[ 25 ] 



II. A Supplementary Memoir on the Theory ofMatricesi 

 By A. Cayley, Esq., F.RS. 



Received Octolier 24, — Read December 7, 1865. 



M. Hermite, in a paper " Sur la theorie de la transformation des fonctions Abeliennes," 

 Comptes Rendus, t. xl. (1855), pp. 249, &c., establishes incidentally the properties of 

 the matrix for the automorphic linear transformation of the bipartite quadric function 

 xw' -{-y^ —zf—wa^ , or transformation of this function into one of the like form, 

 XW+YZ'— ZY'— WX'. These properties are (as will be shown) deducible from a 

 general formula in my" Memoir on the Automorphic Linear Transformation of a Bipar- 

 tite Quadric Function," Phil. Trans, vol. cxlviii. (1858), pp. 39-46 ; but the pax-ticular 

 case in question is an extremely interesting one, the theory whereof is worthy of an 

 independent investigation. For convenience the number of variables is taken to be four ; 

 but it will be at once seen that as well the demonstrations as the results ai-e in fact 

 applicable to any even number whatever of variables. 



Article Nos. 1 & 2. — Notation and HemarTcs. 



1. I use throughout the notation and formulae contained in my " Memoir on the Theory 

 of Matrices," Phil. Trans, vol. cxlviii. (1858), pp. 17-37, and in the above-mentioned 

 memoir on the Automorphic Transformation. With respect to the composition of 

 matrices, the rule of composition Is as. follows, viz., any line of the compound matrix is 

 obtained by combining the corresponding line of the first or further component matrix 

 with the several columns of the second or nearer component matrix ; it is very convenient 



to indicate this by the algorithm, 



(g, g', g"), (/3, /3', /3"), (y, y\ y ") 

 { a , b , c X a, , ji , y )=(« , b , c) 



a!,b',d u',(3',y' {a',V,d) 



a", b\ c" a", /3", y" {a", b", c") 



which exhibits very clearly the terms which are to be combined together ; thus in the 

 upper left-hand corner we have (a, b, cjjt, «', «s"), and so for the other places in the 

 compound matrix. 



2. It is not in the Memoir on Matrices explicitly remarked, but it is easy to see that 

 sums of matrices, all the matrices being of the same order, may be multiplied together 

 by the ordinary rule ; thus 



(A+B)(C+D)=AC+AD+BC+BD: 



this remark will be useful in the sequel. 



MDCCCLXVI. K 



