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

 and hence, Theorem, 



coeff. x m if in 



(l-*«y«)(l-arV)(l-tf^).. 

 1 



= coeff. x am - a > x in — r- a 



(1— x ab - a ^) (I— x ac - a v) . 



1 



+ coeff. a^ m_ *' i in 

 + coeff. a?v»-<v in 



(l-^°-* a ) (l—^ c -*y). 



1 



(1—^ya -) (1 -#**-<*).. 

 + &c, 

 the fraction of the left-hand side being expanded in ascending 

 powers of x, y, and those on the right-hand side being expanded 

 in ascending powers of x, and the data satisfying the above- 

 mentioned conditions. The number of partitions of (m, (j,) is 

 thus found to be equal to the expression on the right-hand side. 

 It is to be noticed that on the right-hand side, when any of the 

 indices um—afi, fim—bfi, . . is negative, the corresponding co- 

 efficient vanishes ; and that when the index of the power of x in 

 any factor of a denominator is negative, e. g. if ub—a/3=—p, 

 then (in order to develope in ascending powers of a?) we must in the 



1 1 . x p x 



place of - -t— s = -= write — , = — , and de- 



1 1 — x ab ~ afi 1—x-p x p — V l—x p 



velope in the form — (x p + x 2p 4- % 3p + . . . ). The right-hand side 



is thus seen to be the sum of a series of positive or negative 



numbers, each of which taken positively denotes the number of 



the single partitions of a given partible number into given parts. 



If, using a term of Professor Sylvester's, we say that 



coeff. m in (1 _ aa)(1 1 _^ ) .,. 



(where m, a, b, c . . are positive or negative integers, and the 

 fraction is developed in ascending powers of x) is 



= Denumerant of m in respect to the elements (a, b, c, . ..), say 

 = Denum* (m ; a, b, c . . ). 

 Then when m, a, b, c. are positive, but not otherwise, Denu- 

 merant (m ; a,b,c . . ) denotes the number of ways in which m 

 can be made up of the parts a,b,c... And the foregoing result 

 showa that the number of ways in which (m, fi) can be made up 

 of the parts (a, a), (b, /3), {c, 7), &c. is equal to the sum 

 Denum* (am—afi; ab—aj3, etc— ay, . . ) 

 + Denum* {ftm—b^; fta—boc, /3c— by, . . ) 

 4- Denunr* (ym—cft; ya—cu, yb—cfi,..) 

 + &c. 



