130 HISTORY OF THE THEORY OF NUMBERS. [CHAP, in 



P. Mansion 87 noted that the /cth power of an integer n is the sum of 

 n consecutive odd numbers (those nearest n*" 1 ), as 3 4 = 25 + 27 + 29. 



J. W. L. Glaisher 88 stated that, if C m is the number of compositions of 

 N into m triangular numbers, and A is the sum of the reciprocals of those 

 divisors of N whose conjugates are odd, B if even, then 



GI 562 T~ "3^3 -^ GAT = A C. 



Glaisher 89 noted that, if P(x) is the number of partitions of x into 1, 2, 

 3, , repetitions allowed, and Q(x) is the number of partitions of x into 

 1, 3, 5, 7, -, repetitions excluded, then Q(x) = 2P{(x 0/4}, summed 

 for the triangular numbers t < x such that t = x (mod 4). 



Glaisher 89 " used an identity due to Jacobi, 226 p. 185, to show that 



P(x) + 2P(x - 1) + 2P(x - 4) + 2P(x - 9) + 



= Q(x) + Q(x - 1) + Q(x - 3) + + Q(x - %n(n + !))+, 



if P(x) is the number of partitions of x into even elements without repeti- 

 tions, and Q(x) the number into odd elements without repetitions. 



A. Cayley 90 denoted by u n the number of partitions of n with no part < 2 

 and order attended to. Then u* = u s = l,u n = u n -\ + u n - z . 



E. Laguerre 91 started with Euler's result that the number T(N) of 

 sets of positive integral solutions of ax + by + = N is the coefficient 

 of n in 



decomposed the latter into partial fractions, and called the result 

 where $() is the sum of the simple fractions whose denominator is a power 

 higher than the first of one of the factors in the denominator of F(). Let 

 0(AO denote the coefficient of N in the expansion of $(). Then 



T(N) = Q(N), 



with an error which is independent of N. For example, if ax + by = N 

 and a, b are relatively prime, Q(N) = (N + l)/(a&), so that T(N) = N/(db) 

 approximately [Paoli 117 of Ch. II], the error being < 1. For 



ax + by + cz = N, 



the approximation is N(N + a + b + c)/(2a6c). 



F. Faa di Bruno 92 gave an exposition of Brioschi's 47 work and noted 

 that his linear equations (7) are of the same form as Newton's identities if 

 the sign of s,- be changed. Hence, by Waring's formula, 



87 Messenger Math., 5, 1876, 90. Cf. Fre"gier. 22 

 **Ibid., 91. 



89 Ibid., 164-5. 



89ft Math. Quest. Educ. Times, 24, 1876, 91. 



90 Messenger of Math., 5, 1876, 188; Coll. Math. Papers, X, 16. 



91 Bull. Math. Soc. France, 5, 1876-7, 76-8; Oeuvrcs, 1, 1898, 218-20. 



92 Th6orie des formes binaires, 1876, 157; German transl. by T. Walter, 1881, 127. 



