500 



NA rURE 



[March 26, 1896 



platinicyanides and the platosaniine salts are less fluorescent. A 

 study of the discharge phenomena observed during the exhaus- 

 tion of the tube shows that the rays proceeding from the concave 

 kathode meet at the centre of curvature of the latter, and then 

 diverge in a solid cone ; as the vacuum becomes higher, this cone 

 gradually narrows until it becomes at length a straight line. It 

 IS interesting to note that this latter would be the behaviour of 

 non-elastic particles emanating normally from the concave 

 kathode.— The union of carbon and hydrogen, by W. A. Bone 

 and D. S. Jordan. On heating carefully purified sugar charcoal 

 to a white "heat in hydrogen, i to 2 per cent, of the latter is con- 

 verted into methane. . During the burning of an electric arc 

 lamp in hydrogen, acetylene and another hydrocarbon, probably 

 methane, are produced. — Note on the ctoi-dimethylglutaric 

 acids, by W. A. Bone and W. H. Perkin, jun.— The sym- 

 metrical dimethylsuccinic acids, by W. A. Bone and W. H. 

 Perkin, jun.— The cis- and trans- methylisopropylsuccinic acids, 

 by VV. H. Bentley, W. W. Perkin, jun., and J. F. Thorpe. In 

 these three papers the preparation and properties of the acids 

 named are described. 



Linnean Society, March 5. — Mr. W. Percy Sladen, Vice- 

 President, in the chair. — On behalf of Capt J. Marriott, Mr. 

 Harting exhibited an antler of the Burmese deer [Cervtis Eldi), 

 and described a singular condition in another example which for 

 eight years had continued to exude a blood-coloured liquid from 

 a puncture on the under surface of the brow-tine. Prof. Stewart, 

 to whom some of the substance had been submitted for examina- 

 tion, had found no blood-corpuscles therein, and considered it 

 to be grease in a semi-fluid condition, the nature of the colouring 

 matter being as yet undetermined. Mr. Druce thought the sub- 

 stance exuded might be the excretion of the larva of some insect 

 feeding upon the internal surface of the horn, and suggested the 

 examination of a section, if possible. — Mr. Harting exhibited a 

 drawing from life of a Klipspringer antelope ( Oreotragus saltator), 

 lately received (for the first time in this country) at the Zoo- 

 logical Society's Gardens. — Mr. Thomas Christy exhibited several 

 cases of butterfles collected by Mr. Horace Billington in Old 

 Calabar, on which remarks were made by Messrs. W. F. Kirby 

 and H. Druce. — Mr. B. D.Jackson, in directing attention to an 

 English translation by Mr. J. Lucas of that portion of Pehr 

 Kalm's " Travels" which relates to England, remarked that few 

 persons were aware that Kalm, a pupil of Linnieus, had in 

 1748 spent six months in this country and had diligently noted the 

 plants which he met with. Thus he had recorded no less than 

 sixty plants for Hertfordshire alone, deriving some of his informa- 

 tion from an examination of the contents of two haystacks in 

 that county — in this way anticipating by more than a century 

 one of the methods employed by Sir John Lawes and Sir J. H. 

 Gilbert, and by Prof. Fream. — On behalf of Prof. Gustav (jilson, 

 ofLouvain, two papers, entitled " Studies in insect morphology," 

 were communicated by Prof. Howes. In the first of these, on 

 segmentally disposed thoracic glands in the larva; of Trichoptera, 

 the author found that in Linuiophih(s flavicornis the prothoracic 

 prominence gives exit to an underlying tubular gland. In 

 Phryganea grandis each thoracic sternum gives exit to a 

 glandular apparatus of the same category, the prothorax alone 

 developing a prominence. — In the second paper by Prof. Gilson 

 and M. J. Sadones, on the larval gills of Odonata, the authors 

 described in each branchial lamella of LibeUula depi-essa three 

 conical processes which are functional in preventing adherence 

 of the lamella to its fellows, and in maintaining full exposure to 

 the surrounding medium. 



Mathematical Society, March 12.— Major MacMahon,K.A., 

 F.R.S. , President, in the chair. — The President read the follow- 

 ing abstract of a paper by Prof. Lloyd Tanner, on the enumera- 

 tion of groups of totitives. The paper explains a method of 

 determining how many groups of given order can be formed 

 with the totitives of any integer, n. In the investigation use 

 is made of a function formed from a binomial coetBcient by 

 replacing each factor, say r, of the numerator or denominator by 

 /"■-I, so that the binomial coefficient is in fact the limiting 

 value of the function as/ approaches i. There are indications 

 of the existence of a reciprocity theorem (viz. that the number of 

 groups of order v is equal to the number of groups of order 

 t(«)/v), but this theorem is not proved. The attempt to establish 

 the theorem has led to the discovery of some notTble properties 

 of the functions — a Van der Monde-theorem, for instance. The 

 functions in question are well known. They were used by 

 Euler as generating functions for the number of partitions, and 



! by Cayley (" Researches on the Partition of Numbers," Phil- 

 Trans., cxlv.). Jacobi in a memoir ( D'e//i?, xxxii., 1846), starting 

 with a more general function, obtained a number of formuke 

 which appear to be different from those used in this paper. 

 Gauss, in the Suiitmaiio scrientin ijuarundavi singularmm, used 

 these functions, the base being a complex number of modulus i. 

 They have been used too (in Schellbach's treatise) as a means of 

 forming the theta-functions. The present application is of a 

 different kind. As in Euler's theory, they are used for enumera- 

 tion ; but the number sought is given by the actual value of 

 the function when the base / is a prime factor of in.— 

 Prof. Greenhill, F.R.S., next read a paper on the as- 

 sociated dynamics of a Top, and of a Body under no Forces. 

 Jacobi's theorems ( /rd?;-Xv, ii. p. 480) flow naturally from Darboux's 

 representation by means of the deformable hyperboloid (Despey- 

 rous, Mc'canique, ii. Note xx.). The hyperboloid is constructed, 

 in Henrici's manner, flattened in the plane of the focal ellipse, 

 by placing the generating lines tangential to the focal ellipse, 

 and knotting together at the points of crossing the generators of 

 opposite systems. Planes are drawn through any point H per- 

 pendicular to the generators HP], HPg, through H (the tangents 

 to the focal ellipse through H), the perpendiculars OG. OC are 

 drawn from the centre O upon these planes, and the perpen- 

 diculars OVi, OV2 on the generators IIPj, HPj. Then, during 

 the deformation of the hyperboloid, the lengths OG, OC, or 

 HV], HVo remain constant, and the points V, T, P in which a 

 generator "meets the principal planes are fixed points on the 

 generators ; so that the planes through H perpendicular to 

 the generators are tangent planes at H to two fixed coaxial 

 quadrics, the squares of whose semi-axes are numerically equal 

 to the rectangles HV.HV, HY.HT, HY.HP, the sign being 

 taken positive or negative according as Y and V, or T, or P are 

 on the same or opposite sides of H. These quadrics are the 

 momental quadrics of Jacobi's two associated bodies moving 

 under no forces ; but as the quadrics are unrestricted in shape, 

 the bodies must be composed of matter which is capable of 

 having a negative density, as is the two-fluid theory of elec- 

 tricity. The curve described by H is a polhode curve common 

 to the two momental quadrics ; it is also a line of curvature 

 formed by the intersection of a confocal ellipsoid and a hyper- 

 boloid of two sheets ; thus any such line of curvature may be 

 taken as a polhode on either of two momental quadrics, the 

 generating lines of the confocal hyperboloid of one sheet through 

 any point being the normals of the quadrics. If OG is held in 

 a vertical position, OC will imitate the associated motion of the 

 axis of a top, if H is moved always in a direction perpendicular 

 to the plane OGC, and OH will represent the resultant angular 

 momentum. If the momental spheroid of the top at the fixed 

 point O is a sphere, then OH will also represent the resultant 

 angular velocity ; but in the general case the resultant angular 

 velocity is represented by the vector OI to a point I fixed in 

 the generator HP.^. In constructing pseitdo-clliptic cases of 

 motion, the ratio of the axes of the focal ellipse is taken as the 

 modulus of the elliptic functions, and the position of P corre- 

 sponding to a parameter one-/nh part of a period will be deter- 

 mined geometrically by means of the poristic relation of a 

 polygon of ;/ or 211 sides, circumscribed to the focal ellipse and 

 inscribed in a confocal. The secular term, associated in general 

 with the azimuth, can be cancelled by placing H in the tangent 

 at Pj in a position given by a simple relation ; and now the cone 

 described by OC is algebraical, as also the herpolhode described 

 by H in the plane perpendicular to OG. Thus for Halphen's 

 algebraical herpolhode, P, is at Fagnano's point, and H is the 

 mid-point of PiYj. If I\ is at the end of the minor axis of the 

 focal ellipse, the axis OC" of tlie top describes cusps. If H is 

 placed at Yj, then OC represents the motion of the thread of a 

 spherical pendulum. After a brief discussion, in which the 

 President and Mr. Love, F.R.S., took part. Prof. Greenhill made 

 a communication on the Catenary on the Paraboloid and Cone. 

 Clebsch's equations for the form of a chain wrapped on a sphere, 

 which is revolving about a vertical axis with sufficient rapidity 

 for the attraction of gravity to be negligible, are here shown to 

 be immediately applicable to the case of a chain on a vertical 

 paraboloid, when gravity is again taken into account. An elliptic 

 integral of the third kind is required, with a pole at the vertex 

 of the paraboloid, and this integral can be compared immediately 

 with the standard form of the pseudo-elliptic integral, by the 

 solution of a certain Jacobian quartic. The arc of the catenary 

 is also directly reducible to the form employed by Mo*t\{CEuvres, 

 ii.). The motion of a little ball, rolling on the paraboloid, is re- 



NO. 1378, VOL. 53] 



