Deckmbkr 29. 1S92] 



NA TURK 



»5 



degree are perfect squares. The reasoning which establishes 

 these propositions does not apply to skew symmetrical deter- 

 minants of odd class. By a different method it is shown that 

 they vanish identically whether the class be even or odd Tt is 

 next shown that if we form any determinant of even class 2/ from 

 2p ordinary determinants, in a manner analogous to that in the 

 rule for the mul.'iplication of two ordinary determinants, the 

 determinant so formed is the product of the 2/ determinants ; 

 and if any determinant of odd class 2/ + i is formed from 

 2/ + I ordinary d^-terminants, the determinant so formed is the 

 product of the last 2/ of these ordinary determinants into the 

 first taken, with all its signs positive. A somewhat similar 

 result is shown to hold for determinants of alternate numbers. 

 As an application, let 



{X - «i) {y - 

 and let (/, q) denote 



{X - a„) {y - fi„) 



\p\ldxPdyi' 



By multiplying the arrays 



we get 



I (o, o), (o, i; I ^2 



apaq {ftp - Oyl {Bp - Bq) 



I (I, o). (I, I) I {x- apY {X - BpY (•*■ - dqf {X 



Suppose, now, « = i, we get that the primitive of 

 I (oo), (01) I is2 = ^_fi_ 

 I (10), (11)1 {x - a^) {y - Bi)' 



Similarly, by multiplying, 



{X - «J 



{X - aj' 

 a. 



^1 



we get that the primitive of 



(00), (01), (02) 

 (10), (II), (12) 

 (20), (21), (22) 



Cy - 01)' 



iy - fiif 



(X 



{y - )3i) {x -a^){y- 3.,) 



Similar primitives are obtained for differential equations, which 

 are in the form of determinants of higher class. A further 

 application is obtained by taking powers of different invariantive 

 symbols, of which (123) is the simplest for the ternary quantic. 

 The resulting invariants are seen to be determinants of some 

 even class. — A geometrical note, by Mr. R. Tucker. — The Presi- 

 dent (Major MacMahon, F.R.S., in the chair) made an im- 

 promptu communication of a problem, the solution of which 

 he thought would be subsidiary to the sought-for solution of 

 the "stamp folding" problem. 



Linnean Society, December 15. — Prof. Stewart, President, 

 in the ch^ir. — The President announced the recent death of Mr. 

 H. T. Stainton, a Fellow and former Vice President of the 

 Society, and of European reputation amongst entomologists, by 

 whom his loss would be widely felt. — Mr. D. Morris exhibited 

 a series of botanical photographs from the west coast of Africa, 

 and gave some interesting details about the appearance and 

 mode of growth of some of the more remarkable forest trees and 

 plants of that region. — The Secretary exhibited a large collec- 

 tion of photographs of Lichens, very neatly mounted and labelled, 

 which had been recent 1> presented to the Society by Pr.f. 

 Arntild, of Munich.— On behalf of Mr George Swainson, of St. 

 Annes-on-Sea, Lancashire, Mr. A. R. Hammond exhibited an 

 aquatic dipterous larva, belonging probably to the genus Dixa, 

 of which by means of the oxyhydrogen lantern, Mith microscopic 

 attachment, a good figure was projected on the screen. He 



NO. 1209, VOL. 47] 



referred to the different views which prevailed concerning the 

 dorsal and ventral aspects of this larva, and pointed out that the 

 tail -plates possessed features which in allied forms were charac- 

 teristic not so much of the larval as of the pupal stage. — A paper 

 was then read by Dr. Maxwell T. Masters, F. R.S., on the 

 classification and geographical distribution of the Taxacea and 

 Conifent, his remarks being illustrated by a specially prepared 

 map, lent by Mr. C. H. Clarke, and by specimens of the fruit 

 and leaves of some of the more notable forms. — Mr. George 

 Brook followed with a paper on the affinities oi Madrefora, and 

 here again, by means of the oxyhydrogen lantern, an excellent 

 series of coral sections was projected, which illustrated very 

 clearly the author's remarks on comparative structure. —A short 

 note on the abnormal form of the lens in the eyes of an albino 

 rat, by Prof. R. J. Anderson, was read on his behalf by the 

 Secretary. The meeting then adjourned to January 19, 1893. 



Zoological Society, December 6. — Dr. St. George Mivart, 

 F. R. S., Vice-president, in the chair. — The Secretary read a 

 report on the additions that had been ma-le to the Society's 

 menagerie during the month of November 1892. — Dr. Hickson 

 read a paper entitled "A Revision of the Genera of the 

 Alcyonaria Stolonifera, with a description of one new genus and 

 several new species." The author commenced by stating the 

 grounds upon which it might be considered desirable to retain 

 the suborder Stolonifera, and criticized the views of those who 

 place these Alcyonarians in the suborder Alcyonida. Of the 

 genera that had already been proposed only four could now be 

 retained, namely, Tiibipora, Clavularia, Cornularia, and 

 Sympodittm, and the author proposed to add one more, namely, 

 Stereosoma. The genera Sarcodictyon, Rhizoxenia, Cornu- 

 larieila, Anthelia, and Gymnosirca must be abandoned, and 

 the species incorporated in the other genera. A description 

 was then given of the new genus Stereosoma, a form found on 

 the coast of North Celebes, distinguished from all other titolo- 

 nifera by cettain characters of its tentacles and by the absolute 

 mm retractability of its polypes. Several new species of 

 Clavularia were then described from North Celebes, Diego 

 Garcia, and Australia. This was followed by a summary of ail 

 the species of the genus known to science. — Mr. F. E. Beddard, 

 F.R.S., reada description of the c )nvolutions of the celehral 

 hemispheres in ceita n rodents. The paper referred chiefly to 

 Dasyprocta Ccelogenys, LagostomuSy PJydrochoerus, and Doli- 

 chotis, being the genera of rodents in which the brains s-how 

 the greatest development of convolutims. — A communication 

 was read from Prof. Collett, containing a description of a new 

 monkey from S. E. Sumatra, for which he proposed the name 

 Semnopithecus thomasi. — Mr. H. J. Elwes read the second 

 portion of an account of the butterflies collected by Mr. W. 

 Doherty in the Naga and Karen Hills and in Perak. 



Paris. 

 Academy of Sciences, December 19. — Annual Public 

 Meeting. — The President, M. d'Abbadie, gave a brief survey of 

 the life and work of those lost to the Academy by death during 

 the year. Among these were the following members : M. D. 

 D. A. Kichet, distin;^uished for his medical disc'Veries ; M. de 

 Quatrefages de Breau, the naturalist ; M. lurien de la Graviere, 

 Vice- Admiral under the Empire ; M. Pierre Ossian Bonnet, 

 geometrician; Admiral Mouchez, late Diiecor of the Paris 

 Observatory. Foreign Associate : Sir G. B. Airy. Acadimicien 

 litre: yi. Lalanne. Correspondents : MM. Gilbert, Abria and 

 Adams. The prizes were awarded as follows : The Grand 

 Prize of the mathematical sciences to M. Hadamard for his 

 solution of the problem of determining the number o( primary 

 numbers inferior to a given quantity. One Prix Bordin to M. 

 Gabriel Koenigs for his solution of a problem concerning 

 geo<lesic lines ; another to M. Humbert for his work 011 hyper- 

 eliiptic surfaces. The Prix Poncelet to the builders of the 

 Fonh Bri<lge, Sir John Fowler, and Sir Benjamin B^ker ; the 

 Extra Prize of 6000 francs to M. Hedouin lor his work on the 

 Channel currents ; the Prix Montyon to M. N. J. Raffard, 

 civil engineer ; the Prix Plumez to M. Augustin Norman d, for 

 his g. ometry of ships. In Astmnouiy, the Prix La lande 

 was doubled, and awarded to Mr. Barnard and Mr. Max 

 Wolf; the Prix Damoiseau to M. Radau for his work 

 on lunar inequalities of long period caused by planets; 

 the Prix Valz to M. Puiseux for his researches on ihe equatorial 

 ft?M<// and other instruments; the Prix Janssen to M. 1 acchini 

 for his solar work. Statistics : The Prix Montyon to MM. M. 



