380 



May 24, 1855. 

 The LORD WROTTESLEY, President, in the Chair. 



The following communications were read : 



I. "A Second Memoir upon Quantics." By ARTHUR CAYLEY, 

 Esq., F.R.S. Received April 14, 1855. 



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

 ductory memoir upon Quantics (vide Proc. R.S. p. 58, and Phil. 

 Trans. 1 854, 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 covariants of a binary quantic. The author, after some 

 remarks as to the asyzygetic integrals and the irreducible integrals of 

 a system of partial differential 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 {xdy} 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 covariants 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 

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

 a quantic 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 covariants does in fact become infinite for quantics of 

 an order sufficiently high ; the number of the irreducible invariants 



