ON THE THEORY OF NUMBERS. 369 



2 contained in 2 a m-2m' 2 . Let/(.v)=l, if .r=0, but let/(.r)=0 for every 

 other value of a? ; the sum 



{^HW] 



will then represent the number of solutions of the equation (A) ; the sum 

 2 v ~ l 2,df(2m') will become 2 a ~ 1 2.d, the summation extending to all solutions 

 of the equation m=dS ; and the sum 2 y ~ i 2,df(2m' + 2 y d) will become 22 y_I i7, 

 the summation extending to all solutions of the equation 



2 a - l m = 2y- 1 d(2 y - i d + 2). 



Of these sums 2 a ~ l 2d is evidently A(2 a_1 m)=|A(2 a+1 m) ; and 22 y_1 disthe 



sum of those divisors of 2 a ~ l m, which arc less than v2 a ~'m, and which are 

 not of the same parity as their conjugates, a sum which is identical with 

 |r(2 a+1 «i) + |r'(2 a+1 m) ; as may be seen by considering separately the cases 

 in which a=l, and a>l. 



A second determination of the number of solutions of the equation (A) is 

 obtained as follows. Write 29 + 1 for i(d, z + 8 2 ) and 4« for d 2 — $ 2 ; it 

 becomes 



2 a+, m-mi=(2d + lX2d + l + 2Z 3 )-4« 2 , 



which is of the same form as that considered by M. Hermite (see art. 134). 

 If then we attribute to m x any particular value, the number of solutions of 

 the equation (A) is F(2 a+, »» — mf) ; its whole number of solutions is therefore 



F(2 a+1 m-l 2 )4-F(2 a+, m-3 2 ) + F(2 a+, m-5 2 ) + . . ; 



equating this expression to that which we have already found, we obtain the 

 formula (XI.)-(XIL). 



M. Liouville tells us that, until M. Hermite's discussion of the equation 

 4H-r-3=(20 + l)(20 + l + 2S 3 )-4« 2 , 

 he had not observed that the number of solutions of the equation 

 2 a+1 m-ml=d.&+(d 2 + S 2 )S i , tf 2 =S 2) mod 4, 



is equal to the number of classes of quadratic forms of det. mf— 2 a+l m; but 

 that with this exception all the principles of the preceding demonstration were 

 in his possession ; so that he had already arrived at formulae identical with 

 those of M. Kronecker, but referring to the numbers of solutions of certain 

 indeterminate equations instead of to the numbers of quadratic forms of certain 

 determinants. We also learn from him that formula? exist, analogous to 

 those of M. Kronecker, in which the series of determinants are of the type 

 2s 2 — n, 3s 2 — n, . . . instead of s 2 — n. 



137. Equations satisfied by the Modules which admit of Complex Multipli- 

 cation. — We have already observed (art. 120) that the 6G(A) values of <j> (o>), 

 corresponding to the quadratic forms of det. — A, are the roots of an equation 

 of that order, having rational coefficients. Several important properties of 

 this equation have been indicated by M. Kronecker ; but, notwithstanding 

 their intimate connexion with the theory of quadratic forms, we can only 

 offer an imperfect account of them. 



We resume the notation of art. 131 ; and we shall begin b} T showing that 

 if n is an uneven number, greater than 3, the values of ^(io), corresponding 

 to the properly primitive classes of det. — n, satisfy one or other of three 

 equations, each of the order 2h(n), and each having rational coefficients. We 

 have already seen in art. 131, that every value of © 3 (o>), corresponding to a 



1865. 2 c 



