ON THE THEORY OF NUMBERS. 329 



residues (mod j?) of a; and y\ or, if j9" be the highest power ofj9 dividing D, 

 for j9-"~> (j5 — 1) combinations of the residues of x and y, mod p". Again, 

 the 4 combinations of residues for the modulus 2 will give | {a,b,c) the va- 

 lues 0, I a^' 2 '^2' 3 0! + ^+ 2 '^j of which it is easily seen that one or three will be 

 uneven, according as ac =s 0, or '^, mod 8 ; i. e. according as D ^^ 1 , or 5, mod 8. 

 The combination of these results will give Dirichlet's theorem. 



101. Series expressing the number of Primitive Classes. — The sixth 

 section of the memoir contains the demonstration of the formulae which 

 express in the form of an infinite series the number of classes of properly 

 and improperly primitive quadratic forms of a given determinant. We shall 

 abbreviate the demonstration of these formulae by using the theorem of 

 art. 87. 



Let h be the number of properly primitive classes of determinant D; we shall 

 firgt suppose D to be negative, and = — A ; let also {a^,b^,c^), (a^, b^, c^), .... 

 (ok, b\, Ch) be a system of forms representing the properly primitive classes 

 of that determinant ; and let us consider the sum 



S = S. ..„..oA n. +^: ^ 



(a,x'+2b,xy+cy-y ' \a,a;'+2b,xy+c,yy^ ' ' ' " 



the sign of summation 'Z^ extending to all values of x and y from — oo to 

 + 00 , which give the form (%, h^, c^, a value prime to A. By the theorem 



of art. 87, any uneven number n prime to A is capable of 22( — ) repre- 

 sentations by the properly primitive forms of determinant D (for there are 

 2| -— J sets of representations, and each set contains two*). "We have there- 

 fore the equation 



«=-[K?)y w 



(the inner sign of summation referring to every divisor c? of w ; and the 

 outer sign extending to every positive value of « prime to 2A). If we write 

 n for d, and nn' for n, so that n and n' each represent any positive number 

 prime to 2A, this equation assumes the simpler form 



S=2S 



\Ji)(nniy ^^^ 



the sign S indicating two independent summations with respect to n and n' : 

 or, if we perform the two summations separately, and omit the accent, 



S=2S -S 

 n' 



(nji <'^> 



To deduce an expression for h from this equation, we write 1 +p for s, and 

 multiplying each side by p, we suppose p to be positive and to diminish 

 without limit. In order to find the limit of pS on this supposition, we con- 

 sider separately the partial sums, such as p2^ — 5 , , of which it 



(o.r- + 2oa:y + cy") • + p 



is composed. 



. * If A = l, each set contains four representations. To obtain a correct result in this case, 

 we must therefore double the right-hand members of the equations (a), (b), (c), and (A). 



