation is, that the facial rigidity, relatively to the primitive cube of § 52, is double the diagonal rigidity in the case in which each atom ex- periences force only from its twelve nearest neighbours. The law of force may he so adjusted as to make n^ii] and in this case we have 3 k — 5n, which is Poisson’s relation. But no such relation is obligatory when the elastic solid consists of a homogeneous assem- blage of double, or triple, or multiple Boscovich atoms. On the contrary, any arbitrarily chosen values may he given to the bulk- modulus and to the rigidity, by proper adjustment of the law of force, even though we take nothing more complex than the homo- geneous assemblage of double Boscovich atoms above described. Boscovichian Kinetic Theory of Crystals , Liquids , and Gases. § 29. The most interesting and important part of the subject, the kinetic, must, for want of time, he hut slightly touched in the present communication. I hope to enter on it more fully in a future communication to the Royal Society of Edinburgh. § 30. To avoid circumlocutions, I shall call any velocity moderate , which is comparable with the maximum velocity acquired by two atoms attracting one another from rest, at distance I. It is the velocity that in the circumstances each would have when their * See §§ 62-71 below. + Thomson and Tait’s Natural Philosophy, 2nded., vol. i. part 2, §680; also reprint of Mathematical and Physical Papers , vol. iii. art. xcii. part 1. 702 Proceedings of Boy al Society of Edinburgh. [sess. distance becomes diminished to £. When I speak of atoms or groups moving “ rapidly,” I mean that the velocities are moderate as thus defined. § 31. Let us consider what would follow if we had given at any time, scattered randomly but equably all through space, simple Boscovich atoms moving with velocities randomly equal in all directions. As we are supposing the masses of all the atoms equal, we may call the mass of each unity: thus v 2 for all the atoms in any part of space at any time, is the total of their kinetic energy. Both the number of atoms and their total energy we shall suppose to be equal in all very large equal volumes. § 32. The result of a collision between two atoms is essentially the same as that of the collision of two equal balls supposed simply repellent at contact, as in the elementary kinetic theory of gases as worked out by Maxwell and Tait ; * but the size of the balls that would give the same result depends, for each collision, very com- plexly on the law of force, and on the velocities and lines of motion of the atoms before the collision. As long as there is no case of collision between more than two atoms, the average energy of the free atoms at any time, and the law of the distribution of energy among the multitude in their free paths between collisions, is not affected by this complication, and is the same as if the atoms were equal hard globes merely repellent at contact. It is only when the results of unequal distributions of density, of energy, or of components of momentum, are to be traced, and the laws of the relation of pressure to density, or of thermal conduction, or of viscosity are to be investigated, that we can take into account the law of force, and can find differences from what the results would be if we had merely the hard equal balls to deal with. § 33. But now suppose, while two atoms are in collision, a third to come within their influential distance, so that three shall be in collision at the same time. All three may go clear, or two of them may remain in collision, or in other words, fall into combination, and one go free. It is scarcely possible that all three can remain in * Maxwell, Philosophical Magazine, 1860, and Philosophical Transactions, 1867 and 1878 ; Tait, “ On the Foundations of the Kinetic Theory of Gases,” Trans. Roy. Soc. Edin., vol. xxxiii., read May 14 and December 6, 1886, and January 7, 1887. 1888-89.] Sir W. Thomson on Constitution of Matter. 703 collision — that is to say, can combine. It will certainly be a very rare incident that they remain for any considerable time in collision ; but I cannot prove that the case may not occur in which none will go free, and the three will remain in combination. § 34. If the initially-given velocities are very great, the general result, even of triple collisions, will be to leave the individual atoms free. The comparatively rare double atoms resulting from triple collisions, and the still rarer triplets, will be liable to be separated again into single atoms by all fresh collisions. This is the case of a perfect monatomic gas, at a temperature much higher than the Andrews’ critical point. § 35. But if the originally-given velocity be exceedingly small, the result of exceedingly nearly every triple collision will be to form a combination of at least two of the three colliding atoms. Imme- diately after the collision by which it was formed, each doublet will generally have considerable relative motion of its two atoms ; that is to say, the two will describe orbits round their common centre of inertia : or, in the extreme case of no moment of momentum round this point, they will oscillate relatively to their centre of inertia to and fro in a straight line ; the centre of inertia itself generally having a considerable velocity. Still supposing the average velo- cities of the free atoms to be very small, and their number to be very great in comparison with that of the double-atoms, we now see that the general effect of the collisions between double and single atoms must be to diminish the energies of the relative and absolute motions of the constituents of the doublets, and so reduce the doublets more and more nearly to the condition of pairs of atoms in relative equilibrium (§ 16 above), a,t distance £ asunder, with centre of inertia of each pair moving very slowly through space. § 36. But now consider the effect of a collision between two doublets each with little or no intestine commotion before the collision, and with its centre of inertia moving very slowly through space. The case in which the same description would be applicable to the four atoms after the collision, whether in the same pairs or in interchanged pairs, would be exceedingly rare. So also would be the case of the four atoms remaining combined. The result in exceedingly nearly every case would be a triplet with considerable intestine commotion, and its centre of inertia moving rapidly 7 04 Proceedings of Royal Society of Edinburgh. [sess. through space, and a single atom moving rapidly through space. The general tendency of subsequent collisions between these rapidly- moving triplets and single atoms, with the multitude of slowly-moving single atoms throughout space, would he to diminish the energy of the intestine commotions of the triplets, and of the motions of the centres of inertia, both of the triplets and of the single atoms, reducing each triplet to very nearly the condition of equilibrium (§16 above) at the corners of an equilateral triangle of side £ with a slow translatory motion through space. § 37. By similar dynamical considerations we see that the general tendency of collisions between doublets and triplets, or be- tween triplets and triplets, must be to form quartets, quintets, and sextets of atoms ; and that when such groups, carrying away large kinetic energies from the generative collisions, subsequently collide with slowly-moving single atoms, the general tendency must be to diminish their kinetic energies, and reduce them more and more nearly to groups in one or other configuration of equilibrium, with slow motion of their centres of inertia through space. § 38. But now consider a collision between a slowly-moving triplet or quartet or more-multiple group, and a slowly -moving single atom. Even with the triplet the case will not be rare in which the single atom will remain in combination, and the result yielded be a quartet having considerable intestine commotion, and moving slowly through space. In collisions between a quartet and a single atom, the case will be relatively less rare, and with a quintet and single atom, still less rare for a single atom to remain in combination, and form a quintet or a sextet. § 39. If groups of large numbers of atoms in equilibrium, or slowly vibrating, have been thus formed, or are given ready formed, with single atoms slowly moving in the space all around them, each single atom colliding with a group will very frequently remain in the group ; and in virtue of the exhaustion of potential energy thus effected the vibrational energy of the group will be slightly aug- mented. But in not rare cases either the single atom which collided, or one of the atoms of the group in the neighbourhood of the collision, will be driven off, and generally with much greater velocity than the colliding atom had before the collision. Thus the average kinetic energy of vibration per atom of the group may be 1888-89.] Sir W. Thomson on Constitution of Matter. 705 kept constant, while the group is gaining by the accession, to it of more and more single atoms from without. But the exhaustion of potential energy due to the greater number falling into, than being thrown out from, the group would cause an augmentation of kinetic energy in the surrounding atmosphere of free atoms. To obviate this, let the atmosphere around the group be contained in a finite closed vessel, which, when left to itself, repels each atom that comes near enough to it, and sends it hack inwards with unchanged energy. Now let portions of this hounding surface he movable, and let them he so moved hy proper external appliances, that work shall be done upon them hy the impinging atoms to just such a degree as to keep the average kinetic energy of the free atoms constant. We have thus a Boscovichian realisation of a crystal of ice (hoar-frost) or other substance growing by condensation of a surrounding atmosphere of the same substance. The process in nature requires the abstraction of what is called the latent heat of the vapour to allow it to condense. This in our Boscovichian system is performed hy the arrangement for letting work he done outwards hy the moving parts of the boundary. § 40. Even if there were no surrounding atmosphere of single moving atoms, our group, unless quite free from intestine com- motion, would occasionally throw off an atom in virtue of the chance concurrence of different sets of component vibrations at some of the outlying atoms. Now let there he just enough of atoms moving about in the space around the group to cause as many fallings-in as throwings-out of atoms, and with just enough of kinetic energy to neither gain nor lose energy in the surrounding atmosphere through these changes. This will also cause the average kinetic energy of the group to remain constant. Thus we have a crystal surrounded hy an atmosphere of vapour at its own tem- perature, and at the proper temperature to cause neither condensa- tion of the vapour nor evaporation of the solid. § 41. Now hy somehow applying force to the atoms of the group increase their vibrational energy. We must, hy introducing atoms from the boundary, increase the density of the atmosphere around it to cause as many atoms to enter the group as are thrown off from it. Continue this process until the inter-atomic oscillations in the group become so great that the atoms begin to pass from one con- 706 Proceedings of Royal Society of Edinburgh. [sess. figuration of equilibrium to another, and back ; as, for instance, the two configurations of § 46 (footnote) below. The group may still retain its form as a solid, and something of its rigidity as a solid. § 42. Now reverse the operations at the boundary so as to diminish the inter-atomic oscillatory energy of the group. The atoms may fall back into their previous positions of equilibrium. But they may not ; and instead they may fall into another configuration more readily taken in a settlement from internal agitation than the previous configuration which was arrived at by growth from the bound- ary. This (with true molecules of matter instead of the ideal Boscovich atoms) seems to me, without doubt, the explanation of Madan’s* beautiful discovery regarding chlorate of potash, and the change of crystalline structure, by which Lord Bayleighf has shown that the optical phenomena presented in it are to be explained. Virtually the same view to explain other changes of crystalline structure by differences of temperature or applications of pressure seems to have been given by M. Mallard, J who is quoted by Madan in the article above referred to. In a future communication to the Royal Society I hope to include considerations regarding the effect of inter-atomic forces and motions in guiding to one or other of the two configura- tions described in § 54 and footnote on § 46 below. § 43. Once more communicate and continue communicating energy to the group by forces applied directly to its constituent atoms, and, at the same time, keep introducing fresh atoms from the outer boundary into the atmosphere surrounding the group to prevent the number of atoms in the group from diminishing. The intestine com- motion will become so great that all configurations of equilibrium are utterly departed from, but still the atom is surrounded by neigh- bours well within the region of its attractive influence (a shell bounded by two concentric surfaces of radius I and £ respectively) and constantly crossing and recrossing the spherical surface of radius £, or into and out of the sphere of repulsive force. If the region of attractive force be sufficiently thick, and the augmentation of the repulsive force from zero towards infinity be sufficiently rapid, it is * “On the Effect of Heat in Changing the Structure of Crystals of Potassium Chlorate,” Nature, May 20, 1886. t Philosophical Magazine, 1888. + Bulletin de la Societe Mindralogique , 1882, and December 1885. 1888-89.] Sir W. Thomson on Constitution of Matter. 707 clear that our original group which was a crystal and is now fluid will remain more dense than the surrounding atmosphere of free atoms until we have imparted to the group far more of energy than was required to dislodge its constituent atoms from configurations of equilibrium. There then is a mass of liquid surrounded by an atmosphere of its vapour, and in thermal equilibrium with the vapour if we cease the action on its atoms by which we imparted energy to it. A little farther consideration would no doubt give us the virtual surface-tension of the liquid exactly according to Laplace’s theory of capillary attraction ; hut we must not pause over this at present. § 44. Recommence applying forces to the atoms of the group, now liquid, and introducing fresh atoms into the surrounding atmosphere. The density of the atmosphere becomes greater, while that of the group becomes less. Go on till the two densities become equal: thus we reach the Caignard de la Tour and Andrews’ critical point. If we continue now imparting energy to our original group, or to any of the atoms of the assemblage, we simply have a homogeneous assemblage in a state of homogeneous intestine commotion all through ; the Boscovich realisation of a fluid raised higher and higher above its critical temperature. On Molecular Tactics of Crystals and of the Artificial Twinning of Iceland Spar (§§ 45-60). § 45. (a) . . . (j). Summary of Bravais ’ Doctrine of a Homo- geneous Assemblage of Bodies. (a) The bodies must he equal, similar, and homochiral.* ( b ) They must he all similarly oriented. (c) They must he so distanced mutually that any point in one * This will be more easily and not less thoroughly understood from illustra- tions than from a definition in general terms. Of an externally symmetrical man, the two hands are allochirally similar. Either is the 'pervert of the other ; or they are mutual perverts. Two men of exactly equal and similar external figures would be allochirally similar if one holds out his right hand and the other his left ; homochirally similar if each holds out his right hand or each his left. (We ignore at present the monochiral anti-symmetry of one heart on one side ; of interior structure of intestinal canal not in the plane bisecting the exterior symmetric figure, &c., &c.). Looking to § (•£) below, we 708 Proceedings of Royal Society of Edinburgh. [sess. of them, and the corresponding points in all the others form a homogeneous assemblage of points. If this condition is fulfilled for any one chosen point of one body, (a) and ( b ) imply it for any other ; and vice versa if this condition is fulfilled for three points of one body chosen arbitrarily hut not in one line, ( b ) is a necessary consequence. (d) A homogeneous assemblage of points means, and cannot mean other than, an assemblage which presents the same aspect and the same absolute orientation when viewed from different points of the assemblage. Some confusion of ideas has been introduced by leaving the generalised simplicity of Bravais, and considering an assemblage of double points, or triple points, or quadruple points, without noticing its being resolvable into two, or three, or four similar homogeneous assemblages of single points. (e) Rows of Points in a Homogeneous Assemblage. — Through any two points of the assemblage draw a straight line, and produce it indefinitely in both directions. All points on this line at intervals successively equal to the distance between the two chosen points, are points of the assemblage. The interval between each point and the next to it on either side in the line is called by Bravais the parameter of the row. (/) Planes of Points (“reseaux”) in a Homogeneous Assemblage. — Take at random any three points of the group. The case of there being other points of the assemblage on the sides or within the area of the triangle of the chosen points may he excluded. Along the line of each side of the triangle produced in both directions, mark off in succession lengths equal to the side, and through each division draw parallels to the other two sides. The plane of the triangle extended indefinitely in all directions is thus divided into equal and homochirally similar triangles turned alternately in opposite directions. At every angle of see two tetrahedrons, OPQR, OP'Q'R', which are equal, and allochirally similar, being parallel perverts, either of the other, or parallel mutual perverts. From every point P of a body or group of points, draw a line through any one point 0, and produce to P', making OP' = PO. The group of points (Pr) is a parallel pervert of the group (P). The groups (P) and (P') are parallel mutual perverts. Turn (P') 180° round any line OK. In the position thus reached, it is the image of (P) in a plane mirror through 0, perpendicular to OK. In their present positions they are mutual perverts inverted relatively to the line OK. Mutual perverts are allochirally similar. 1888-89.] Sir W. Thomson on Constitution of Matter. 709 each of these triangles a point of the assemblage is found. ISTo point of the assemblage is to he found elsewhere in the same plane. Fig. 1 shows a homogeneous distribution of points in a plane. In the diagram they are joined by lines, determinately chosen accord- ing to § (i), so that all the angles of triangles formed by them are acute. Closely related to this triangular arrangement are three others. One of these is obtained by omitting PQ and its parallels and taking instead the other diagonal OD of the parallelogram QOPD and drawing parallels to it through all the points. The two others are obtained similarly by omitting OQ and taking instead the other diagonal PE of the parallelogram QPOE; and by omitting OP and taking instead the other diagonal QF of the parallelogram PQOF. (g) All the points of the assemblage lie in equidistant planes parallel to the plane of (f); similarly placed at the angles of tri- angles equal, similar, and similarly oriented to the triangles of (/). The distance between each of these planes, and the next plane to it, is easily proved to be equal to the reciprocal of the product of twice the area of the triangle into the number of points per unit volume. vol. xvi. 23/11/89 2 z 710 Proceedings of Royal Society of Edinburgh. [sess. In fig. 2 the points PQOP'Q' and their congeners represent a homogeneous distribution in one plane. The orthogonal projection on this plane of the points in the two nearest parallel planes are represented respectively by R and its congeners, black dots ( • ), and by R' and its congeners, white dots ( 0 ). Thus explained, the dia- gram (fig. 2) is a complete specification of the whole homogeneous assemblage throughout space. (h) Tetrahedronal Grouping. — Choose any one of the triangles (OPQ), and any point (S) in the nearest plane of points on either side of it ; and imagine a tetrahedron of which these (OPQS) are the four corner points. By similarly dealing with all the triangles of all the planes, con-orientational with the first chosen triangle, and the points corresponding to the first chosen point in the neighbouring plane, we form a homogeneous assemblage of equal homochirally similar, samely oriented, tetrahedrons. Thus, for example, take the triangle FGH which is con-orientational with QOP. The tetrahedron on the base FGH corresponding to SQOP is RFGH. Each point of the distribution is the common corner point of 1888-89.] Sir W. Thomson on Constitution of Matter. 711 eight of those tetrahedrons ; of which the twelve edges meeting in it lie in the lines of six rows of points which intersect in that point. (?) Best conditioned Tetrahedronal Grouping. No Obtuse Angles. — Instead of choosing our first two points and our first triangle at random, take any point 0 and its nearest neighbour on either side, P ; and its next-nearest neighbour Q on the side making the angle QOP acute. The two other angles of this triangle are obviously, as Eravais remarks, acute. The only other way of thus finding best conditioned triangles is by taking O’s other nearest neighbour, P', and its other next-nearest, Q'. The triangles Q'OP' and QOP are equal, homochirally similar, and oppositely oriented ; and thus we find the only other possible best conditioned triangular grouping. Every other triangle of the points in the same plane, having none of the points within its area, has, as Bravais remarks, an obtuse angle. Consider now the nearest parallel plane of points on one side of the plane of QOP. Let E and its congeners ( • black dots) be the orthogonal projections of its points on the plane of QOP. Let R' and its congeners ( ° white dots) be the projections of the points of the nearest parallel plane on the other side of QOP. These projections will be situated relatively to the triangle P'OQ' and its congeners as are the former projections ( • black dots) relatively to the triangle QOP. R being, of the projections on the plane of POQ of all the points of the two parallel planes, the one which lies within the area of the triangle QOP, we have in OPQR a best conditioned tetrahedronal grouping. OP'Q'R' is another and the only other best conditioned tetrahedronal grouping. It is a parallel pervert of OPQR [ see foot- note on § 45 a above]. Hence a homogeneous assemblage of single points is essentially free from monochiral anti-symmetry ; or it is dichirally symmetrical. (j) The tetrahedron found by taking, with 0, P, Q, any other point than R in the plane through it parallel to QOP, has an obtuse angle along one, or obtuse angles along two, of its three edges, OP, PQ, QO : and so with 0, P', Q', and any other point than R' in the other parallel plane. 712 Proceedings of Royal Society of Edinburgh. [sess. Closest Packing of one Homogeneous* Assemblage of Equal and Similar Globes or Ellipsoids. §46. Take our tetrahedron OPQR, and by homogeneous distor- tional strain convert it into an equilateral tetrahedron ABCD, of equal volume. Take four globes, of diameters equal to the edges of this tetrahedron and place them with their centres at its corner points A, B, C, D. Alter this assemblage of globes by homogeneous strain till their centres, ABCD, become again the corner points of the original tetrahedron, OPQR. The globes have now become ellip- soids. Dealing thus with the whole original homogeneous assemblage of points, we find a closest packed homogeneous distribution of equal and similar ellipsoids through space. § 47. To find every possible closest packed homogeneous assem- blage of given equal and similar ellipsoids, take a tetrahedron of four equal globes. Choose any three mutually perpendicular directions, and, by elongations and shrinkages of the group parallel to these directions, convert each globe into an ellipsoid equal and similar to * There is another closest packing of globes or ellipsoids which has the same density as, and might without careful attention be mistaken for, the closest homogeneous packing. For simplicity think only of globes, and take a plane covered with globes touching one another in equilateral triangular order. Look at the accompanying diagram, fig. 6 of § (55) below, and see that there are two ways of placing a second layer on the first to continue the formation of an assemblage. The globes of the second layer may be placed, all of them over the black dots ( • ) or all of them over the white dots ( ° ). But having once chosen the position of the second layer there is no more freedom to choose in adding on layer after layer if we are to make a single homogeneous assemblage. Of the two positions which might be chosen for the third layer we must choose the one in which the globes are not over the globes of the first layer. The position of the fourth layer must be the one of which the globes are not over the globes of the second layer, but are over those of the first layer, and so on. If on the contrary we place the globes of the third layer over the globes of the first, the globes of the fourth layer over those of the second, and so on, we have a peculiar and symmetrical grouping which was first, so far as I know, described by Mr William Barlow ( Nature , December 20 and 27, 1883). This grouping is not one homogeneous assemblage. It consists of two homogeneous assemblages, one of them constituted by the first, third, fifth, seventh, &c. , layers ; the other the second, fourth, sixth, eighth, &c., layers. The considera- tion of this peculiar mode of grouping may be of great interest in the dynamical investigations to form the subject of my next communication to the R.S.E. (July 15), and, as Barlow has pointed out, may be of great importance in the theory of natural crystalline structure. I must, however, leave it for the present. 1888-89.] Sir W. Thomson on Constitution of Matter. 713 the given ellipsoid. Every possible configuration of closest homo- geneous packing of the given ellipsoids is clearly to he thus found ; and is specified in terms of three independent variables, — the three orientational coordinates, relative to the equilateral tetrahedron of the system of rectangular lines. § 48. In §§ 46, 47 we have a solution of the problem given four points , 0, P, Q, R, not in one plane , to place con-orientationally four equal and similar ellipsoids with their centres at the four points , and the surface of every one touching the surface of each of the three others. From it we have the following perfectly simple construction for the answer. Bisect OP, OQ, OR, in F, G, H, and QR, RP, PQ, in F', G', H', and join FF', GG', II H'. These three lines meet in one point S. The planes GSH, HSF, FSG are parallel to conjugate diametral planes of the required ellipsoids. These ellipsoids touch one another in the points F, F', G, G', H, H'. To construct them, first make four parallelepipeds, having a common corner at S, and their half-edges which meet in S, and their- centres, as follows : — Half-edges. SF SG SH SG7 SIT SF Inscribe within the twelve edges of each parallelepiped an ellipsoid, touching them at their middle points. This construction is interest- ing as showing, in the middle points of the twelve edges of the parallelepiped, the twelve points of contact of the ellipsoid with its twelve next neighbours. The ellipsoid touching the twelve edges is, it need scarcely be remarked, similar to the inscribed ellipsoid touching the surfaces, but of J 2 times the linear dimensions. § 49. To understand the configuration of a closely packed homo- geneous assemblage of ellipsoids, it is convenient to consider the assemblage of globes to which it is reduced by strain (geometrical distortion), in § 46. The assemblage of ellipsoids has all characteristic features the same, except the inequalities of lines and angles involved in the distortional transition from one configuration to the other. Centres. Half-edges. Centres. I SH' ) - o SF' r f SG SF' I i p SG' y e f SH ( 714 Proceedings of Royal Society of Edinburgh. [sess. § 50. In the close homogeneous assemblage of globes, we may first remark, that each globe is touched by its neighbours, at twelve points, being the points in which its surface is cut by diameters parallel to the six edges of the tetrahedron. If we place a number of small globes (hoys’ marbles, or billiard balls) on a table in close triangular order, and three as close as they can be together above them, we see nine of the twelve points of contact on the ball below the middle of the triangle of these three ; six points on the circle in which it is cut by a horizontal plane through its centre, and three symmetrically ranged on a small circle above it. The other ends of the diameters* through these three are the remaining three of the twelve. Or if we join the upper three by great circles, making a spherical triangle of 60° side, and complete these circles, they make another spherical triangle of 60° side, whose angular points are the lower three of the twelve contact points. The three great circles thus drawn cut the horizontal great circle in the first six points. Thus we see that the * In the compound assemblage of two homogeneous assemblages described in the preceding footnote, there are twelve points of contact on each globe, of which nine are placed as those described in the text for the homogeneous single assemblage, and the remaining three are not “at the other ends of the diameters” as described in the text, but are at the opposite points of the small circle on which lie the ends of the diameters referred to. 1888-89.] Sir W. Thomson on Constitution of Matter. 715 twelve points are the intersections of four great circles, which divide the spherical surface into eight equilateral triangles, and six squares ; all with arcs of 60° for boundaries. Fig. 3 shows an orthogonal projection of these circles on the plane of one of them; each an ellipse whose minor axis is J of its major axis. The eight equi- lateral spherical triangles are abc, ahg', bfh' , cgf, a'b'c', ali'g , b'fh, cg'f. The six squares are begh calif , abfg', b'cg'h, calif, a'b'fg. § 51. Draw planes through the centre of the sphere, parallel to the pairs of planes of the angular points of the eight spherical triangles ; these are four planes, the four planes in which the assemblage is found in close triangular order. They are parallel to the sides of the tetrahedron A BCD. § 52. Draw planes through the centre of the sphere, parallel to the pairs of planes of the angular points of the six spherical squares ; these are three planes, the three planes, in which the assemblage is found in square order. They are parallel to the pairs (AB,CD), (AC,BD), (AD,BC) of the edges of the tetrahedron ; and are mutually orthogonal. § 53. Take a cube of the assemblage, having its sides parallel to the planes of § 51. It will present on every side, arrangement of the globes in square order, with rows along and parallel to the diagonals of the square sides of the cube. This I call the primitive cube of Fig. 4. Fig. 5. a homogeneous assemblage of closely packed globes. It is seen in fig. 4 taken from a paper published in Nature (Dec. 20, 1883), by Mr Barlow, who, so far as I know, was the first to show a cubic part of the close-packed homogeneous assemblage of equal globes. § 54. Bevel the corners of the primitive cube perpendicularly to its four line- diagonals as shown for one only of the corners bevelled in 716 Proceedings of Royal Society of Edinburgh. [sess. fig. 5, which also is taken from Mr Barlow’s paper. "We thus get eight equilateral triangular facets, each showing close triangular grouping of the globes appearing in it. The four pairs of planes of these facets are, of course, parallel to the four faces of the tetra- hedron, ABCD. If we make the bevelling of each corner deep enough, nothing is left of the cube but a regular octohedron, whose eight faces are parallel to the eight faces of the tetrahedron. § 55. If in building a triangular pyramid we commence with globes in close triangular order on a horizontal plane, and place the second layer above it over the white dots ( ° ) of the diagram (fig. 6), the third layer over the inner triangle of black dots ( • ), and the fourth a single globe over the centre of the diagram, we build up precisely the portion bevelled off the primitive cube in § 54. Thus we have a triangular pyramid whose three sides are isosceles right- angled triangles meeting at right angles along the three slant edges. The globes in these three faces are in square order. The lines of globes in contact in these faces are parallel and perpendicular to the bounding edges of the base. In the pyramid corresponding to the actual diagram, or any other with an odd number of globes in each 1888-89.] Sir W. Thomson on Constitution of Matter. 717 edge of the base, there are three lines of globes in contact along the lines bisecting the three vertical angles of the sides of the pyramid and ending in the single crowning globe. § 56. If instead of building the second layer as in § 55, we place a second layer over all the black dots ( ° ), a third layer over all the white dots ( ° ), a fourth layer over centres of globes of the first layer, a fifth over black dots ( * ) again; a sixth over white dots ( ° ), and the last a single globe as in § 55, we make an ordinary triangular pyramid having three equilateral triangles for its slant sides and a fourth for base ; and having the globes arranged in equilateral triangular order not only in the base as in § 55, but also in each of the three slant sides. § 57. The ordinary square pyramid of globes has for its base the same square order structure as the slant sides of the triangular pyramid of § 55, while its four slant sides have the same equilateral triangular structure as each of the three slant sides and the base of the pyramid of § 56. If we divide the ordinary square pyramid into four parts by two diagonal vertical planes through its centre, and turn one of these parts over till it rests on its triangular slant side, it becomes the triangular pyramid of § 55. § 58. In considering Baumhauer’s splendid discovery of the arti- ficial twinning of Iceland spar, by means of a knife, published about 22 years ago, soon after Beusch’s fundamental discovery (1867) of the artificial twinning of Iceland spar by pressure, I endeavoured to picture to myself the molecular tactics called into play in the wonderful change of shape thus produced. It was necessary first to suppose known the molecular arrangement in the natural crystal. Two distinct hypotheses presented themselves, each perfectly definite ; and it seems certain that the structure is one or other of these two. Hypothesis (1). — Imagine an equilateral tetrahedron of a close packed homogeneous assemblage of globes. To avoid circumlocution let one of its faces rest on a horizontal plane. Let the whole system be shrunk homogeneously in lines perpendicular to this plane till the originally acute trihedral angle of the triangular pyramid of globes becomes the obtuse trihedral angle of the rhomb of Iceland spar. The shrinkage ratio required to do this would be exactly JS to 1 if the inclination of each slant face to the base were exactly 718 Proceedings of Royal Society of Edinburgh. [sess. 45°* in the triangular pyramid obtained by truncating the obtuse trihedral angle of Iceland spar perpendicularly to the “ axis ” (or line equally inclined to the three edges meeting in the trihedral angle). Hence if, instead of globes to begin with we take oblate ellipsoids of revolution, each having its equatorial diameter J 8 (= 2*83) times its polar axis, and make a pyramid of them by laying a number of them flat on a horizontal plane and putting them to- gether and building others up on them according to the rule of § 56, we have an obviously conceivable structure for Iceland spar. This is Hypothesis (1). This Hypothesis I now find was given 200 years ago by Huyghens in his Traite de la Lumiere (Leyden, 1690), and independently by Wollaston in the Bakerian Lecture for 1812, Philosophical Trans- actions Royal Society for the year 1813, Part 1, but with priority attributed to Huyghens. I had thought of it independently, but did not feel altogether satisfied with it, in the first place because of the great internal commotion which it would imply in the tactics of Baumhauer’s twinning. Then it occurred to me to think of the subject thus. It seems as if the aeolotropic quality of Iceland spar, according to which there are differences of quality for directional actions along and perpendicular to the shortest line-diagonal of the rhomb, may be naturally supposed to depend on the rhomb not being a cube ; and that the change from a cube to the Iceland spar rhomb should be looked to as the cause of the seolotropy. If this is so we must begin with a cube which is isotropic in respect to its four line- diagonals. This is the case with the cube described in § 53, but it is not the case with the cube which we find if in the shrinkage! of Hypothesis (1) we pause at the stage in which the acute trihedral angle of the equilateral tetrahedron is rectangular on its way to be- coming obtuse ; on the contrary, in this configuration each globe is * At ordinary temperatures the angle is 44° 36'’6 (Phillips, Brooke, and Miller’s Mineralogy, § 407); and at temperature 300° it is almost exactly 45°. Huyghens must have taken it as exactly 45°, as he gave \/8 for the ratio of the equatorial to the polar diameter in the statement of his hypothesis. t The shrinkages to pass from the equilateral triangular pyramid to the pyramid with rectangular vertex and to the triangular pyramid for Iceland spar, will be understood in a moment by remarking that the tangents of inclinations of slant sides to base in the three cases are respectively \/8, v% and 1 ; and therefore the distances of vertex from base are as these numbers, the base being unchanged in the simple shrinkage specified in the text. 1888-89.] Sir W. Thomson on Constitution of Matter. 719 an oblate with equatorial diameter twice as long as the polar axis. Hence I have been led to think it probable that the molecular struc- ture of Iceland spar is not that of Hypothesis (1), but is as follows. § 59. Hypothesis (2). — Take a primitive cube § 53 of the assemblage and distort it by shrinkage along any one of its four line-diagonals, with (for simplicity) no change of length in directions perpen- dicular to it. This reduces each globe to an oblate ellipsoid of revolution, and the cube to a rhomb, which is the rhomb of Iceland spar if the shrinkage-ratio is J2 to 1. § 60. Let FG (fig. 7) be one edge and G one of the two obtuse trihedral angles of a rhomb of Iceland spar ; and F', G' the corners Fig. 7. opposite to F, G (so that GrG is the optic axis). Let HKK'H' be a diagonal plane parallel to FG and F'G'; HK, H'K' being parallel to FG, F'G'. How consider rows of oblates parallel to KK' and HH\ The oblates are in contact at the ends of equatorial diameters in the lines of these rows. Turn the oblates of each row round the line of the row, in the half of the assemblage above HKK'H' (suppos- ing this plane horizontal and G to the right) all through equal angles against the motion of the hands of a watch till their equators become horizontal. The assemblage of centres shears to the right (with rota- tion in the direction of the hands of a watch), FG moving rightwards, till the angle between FG and the end face through F becomes a right angle. Simultaneously with this shearing motion there is a shrinkage of the assemblage in the direction perpendicular to the plane HKK'H', entailed by the fact of the equatorial planes of the oblates turning from their primary inclined positions, with equatorial planes perpendicular to the optic axis, to their present horizontal posi- tions. This is most readily seen by confining attention to the single row of oblates which initially had their centres and points of mutual contact in the short diagonal GF' of the right-hand end face. The 720 Proceedings of Royal Society of Edinburgh. [sess. shrinkage of the assemblage perpendicular to the plane HKK'H' implies elongation in lines parallel to FG, because the volume remains constant, and there is clearly neither elongation nor shrinkage perpendicular to the plane of the diagram. Now to fit the tactics of Baumhauer’s twinning by the knife, we must have no change of dimensions of the assemblage in the plane HKK'H'. Hence while the turning and shearing motions described above are taking place, there must be a continual elongation of the sub- stance of each oblate perpendicular to this plane, and shrinkage parallel to FG,* to just such an extent as to prevent the centre of each oblate from coming nearer to the plane HKK'H', but instead to cause all the centres to move in lines parallel to FG. The oblates are now no longer figures of revolution but are ellipsoids with three unequal axes : the shortest, vertical ; the longest, perpen- dicular to the plane of the diagram ; and the mean axis parallel to FG. To complete the process, proceed as follows : — § 61. Turn the oblates farther on in the same direction (opposite to the motion of the hands of a watch, as that in which they were turned in § 60), and through the same angle; and while, in conse- quence, the assemblage of centres shears to the right, give to the substance of each oblate a gradual shrinkage perpendicular to the plane FIKK'H' and elongation parallel to the line FG, so as to cause the rightward shearing motion of the assemblage of centres to be still exactly parallel to the initial position of the line FG. The whole movement of which the first half has been described in § 60, and the second half in § 61, constitutes exactly what is done in Baumhauer’s artificial twinning of an end portion of a prism of Iceland spar, by a knife applied at F, with its edge perpendicular to the plane of the diagram, and pressed against the edge FG of the obtuse angle between the two upper faces of the prism before and behind the plane of the diagram. * Perhaps the simplest way of looking at the affair is found by considering that the elliptic section of each ellipsoid in the plane HKK'H' must remain con- stant ; and so also must the horizontal and vertical axes of the elliptic section in the plane of the diagram. Hence, while the principal axes of the elliptic section turn in the manner described in §§ 60, 61, the ellipse itself must remain inscribed in a constant rectangle of vertical and horizontal sides in the plane of the diagram, while the third axis of the ellipsoid, which is perpendicular to the plane of the diagram, remains constant. 1888-89.] Sir W. Thomson on Constitution of Matter. 721 On the Equilibrium of a Homogeneous Assemblage of mutually Attracting Points (§§ 62-71). § 62. The chief object of this communication is to find the simplest possible way of realising, by means of an assemblage of points act- ing upon one another with forces in the lines joining them, and depending merely on the lengths of the joining lines, an elastic solid which shall not be subject to Poisson’s restriction of the bulk- modulus to be exactly -f of the rigidity-modulus ; but which may on the contrary have, with given rigidity, any magnitude of bulk- modulus through the whole range from - of the rigidity to + oo, shown to be imaginable by Green. That the thing can be done I showed in my Baltimore Lectures (1884), and I gave an easily con- ceived although a somewhat complex way of doing it. I now find that the next-to-the-simplest-possible mode of arranging an assem- blage of points to produce an elastic solid realises Green’s ideal; while the very simplest possible is restricted by Poisson’s limitation. § 63. The simplest possible arrangement of points to make a homo- geneous elastic solid, is a single homogeneous assemblage as defined in § 45 a-d above. In the first place, for simplicity we shall suppose it to be elastically isotropic, or as nearly isotropic as we can make it. § 64. To make the solid as nearly as may be isotropic, the unstrained equilibrium distribution must be the equilateral homogeneous assem- blage of § 21 above. Consider now a finite assemblage containing a very great number of points thus distributed. To take the very simplest possible case, let there be no force exerted between others than nearest neighbours. For the case of equilibrium, no force acts from without on any of the points, whether on the boundary or in the interior; and therefore clearly there is no mutual action between any of the points according to our present supposition of forces between nearest neighbours only. Suppose now the assemblage to he in equilibrium under the influence of forces acting on points in the boundary, giving rise to infinitesimal deviations from the equi- lateral homogeneous grouping. Instead of zero force in each shortest distance, there will now he a force which, for stability of equilibrium, must be pull or thrust, according as the distance is greater or less than that which we had in the zero-equilibrium. 7 22 Proceedings of Royal Society of Edinburgh. [sess. Thus if, to help ideas, we look to a Boscovich curve, the distance between nearest neighbours for zero-equilibrium, which for brevity we shall call £, must be a point in which the curve cuts the line of abscissas with slope corresponding to repulsions for less distances and attractions for greater, and shows zero-force for all distances not less than £ J 2. §65. To investigate moduluses of elasticity, we must suppose the forces applied from without to the points on the boundary to be such as to produce homogeneous strain throughout the assemblage. The working out of this statical problem, to be given in a future communication, shows that the solid so constituted is not elastically isotropic; but that, on the contrary, it has essentially two different rigidities. It is in fact a cubical isotropic body with its two rigidities (article “Elasticity,” . Encyclopaedia Britannica , ninth edi- tion, or vol. iii. of my Collected Papers) not equal. An extension of the investigation to include the supposition of forces not only between nearest neighbours, but between nearest and next nearest neighbours and none farther, gives of course the two rigidities generally not equal; but it allows them to be equalised by a certain definite relation between forces and variations of forces at the two distances £ and £ J 2. Imposing this condition, we have elastic isotropy; and I find the compressibility to be essentially of the rigidity. The solid thus constituted is therefore subject to Poisson’s restriction; and it will no doubt be found that this restriction is valid for any single equilibrated homogeneous distribu- tion of points, with mutual forces according to Boscovich, and sphere of influence not limited to nearest and next-nearest neigh- bours, but extending to any large, not infinite, number of times the distance between nearest neighbours. § 66. Having thus failed to produce a solid free from Poisson’s re- striction, go back to the very simplest case, and try for another way of leaving its simplicity by which we may succeed. Try first to realise an incompressible elastic solid. When this is done we shall see, by an inevitably obvious modification, how to give any degree of compressibility we please without changing the rigidity, and so to realise an elastic solid with any given positive rigidity, and any given positive or negative bulk-modulus (stable without any surface constraint, only when the bulk-modulus is positive). 1888-89.] Sir W. Thomson on Constitution of Matter. 723 § 67. To aid conception, make a tetrahedronal model of six equal straight rods, jointed at the angular points in which three meet, each having longitudinal elasticity with perfect anti-flexural rigidity. These constitute merely an ideal materialisation of the connection assumed in the Boscovich attractions and repulsions. A very telling realisation of the system thus imagined is made by taking six equal and similar bent bows and jointing their ends together by threes. The jointing might be done accurately by a hall and double socket mechanism of an obvious kind, hut it would not he worth the doing. A rough arrangement of six hows of bent steel wire, merely linked together by hooking an end of one into rings on the ends of two others, may he made in a few minutes ; and even its defects are not unhelpful towards a vivid understanding of our subject. We have now an element of elastic solid which clearly has an essentially definite ratio of compressibility to reciprocal of either of the rigidities (§ 27 above), each being inversely proportional to the stiffness of the hows. Now we can obviously make this solid incom- pressible if we take a boss jointed to four equal tie-struts, and joint their free ends to the four corners of the tetrahedron ; and we do not alter either of the rigidities if the length of each tie-strut is equal to distance from centre to corners of the unstressed tetrahedron. If the tie-struts are shorter than this, their effect is clearly to augment the rigidities; if longer, to diminish the rigidities. The mathe- matical investigation proves that it diminishes the greater of the rigidities more than it diminishes the less, and that before it annuls the less it equalises the greater to it. § 68. If for the present we confine our attention to the case of the tie-struts longer than the un-strained distance from centre to corners, simple struts will serve ; springs, such as bent hows, capable of giving thrust as well as pull along the sides of the tetrahedron, are not needed ; mere india-rubber elastic filaments will serve instead, or ordinary spiral springs, and all the end-jointings become much simplified. A realised model accompanies this communication. § 69. The model being completed, we have two simple homo- geneous Bravais assemblages of points ; reds and blues, as we shall call them for brevity ; so placed that each blue is in the centre of a tetrahedron of reds, and each red in the centre of a tetrahedron of blues. The other tetrahedronal groupings (Molecular Tactics, 724 Proceedings of Royal Society of Edinburgh. [sess. §§ 45, 60) being considered each tetrahedron of reds is vacant of blue, and each tetrahedron of blues is vacant of reds.* § 70. Imagine the springs removed and the struts left; but now all properly jointed by fours of ends with perfect frictionless ball-and- socket triple-joints. We have a perfectly non-rigid three-dimensional skeleton frame-work, analogous to idealised plane netting consisting of stiff straight sides of hexagons perfectly jointed in threes of ends. § 71. Leaving mechanism now, return to the purely ideal mutually attracting points of Boscovich. The group is placed at rest in simple equilateral homogeneous distribution : — shortest distance £. It will be in stable equilibrium, constituting a solid with the compressibility, and the two rigidities referred to in § 27 above. Condense it to a certain degree to be found by measurements made on the Boscovich curve, and it will become unstable. Let there be some means of consuming energy, or carrying away energy ; and it will fall into a stable allotropic condition. The Boscovich curve may be such that this condition is the configuration of absolute minimum energy ; and may be such that this configuration is the double homogeneous assemblage of reds and blues described above. Though marked red and blue, to avoid circumlocutions, these points are equal and similar in all qualities. The mathematical investigation must be deferred for a future communication, when I hope to give it with some further develop- ments. * An interesting structure is suggested by adding another homogeneous assemblage, marked green ; giving a green in the centre of each hitherto vacant tetrahedron of reds. It is the same assemblage of triplets as that described in § 24 above. It does not (as long as we have mere jointed struts of constant length between the greens and reds) modify our rigidity-modulus, nor otherwise help us at present, so, having inevitably noticed it, we leave it. 1888-89.] Mr D. M‘ Alpine on Bivalve Molluscs. 725 Continued Observations on the Progression and Rotation of Bivalve Molluscs and of detached Ciliated Portions of them. By D. M ‘Alpine, Esq. Communicated by Dr Sims Woodhead. (With two Plates.) Part II. — In Fresh-Water Mussel ( Unio ). {Abstract.) (Read May 6, 1889.) In the fresh- water mussel the general results are much the same as in Mytilus. Movement of the animal is as a whole right-handed and slightly forward, though this is not invariably the case. The movement appears to he brought about by the contraction and expansion of the foot, which is wedge-shaped, and larger in proportion than in the sea-mussel. When portions are detached, however, the same four parts, or pieces of them, exhibit decided movement as in the sea-mussel. The palps rotate, the gills and mantle-lobes also move, but remarkably slowly, while the ventral margin of the foot is pre-eminently active ; it is par excellence the highly motile detached portion. Where not otherwise stated, the palps, gills, and mantle-lobes are always laid out with their inner surface uppermost, and for convenience of diagrammatic representa- tion, as well as for clearness of explanation, the palps are always supposed to start from the half-round of the clock face. Thus right- handed rotation will have the successive positions of quarter to, hour, quarter past, ending with original position at half past. I. The Labial Palps are somewhat transverse, triangular flaps on each side of the mouth, situated between the anterior adductor. Each pair of polyps, on either side of the mouth, is united along the line of attachment of the body, and this forms a groove between them leading to the mouth ; they are fawn-coloured, and finely striated on their apposed faces. 1. Right Labial Palps, together rotate slowly to the left at different rates, from one round in 6 hours to one in 21 minutes ; there is also slight forward movement. 2. The Inner Palps rotate in opposite directions, the right rotating left-handed, and the left right-handed ; the left inner palp VOL. XVI. 25/11/89 3 A 726 Proceedings of Royal Society of Edinburgh. [sess. "being relatively the weaker, — the right moving at an average rate of about one round in about 1 4 minutes, and continuing for about 10 days,' the left moving considerably more slowly, and continuing for over 7 days. 3. Outer Palps. — The right is very fitful in its movement, and varies considerably in different cases, both as regards rate of move- ment and direction of rotation ; but the typical specimen rotates slowly left-handed, at an average rate of one round in 8f minutes ; this movement going on for 52 hours, the rate of rotation diminishing gradually to one round in 3 hours and 10 minutes. The left invariably rotates right-handed, and the average rate was one round in 5 minutes. 4. Labial Palps generally.— The palps, normally, rotate inwards or towards their attached margin, and both on the same side rotate in the same direction. In some cases abnormal rotation occurs away from the attached margin. The rate varies from 5 minutes to 14J minutes per round. The two outer palps seem to he possessed of nearly equal motive power, while of the two inner, the right is relatively twice as powerful as the left. The duration of movement is usually about a week, hut the right inner continued for 10 days. It must be remembered, however, that when visible or even micro- scopic movement of the palp has ceased, the cilia are still in active motion. The constancy of direction of the palps was well maintained. The right inner and left outer never varied from their normal. The right outer, in four recorded specimens, rotated in its normal left-handed direction, hut one of them (the first too) began right- handed. The left inner also rotated normally right-handed in four specimens, hut one of them on the fifth day was found moving in the reversed direction. It would he interesting to determine how far the rotation of the palps of the Swan Anodon of Britain (Anodonta cygnea) agrees with the above. II. Gills. — The entire right outer gill moves extremely slowly in the direction of its cut surface ( J inch in 5 days), and there is slight rotation of the anterior end. There is no movement of the left gill at all. A piece of the left inner gill in one case exhibited slight forward movement (1 inch in 24 hours), a second moved forwards 727 1888-89.] Mr D. M‘Alpine on Bivalve Molluscs. and rotated through, a quarter of a circle on its posterior end in 2J hours. A piece of the left outer gill rotated round the posterior end, whilst a second portion travelled about J inch in 3 hours. The sluggish movements of the entire gill in Unio contrast strongly with those in Mytilus. It may he that more extended observation will reveal a greater capacity for movement, but the gill certainly does not possess that readiness of movement characteristic of the sea-mussel. III. Mantle-Lobes. — Both right and left mantle-lobes were detached and laid down with their inner or ciliated surface upper- most, but no movement of any kind was detected. Then the right from another mussel was laid down with the outer or non-ciliated surface uppermost, and it moved away gently at once. It progressed in the direction of its cut surface, rotated around its posterior end (the opposite of the right outer gill), and travelled \ inch in 6 hours. Finely powdered charcoal placed upon the inner surface of the mantle was carried towards the free ventral margin and posteriorly. Two * small pieces were taken from the body of the mantle, the one {a) with its inner surface uppermost, the other ( b ) with its outer surface uppermost. The latter was taken from a thin part of the mantle, and moved ; while the former (a), which was thicker, did not move. The movement (b) was exceedingly slow, and could only be registered at distant intervals. In 15 J hours (at night) it had only covered of an inch, but in three hours during the day it travelled an equal distance. In 27 hours from the start it had progressed exactly half an inch altogether. After this it moved laterally as well as forward, so the exact distance traversed cannot be given. A strip of the muscular free margin, about one inch in length, was next detached from the posterior end. The two ends immediately came together, and a coil was formed, but there was no further movement ; while a similar piece from the anterior end remained perfectly still. The muscular margin was then cut up into very small pieces, about of an inch in length. These fragments were observed for several days, and only exhibited a very slight change of position from day to day. There was an evident absence of that motive- 728 Proceedings of Royal Society of Edinburgh. [sess. power which enabled similar pieces of the sea-mussel to roam about in all directions. It will be observed that the mantle-lobe, in whole or in part, moved with its outer or non-ciliated surface uppermost. IV. Foot. — The foot is dark grey in colour and keel-shaped. It is capable of considerable expansion, and possesses great flexi- bility. Within the shell it is wedge-shaped, but when protruded it may enlarge to almost the size of the shell, 2J inches from a shell measuring 2J inches in length. Particles placed on the free ventral margin are carried inwards towards the body and then posteriorly. Although the entire detached foot does not move, the ventral margin is highly motile. A small strip from the anterior end moved forward 1J inch in 5 minutes. A piece detached from the central portion of the margin moved and wriggled about in various directions. Both ends were very sensi- tive, and in its contortions it travelled all over the plate. A strip from the whole margin moved irregularly over the plate, covering 7J inches in 7J hours. Every part of the ventral margin — anterior, middle, and posterior — was seen under the microscope to be richly ciliated. The anterior end of the foot moved in the direction of the free margin, rotating right-handed, 1 inch forward and f inch to the left. It would follow from the ciliary current being inward and posterior on the foot, that it would, when detached, move in the opposite direction, or towards the free margin. It is probable that the direction, when necessary, can be reversed, as indicated in the forward and backward movement mentioned above. And this applies not only to the palps and foot, but possibly to the gills and mantle-lobes ; for Tryon, in his Structural and Systematic Gonchology , mentions a common case of the “ inhalent ” becoming an “ exhalent ” current : — “ If an Anodonta be placed in a vessel of water into which some fine sand is introduced, the particles will be seen entering the incurrent siphon, and repelled from the orifice of the excurrent one; but after the animal has had enough of the unpalatable and irritating food, it will close its valves, forcing out the water, and with it the sand.” 1888-89.] Mr D. M‘ Alpine on Bivalve Molluscs. 729 Part III. — In the Oyster. Ostrea glomerata , Glas.; 0. edulis , Linn., var. purpurea , Hanley. In the oyster the left valve of the shell is thick and convex ; it is on this side that the oyster usually rests. When unattached, however, there is some difference of opinion as to its position. In a recent number of Nature (12th April 1888) a summary of a Report from the British Consul at Baltimore on the Oyster Fisheries of Maryland is given, in which the oyster is said to feed twice a day, always at the still moment preceding the turn of the tide; and at no other time, except when feeding, or rather taking in food, does it open its shell. It feeds on the liquor in the shell, this habit necessitating the convex or left valve being lower, to retain the liquor in sufficient quantity. Even when the shell is closed the liquid inside will be in circulation, owing to the action of the cilia, but with the im- portant difference that the energy of the cilia, under these circum- stances, is probably less, for, as will be fully shown in the sequel, the cilia-bearing part can increase or decrease the expenditure of energy. Professor Semper mentions ( Animal Life , p. 147) having eaten oysters, which although with a salt flavour and bathed with brackish water at high tide, at ebb tide were surrounded with a rapid stream of drinkable fresh water, and opened their shells to it. He gives a drawing of this oyster living in spots where the water is quite fresh, and it is noteworthy that he should find that these oysters opened their valves at the turn of the tide. The most highly prized oysters in New South Wales are likewise those taken from beds where fresh and salt water mingles at certain seasons of the year. The movements of the parts of the oyster may help to explain some of its habits and to show the differences between the lower and upper side. In the following brief account of the three motile parts — palps, gills, and mantle-lobes — as well as of the entire shell-less animal, the rock oyster will always have precedence, since the results were decided and satisfactory on the whole ; the movements of the mud oyster will be added as con- firmatory. Any general description of parts is likewise given from the rock oyster. The rock oyster of Sydney is 0. glomerata , Gould. “The 730 Proceedings of Royal Society of Edinburgh. [sess. rock oysters, although usually known under several different names, are now by most conchologists admitted to he only localised varieties of one and the same species, Ostrea glomerata.” The best oysters are obtained from the shallowest beds, and the very best are dry at low water. The mud oyster of New Zealand, as well as of Australia, is 0. edulis, Linn., var. purpurea , Hanley; 0. chiloensis, Sowerby, is identical with the small form known as the “native” in the London market. I am informed by Sir James Hector of New Zealand, that the common name of mud oyster given in Hutton’s Manual is misleading, and ought rather to be the deep-water oyster of New Zealand, since it is found in from 3 to 12 fathoms of water, and usually on a shelly bottom. Movements of the Oyster as a whole. — A specimen of the oyster laid out in pure salt water, with the gills fully expanded in front and posteriorly, was found to have moved to the left at the posterior end 1 inch in 8 J. hours; while the hooded head, or anterior end, remained relatively fixed, as it is too shut in to allow the free play of the cilia. After a time the gills were drawn .in, and the whole animal shrunk considerably. Labial Palps. — The labial palps are a pair of roughly triangular or fan-shaped bodies of a pale flesh colour lying on each side of the mouth ; the inner and outer on each side passing respectively into the lower and upper lips. When detached its shape is often lost completely. The line of attachment to the body is somewhat inclined to its long axis, and is just a continuation to the angle of the mouth of the line of attachment of the gills. Of the palps together, the left was comparatively inactive, whilst the right exhibited considerable activity. The outer rotated right-handed, taking 42 J minutes for the first round, and an average of 23 minutes for the whole, and 3 minutes for the quickest round. The inner began to rotate right-handed very slowly. It then reversed its direction of rotation several times, always moving very slowly. The left inner palp rotated right- handed, and gradually left the outer palp, which was stationary. The first round took 36 minutes, after which the motion was very slow indeed, and was not recorded. The two inner palps together with left uppermost gave evidence of very slight left-handed rota- tion. Of the right palps, the inner rotates left-handed pretty 1888-89.] Mr D. Mc Alpine on Bivalve Molluscs. 731 constantly, but the rate varied considerably in the different speci- mens. One specimen with the outer surface uppermost began to move bodily to the right and to rotate right-handed; the first round was as usual the slowest, taking 19J minutes, the 30th taking only 3 minutes. Next morning it took 16 minutes. On the 3rd day, 38 minutes; on the 4th, 10 J and 13 J minutes; on the 6th day three rounds took 5, 9|, and 10 J minutes respectively; on the 8th day lateral movement was observed, but no definite rotation and no further movement was noted. Another specimen, with the inner surface uppermost, on the 2nd day was rotating left-handed at a rate of a round in 2J minutes ; on the 6th day in If to 2J minutes, now right-handed; on the 7th day the movement became much slower, and was again left-handed and more irregular, and it now began to move in a straight line. The move- ments ended on the 9th day. Reckoning from the last of the continuous rotation to the last of the recorded rounds, after which there was a little left-handed rotation, the time was exactly 1 day 6 hours 52 minutes, so that the palp had been rotating more or less continuously for nearly 8 days of 24 hours each. On the 10th day, examining under the microscope, the cilia were seemingly as active as ever, but there was no movement of the palp; on the 11th day, however, left-handed rotation was again resumed after renewal of the sea W’ater, and half a round was completed, and the palp had moved § of an inch higher up. On the 12 th day another round was completed, one quarter during the night, the other during the day, that of the night taking about 10 hours, and that of the day about 9 hours. During the night there was little change of position while rotating, but during the day, or last quarter round, there was a deal of progressive move-, ment combined with the rotation. In less than 2 hours the half of the last quarter round was about completed when the palp began to move in a curve in the direction of "the tip, with very little if any rotation for some time. The distance thus traversed varied for different periods of the course. The entire curve measured about 2-J inches, while for a given period of 1J hour the distance was of an inch, and for the last 2 hours 22 minutes it was | of an inch. At the close of the round the palp was almost exactly 1J inch to the left of its first position, and the same lower down. 732 Proceedings of Royal Society of Edinburgh. [sess. On the 13th day the curve of progression was not passed through, and movement ceased altogether on the 14-th day. The right inner palp, with a renewal of water once, was actually engaged in rotation for the space of 11 days, and, if the rotating detached palps of oyster and sea-mussel were kept in constantly moving sea water, it is hard to say how long they might not continue their movements. The day after rotation ceased the palp was examined under the microscope, and the cilia were still in active motion except at the tip end, where the long cilia moved hut sluggishly, and were almost still. Now that rotation had ended the next point to determine was, how long the “ ciliary motion” might last after such a display of energy. On the 15th day, as just mentioned, the cilia were in motion all round, hut most actively in the curved part. Here material was being passed along in the direction of the arrow and thrown off at the anterior end. Later on, this end was found to move a little backward, while the tip end remained fixed. On the morning of the 16th day the anterior end was found to have moved hack nearly inch, while the tip end was stationary. Microscopic appearances explained this, for ciliary motion was now seen to he confined to the curve a, where the cilia were still working actively. On the evening of the same day no further movement had taken place, only a slight movement at the tip end. Under the microscope, the tip was seen to have tucked itself in a little (hence the motion), and the cilia were still active in the curve, more particu- larly towards the very anterior end. On the morning of the 17th day, there was no change of position, and under the microscope no ciliary motion was observed. The cilia stood out like a fringe in the curve, perfectly still ; the cilia had thus ceased to move not all round at once, but in patches, as it were, and as far as observed in the specimen in the following order: — At the tip end first, round the outer margin next, and on the attached margin nearest to the mouth last of all, where the cilia had a powerful appearance. The ciliary motion lasted at least up to the evening of the 16th day, while the rotatory motion ended on the 14th day. And taking the last observation for the ciliary motion, it exceeded the other by 2 days 9 hours. The palp, as it lies in the clear sea water, still retains its colour. 1888-89.] Mr D. M‘Alpine on Bivalve Molluscs. 733 a pale brown, and there are no external indications of disruption or decay. Three specimens were taken from the mud oyster, but the move- ment was quite insignificant, though it was always left-handed, •commencing slowly, and gradually attaining the speed of one round in 2f minutes, after which the movement was gradually lost, the same irregularities as in the left inner palp, though less marked, being still observed. A specimen from a mud oyster exhibited no rotatory motion, but it progressed about \ an inch, and altered its shape most materially. A left inner palp rotated right-handed at once, and after moving upwards for a little, the tip became stationary, and it rotated regularly, the tip being elevated and bent back upon the body of the palp, pointing in an opposite direction to that of the movement. The slowest of fifty rounds was done in A\ minutes, and the quickest in 2 J minutes, the general average giving one round in 3J minutes. At the end of fifty rounds the palp was only § inch lower down, and \ inch to the left of the original posi- tion. This rotation continued for 24 hours. In the mud oyster the movement was very indefinite. In the left outer path the rotation was right-handed and very irregular, but it continued for over 53 hours. Another specimen, tried with the outer surface uppermost, at first moved laterally to the right and with slight rotation to the left ; the speed steadily increasing, the slowest round taking 62 minutes, the last and quickest (when movement was accidentally stopped) 7 \ minutes. Being again started it began to move bodily to the right, and after a little irregular movement this was continued, attaining a speed of one round in 3 J minutes, and an average of one round in 6-J- minutes. This continued, gradually diminishing in speed, until the fifth day, when two complete rounds were followed, taking 14f and 23 J minutes respectively. Shortly after, the palp reached the margin of the water, where it remained. A specimen from a mud oyster moved upwards, and rotated right- handed, but very slightly. During the second and third days it moved very little to the right, without any turning, and then ceased. Palps generally. — When the palps of either side were laid down together they rotated in the direction of their cut margins, and 7 34 Proceedings of Royal Society of Edinburgh. [sess. when the palps were taken separately they still followed the same direction, with the exception of the right outer palp. But it will he remembered that it also rotated left-handed as well as right- handed, with its inner surface uppermost, so that, generally speak- ing, the palps may he said to rotate in the direction of their cut margins. And since the palps rotate generally in this direction, it might be assumed that the cilia work in the opposite way, and this is found to be the case. Thus it is no uncommon sight to see matter sweeping across the palp, whilst it is rotating from the attached to the free margin, there to be got rid of, and this shows unmistakably in what direction the cilia act, at least on that surface. Having settled these two general facts, that the palps rotate usually in the direction of their cut margins, and that the cilia work in the opposite direction, we may now enter a little into detail. Of the two right palps the inner is particularly active and persistent, going at the average rate of 2J minutes per round, and continuing its movements for 13 days, while the outer has likewise a good record of 3 minutes per round. The direction of rotation in both is slightly variable. Of the two left palps, the inner is more active, going at the average rate of 3 minutes per round ; while the outer is not only slower, but the slowest of all the palps. There was here also varia- tion in direction of rotation, and it may be assumed that all the palps are capable of it. The palps on the two sides of the body are thus pretty evenly balanced, as far as movement is concerned, but the advantage on the whole evidently lies with the right side. Function. — In detached palps it is very easy to observe the direction in which matters are carried by them. In the course of rotation it was observed, more particularly in the left inner and left outer palp, both lying as they naturally do with their inner surface uppermost. In the left outer the dirty matter was sent spinning across from the inner to the outer margin, where it formed into slimy threads along the outer edge. There it gradually became parted off from the body of the palp, separating from the tip end first, and latterly it formed a streamer, carried round by the re- volving palp. Ultimately the matter was got rid of entirely, and 1888-89.] Mr D. M'Alpine on Bivalve Molluscs. 735 the same process was repeated as occasion arose. In the left inner, matter was also seen passing rapidly across from the inner to the outer margin, gradually getting loosened from the hody of the palp, and ultimately being thrown off from the tip backward. There was none of that literal shaking off of rubbish met with in the palp of Mytilus, but it separated almost insensibly. In the face of the above facts, it does sound strange to read in a lecture delivered by Professor Huxley at the Poyal Institution, on i( Oysters and the Oyster Question” (English Illustrated Magazine , vol. i. p. 52) — “The anterior ends of each pair of hemi-branchise are attached between the two palps of the side to which they belong. The applied surfaces of the palps, between which lies the commencement of the mouth-cleft, are ridged and richly ciliated, so that anything brought by the ciliary current of the gills is led directly into the oral cavity.” Instead of that, it is generally led directly away from it, and we have already seen that in the sea-mussel it is the same when attached, so that the onus of proof must rest with those who make such a statement as the above, in future. The larval oyster is unprovided with palps, and so it would seem to have its mouth unprotected at an age when such protection was most needed. But there is an apparent substitute for them on the relatively large oval ciliated disc or velum, which overlies the larval mouth. Whatever development may say as to the future of the velum, it probably functions partly in the larva, as do the palps in the adult, i.e., in addition to the locomotive, and pos- sibly respiratory, function, it has the function of guarding the mouth against unsuitable food as it swims about with its velum in front. And thus the suggestion of Loven, that the velum becomes the palps, may at least have a functional basis. It may be suggested that the palps act both as guards and guides to the mouth, seeing that they can vary their direction of rotation, and consequently the direction of the ciliary current, the latter when feeding, and the former at other times. Still the unguarded statement must not henceforth be made, that the cilia of the palps act constantly and mechanically in the direction of the mouth. Since the palps are all capable of reversing the direction Of their rotation, and since a palp (the right inner) has been actually 736 Proceedings of Royal Society of Edinburgh. [sess. observed to send materials towards the mouth end when detached, it is rendered extremely probable that the palps in the oyster exercise the double function of guarding the mouth, and of guiding food materials towards it, at the proper time. Right Inner Gill-Plate. — In the rock oyster the entire gill exhibited only faint indication of movement in the direction of its cut surface. Pieces of the gill-plate of the mud oyster laid on the inner surface moved forward and to the left, then one began to rotate on its anterior end, then the posterior end became the pivot, and in 57 minutes it had returned to its original transverse position, but to the left and about one inch higher up. Placed in its original position again, the movements were exactly reversed. It retained very slight power of movement for five days. Right Outer Gill-Plate — (a) Entire rock oyster. — The move- ments were very irregular and slight, the only definite result being that the gill-plate moved as a whole ^ inch in 6 hours. ( b ) Pieces of the corresponding gill-plate of the mud oyster exhibited no movement of any kind. Left Inner Gill-Plate — (a) Entire rock oyster. — One specimen moved in the direction of the cut surface, moving more at the ends than at the centre. It thus moved forward in this irregular manner Iff of an inch in 4 hours 50 minutes. The free edge travelled Iff inch, whilst the cut edge had worked through only If inch during the same period, (b) Pieces (of mud oyster) gave absolutely negative results. Left Outer Gill-Plate — (a) Entire rock oyster. — Beyond slight movement in the direction of the cut surface, there was no change in position. ( b ) Pieces (mud oyster). A small piece half an inch in length, completed a revolution in 3 hours 1 2 minutes, and moved slightly forward as a whole ; the third and fourth quarter revolu- tions were performed in 30 and 32 minutes respectively. Mantle-Lobes. — Each mantle-lobe of the oyster contracts consider- ably on being separated from the shell, hence on the opened side it is greatly shrunken, but it may be removed from the remaining valve in a fair condition for laying out. Thus the mantle-lobe when detached has not the compact form of that of Mytilus, but is elongated, curved, more or less puckered, and the portion extending from the mouth end, and sweeping round the adductor 1888-89.] Mr D. M ‘Alpine on Bivalve Molluscs. 737 muscle, is much broader than the remainder. The one end will be called the anterior and the other ' the posterior, for purpose of description. The mantle-lobe is attached during life to the inside of the valve. In the right mantle-lobe of the rock oyster there was not the slightest movement, whilst there was only slight movement of the two ends of the left when its lobe was laid down with its cut surface uppermost. It appeared as though the contraction of the centre brought about curling inwards of the two ends. Pieces of the mascular margin about a quarter of an inch in length can turn on themselves and change their position to a very slight extent. If the movements of the three forms, Mytilus, Unio, and Ostrea, be compared in their natural condition, Unio possesses the greatest activity, and Ostrea, as far as known, the least ; but if the progres- sive and rotatory movements due to cilia are in question, then Mytilus undoubtedly takes the lead. Unio, with its relatively large keel-shaped foot, can move along, either upright or on its side, in a way that neither Mytilus, with its byssus-secreting foot, nor Ostrea, without a foot at all, can approach ; but when the animal, divested of its shell, is placed in its native element, there is, perhaps, contrary to expectation, a power of progression and rotation in Mytilus only slightly shared by Unio, and almost absent from Ostrea. The general table will show at a glance the contrast between the nature, direction, and rate of movement in each of the various parts ; and it will likewise be noticed that each of the three forms has a distinct and specially active part, suggestive of underlying differ- ences. There is the gill in Mytilus, the ventral margin of the foot in Unio, and the labial palp in Ostrea. In Mytilus, no doubt, the palps are specially active, but they are eclipsed in rapidity and readiness of movement by the gills, so that the latter form the more prominent motile parts. Palps. — The palps have all a combined rotatory and progres- sive motion, normally, in the direction of the cut margin. As between Mytilus and Unio, the main difference lies in the two fellow palps on either side of the former, rotating in opposite direc- tions, and in the latter in the same direction. Ostrea may be said to agree with Mytilus, only the left outer or lowermost palp rotates 738 Proceedings of Royal Society of Edinburgh. [sess. like its fellow, or it may be said to agree with Unio, in the palps of both sides rotating alike, only the right outer or uppermost can turn either way, by preference, the opposite to that of Unio. The rate on the whole is in favour of the oyster, the general average being 3J- minutes per round ; next comes Mytilus, with a general average of 4 ; and lastly Unio, with a general average of 8J. It will be remarked that the average of the corresponding palps agree pretty closely in Ostrea and Mytilus, and that it is the left outer which is the slowest in both. Gills. — As regards the movement of the gills in their entirety, . there is hardly any comparison between those of Mytilus and the other two. All agree indeed in moving, and in moving in the direction of the cut margin, but beyond that they have little in common. The noticeable feature in the gill of Mytilus is its readiness, when detached and laid out, to move away ; and so energetic is it that even the anatomists have not failed to notice the swimming movement of parts of it, and as far as known to me,, this is the only mollusc in which such has been noticed. It is the only part of any mollusc examined so far which can be absolutely relied upon to move steadily and immediately when detached, hence its great suitability for investigations concerning cilia and their movements. The palps as a rule could be trusted to rotate after a longer or shorter resting period; but it was noticeable in the oyster, that of the two fellow palps detached from either side, only one of them usually rotated, so much so, that after spending much time at first in detaching single specimens, I got into the habit latterly of detaching and laying out two together, with the sure and certain hope that one of them, at least, would not disappoint me. The average rate of movement for the gills of the sea-mussel is 2 minutes per inch for those of the fresh-water mussel ; there is no general average, but -J- inch done in 3 hours; and for those of the oyster the average of about 13 minutes per of an inch, or 85 minutes for one measured inch; or, to express it in a comparative way, 1 inch is traversed by the gill of Mytilis, Ostrea, and Unio in 2 minutes, lj hours, and 24 hours respectively. As there was no definite progressive movement of the entire lobe observed in the oyster, the comparison lies between the sea and fresh- 1888-89.] Mr D. M‘ Alpine on Bivalve Molluscs. water mussel. There was rotation as well as progression in both, but if we compare the latter, then the mantle-lobe of Mytilus has the decided advantage. That of Mytilus went at the rate of 1 inch in 50 minutes, and that of Unio an inch in 24 hours. The foot of Mytilus is so entirely different from that of Unio that comparison will not be attempted. Suffice it to say, that the slow-moving, definite- shaped, steadily directed foot of Mytilus, when attached, is a perfect contrast to the same when detached, while the free margin of the foot of Unio appears to possess all the activity of the keel-shaped mass. The cilia of the sea-mussel are almost instantly arrested in their movements by fresh water, just as those of the fresh- water mussel are by salt water, but it is interesting] to note that each may be gradually accustomed to the changed conditions. Professor Semper, in his Animal Life , mentions the case of a Unio living within reach of the flood tide, of the sea-mussel in perfectly fresh water, and of the edible oyster in brackish water. This may help to explain how the cilia originally became gradually adapted to salt water in the one mussel, and fresh water in the other. There are still two important considerations suggested by the pre- ceding observations — the resting of the cilia and the relative and absolute rate of movement of detached parts. It is pointed out that the cilia are supposed to continue their work without any rest, and it may be conceded that microscopic examination goes to prove this, but from the observations made on the variation in direction of the movements of pieces of gills, &c., within very short periods, such can scarcely be the case ; and it must be argued that different series of cilia are brought into play at different times. It may be imagined, in a structure like the gill, with its innumerable cilia, that they rest in relays without inter- fering much, if at all, with the general effect ; and I have often had to point out, as in the rotating palps of the oyster, that in the course of long spells of rotation they “ rested half a minute ” or so. To the observer this looks exactly like “ stopping to take breath,” or the wearied rower laying down his oar for a minute. In the course of these investigations an important distinction was noticed between the action of the cilia and the movement of the cilia-bearing mass. The movement of the mass might cease, and 740 Proceedings of Royal Society of Edinburgh. yet the cilia themselves, when examined under the microscope, would be in active motion. The cilia in themselves are thus not the cause of movement ; there has to he co-operation or co-ordination of some sort before the ciliary motion can give rise to movement of the part bearing the cilia. There are, therefore, two motions con- nected with cilia to be distinguished — one, the ordinary so-called “ ciliary motion,” which creates currents in the liquid, and keeps up a constant stream ; another, which may be called ciliary motive- power, which is sufficiently powerful to move the cilia-bearing mass. The practical bearing of the distinction is evident in the investiga- tion of the action of drugs, &c., upon ciliary movement. It will be necessary in future, not only to determine what arrests ciliary motion, but also what affects their motive power, apart from their own proper movement. The absolute and relative rate of movement of the detached parts is a subject replete with interest, and the movements of the gill of the sea-mussel, slow as they may seem when compared with the dashing Infusorian, can actually hold its own; but, as pointed out by JSTageli, quoted by Sachs in his Lectures on the Physical Plants — c Whether the movements of a body appears to us rapid or slow, however, depends also on the relation between its size and the space passed over in a definite time. If an elephant and a mouse travel an equal distance in the same time, we call the first slow, the second quick. A man in walking passes over somewhat more than half his length in one second. The most . rapid swarm-cell travels, in the same time, a distance which is 2J times as great as its diameter. Judged by this standard, the gill of Mytilus only traverses its own height in about a minute, and so far is relatively slow. In both the Odontophora and the Lamellibranchiata the cilia on the velum cause a rotation of the embryo within the egg-capsule, but it is a curious fact that the cilia do not always act. Thus on the development of Sepia, it is noted that “ the whole embryo now be- comes ciliated, though the ciliation does not cause the usual rotation; while, in such a closely allied form as Loligo, it does occur. Loligo differs mainly from Sepia in the early enclosure of the yolk by the blastoderm and in the embryo exhibiting the phenomena of rotation within the egg-capsule so characteristic of other Mollusca.* The * Balfour’s Embryology , p. 247. 741 1888-89.] Mr D. M‘ Alpine on Bivalve Molluscs. ciliated non-rotating embryo of Sepia recalls the cases of the non- rotating ciliated detached parts, even although the cilia themselves are in active motion. After these general references, the rotation of the embryos of Mytilus, Unio, and Ostrea will now be considered. The develop- ment of Mytilus edulis, L., has been recently and specially studied by J ohn Wilson, Demonstrator of Zoology, University of St Andrews, and he has found that cilia cover the greater part of the surface of the embryo, causing it to rotate actively, but no idea is given of rotation.* * * § In Anodonta and Uhio there is likewise rotation of the embryo, j In Unio litoralis, wrhen the rotation is most active, 7 or 8 revolutions are said to be observed per minute,! but this is presumably, as seen under the microscope, and therefore of no value without the magni- fication. In Anodonta intermedia the rotation is at the rate of from 4 to 1 rounds per minute (15 to 79 seconds), § but the same remark probably applies. Although the development of the oyster has been recently and carefully studied, there is no mention of rotation of the embryo within the egg, although there is a double oval ring of cilia. Probably it does occur, as in the larva of Cardium, which it other- wise closely resembles. The rotation of the embryo in the ovum of the frog is described as being from right to left, at a rate of from 5 to 1 2 minutes per round (Pfluger’s Archiv , 1870, Heft 2 and 3). Even at the highest rate, it is slow, as compared with some of the detached parts already con- sidered, and with the embryo of Unio, for instance. Eor proper comparison, however, the relative sizes would require to be taken into account. The rate of rotation, so seldom given, is worthy of attention, particularly in a case like Mytilus, where the adult still retains a power only possessed by the embryos of higher forms. The movements of the detached parts of the three chosen forms have thus yielded valuable results, and to a certain extent these forms are representative for our present purpose. There is the free * First Annual Report of Scottish Fisheries Board. t Balfour’s Comparative Embryology , vol. i. p. 266. + Owen, Lectures on Invertebrate Animals , p. 526. § Bronn’s Thier-reich. VOL. XVI. 21/1/90 3 B General Table for Comparison of Results as to Direction and Rate of Movement. 742 Proceedings of Royal Society of Edinburgh. tr a 03 pH a a o o ?H t-P e£ H|C3 2 PH ~ ~ " . . r— 1 ta o CO ■ : .a ■ rP a C3 a H rP '&C o CO o . R 6 per r ? ? > ? " o > . c3 . . . .a pH . 03 • © a Pi a *3 U> zn a : £ 2 : : rP g : ■ Ph : 03 Ph . ra 2 -R .a a s SrP a o a 2 .3 co ,a H|(M «|iO -w ©NTUfl 00 »H|CD -sH 30 1 — 1 PR ■d a rP* a rP" © • o O .a P cd Rl „ .a Ph *-3 03 ~ ~ ~ Ph 03 03 Ph ~ © ; P 03 03 ’ P % R w a .a a a a a o h|in h|o>h1« ^ H|* OR t— l CR OR OR O i-i Ph 4 m p4 p4 WKWW 4 4 p4 4 *73 73 03 <73 Tj ^ S S3 73 cjRRrR ci rR+l^rR | 'i r P rP H-H Xn-1 3 752 Proceedings of Royal Society of Edinburgh. [sess. pour la seconde suite, par Voi Vu • • • • J Vn- 1 j pour la troisieme suite, par ^0’ zi zn- 1 > etc., . . . . • et soit f • • • • J »«-!■ j 2/o> 2/lJ * • • • 1 Vn- 1 ; 20» 21» * * * * J ^»-l >•••*) une fonction donnee de ces divers termes. Si a cette fonction Ton ajoute toutes celles que l’on peut en deduire, a Taide d’un ou de plusieurs echanges operes entre les lettres y,z}. . . . prises deux h deux, chacune des nouvelles fonctious etant prise avec le signe + ou avec le signe - , suivant qu’elle se deduit de la premiere par un nombre pair, ou par un nombre impair d’echanges; le- resultat de cette addition sera une somme alternee par rapport aux suites dont il s’agit.” It is a little unfortunate that this definition proceeds on different lines from the others, being rather indeed a rule for the formation of an alternating function with respect to several sets of variables than a definition of such a function. It would have been much more appropriate and instructive to have said that a function was called alternating with respect to two or more sets of the same number of variables when the interchange of each member of a set with the corresponding member of another set altered the function in sign merely. Examples like the following could then have been given to make the two usages of the term perfectly clear, and to show the exact relation between them. To illustrate the first usage, the expressions ac-bc , (< a - b){c - d) , (a - b)(a - c)(b - c ) , might be taken, where ac - be is an alternating function with respect to the variables a, b ; (a - b)(c - d) an alternating function with respect to «, b, and also with respect to c, d ; and (a - b)(a - c)(b - c) an alternating function with respect to «, b , with respect to a, c, and 753 1888-89.] Dr T. Muir on the Theory of Determinants . with respect to b, c, or, shortly, an alternating function of all its variables. On the other hand, the expressions a2b - c2d , ab -c d , would illustrate the second usage ; a2b — c2d being an alternating function with respect to the sets of variables ab, cd ; and ab - cd an alternating function with respect to the sets ab, cd, and also with respect to the sets ae, bd. In a word, the alteration which produces change of sign is, in the case of the first usage, interchange of two individual elements ; in the case of the second usage it is interchange of two ranks or sets of elements. The entity to which the new name somme alternee is given is explained as follows (p. 1 60) : — “ Soit f(x, y, z, ... .) une fonction quelconque de n variables x, y,z, ... . et ajoutons a cette fonction toutes celles qu’on peut en deduire par la transposition des variables, ou, ce qui revient au meme, par un ou plusieurs echanges operas chacun entre deux vari- ables seulement, chaque nouvelle fonction etant prise avec le signe + ou le signe - , suivant qu’elle se deduit de la premiere a l’aide d’un nombre pair ou impair de semblables echanges. La somme s ainsi obtenue sera la somme alternee que nous representons par la notation S [±%, y,z,... )]. Ou trouvera, par exemple, en supposant n = 2, s = f(x, y) - %, x)] en supposant n = 3, s = i{x,y,z) - i(x,z,y) + i(y,z,x) - f(y,oc,z) etc.” + Kwj) - The only matter now remaining for explanation is the mode of transition from sommes alternees to resultantes, the difficult point 754 Proceedings of Royal Society of Edinburgh. [sess. being, as in the memoir of 1812, to include all kinds of the latter as special cases of the former. The two pages which Cauchy devotes to the subject are curious to read, and deserve a little attention. He says (p. 161) : — “ Concevons maintenant que la fonction Kx>y& • • • ) se reduise au produit de divers facteurs dont chacun renferme une suite des variables x, &«,•••■ en sorte que l’on ait, par exemple, i{x,y,z, . . . ) = ^{x)x{y)^(z) .... alors, pour obtenir la somme alternee *= s[±#%G#(z) • . • ] il sufhra ...” and having shown the mode of formation, and given the examples s — (x)x(y) ~ 4>{y)x(x)’ s=4,{x)x(y){x)x(z)'l'{y) + • • • he adds “Les sommes de cette espece sont celles que M. Laplace a designees sous le nom de resultantes.” In regard to this the first comment clearly must be that it is not a little misleading. The sums referred to are only a very special class of those functions which Laplace called resultants ; they belong, in fact, to that peculiar type for which in later times the name alternant was coined. In the second place, Cauchy’s virtual renun- ciation of his own word “ determinant ” must be noted, — a renuncia- tion all the more curious when we consider that the word had now been adopted by Jacobi, and had thereby become the recognised term in Germany. It may be that Laplace’s word “ resultant ” had proved more acceptable in France, and that Cauchy merely bowed to the fact; but there is little or no evidence to support this.* * Liouville, in a paper published in the same year as Cauchy’s memoirs, uses resultant, but adds in a footnote, ‘ ‘ Au lieu du mot rlsultante, les geometrgs emploient souvent le mot determinant ” {Liouville' s Journ., vi. p. 348). 1888-89.] Dr T. Muir on the Theory of Determinants. 755 In the paragraph following the above Cauchy proceeds, as it were, to rectify matters. He says (p. 162) : — “ Les formes des fonctions designees par $(x)> x(x\ etc- etant arbitrages, aussi bien que les variables x,y,z,. . . , permettent aux divers termes qui composent le tableau (2) d’acquerir des valeurs quelconques, et repr^sentons ces vari- ables a 1’aide de lettres diverses xfy,z, .... ,t affect4s d’indices diffdrents 0, 1, 2, . . . , ra-1, dans les diverses lignes verticales. Alors, au lieu du tableau (2), on obtiendra le suivant if* w nt* o/* '*'2’ * ‘ '• • 3 1 Vm Vv Vv • • • * > Vn-i (5) 1 4 • • • • r 4-1 A> ° ' 5 4-i et la resultante s des termes dans ce dernier tableau sera ^ = S[ + ^o2/i4* • • 4-i] • The general determinant is doubtless here reached, but the transition requisite for the attainment of it, viz., from x(x)> . to the perfectly independent x0, xv x2, .... is not made without considerable strain. This is all the more surprising, too, when we consider, that a much less troublesome and less objectionable mode of bringing determinants under alternating aggregates lay ready to Cauchy’s hand. Bearing in mind the definition given above, of fonctions alternees par rapport a diverses suites , we see that a determinant of the %th order could have been made to appear as an alternating function with respect to n ranks of n variables each. For example, the determinant 756 Proceedings of Royal Society of Edinburgh. [sess. could have been introduced as a function alternating with respect to any two of the three ranks, a§x + b^y + c^z = £ J as giving each of the three variables £, y, £, in terms of the other three a?, y, z, we see that on solving for x, y, z , we obtain a con- verse system, that is to say, a system giving each' of the three x , y, 0, in terms of y, £. The latter system is, as we know, A, , -A.o -A-Qc, X=A(+ A71 + B, . B„ B,. y=^+^v + -ft. Ci> , C2 , c3? z=-££+ a’+ where A is the determinant of the original system, and -A-l> ®1> Ql» * * * * are the cofactors in A of av bv cv a2, .... f respectively. Multi- plying the determinants of the two systems, we obtain the determi- nant of the quantities 1 0 0 0 10 0 0 1 . Hence (p. 176) : — - “ Si, n variables #, y, z, ... ,t, etant liees k n autres variables x, y, z, . . . , t, par n equations lineaires, on suppose les unes exprime es en fonctions lineaires des autres, et reciproquement ; les deux r^sultantes formees avec les coefficients que renfermeront ces fonctions lineaires dans les deux hypotheses, offriront un produit equivalent a l’unite.” (xxi. 6) VOL. xvi. 21/1/90 3 c 758 Proceedings of Royal Society of Edinburgh. [sess. Retrospect of the Period, 1813-1841. The characteristics of this period are best brought out by com- parison with those of the preceding period, it being carefully borne in mind, in making the comparison, that the two are markedly unequal in length, the period of pioneering, as we may term it, extending to 120 years, and the next to only about 30. In the first place, then, the evidence shows that as time went on there was considerable increase of interest in the subject, and a more widely spread knowledge of it; for, whereas to the longer period there belong 20 papers by 13 writers, for the shorter period the corresponding numbers are 35 and 18. Among the 18 writers, too, are represented nationalities which had previously not put in an appearance, viz., English, Italian, and Polish. In the second place, we have proof that the early period was by far the more fruitful in original results. The pioneers had mapped out most of the prominent features of the new country; their successors had consequently to concern themselves in a considerable degree with filling in the details. During the second period one finds the fundamental propositions of the first period reproduced in new varieties of form ; also, there are not awanting new proofs, extensions, and specialisations of old theorems; but of absolutely fresh departures there are comparatively few. An examination of the results numbered xlv.-lviii. will show the character of these departures. It will be seen that they are due to Desnanot, Scherk, Schweins, Jacobi, Sylvester, and Cauchy. The most notable name of the period is Jacobi’s, and next to it that of Schweins. There is no one name, however, which stands out in this period so con- spicuously as Cauchy’s does in the first period. Sylvester, unlike the others, it must be remembered, was only beginning his career, and we have yet to see him in the fulness of his power. In the next place, the second period contrasts with the first in that during it important work was done on the subject of special forms of determinants. Here, again, the noteworthy names are those of Jacobi and Schweins. Lastly, it having been noted in the retrospect of the first period that the subject of determinants was almost entirely a creation of the French intellect, we must not fail to take cognisance now of the 1888-89.] Dr T Muir on the Theory of Determinants. 759 fact that in the second period the pre-eminence belongs to Germany, France, however, taking still a fairly good second place. CAYLEY (1841). [On a theorem in the geometry of position. Cambridge Math. Journ ., ii. pp. 267-271 ; or Collected Math. Papers , i. pp. 1-4.] Of the two English mathematicians whose names are inseparably associated with the development of what has been called Modern Higher Algebra , Sylvester, as we have seen, was the first to direct public attention to the functions then partially known as deter- minants, but called by him in the heat of supposed discovery “zetaic products of differences.” Cayley it was, however, who gave the great impetus to the study of them — an impetus due to two different causes, the choice of an exceedingly apt notation and the masterly manner in which he put the functions to use. How he obtained his knowledge we know not. It may be that Sylvester’s two early papers had directed his attention to the matter, and that he had then read some of the authors who preceded Cauchy ; but, whether this be true or not, it is certain that by his own inde- pendent research he had attained in 1841 a powerful and compre- hensive grasp of the subject. The little paper to which we have now come is ample evidence of this. A peculiar interest attaches to it also, as being the first fruits of Cayley’s genius, the earliest of that long and varied series of papers which has done so much to extend the bounds of pure mathematics.* With characteristic directness and concision he opens as follows : — “We propose to apply the following (new 1) theorem to the solution of two problems in Analytical Geometry. * In a strictly chronological arrangement Cayley’s paper would not follow, but precede the papers of Craufurd, Cauchy, and Jacobi of the same year. It was published in February : Cauchy’s note was presented to the Academy on 8th March, and Jacobi’s memoir bears the date 17th March, though not pub- lished for more than two months afterwards. As Cayley’s first appearance, however, marks the beginning of a newT epoch, and as the other papers referred to belong by their character to the preceding epoch, a slight deviation from the chronological order seems warranted. 7 60 Proceedings of Royal Society of Edinburgh. [sess. a , “ Let the symbols a, ft a', P' a> P, 7 a , P\ y tt a ,P',y" &C. (VII. 10) denote the quantities a P' - a'P , ap'y" - a/3"y' + a P"y - a Py" + a ' py - a "P'y , &C. the law of whose formation is tolerably well known, but may be thus expressed, a , p a , P = a\P'\ - a \P\, a } P , y a, P’, y tt r\tt tt a y P , y p. y \P'> y" + a + a" A 7 /3", tt 7 1 ft 7 P> y &C. the signs + being used when the number of terms in the side of the square is odd, and + and - alternately when it is even. Then the theorem in question is p a + a P + t y ... y p a' + c j8' + T y ... , p a" + v P" + r y" . . . p a + cr P + t y . . . , p a + or' P' + r y . ■ . , p a" + (x), but unable to symbolise the differential coefficients of such a special function as ax 3 + bx 2, or log (1 — a?)/(l + 3?) would be in the exact predicament of the theory of deteminants prior to Cayley. Of less importance is the fact, which the quotation indicates, that Cayley had discovered for himself the multiplication-theorem, but characteristically hesitated to proclaim it new : also, that, probably following Yandermonde, he took the recurrent law of formation for his definition, making the signs all + in one case and + and - alternately in the next, exactly as Yandermonde did. He then proceeds to the seemingly geometrical problem : — “ To find the relation that exists between the distances of five points in space. “ We have, in general, whatever aq, ylrzv &c., denote, 762 Proceedings of Royal Society of Edinburgh. [sess. xi + Vi + z\ + wi2> ~ 2xv ~ 2Sv ~ 2zj xf + yt + zf + wf, -2*2, -Zy* - 2s, + + + -2*6, -2 yw 1 , 0 , 0 , multiplied into '2%. 0 , - 2 wv 1 -2w2, 1 - 2 W, 0 , 0 1, XV Vx> zi> wi> x\ + V\ + zi + w\ 1, X2, ?/2, 22, W2, £22 + 2/22 + *22 + W22 1, 2/5> V W5> *52 + ?/52 + 252 + “,52 o, o , o , o , o , #1-*1 +01-^1 +Zl-*l +M>1 -Ml , ^1-^2 +—, ^1-^3 +•••> *l—#4 +•••» ^1-^5+—. 1 ^2—^i + . . , X2~X2 +..., X2-X3 +..., X2 — X± + ..., x2-x5z+. ^5-^1 + • , x5-x2 +..., X5-X3 +..., x5-x5 +..., 1 , 1 1 1 1,0 Putting the w’s equal to 0, each factor of the first side of the equation vanishes, and therefore in this case the second side of the equation becomes equal to zero. Hence xvyvzv x2,y2iz2, &c., being the coordinates of the points 1, 2, &c., 2 - 2 situated arbitrarily in space, and 12 , 13 , &c., denoting the squares of the distances between these points, we have imme- diately the required relation 0, l22, I32, 1 42, l52, 1 2T2, 0, 232, 242, 252, 1 3l2, 322, 0, 3i2, 352, 1 £I2, 422, 432, 0, 452, 1 5l2, 522, 532, 542, 0, 1 1 , 1, 1, 1 , 1 , 0 0, 763 1 888-89.] Dr T. Muir on the Theory of Determinants. which is easily expanded, though from the mere number of terms the process is somewhat long.” Than this no better example could have been chosen to illustrate what has just been said above regarding the great advantages of Cayley’s notation. As is well known, the result arrived at had been given in forms, lengthy and forbidding, many years before by Lagrange and Carnot. What Cayley did was to rob it of all disguise, by expressing it as the vanishing of an elegantly formed determinant ; and secondly, to show that the said determinant vanished because it was eight times the square* of another deter- minant whose zero character could not be overlooked. As has been implied, the result is purely algebraical, its geometrical character only appearing when x , y, z are taken to denote the coordinates of a point. The corresponding identities for the cases of four points in a plane and three points in a straight line are given ; and the latter of the two is most interestingly shown to be deducible also from the general theory of elimination. This is done as follows : — “ Let xu - xtit = a, xtn - xt = /3, X,-Xn = y; then 122 = y2, 232 = a2, ^2 31 =/32, and a + fi + y = 0 ; from which a, (3, y are to be eliminated. Multiplying the last equation by (3y, ya, a/3, and reducing by the three first, O.a -f 122./3 + f22.a +• 0./3 + -f a + (3 312.y + 312.a 23 + 23 .y + 0 .y + y + from which, eliminating a, j3, y, aj3y by the general theory of simple equations a/?y = 0, a/?y = 0, a/3y = 0, O.a (3y = 0; 0, si*. 312, 1 , 122. 0. 322, 1 , 132, 232, 0, 1 . 0.” The first factor being 16 times the second, and the w’s unnecessary. 764 Proceedings of Royal Society of Edinburgh. [sess. The conviction that the identity ought to come out as a result of elimination, and the ingenious fulfilment of it by using the identity a + /? + y = 0 after the manner of Sylvester’s paper of 1840 are very noteworthy. It is finally noticed that “ the additional equation that exists between the distances of five points on a sphere ” can be similarly obtained, and the process is given. GRUNERT (1842). [LTeber die Theorie der Elimination. Archiv der Math. u. Phys., ii. pp. 76-105, 345-377.] This paper, extending to more than sixty pages, is little else than an amplified reproduction of work by Cauchy. Nine pages at the beginning concern simultaneous linear equations; the rest is entirely taken up with the various modes of eliminating x between two algebraical equations, <£(#) = 0, \]/(x) = 0. In the former part, which seems based on the third chapter of the Cours d’ Analyse, the only fresh matter is a lengthy proof of the proposition that the difference-product of any number of quantities changes sign when two of the quantities are transposed. It will suffice to note in regard to it that the so-called inductive method is followed, and that two cases have to be considered, viz. (1) when the new quantity is not one of the two which are interchanged, (2) when it is. (hi. 38) The second part follows closely Cauchy’s memoir of 1840. TERQUEM (1842). [Notice sur l’elimination. Formules de Cramer. Nouv. Annates de Math., i. pp. 125-131.*] This is merely a simply written exposition of Cramer’s rule, and of Bezout’s rule of 1779, and contains nothing noteworthy. It is curious, however, to observe the reason given for directing attention to Cramer’s rule, — “ Comme ce procede ne se trouve * The continuation intimated at the close (p. 131) was never made. 1888-89.] Dr T. Muir on the Theory of Determinants. 765 decrit, que je sache, que dans un seul onvrage elementaire frangais, peu repandu ( Manuel d'Algebre , p. 80, 2e edition, 1836).” This indicates a sad contrast to the state of matters attested to by Gergonne,* showing that there is a fashion which changeth even in things mathematical. The new favourite, it also appears, was Bezout’s rule of 1764 ; for in passing this over, in order to give an account of the same author’s rule of later date, Terquem says in regard to it, “ Comme ce procede est decrit dans tous les ouvrages a l’usage des classes, nous ne nous y arreterons pas.” CAYLEY (1843). [Demonstration of Pascal’s Theorem. Cambridge Math. Journ., iv. pp. 18-20 ; or Collected Math. Payers, i. pp. 43-45.] At the outset of this paper two lemmas are given, the second of which stands as follows : — “ Lemma 2. Representing the determinants Vv *^2> Vv X8> Vv by the abbreviated notation 123, &c. ; the following equation is identically true : 345. 126 - 346. 125 + 356 . 124 - 456 . 123 = 0. This is an immediate consequence of the equations % «4, X6 xv xv XQ * yv y# y5> y6 • • y*> y « 2/5> y 6 • • h. Z5> Z6 *5> Z6 xv x2, *5> x6 xY, x2’ * Vv Vv Vv Vv y& y6 Vv> y» • • h> 22> *5> Z6 zi> z2'> • . . (xxiii. 13) * The passage in question, which we quoted under Cramer, is to he found in the Annales de Math., xx. p. 45. 766 Proceedings of Royal Society of Edinburgh. [sess. The identity is readily recognisable as Bezout’s (1779). The mode of arriving at it, however, is fresh, and worthy of every attention. The determinant of the sixth order on the left is shown to he equal to zero ; and it is implied that the identity is got by transforming the said vanishing determinant into an aggregate of products of pairs of determinants by means of Laplace’s expansion-theorem. The method is far-reaching in its application, and manifestly Cayley could have used it to produce a host of identities of similar kind. The equatement of the two determinants of the sixth order deserves also to be noted, and may be taken as evidence that Cayley was familiar with the theorem that a determinant is not altered if each element of one row be diminished by the corresponding element of another row. No such theorem had been formulated or used before his time. (lix.) Lastly, it may be pointed out that we have here the first instance of a practice which afterwards became very general, viz., putting a dot instead of a zero element when writing a deter- minant. The other lemma and the main body of the paper are geometrical ; but as an important determinant identity is implicitly established in the course of the investigation, and as it is of the greatest histori- cal importance to make evident the wonderful command which Cayley with his new notation had suddenly obtained over deter- minants, we shall give the full text of these portions also, at least up to a certain point. “ Lemma 1. Let U = kx + ¥>y + Cz = 0 be the equation of a plane passing through a given point taken for the origin, and consider the planes U^O, U2 = 0, U3 = 0, IX4 = 0, U5 = 0, U6 = 0; the condition which expresses that the intersections of the planes (1) and (2), (3) and (4), (5) and (6), lie in the same plane, may be written down under the form * * The commas which Cayley prints after the elements in a determinant we omit here and henceforth. 1888-89.] Dr T. Muir on the Theory of Determinants. 767 Aj A2 Bj B2 c, c2 B3 B4 . c3 c4 . A3 A4 A5 Ag B3 B4 B5 B6 C3 c4 C5 Cg “Consider now the points 1, 2, 3, 4, 5, 6, tlie coordinates of these being respectively xv ylt zv , x6, yQ, z6. I represent, for shortness, the equation to the plane passing through the origin, and the points 1, 2, which may be called the plane 12, in the form x 12* + y 12, + z 122 = 0; consequently the symbols 1 2Z , 1 2, , 1 22 denote respectively ?/iZ2 ~ y& , zi%2 ~ z2xi j X\V* ~ > an(i similarly for the planes 13, &c. If now the intersections of 12 and 45, 23 and 56, 34 and 61 lie in the same plane, we must have by lemma (1) the equation 12, 45, 23, 56x 12, 45, 23, 56, 12. 452 23. 56a . . 23, 5 6* 34, 61, 23, 56, 34, 61, 23, 56a 342 61* Multiplying the two sides of this equation by the two sides respectively of the equation Xq x1 x2 Ve V\ y-> • - ■ % «l 2, . • • . . . #3 x4 x5 • • • y3 y± y§ • % S4 Z5 = 612 . 345, 7 68 Proceedings of Royal Society of Edinburgh. [sess. and observing the equations xqV2x + y6Uy + z612z = 6l2 , 112 = 0, »' The method is thus seen to consist in the deduction of a new equa- tion by addition, and in the elimination of all the unknowns, except one, from the equation, by multiplying both sides by the product of the coefficients of the other unknowns, — the multiplication in question being “ outer,” and for the purposes of the multiplication, any two coefficients of one and the same equation being considered as “ like,” and any two belonging to different equations as “ unlike.” For example, in the case of n = 3 we have x _004-ao + Co).(g2 + &2 + <;2).(a3 + 63 + C3) 1 («! + + cj . (a2 + b2 + c2) . (a, + b3 + c3) ’ {a^a2 + af>2 d* ^0^2 d~ byi2 d- bfo2 d- . . .) . (a3 + bo + c3) — (eqa2 + af>2 + axc2 + \a2 + bf2 + . . .) . (n3 + b3 + c3) ’ _ (a0b2 + a0c2 + b0a2 + \c2 + c0a2 + c0b2) . (a3 + b3 + c3) (a1b2 + a1c2 + bxa2 + bxc2 + cYa2 + cx62) . (a3 + b3 + c3) ’ since a0a2 = b0b2 = ... = cYc2 — 0 ; and finally x _ af)2c3 ~ a(fisc2 d* a2^3,C0 ~ a2^0C3 d~ af)0C2 ~ aS^2C0 1 af)2c3 - af>3c2 4- af)3cx - ajb 1c3 -1- af)Yc2 - af)2cY ’ “ worin wir, da alles entsprechend geordnet ist, wieder die gewohn- liche Multiplicationsbezeichnung einfuhren konnten.” (in. 39) All this semblance of demonstration is of little moment compared with the fact sought to be demonstrated, viz., that a determinant is expressible as the outer product of the sums of the elements of its columns. Grassmann, however, makes no reference to determinants. In a paragraph of a subsequent chapter (p. 129), he takes up the problem of elimination between two equations of the mth and wth degrees. What it contains is a reproduction of Sylvester’s dialytic method, without any reference to the author of the method. 1888-89.] Rev. M. M. U. Wilkinson on Scalar Relations. 773 On the Scalar Relations connecting Six Vectors. By the Rev. M. M. U. Wilkinson, Reepham Vicarage, near Nor- wich. Communicated by Professor Tait. (Read July 15, 1889.) A. Introduction. 1. In the case of two Vectors, a, J3, the Scalars a2, /32, Sa/? are connected hy no relation. In other words, the Tensors of two straight lines, and the angle between them, are three independent quantities. In this case every other Scalar Function of a and j3 can he expressed in terms of a2, ft2, Sa/?. Thus, Sa/?a/?=2S2a/?-a2/?2. For convenience we shall call Scalars of the form - a2, Tensor Scalars, and Scalars of the form Sa /?, Primary Scalars. 2. The introduction of a third Vector, y, introduces three fresh Scalars, as the Tensor of y, and its two inclinations to the Vectors a, /?. In this case we have six independent Scalars, in terms of which every other Scalar involving only the three Vectors can he expressed. Thus, a2, /?2, y2, Sa /?, S /?y, Sya, are independent Scalars. All other Scalar functions of a, /?, y, connect themselves with these by equations. Thus, S 2a/?y = Say , Sa /? , a2 S/?y, /?2 , Sa/? y'2 , S/?y , Say . . (1) 3. In general, if we have n Vectors, we have 3 (n - 1) independent Scalars, as each fresh Vector introduces three fresh Scalars, namely, its Tensor and its inclination to any two of the other Vectors. All other Scalars involving the n Vectors can he expressed in terms of the 3(?&-l) independent Scalars. Now, with n Vectors we have n Tensors, and n^n ~ ^ Primary Scalars, making Scalars I . Z 1 . z m all . As of these 3 (n - 1) are independent, it follows that ^ independent equations connect the Tensors and Primary Scalars. VOL. xvi. 13/2/90 3 d 774 Proceedings of Royal Society of Edinburgh. [sess. All Scalars, it is clear, express in terms of Tensors and Primary Scalars. Thus, when n — 4, we have one such independent equation, as a2 , Say8, Say , SaS = 0 . . . (2) Safi, p , S/3y, S£S Say , S/8y, r2 - SyS SaS , S/3S, SyS , S2 Equations such as SaSSa/?y = a2S/?yS + Sa/3SyaS + SaySa/3S ... (3) are not independent equations, as they can be obtained from (2) by means of equations of the form (1). 4. In the case of n = 5 there are three relations connecting the five Tensors and tenTPrimary Scalars. Here various problems present themselves, of this character ; having given twelve of these Scalars, to find equations connecting them with the other three. Of course the twelve must be so selected as not to contain ten which are functions of only four vectors, and which would, therefore, be con- nected by an equation (2). In the case of w = 6, we have six Tensors and fifteen Primary Scalars, connected by six independent equations. So if fifteen of these, so selected as to be a set of fifteen independent Scalars, are given, six equations sufficient to determine the remaining six, can be found. The problem we aim at discussing is, in the case of six Vectors, having given the fifteen Primary Scalars to express in terms of them the Tensors, and other Scalars. B. Principal Formulae. 5. Our Vectors are a, j3, y, S, e, £. We have at once, Sa/?yS = Sa/?SyS - SayS/38 + SaSS/3y ; . . . (4) and, since, Sa/?yS Se£ = SajG(VScSy£ + Ve£SSy + V£SSye) ; 1888-89.] Rev. M. M. U. Wilkinson on Scalar Relations. 775 we have the important formula, Sa/?yS8e£ = Sa£ , Sac , SaS S Pt, S/?c, S/38 SyC, Syc, Sy8 of which (1) is a particular case. Also, since, . . (5) Sa/3y8c£ = Sa/?ySSc£ + Saygy(8Se£ - cS8£ + £S8c) ; . (6) we see that the expression for Sa/?y8c£ in terms of Primary Scalars contains 6 + 3x3=15 terms in all. 6. Representing the 10 determinants (of which Sa/?yS8e£ is one) as follows : — Sa/}yS&£ = V ; 1 Sa/3fSySc = A.?V; Sa/?6Sy£S = A!V; Saj3SSy4 = \%V ; I SyafSySSc^/xfV; SyaeS y8£S = /4V; SyaSS^ef = tfY ; ’ ^ S/3y£SaSe = yfV ; S/3yeSa£S = v^V ; S/3ySSae£ = VgY ; _ we have, at once, 1 = A2 + XI 4. A.2 = y^2 + y^ + ^2 __ v2 + „2 + v2 __ | = A-f + p-1 + Vi = A| + /i| + V2 = + fA + v\ ) i Now define as follows, W = SaycSaS£S/3e£S/3y8 — Say8Sae£S/3S£S/?yc ; . . (9) a little consideration will show that, by permuting the Vectors, there are only two expressions of the form W, and that W2 is a sym- metrical function of the Vectors. Thus, since, Sy3y8S/?c£ + Sy8yeSy8£8 + S/?y£S/?8c = 0 ; Say8Sae£ + SaycSa£8 + Say£Sa8c = 0; we have, Sy8y8Sy8e£SaycSa£S — Say8Sae£S/?ycS/3£8 = Say£Sa8cS/3ycS/?£8 - Sygy£S/38eSaycSa£8 . Calling the permutation of any two Vectors one permutation, an odd number of permutations changes the sign of W merely. The formulae (7), (8), readily show that W2 is symmetrical. 776 Proceedings of Royal Society of Edinburgh. [sess. 7. We have, W2 + 4 = V VM - - g-A)2 ; or, W2 = VtyM + iAA + /4*4 - m&lvl - ZgliAyA - • (10) The symmetry of this is obvious, for {^y - (/Vi + hv2)2} {py - (wi - /x2v2)2} = = { 1 - f4 - A - Vl - v\ + {fljV2 ~ /*2iq)2} { 1 - (A - 14 - v\ - v\ + (lXYV2 + /^iq)2} ; 1^3 ~ (v^ + V^)2} {AJ - (vlfX2 - V^)2} = = { - 1 +/^i + /x1 + v? + ^-(/a1v2 + /a2v1)2}{- 1 +^? + ^2+v? + ^-(/x1i/2-/x2v1)2} When W vanishes, be it observed, equations (8) and (10) give us the well-known rectangular system. 8. It will assist brevity in the calculations to define as fol- lows : — \ = So£S£cSy8; c4 = SofSjBSSyc; ex = Sa£S/?yS8e ;\ \ = SacS^fSyS ; C2 = SaeS/?yS8£ ; e2 = SaeS/3SSy£ • I 63 = SaSS/?ySe£; C3 = Sa3S/3£Sy€ ; e3 = SaSS£eSy£;V. (11) 64 = SayS/?SSe£ ; C4 = SayS/ScS S£ ; e4 = SayS/?£SSe • i &5==Sa/?SySSe£; C5 = SajSSyeSSf ; e5 = Sa/?Sy£SSe / Bi — Cj - ; Ci = Ci — ; Ex = 64- C1 > 1 ii B-2 “ ^2 C2 ’ = C3 — e3 > C3 — e 3 - bs ‘ E3^3 _ C3 > B4 = c4 - e4 j C4 = c4 - &4 ; E4=64- c4; •^5 = e5 — C5 j C5 = ^5 — ^5 ; E5 = C5- h\ so that 0 = B1 + C1 + E1 = B2 + C2 + E2 = . . . — B5 + C5 + E5 . (13) 9. Then we have, Y = E1 + C2 + B3; ] XfV= -B2-C3-E4; X|V= -E3-B1-C4; A|V= -C4 - E2- B4 ; ! ^?Y=-E5-E2-E3; ^-C^-Cs-C,; ^V=--B,-B,-B,; ^ (U) vfV= Ej + E4 + E6;4V- C5 + C2 + C4; vfV = B4 + B5 + B3 1888-89.] Rev. M. M. U. Wilkinson on Scalar Relations. 777 Whence, among many other expressions for W2, we have, w2= (B5 + B1 + B2)2(B5+B3+B4)2 + (C5 + C1+C3)2(C5 + C2 + C4)2 + (E5 + E1 + E4)2(E5 + E2 + E3)2 - 2(C5 + C4 + C3)(C5 + C2 + C4)(E5 + E, + E4)(E5 + E2 + E3) - 2(E5 + Ej + E4)(E5 + E2 + E3)(B5 + B4 + B2)(B5 + B3 + B3) - 2(B5 + B1 + B2)(B5 + B3 + B4)(C5 + C4 + C3)(C5 + C2 + C4) ; (15) we postpone for the present the consideration of the expansion of this, which will, of course, he symmetrical in form as well as in reality. 10. From the known formula for 8 vectors, Sotot-j^ j S/toj , Syax , S8a4 = 0; . • (16) Sa^j, mi. Syft. > Sayj , S/5ri, sryi » SSyi SaS4 , m , SyS15 sasx putting a4 = a, /?! = /?, “ft = c, S4 = t, we find where Now, IV/32 + D2a2 + Dg/32 + D4 = 0 ; Di = Sye , SSe Sy£, Sy ; I>2 = 0 , Spy, S/J8 j ; S fie , Sy€ , SSe SyC, SSC S0£SjS8ye-S0eSj88y£ = • - (O) • • (18) SPZSP&Syt - Bpt&PySSi - SfcSpSSyZ + S/? Pi -9.1- - B5 + B3 + B4 ; 1 1 P-2 + 1-2 = + vO O 11 Cx + C3 3 P2 -92 = C5 + C2 + C4; h • (31) p3+q 3= + VO w II Ex + E4 ; Ps = E5 + E2 + E3 ; J have, by (13), • • (32) Pi+Pz + Pz = 0; . . . . so that pi + pi +pi - 2 vlv\ - 2 vlp\ - Zplvl = 0 , and, W2 = qt + q42 + qt - 2 q\q\ - 2 q\q\ - 2 q\q* .... (33), from which terms of the form Eg, 2BiBg, &c., have disappeared. 14. It is now evident that we may write, W2 = 5o)r,s + ^ + ^; ..... (34) where = <**5,1 + (t)6>2 + <*>5,3 + <05>4 4- <04,x + 0>4,2 + <*>4,3 + 0>3,i + <03,2 + <*>2,l ; , . . (35) ^<*>* = oq 4- o>2 + <*>3 + <*>4 + o>5 j (36) So) rtS)t = <*>5,1,2 + <*>5,i,3 + <*>5,1,4 + <1)5,34 + 0)53,4 + <*>5,2,3 + <**4,1,2 + <**4,1,3 + <**4,1,5 + <**4,3,5 + <**4,2,5 + <**4.2,3 + • . . + <**1,2,3 + 01x 2,4 + 0>x,2,5 + <<>1,4,5 + <*>1,3,5 + <*>1,3,4 ; . (37 ) Any one of the terms in each of these equations being found, all the rest may he found by simple permutation in various ways. But, as in such process mistakes are likely to he made, a table of the values o>g,i, &c.; oq, &c.; and o>5,i,2, &c., will he given. 15. The table may be constructed as follows, or in many other ways : — %,i - BfBl + QCl + E?Ef - 2C1E1C6E5 - 2E1B1E5B5 - 2B1C1B5C5 = (BjB5 - Cfij* + (Bj + CtXB, + C6)[(BX + CX)(B5 + C5) - 2C4C5 - 2B1B5] = (BjB- - Cjty* - (Bf - C?)(B§ - Cl) = (BA - BA)2 ; 2 = 2BABI - 20^0^ - 2BJE2E4B4 - 2B2C1Bfi4 = - 2B1B2B4C4 - 2B4B2E4B4 - 20^2^4 - 2B1E2E,B4 - 2B2C1B,C4 = 2E4B2B4C4 - 20^0^4 + 2B1C2E4Bi. 16. So we have, <%.. = (BiC5 - B/A)2 = (C^ - C5E,)2 = (EjB5 - EjBj)2 -| <0,., = (B,B4 - CXC4)2 = (EjC4 - BjE4)2 = (CjE4 - EjB4)2 W4,2 = (B2B4 — E2E4)2 •»4,3=(c3c4-e3e4)2 1- • (38) “>3,1 = (B'l B3 — EjE3)2 “3,2 = (B2B3 - C2C3)2 “2,1 = (CA - EjEj)2 “5 = 2(BjB2B3B4 + C1C2C3C4 + EjE2E3E4) ; • <04 = 2(B1C2E3C5 + C1E2B3B5 + EABA) ; 2 = 2(E1B3C4C5 + CjE3B4E5 + B,C3E4B6) y “i = 2(E2C3B4C5 + B2E3C4B5 + C2B3E4E5) ; . Where we observe the identity, “5 = 2(0,0, - EjEjXCjC* - EsE4) + 2(EjE3 - B,B3)(E2E4 - B2B4) + 2(B,B4 - CjC4)(B2B3 - C2C3) ; (40) &c. 1888-89.] Rev. M. M. U. Wilkinson on Scalar Relations. 783 17. The other values are, <*>5,1,2 = 2(E1E2B5C5 — -^BgCgEg + CjC^Egl^) j <*>5,i,3 = 2(RiR3^5^5 “ C1C3E5B5 + EjEgBgCg) ; <*>5,if4 = 2(C1C4E5B5 - E^BgCg 4- B4B4C5E5) ; <*>5,3,4 = 2(E3E4B5C5 — B3B4C5E5 + C3C4E5B5) ; <*>5,2,4 = 2(B2B4C5E5 — C2C4E5B5 + E2E4B5C5) y <*>5,2,3 = 2(C2C3E5B5 — E2E3B5C5 + B2B3C5E5) , <*>4,i,2 = 2(E1B2B4C4 — C1E2C4E4 + B1C2E4B4) ; <*>4,!, 3 = 2(E1C3B4C4 - B4E3E4B4 + C^gC.E,) ; <*>4,4,5 = 2(C1B5C4E4 - EXE5B4C4 + B4C5E4B4) ; <*>4,3,5 = 2(C3E5B4C4 — b3b5c4e4 + E3C5E4B4) y <*>4,2,5 = 2(B2E5B4C4 — C2C5E4B4 + E2B5C4E4) y <*>4,2,3 = 2(E2B3C4E4 - B2C3B4C4 + C2E3E4B4) ; <*>3,i,2 — 2(B1E2B3C3 — EjC^CgEg + C^EgB,) ; <*>3,1,4 = 2(0^^363 - B^BgCg + EjB.CgEg) ; <*>3,1,5 = 2(B1E5B3C3 — CjCgEgBg + EjBgCgEg) \ <*>3,4,5 = 2(C4E5B3C3 — B4B5C3E3 + E4C5E3B3) ; <*>3,2,5 = 2(B2C5E3B3 — E2E5B3C3 + C2B5C3E3) y <*>3,2,4 = 2(C2B4C3E3 — B2E4E3B3 + E2C4B3C3) ; <*>2,1,3 = 2(B1C3C2E2 — E1B3E2B2 + C1E3B2C2) ; <*>2,1,4 = 2(B1E4C2E2 - C1B4B2C2 + EjC^B,) ; <*>2,1,5 = 2(C1E5B2C2 — B^C* + E^B,) ; <*>2,4,5 = 2(B4E5B2C2 — C4C5E2B2 + E4B5C2E2) j <*>2,3,5 = 2(B3C5E2B2 — E3E5B2C2 + C3B5C2E2) ; <*>2,3,4 = 2(E3B4B2C2 - C3E4C2E2 + B3C4E2B2) ; <*>1,2,3 = 2(B2E3C1E1 — CgBgBj^Cj + E2C3E1B1) ; 0)1>2>4= 2(C2E4B1C1 - E2B4E1B1 + B^C^) ; <*>1,2,5 = 2(C2E5B101 — BgBgCjEj + E^gEjBj} ; <*>i,4,5 ^ 2(B4C5E1B1 - E4E5B1C1 + C.BAEJ ; <*>1,3,6= 2(B3E5B1C1 - C^E^ + EgBgC.E,) ; <*>1,3,4 = 2(B3E4B1C1 - E3C4C1E1 + CgB.EjBj) . 7 84 Proceedings of Royal Society of Edinburgh. [sess. In connection with which equations we observe the identities, 2(B1C, - C]B5)(E1E2 - CA) = 2(E2C5E1B1 + 0^ + B^EJ ; &c. In writing down the complete out-spread of W2, it will not be necessary to avail ourselves of more than a few of these results. 18. From what proceeds we have W2 = U0 + U1 + U2 + U3 + U4 + U5 + U6 + U7 + U8 + U9; where U0 = VS2y8yS2ySS2S€S2^S2Ca ' Uj = - 22S2a£S2a£S2/3yS2S€SySS6£SS£Sye ; U2= - 2^S2ay8SVS2S€S2y8SeCS/36SyCS/?y ; U3 = 22S2a/?S2a£S2S€S£yS/?8Sy<:Sy£SySSc£ ; U4 = 22S2a/?S2y8S2e£SaeSSeS/?SS£ySy£Sa£ ; U5= - 4^S2ay8S2y3S2e^Sa^S3CSaSS^ySy€S^€ ; U6 = 22S2a£S2yeS23£SaeSS€S£SS/?ySy£Sa£ ; Ur = 45SVS2y3SacSa^S/3€S^Sy€Sy^S8€S3^ ; Ug= - 2^S2a/3S2ySSaeSa£Sy€Sy£S/3eSS£S/?SSe£ ; U9 = 4SSaiSo8SaySaj8S^S«J SycSSf^SfS^cSySSSe . In this expansion for W2 it is to be observed that U9 contains 15 terms, U0, U5, U6, each contain 60 terms, U7 contains 45 terms, U1, U4 each contain 180 terms, U2 contains 90 terms, and U3, U8 360 terms each. Counting the weight of each term as 1, 2, or 4, according to its coefficient, we have, weight of + terms = 60 + 2 x 360 + 2 x 180 + 2x 60 + 4x45 + 4x15 = 60(1 + 12 + 6 + 2 + 3 + 1) = 1500 ; weight of - terms = 2xl80 + 2x 90 + 4x 60 + 2x 360 = 60(6 + 3 + 4 + 12) = 1500. The same number, as should manifestly be the case. 19. If we had expressed W2 in terms of bl9 cl9 elf &c., it should have been borne in mind that these Scalars are connected by five 1888-89.] Rev. M. M. U. Wilkinson on Scalar Relations. 785 relations, contained in the following 10 equations, five of which are easily obtained from the other five. eie3^5 = ^1^3e5 > eie2,Co ~ ClC2e5 > e2e4^5 = ^2^4e5 i C2e2pA ~ e2^3C4 } C2C3^5 = ^3C5 ) C1^3e4 = eiC3^4 3 e3e4C5 = C3C4e5 > ^lC2e4 = ei^2C4 ) ^AC5 = C1C4^5 > ^ie2C3 = C1^2e3 * 20. It can be readily shown that the expression for W2 cannot be square rooted so as to express W in the form S( - 1 )rSa/3S/3ySySS8eSt£S£a . For a single permutation of any two letters changes the sign of W, while the successive permutations of a, ft; a, y; a, 8; a, e; a, £, being 5 in number, do not change the sign of Sa/3S/?ySySSSeSe£S£a. So the expression for W would not change its sign for a permuta- tion which would change the sign of W. E. Formulce for Sa/3y, &c. These may be obtained in various ways. Thus we have SaySSae£ = a2SySVe£ + SaySaSV£e 4- SaSSayVe£ • SaycSa^S = a2SycV{8 + SaySaeVS£ + SacSayY^ ; Say£Sa8e = a2 Sy£VS€ + SaySa^VcS + Sa£SayVSe ; whence SySSe^SaySSae^ + Sy€S£SSay