CHAP, ill] PAETITIONS. 119 



partitions of x into k parts > 0, for k = 2, , 6, in terrns of the circulator 

 s e , which is unity if e/s is a positive integer, zero if e/s is fractional or negative. 

 For k = 5, his results are identical with those of Herschel, 33 but were ob- 

 tained by more elementary methods. Kirkman 40 corrected his expression 

 for (x, 6) and found (x, 7). He 41 found (r 2 - r + 1, r). 



J. J. Sylvester 42 called the number of ways of composing n with given 

 positive integral summands a\, -, a r the quotity Q of n with respect to 

 Oi, , a r . Thus Q is the number of sets of integral solutions ^ of 



He stated that Q = A + U, the periodic part U (depending on roots of 

 unity) not being discussed, while the non-periodic part A is the coefficient 

 of 1/t in the expansion of 



e nt (l - e-^O" 1 - (! - e-*')- 1 . 



Other formulas for A are given. But all these formulas were provisional 

 and were replaced in his next paper by others more expeditious for com- 

 putation. 



Sylvester 43 stated that Q = 2TF g , where W q (called a wave) is the 

 coefficient of 1/t in the development in ascending powers of t of* 



summed for the various primitive qth roots p of unity. Thus W q = 

 except for a q which divides one or more of the a*. Thus W\ is his former A. 

 Taking the a's to be 1, , 6, Sylvester computed Wi, -, TF 6 initially in 

 terms of certain Sp fc and finally in terms of HerschePs 34 circulating functions, 

 obtaining results agreeing with Cayley's. 44 



But Sylvester did not give a full account 107 of his theorem until 1882. 



A. Cayley 44 employed P(a, b, )<?> i n the sense of Sylvester's Q, to 

 denote the number of partitions of q into the elements a, b, , with 

 repetitions allowed. As known, it is the coefficient of x q in n(l a: )" 1 . 

 By decomposing the latter into partial fractions, it is shown that 



P(a t b,---)q = Aq k ~ l + Bq k ~ 2 H ---- + Lq + M + 2q r (A Q , A lt - -,A^pcrl q , 



where k is the number of the elements a, b, , and I is any divisor > 1 of 

 one or more of these elements, and the summation extends, for each such 

 divisor, from r = Qtor = x 1, if a; is the number of elements a, b, 

 having I as a divisor. Also 



(A Q , , Ai^pcrlq = A a q 



40 Mem. Lit. Phil. Soc. Manchester, (2), 14, 1857, 137-149. 



41 Proc. and Papers Lancashire and Cheshire Hist. Soc. Liverpool, 9, 1857, 127. 



42 Quar. Jour. Math., 1, 1855 (1857), 81-4; Coll. Math. Papers, II, 86-9. 



43 Ibid., 141-152; Coll. Math. Papers, II, 90-99. An Italian transl. of an extract appeared 



in Annali di sc. mat. e fis., 8, 1857, 12-21. 



* Sylvester's first factor p n has been changed to p~ n to accord with Battaglini, 48 Brioschi, 49 

 Roberts, 61 and Trudi. 66 



44 Phil. Trans. Roy. Soc. London, 146, 1856, 127-140; Coll. Math. Papers, II, 235-249. 



