338 Mr. A. Cayley on a Problem of Double Partitions. 

 or what is the same thing, we have 



A (*> *"■«)= ~aj a~y ' 



This in fact determines A(x, y) ; for the right-hand side of the 

 equation may be rdduced to the form 



_o (a-l)a 



A 4- A,a? « i . . + A a _iic « 

 (1— #«*-«£) (1 — a?« c -«y) . . . ' 



where A , A x . . . A a _! are rational functions of x : to do this, it 

 is only necessary (taking o> an imaginary a-th root of unity) to 

 multiply the numerator and denominator by 



n{l-cox b -f)II(l--cox c - a i) 



where II denotes the product of the factors corresponding to the 

 a — 1 values of m \ the denominator is thus converted into 



(i-A b - a £)(i-A c - a «)) . . . , 



which is of the form in question ; and the numerator becomes 



a a 



a rational function of x and af~«, integral as regards #~«, and 

 therefore at once expressible in the form in question. And the 

 equation, viz. 



_a (a— l)a 



Ate S -K A ° +A ' a ' ; --+A,-.» -s" 



_a a 



remains true if instead of x « we write <w# _ « ; in fact, instead 



_a 



of writing in the first instance y=x «, it would have been allow- 



m 



able to write y = oox «, to being any a-th root, real or imaginary, 

 of unity. Hence recollecting that A(x, y) is a rational and inte- 

 gral function of the degree a — 1 in y, the equation 



A(x ,_ A +A,y...+A«. li r- 1 



which is true for the a values <»# « of y, must be true identi- 

 cally; or this equation gives the value of A(x, y). And the 

 values of B(x, y), C(x, y) } &c. are of course of the like form. 

 Now consider the term 



A(x, y) 



1 —x a y«' 



where A(x, y) is a rational and integral function of the degree 



