CHAP, ill] PARTITIONS. 159 



proved that the last circulator of G n (x, a p j3 q - ) is the same for all forms 

 a p (3 Q -, and hence need be computed for the form a n only, which case is 

 treated at length. 



Glaisher 199 proved Sylvester's theorem on waves, developed the formulas 

 for waves of periods 3, 4, 5, 6, and treated the non-periodic terms. 



Glaisher 200 noted that his 198 formulas for the number P(l, , ri)x of 

 partitions of x into 1, , n, repetitions allowed, are greatly simplified if 

 expressed in terms of = x + \n(n + 1) instead of x and gave the simplified 

 formulas for n ^= 9, and also those in terms of X = 2% for n = 2, 5, 6, 9. 

 He proved (p. 104) that 



(-l)-ip(l, ..., n )(-aO =P(1, .--,n){aj-ln(n + l)} = Q(l, 2, --.)*, 



where Q is the number of partitions of x into elements 1, 2, , unlimited 

 in number, each partition containing exactly n parts without repetition. 

 He proved (p. 106) that, if in the circulators occurring in the ^-formulas, 

 the order of the elements be reversed, the original circulator is reproduced 

 except as to sign. Finally, he gave the leading circulator in each wave 

 W m (l, 2, ...,ft + r). 



E. Barbette 201 noted that there are exactly 2(2 X ~ 2 1) ways of partition- 

 ing x + a into distinct parts the greatest of which is x, where 



a = S x - R, 1 ^ R ^ %x(x - 1) - 1, S x = 1 + 2 + + x. 



In fact, such a partition of x + a corresponds to a partition of a into distinct 

 parts each < x. Next, to find all the partitions of N into distinct parts, let 

 x be the least integer for which S x ^ N, and convert S x , S x+ i, -, S N -i 

 into sums of distinct numbers of which the greatest is N and such that all 

 the other parts are less than x, x -\- 1, , N 1, respectively. Suppress 

 the parts in common to two members of the resulting equalities. Finally, 

 to find all sets of consecutive integers whose sum is N (as 8 + 9 = N), 

 write 1, 2, 3, along the diagonal of a square; above x in the diagonal 

 write the sum 2x 1 of x and the preceding term x 1 ; above that sum 

 write the sum 3x 3 of it and the number x 2 preceding it in the former 

 list; etc., until 1 is added. Cf. Sylvester. 119 



P. Bachmann 202 gave an extended clear account of the literature on 

 partitions. He inserted (pp. 109-110) a theorem communicated to him 

 by J. Schur: If S is any set of positive integers not divisible by r, and R is 

 the set of numbers obtained by multiplying the numbers in S by 1, r, r 2 , , 

 then any positive integer can be partitioned into equal or distinct parts 

 chosen from S as often as into parts chosen from R, each occurring at most 

 r 1 times. The case r = 2 gives Euler's theorem that any integer can 

 be partitioned into equal or distinct odd integers as often as into any distinct 

 parts. 



199 Quar. Jour. Math., 40, 1909, 275-348. 



200 Ibid., 41, 1910,94-112. 



201 Les sommes de p-iemes puissances distinctes 6gales a une p-ieme puissance, Liege, 1910, 



12-19. 



202 Niedere Zahlentheorie, 2, 1910, 102-283. 



