GENERATING FUNCTIONS IN THE THEORY OF NUMBERS. HO 
Al’t. 10. The coefficients of the terms 
{s^x^y^ {s.2x.2y^ ... {s„x„y\ 
in the expansions of both fractions, are the same. 
Hence, the coefficient of the product 
? 
in the expansion of algebraic fraction 
_1_ 
I (I (1 . • • (1 I 
is equal to the same coefficient in the product 
(cqa?! + . . . + a>,x„y' {b^x^ + . . . + b„x„y^ . . . {n^x^ + . . . + n„x,)^\ 
where this product is a “ particular redundant generating function,” the use of which 
renders the quantities s^, So, . . . s„ unnecessary to the statement of the theorem. 
Art. 11. The theorem regarded as a proposition concerning the coaxial minors of a 
general determinant is very remarkable ; for it will be observed that we are able to 
exhibit the coefficient of 
/Y» ^2 <Y* 
in the “ particular redundant generating function ” as a function of the coaxial minors 
of the determinant of the n quantities. 
^ 2. Arithmetical Interpretations. 
Art. 12. Most of the arithmetical results that can be deduced arise from duality 
of interpretation from algebra to arithmetic in particular cases. In the memoir to 
which reference has been made two particular cases presented themselves. 
Art. 13 . The first one was connected with the matricular relation 
(Xi, Xg, X3 . . . X„) — [k, 1, 1, 
k, 1 , 1 , 
k, k, k, 
1) Xg, Xo . . . .a,,). 
1 
■K 
J 
1 
1 
