Mr. A. Cayley on Quantics. 149 



Bisect BD in E ; join AE, and produce the joining line to meet the 

 circumference in F. Then AF ditFers from the side of a square 

 equal in area to the circle by somewhat less than the one four- 

 thousandth part of that side. 



May 24. — The Lord Wrottesley, President, in the Chair. 



The following communication was read : — 



" A Second Memoir upon Quantics." By Arthur Cayley, Esq. 

 F.R.S. 



The memoir is intended as a continuation of the author's Introduc- 

 tory memoir upon Quantics (vide Phil. Mag. vol. vili. p. 69, and Phil. 

 Trans. 1854, p. 245) ; the special subject of the memoir is the theorem 

 referred to in the postscript of the introductory memoir, and the 

 numerous developments arising thereout in relation to the number 

 and form of the covarlants of a binary quantlc. The author, after some 

 remarks as to the asyzygetic Integrals and the irreducible Integrals of 

 a system of partial difterentlal equations, and after noticing that the 

 number of irreducible integrals is in general infinite, proceeds to 

 establish the above-mentioned theorem, viz. that a function of any 

 order and degree satisfying the necessary condition as to weight, and 

 such that it is reduced to zero by one of the operations {^dy} —xdy 

 and {ydx) —ydx, is reduced to zero by the other of the two opera- 

 tions, i. e. that it is a covariant ; and he shows how by means of the 

 theorem the actual calculation of the covarlants is to be effected. The 

 theorem gives at once (in terms of symbols P, P', which denote a 

 number of partitions) expressions for the number of the asyzygetic 

 covarlants of a given degree and order, or of a given degree only, of 

 a quantlc of any order ; this enables the discussion of particular cases, 

 but to obtain more general results it is necessary to transform the 

 expressions for the numbers of partitions by the method explained in 

 the author's " Further Researches on the Partition of Numbers." It 

 appears by the resulting formulae that the number of the irreducible 

 invariants or covarlants does in fact become infinite for quantics of 

 an order sufficiently high ; the number of the irreducible invariants 

 first becomes infinite in the case of a quantlc of the order 7 ; the 

 number of Irreducible covarlants first becomes infinite in the case of 

 a quantlc of the order 5. In particular, the formulae show that in 

 the case of a quantlc of the order 5, or quintic, there are 4 irre- 

 ducible invariants of the degrees 4, 8, 12 and 18, respectively con- 

 nected by an equation of the degree 36 ; and that in the case of a 

 quantlc of the order G, or sextic, there are 5 irreducible invariants 

 of the degrees 2, 4, 6, 10 and 15 respectively connected by an 

 equation of the degree 30 ; so that the system of the Irreducible in- 

 variants of a sextic is analogous to that of the irreducible invariants 

 of a quintic. The memoir concludes with a table of the covarlants 

 of a quadric, a cubic, and a quartic, and of certain of the covarlants 

 of a quintic. 



