140 HISTORY OF THE THEORY OF NUMBERS. [CHAP, ill 



n = %> m = \ + e > where e is infinitesimal, we get 



a result due to Jacobi 10 of Ch. X in Vol. I of this History. Sylvester wrote 

 Jacobi's initial formula in an equivalent form by setting n m = a, 

 n + m = 6, and discussed at length (here and elsewhere 120 ) the new formula 

 from the standpoint of arrangements of three kinds of elements. He noted 

 (p. 53, p. 70) that Euler's formula (3) is the special case a = 1 of 



i _i 



(1 + ax)(l + ax 2 )(l + ax 3 ) - = 1 + - -xa H 



1 x 



(1 + as) (1 + qgj-1) (1 -f- 



(l-z).--(l-^- 1 ) 1 - a? 



which was given elsewhere by Sylvester 121 and proved also by Cayley. 122 



Chr. Zeller 123 stated Euler's 13 recursion formula for P(n) and expressed 

 the number <r(ri) of divisors of n in terms of the P(j), j < n. [See Vol. I 

 of this History, p. 290, Catalan, 42 p. 292, p. 312, Glaisher, 55 ' m p. 303, 

 Stern. 85 ] 



E. Cesaro 124 noted that aiXi + + a k x k = n has n k ~ l /{ai- *.(& 1)!} 

 sets of positive integral solutions, in mean. 



J. W. L. Glaisher 125 noted that Euler's theorem that there are as many 

 partitions without repetitions as into odd parts follows from the case r = 2 

 of the fact that the number of partitions of n, in each of which a part occurs 

 at least r times, equals the number of partitions of n in each of which either 

 r or a multiple of r occurs. In the proof, a repeated term is replaced by its 

 expression to base r (Glaisher 86 ). If P(ri) is the total number of partitions 

 of n, and Q r (n) is the number of partitions of n in which no part occurs 

 more than r times, 



P(n)-P(n-r)-P(n-2r)+P(n-5r)+P(n-7r') ---- =Q r -i(ri), 



n-7 ---- =0 or -! 



according as n is or is not of the form (3m 2 m)(r + l)/2, and 



P(0) = Q r (0) = 1. 



Write Q = Qi. There are given recursion formulas for Q, and 

 Q(2m) = P(n) + P(n - 3) + P(n - 5) + P(n - 14) + 



involving halves of triangular numbers; similarly for Q(2m + 1). 



M. A. Stern 126 proved that the number of variations [with attention to 

 the arrangement of the parts] with the sum n formed from two elements 1 

 and m equals the number of variations with the sum n + m formed from 

 all elements ^ m. This is the analogue of Euler's 9 second theorem. 



120 Comptes Rendus Paris, 96, 1883, 1276-80; Coll. Math. Papers, IV, 97-100. 



121 Comptes Rendus Paris, 96, 18S3, 674, 743-5; Coll. Math. Papers, IV, 91, 93-4. 



122 Amer. Jour. Math., 6, 1884, 63^; Coll. Math. Papers, XII, 217-9. 



123 Acta Math., 4, 1884, 415-6. 



124 Mem. Soc. R. Sc. de Liege, (2), 10, 1883, No. 6, 229. 

 126 Messenger Math., 12, 1883, 158-170. 



126 Jour, fur Math., 95, 1883, 102-4. 



