CHAP, ill] PARTITIONS. 117 



expansions of P 2 and P 3 . Only those ra-gonal numbers give the remainder 

 1, when divided by m = orb, whose side has the remainder 1 when divided 

 by ab, where a 2 is the greatest square dividing m. 



H. Warburton 32 considered the number [/V, p, 77] of partitions of N 

 into p parts each ^ 77, and proved that 



IN, P, 17] - IN, P,-n + 1] = CAT - * P - i, i?l 



[TV + p, P) 1] = [tf , z, 1], [AT, p, ,] = [^ - p17j z, i], 



p, i] = IN - i, P - i, i] + [AT - p - i, p - i, i] 



to [_Njp~\ terms. He applied these formulas to the construction of a table 

 of partitions and proved that the number of partitions of x into three parts 

 is 3i 2 , ' 3Z 2 d= t, 3t 2 2t, 3t 2 + 3t + 1 according as x = Qt, Qt 1, 6Z 2, 

 6 + 3 [in accord with De Morgan]. 



J. F. W. Herschel 33 recalled his 34 earlier notation s x = s~ l 2o> x , where a 

 ranges over the sth roots of unity, so that s x = 1 or according as x is or 

 is not divisible by s. Then A x s x + B x s x -i +.+ N x s x - a+ i will circulate 

 in its successive values as x increases by units from zero, being A x when x 

 is divisible by s, but B x when x 1 is divisible by s, etc. If A x , etc., are 

 constants, the function is called periodic. He wrote s U(x) for the number 

 (x, s) of partitions of x into s parts > 0. Starting with 



(x, s - 1) = 0(z) -f Q x , 



where <f>(x) is the non-periodic part and Q x the periodic or circulating func- 

 tion, and applying the final formula quoted from Warburton, he obtained 



(x, s) = A + Z, A = 4>(x - 1) + 4>(x - s - 1) + , 

 Z = Q,_i + &_ s _i + , 



each extending to \jc/s~] terms. Then A is expressed explicitly in terms of 

 the numbers A^O", giving the mth order of difference of z n for z = 0, while Z 

 is expressed in terms of these numbers and the above circulating functions s x . 

 He deduced explicit expressions for (a;, s), s = 2, 3, 4, 5, as (x, < 2} = ^(x 2 x , l } J 



( X) 3) = ^{3* _ 6 Z _! - 4-6,_ 2 + 3.6,_3 - 4- 



which, with the expression for (x, 4), are in accord with the results by 

 De Morgan, 28 although the latter was not treating partitions into s parts. 

 While the method of Herschel is laborious, it anticipated to some extent 

 the simpler method of Cayley. 44 



J. J. Sylvester 35 quoted Euler's theorem that the number of partitions 

 of n is the same whether the number of parts is == m or every part is ^ m, 

 and noted that, if we apply the theorem also when the limiting number is 



32 Trans. Cambridge Phil. Soc., 8, 1849, 471-492. 



33 Phil. Trans. Roy. Soc. London, 140, II, 1850, 399-422. 



34 From his paper on circulating functions, ibid., 108, 1818, 144-168. 



35 Phil. Mag., (4), 5, 1853, 199-202; Coll. Math. Papers, I, 595-8. 



