1893.] Generating functions in the Theory of Aumlcrs. 363 



which is a function of the products, 



only, is l/V n , where (putting Si = s- z = r . . . 



a n ijxi, K , 



The proof of this theorem rests upon an identity which, for order 

 3, is 



^iXj, o, o, 



o, i 5 2 X 2 , o, 

 o, o, 1-53X3, 



auff> XQ _ 1 



1 5 x Xi 1 



1 5 2 X 2 ' 



=?)-!, 



^ : v r ~' 



1 S 2 X 2 



1 



v 



^3X3 



X 3 ) , 



-- ' 



and is very easily established. 



An instantaneous deduction of the general theorem is the result 

 that the generating function for the coefficients of xfix^* .... x n % n in 

 the product 



is 



The expression Y M involves the several coaxial minors of the de- 

 terminant of the linear functions. Thus 



The theorem is of considerable arithmetical importance and is also 

 of interest in the algebraical theories of determinants and matrices. 

 VOL. LTV. 2 c 



