This expression is shown to give known particular integrals, such as (1-2r cos 0+r), and

It appears probable, therefore, that the generalization of the result obtained for the limited value of n is legitimate; but the author does not profess to demonstrate this conclusion, believing that the principle of the "permanence of equivalent forms" is not at present established in such a sense as to amount to a demonstration.

VII. "A Memoir on Curves of the Third Order." By ARTHUR CAYLEY, Esq., F.R.S. Received Oct. 30, 1856.

(Abstract.)

A curve of the third order, or cubic curve, is the locus represented by an equation such as U=(*(x, y, z)3=0; and it appears by my "Third Memoir on Quantics," that it is proper to consider, in connexion with the curve of the third order, U=0, and its Hessian HU=0 (which is also a curve of the third order), two curves of the third class, viz. the curves represented by the equations PU 0 and QU=0. These equations, I say, represent curves of the third class; in fact, PU and QU are contravariants of U, and therefore, when the variables x, y, z of U are considered as point coordinates, the variables E, n, ¿ of PU, QU must be considered as line coordinates, and the curves will be curves of the third class. I propose (in analogy with the form of the word Hessian) to call the two curves in question the Pippian and Quippian respectively. A geometrical definition of the Pippian was readily found; the curve is in fact Steiner's curve R, mentioned in the memoir "Allgemeine Eigenschaften der algebraischen Curven," Crelle, t. xlvii. pp. 1-6, in the particular case of a basis-curve of the third order; and I also found that the Pippian might be considered as occurring implicitly in my "Mémoire sur les Courbes du Troisième Ordre," Liouville, t. ix. p. 285, and "Nouvelles Remarques sur les Courbes du Troisième Ordre," Liouv. t. x. p. 102. As regards the Quippian, I

have not succeeded in obtaining a satisfactory geometrical definition; but the search after it led to a variety of theorems, relating chiefly to the first-mentioned curve, and the results of the investigation are contained in the present memoir. Some of these results are due to Mr. Salmon, with whom I was in correspondence on the subject. The character of the results makes it difficult to develope them in a systematic order; but the results are given in such connexion one with another, as I have been able to present them in. Considering the object of the memoir to be the establishment of a distinct geometrical theory of the Pippian, the leading results will be found summed up in the nine definitions or modes of generation of the Pippian, given in the concluding number. In the course of the memoir I give some further developments relating to the theory in the memoirs in Liouville above referred to, showing its relation to the Pippian, and the analogy with theorems of Hesse in relation to the Hessian.

VIII. "On the k-partitions of a Polygon and Polyace." By the Rev. T. P. KIRKMAN, M.A. Communicated by ARTHUR CAYLEY, Esq. Received November 13, 1856.

(Abstract.)

The problem relating to the polyace is the reciprocal of that relating to the polygon, and is not separately discussed. By the k-partitions of a polygon, the author means the number of ways in which the polygon can be divided by (k-1) diagonals, no one of which crosses another; two ways being different only when no cyclical permutation or reversion of the numbers at the angles of the polygon can make them alike: it is assumed that the polygon is of the ordinary convex form, so that all the diagonals lie within its area. The author remarks, that the enumeration of the partitions of the polygon and polyace is indispensable in the theory of polyedra, and that in his former memoir "On the Enumeration of a-edra having Triedral Summits and an (x-1)-gonal Base," Phil. Trans. 1856, p. 399, he has, in fact, investigated the (r-2)-partitions of the r-ace

or r-gon; so that the present memoir may be considered as a completion, or rather an extension and completion, of the investigations in his former memoir. The number of distinctions to be made in the problem of the present memoir is very great; thus, a partition of the polygon may be either reversible or irreversible; and if reversible, then the axis of reversion may be either agonal, monogonal, or diagonal, that is, it may pass through no angle, one angle only, or two angles of the polygon; and in the last case it may be either drawn or undrawn. Again, there may be a single axis or a greater number of axes of reversion: in the case of m such axes, the partition is said to be m-ly reversible; and in like manner an irreversible partition may consist of a single irreversible sequence of configurations, or it may contain such sequence m times repeated, it is then said to be m-ly irreversible. In consequence of this multiplicity of distinctions, the author's final results are necessarily very complicated, and cannot be exhibited in an abstract; they appear, however, to contain a complete solution of the problem, i. e. to afford the means of finding, without anything tentative, the number of the k-partitions of an r-gon when k and r are given numbers.

December 18, 1856.

The LORD WROTTESLEY, President, in the Chair.

The following communications were read :

I. "On the Scelidothere (Scelidotherium leptocephalum, Owen), a large extinct Terrestrial Sloth." By Professor R. OWEN, F.R.S. Received October 30, 1856.

(Abstract.)

The extinct species of large terrestrial Sloth, indicated by the above name, was first made known by portions of its fossil skeleton having been discovered by Charles Darwin, Esq., F.R.S., at Punta "Alta, Northern Patagonia. These portions were described by the

author in the Appendix to the Natural History of the Voyage of H.M.S. Beagle.'

The subsequent acquisition by the British Museum of the collection of Fossil Mammalia brought from Buenos Ayres by M. Bravard, has given further evidence of the generic distinction of the Scelidothere, and has supplied important characters of the osseous system, and especially of the skull, which the fragments from the hard consolidated gravel of Punta Alta did not afford.

The best portion of the cranium from that locality wanted the facial part anterior to the orbit, and the greater part of the upper walls; sufficient however remained to indicate the peculiar character of its slender proportions, and hence Professor Owen has been led to select the name leptocephalum for the species, which is undoubtedly

new.

The aptness of the epithet 'slender-headed' is proved by the author's researches to be greater than could have been surmised from the original fossil; for the entire skull, now in the British Museum, exhibits a curious and very peculiar prolongation of the upper and lower jaws, and a slenderness of the parts produced anterior to the dental series, unique in the leaf-eating section of the order Bruta, and offering a very interesting approximation to the peculiar proportions of the skull in the Ant-eaters.

The original fossils from Patagonia indicated that they belonged to an individual of immature age: the difference of size between them and the corresponding parts in the British Museum, depends on the latter having belonged to full-grown individuals: the slight difference in the shape of the anterior molars seems in like manner to be due to such an amount of change as might take place in the progress of growth of a tooth with a constantly renewable pulp. Professor Owen finds at least no good grounds for inferring a specific distinction between the mature if not old Scelidothere from Buenos Ayres, and the younger specimen from Patagonia.

The author then proceeds to give a detailed anatomical account of the fossil bones in the British Museum, instituting a comparison between them and the bones of other large extinct animals, especially those of the Edentate order.

The Scelidothere was a quadruped of from 8 to 10 feet in length, but not more than 4 feet high, and nearly as broad at the haunches;

the thigh-bones being extraordinarily broad in proportion to their length. The trunk gradually tapered forwards to the long and slender head. The fore-limbs had complete clavicles, and the rotatory movements of the fore-arm. All the limbs were provided with long and strong claws. The animal had a long and muscular tongue, and it is probable that its food might have been of a more mixed nature than in the Megatherium. But it was more essentially related to the Sloths than to the Ant-eaters.

In conclusion the author remarks, that as our knowledge of the great Megatherioid animals increases, the definition of their distinctive characters demands more extended comparison of particulars. Hence in each successive attempt at a restoration of these truly remarkable extinct South American quadrupeds, there results a description of details which might seem prolix and uncalled for, but which are necessary for the proper development of the task of reproducing a specimen of an extinct species.

Professor Owen adds, that he is indebted to an allotment from the Government Grant, placed at the disposal of the Royal Society for scientific purposes, for the means of laying before the Society large and admirably executed drawings of the fossil bones described in his paper.

II. "On the Evidence of the existence of the Decennial Inequality in the Solar-diurnal Variations, and its non-existence in the Lunar-diurnal Variation of the Magnetic Declination at Hobarton." By Major-General SABINE, R.A., D.C.L., Treas. and V.P.R.S. Received Nov. 17, 1856.

(Abstract.)

In a communication made to the Royal Society in the last Session, "On the Lunar-diurnal Magnetic Variation at Toronto," the author had stated that he could discover no trace of the lunar influence of the decennial inequality which constitutes so marked a feature in the solar magnetic variations. He has since read, in a memoir communicated to the Imperial Academy of Sciences at Vienna, entitled "On the Influence of the Moon on the horizontal component of the Mag