304 HlSTOEY OF THE THEORY OF NUMBERS. [CHAP. X sth powers of the divisors of n whose complementary divisors are even, the excess f '8(n) of the sum of the sth powers of the divisors whose complementary divisors -are odd over that when they are even, and the similar 'functions74 A'a, f „ <rg. The seven functions can be expressed in terms of any two: where the arguments are all n. Since D'8(2k) =<rs(k), we may express all the functions in terms of <r8(n) and <r,(n/2), provided the latter be defined to be zero when n is odd. Employ the abbreviation S/F=SjF/ for This sum is evaluated when/ and F are any two of the above seven functions with s = 1 (the subscript 1 is dropped). If then By using the first formula in each of two earlier papers,74'75 we get ' = 5D3'(n) -24Sol)' = 2<r3(n) + (1 -3n)(r(n) + (1 -6n)Z)'(n) +8£>3'(n). Hence all 21 functions can now be expressed at once linearly in terms of or3, D3', <r and D'. The resulting expressions are tabulated; they give the coefficients in the products of any two of the series Z*/(n)af, where /is any one of our seven functions without subscript. Glaisher88 gave the values of So"3<rt- for i = 3, 5, 9 and Sor5cr7, where the notation is that of the preceding paper.. Also, if p = n— r, 12 2 r/xr(r)er(p) =nV3(n) -nV(n), S r/(r)F(p) =^2/F. r-l r=l 2 L. Gegenbauer89 gave purely arithmetical proofs of generalizations of theorems obtained by Hermite70 by use of elliptic function expansions. Let &(r)= S jfc, er= S ;-l x=l Then (p. 1059), The left member is known to equal the sum of the kth powers of all the divisors of 1, 2, . . ., n. The first sum on the right is the sum of the Mi powers of the divisors ^ \/n of 1, . . . , n. Hence if Ak(x) is the excess of the 88Messenger Math., 15, 1885-6, p. 36. "Sitzungsberichte Ak. Wien (Math.), 92, II, 1886, 1055-78.