n" 
300 History of the Theory of Numbers. [Chap, x 
Beginning with p. 425 and p. 430 there enter the two functions 
■.?,(i)-{© -'■'-}• 
in which (A/p) is Legendre's symbol, with the value 1 or —1. 
J. W. L. Glaisher^'* investigated the excess tr{n) of the sum of the rth 
powers of the odd di\'isors of n over the sum of the rth powers of the even 
divisors, the sum A'r(n) of the rth powers of those divisors of n whose 
complementary divisors are odd, wrote f for f i, and A' for A'l, and proved 
A'3(n)=nA'(n)+4A'(l)A'(n-l)+4A'(2)A'(n-2)+. . .+4A'(n-l)A'(l), 
r3(n) = (2n-l)r(n)-4r(l)r(n-l)-4r(2)r(n-2)-...-4r(n-l)f(l), 
nA'(n)=A'(l)A'(2n-l)-A'(2)A'(2n-2) + . . .+A'(2n-1)A'(1), 
(-l)''-H(n)=A'(n)+8r(l)A'(n-2)+8f(2)A'(n-4) + . . ., 
A'3(n)=7zA'(n)+A'(2)A'(2n-2)+A'(4)A'(2n-4) + . . .+A'(2n-2)A'(2), 
-f3(7i)=3A(n)+4{A(l)A(n-l)+9A(2)A(n-2)+A(3)A(n-3) 
+9A(4)A(n-4)+ . . . +A(n-1)A(1)) {n even), 
2^-W2.+i(n) _ [l,2r-l] [3, 2r-3] [2r-l, 1] 
(2r)! l!(2r-l)!"^3!(2r-3)!'^""'^(2r-l)!l!' 
where 
b,?]=(7p(lK(2n-l)+(rp(3K(2n-3) + . . .+(Tp(2n-lK(l). 
For n odd, f (n) =A'(n) =a{n) and the fourth formula gives 
(/i-lM7i)=8{(7(lMn-2)H-r(2M7i-4)+(7(3Mn-6)+r(4Mn-8) + . . .). 
Glaisher'''^ proved that 
5o'3(n) — 6w(7(n) -\-(j{n) 
= 12{(7(l)(7(7i-l)+o-(2)(r(n-2)+...+(7(n-l)(T(l)), 
(r(l)(r(2n-l)+(r(3)(r(2n-3) + . . .+or(2n-l)(T(l) 
=A'3(n)=|{cr3(2ii)-(T3(n)}. 
The latter includes the first theorem in his^^ earUer paper. 
Glaisher'^^ proved for Jacobi's^^ E{n) that 
cr(2m + l)=E(l)E(4m + l)+£;(5)^(4w-3)+E(9)E(4w-7) + . . . 
+£(4m + l)E(l), 
E(0-2^(^-4)+2E(<-16)-2E(«-36) + . . . =0 (« = 8n+5), 
(7(y)-2(r(y-4)+2(7(y-16)-2(7(y-36) + . . . =0 (y = 8n+7), 
<t(w) +(t(?/- 8) H-(r(i/- 24) +(r(w- 48) + . . . =4{(r(w)+2(T(m-4) 
+2(r(rn-16)+2(7(m-36)+. . .) (m = 2n + l, w = 8n+3), 
and three formulas analogous to the last (pp. 125, 129). He repeated 
(p. 158) his^'* expressions for A'3(n). 
^^Messenger Math., 14, 1884r-5, 102-8. 
"•Hbid., 156-163. 
'•Quar. Jour. Math., 20, 1885, 109, 116, 121, 118. 
