122 HISTORY OF THE THEORY OF NUMBERS. [CHAP. Ill 



x m a,ix m ~ l + d= a m = with the roots a;,-, the symmetric function 

 belonging to the partition (pi - p m ) is Sa^ 1 a". Part of a[a' 2 a p m is 

 the symmetric function 



to which belongs the partition (p + + 1, , p} conjugate to 

 (m p - -2'1'). Thus ojfl 3 , belonging to (31 3 ), contains with the coefficient 

 unity the symmetric function belonging to the conjugate partition (41 2 ), 

 and with other coefficients, the symmetric functions belonging to (321), 

 (2 3 ), (31 3 ), (2 2 I 2 ), (21 4 ), (I 6 ), but not (3 2 ). 



J. J. Sylvester 51 stated that the number of ways n can be composed 

 additively of the positive integers ai, , a, relatively prime in pairs, 

 differs by a periodic quantity depending on the remainder of n modulo 

 -a>i from 



1 n + i-l\ Itn + i -1\ 



1/n + i- 

 4V i-3 



4 



where Si, , Si-i are the coefficients oi x, , a;*" 1 in 



(x + 0,1 l)(a; + a 2 1) -(a; + * 1). 



For systems hke (a l} ) = (1, 2, 3) or (1, 3, 4), the residual periodic 

 quantity lies between | and ^, whence the number of partitions is the 

 integer nearest to Q n . 



Cayley 52 proved that the number of partitions into x parts, such that 

 the first part is unity and no part is greater than the double of the preceding 

 part, equals the number of partitions of 2 x ~ l 1 into the parts 1, I', 2, 

 4, ,2*- 2 . 



Sylvester 53 gave an explicit expression for "Sx*^ w*, summed for 

 all N sets of integral solutions of ax + by + + Iw = n, where a, , I 

 are positive integers. The case a = = =X = gives the number 

 N of sets. Let Q(F) denote the coefficient of I ft in the expansion of Ft 

 in ascending powers of t. Let ra be the l.c.m. of a, ,/. Then his 43 

 former theorem may be expressed in the form 



- AZ) r 



I (1 - Ao) ... (1 - AO . 



summed for the primitive mth roots p of unity, where Ap = pe~ pt . Then, 

 for examnlfi. 



for example, 



1 ' = 20 



I A(a)(l + Ao) ." (i - 1 + Aa)A(- n) \ 

 \ (1 - Aa) i+1 (l ' - A6) (1 AQ J 



81 Quar. Jour. Math., 1, 1857, 198-9. 



" Phil. Mag., (4), 13, 1857, 245-8; Coll. Math. Papers, III, 247-9. 



"Ibid., (4), 16, 1858, 369-371; Coll. Math. Papers, II, 110-2. 



