Mr* A. Cayley on a Problem of Double Partitions, 339 



a — 1 in y, and - is by hypothesis a fraction in its least terms. 



The coefficient therein of x m y* (the fraction being developed in 

 ascending powers of x, y) is 



= coeff. ^^ in ^^ 

 1— -x*y 



(the fraction being developed in ascending powers of x, y). In 

 fact the two fractions only differ by a wholly irrational function 



of x, as is at once obvious by developing — in ascending 



1— x*y 

 powers of y. We have, separating the integral part, 



A(x, 





l—x<xy 1—x^y 



where U is a rational and integral function of the degree a — 2 

 in y. But a being by hypothesis ■< fi -f 2, or what is the same 

 thing, a — 2 < p, U does not contain any term of the form x m y*, 

 and therefore 



coeff.a^^in A ^f ) 

 1 — x*y 



., . A(x, x « ) 



= do, in — 



a 



■X* 



And this last is 



\ut 



= coeff. x m in x* A(x, x «), 



= coeff. # w * in A(x, x «), 

 = coeff. ic^-^in A(x«, x~ a ). 



And from the foregoing equation 



_^ 1 



A(#, x «) = - 



(1— a? «)(!—# « 

 this is 



i 

 = coeff. x* m ~ a v- in 



(1— a?* 6 -^)(l— #**'-**) 

 The last-mentioned expression is thus the value ef 



coeff. **# in *&A} 

 u 1 — x a y a ' 



