101 



fractional) of a matrix, and thence to arrive at the notion of a 

 rational and integral function, or generally of any algebraical func- 

 tion of a matrix. I obtain the remarkable theorem that any matrix 

 whatever satisfies an algebraical equation of its own order, the coeffi- 

 cient of the highest power being unity, and those of the other powers 

 functions of the terms of the matrix, the last coefficient being in fact 

 the determinant. The rule for the formation of this equation may be 

 stated in the following condensed form, which will be intelligible 

 after a perusal of the memoir, viz. the determinant, formed out of 

 the matrix diminished by the matrix considered as a single quantity 

 involving the matrix unity, will be equal to zero. The theorem 

 shows that every rational and integral function (or indeed every 

 rational function) of a matrix may be considered as a rational and 

 integral function, the degree of which is at most equal to that of the 

 matrix, less unity ; it even shows that in a sense, the same is true 

 with respect to any algebraical function whatever of a matrix. One 

 of the applications of the theorem is the finding of the general ex- 

 pression of the matrices which are convertible with a given matrix. 

 The theory of rectangular matrices appears much less important 

 than that of square matrices, and I have not entered into it further 

 than by showing how some of the notions applicable to these may 

 be extended to rectangular matrices. 



IV. "A Memoir on the Automorphic Linear Transformation of 

 a Bipartite Quadric Function." By ARTHUR CAYLEY, 

 Esq., F.R.S. Received December 10, 1857. 



[Abstract.] 



The question of the automorphic linear transformation of the 

 function a^+^ + r 2 , that is the transformation by linear substi- 

 tutions, of this function into a function xf+yf+sf of the same 

 form, is in effect solved by some formulae of Euler's for the transform- 

 ation of coordinates, and it was by these formulae that I was led 

 to the solution in the case of the sum of n squares, given in my 

 paper " Sur quelques proprie'tes des de'terminants gauches," Crelle, 

 t. xxxii. pp. 1 1 9-123 (1 846). A solution grounded upon an h-priori 



