CHAP. X] SUM AND NUMBER OF DlVISOKS. 311 He proved two theorems relating to the divisors of 1, 2, . . . , n: all cancel with the exception of —2, — 4,. . ., — ,(p — 2), each taken twice, p taken p — 1 times and — 0, if p be even; but with the exception of 1, 3, . . ., p— 2, each taken twice, and —p taken p—l tunes, if p be odd, where is the triangular number next >n\ g)(d± 1, d±3) +(Gn_10+ . . . +0-14) W, all cancel with the exception of k taken k times, for k = 1, 3, 5, . . . , p— 1, if p be even; and of —A; taken k times, for A; =2, 4, 6, . . ., p — 1, if p be odd; here zeros are ignored. The last two theorems yield (as before) corresponding relations for any even function % and any odd function \f/. Applying them to x(d+l) = (d+l)m and \l/(d)=dm, where m is odd, and in the first case dividing by 2(m+l), and modifying the right members, we get for the respective relations ^m-k(n} -2*+1(^-*(ra +34+1 (next three) - . . . where s= (m+l)/2 and <r,-(0) terms are suppressed; m^(n-2)^2fc (next three) + (!*+ 3*) (next four) 2m-|-1-4m-|-1-6wl+1- . . . -(p-i)^1, if p be odd, where, in each, k takes the values 2, 4, . . . , m — 1. These sums of like powers of odd or even numbers are expressed by the same function of Bernoullian numbers. For m = l, the first formula becomes that by Glaisher,04 repub-lished.67 Three further Gn formulas are given, but not applied to <rn. J. Hammond111 wrote (n; ra) = l or 0 according as n/m is integral or fractional, also r(x) —cr(x) =0 if x is fractional, and stated that 00 00 r(n/m) = 2 (n; jm), a (n/m) = S j(n; jm). y-i j-i "Messenger Math., 20, 1890-1, 158-163.