CHAP, ill] PARTITIONS. 131 



summed for all solutions of Xi + 2A 2 + + p\ p = p. At the end of 

 this 12, he gave other expressions for C p . He 93 later transformed the 

 above formula into 



p\C p = 



\ J. 



(1 - x n+1 ) 



where [X P ~\T denotes the coefficient of x p in r, while, after the expansion, 

 5* is to be replaced by i\. Similarly, for the number W p of sets of positive 

 integral solutions of a^i + + a n x n = p, 



p\W p = [> p ]{5 - log (1 - x ai ) (!- of")}", 



which is much simpler to apply than Sylvester's 43 formula. He stated 

 (p. 1259) the generalization to two variables : 



'* V) = rr^ O p 2/ 9 ]( 5 + (5 + log ^) p } 9 . 



F. Franklin 94 proved that if, in all the partitions of n which do not con- 

 tain more than one element 1, each partition containing 1 be counted as 

 unity and each partition not containing 1 be counted as the number of 

 different elements occurring in it, the sum of the numbers so obtained is 

 the number of partitions of n 1. Application is made to the distribution 

 of bonds between atoms. 



A. Cayley 95 noted that the partition abc-def of 6 letters into 3's contains 

 6 duads ab, ac, be, , while the partition ab-cd-ef into 2's contains 3 

 duads. Hence if a partitions into 3's and /3 partitions into 2's contain all 

 15 duads once and but once, 6a + 3/3 = 15. The solution a = 1, = 3, 

 furnishes an answer of the partition problem: abc-def, ad -be -of, ae-bf-cd, 

 af-bd-ce. Likewise for a = 0, = 5; but not a 2, 1. Similarly 

 for 15 or 30 letters. 



J. J. Sylvester 96 considered the e = (w; i, j) partitions of w into j parts 

 0, 1, , i, the elements of a partition being arranged in non-increasing 

 order, as 3, 2, 2. Without computing e and f = (w 1; i, j) separately, 

 we obtain e f = E F, by counting the E partitions of w in which the 

 initial two parts are equal, and the F partitions of w 1 in which one 

 element is i. Also, 



3=0 



F. Franklin 97 proved this rule of Sylvester's by converting each partition 

 into one consisting of i of the numbers 0, 1, , j. Then e f = e 0, 



93 Comptes Rendus Paris, 86, 1878, 1189, 1259; Jour, fur Math., 85, 1878, 317-26; Math. 



Annalen, 14, 1879, 241-7; Quar. Jour. Math., 15, 1878, 272-4. 



94 Amer. Jour. Math., 1, 1878, 365-8. 



95 Messenger Math., 7, 1878, 187-8; Coll. Math. Papers, XI, 61-2. 



96 IUd., 8, 1879, 1-8; Coll. Math. Papers, III, 241-8. 



97 Amer. Jour. Math., 2, 1879, 187-8. 



