ON THE THEORY OF NUMBERS. 331 



lively. This equation is easily verified ; for if «^a, mod 2*', =^6, modp, 

 ^b', modp', ... we have 



80 that each member of the equation consist-; of the same units. But one at 

 least of the factors of which the right-hand member is composed is zero ; 

 unless we have simultaneously ^=1, e=l, P=l, a supposition which is inad- 

 missible, because it implies that D is a perfect square. We infer there- 

 fore that 2( — j=0, 1, e. that the sum of the terms of a period of the symbol 



/D\ . . /D\ 1 



( — I is equal to zero. If, then, we suppose the terms of the series S ( — )-f:j: 



to be taken in their natural order, it will follow from Dirichlet's second 

 Lemma (art. 99) that its sum represents a finite and continuous function of 



p for all values of p superior to — 1 ; i. e. the limit of the series S( - j_i_, 



for fl=0 is the series S ( - )-, in which the terms are taken in their natural 



\njji 



order. "We thus obtain the equation 



A=i^.(5)l (A) 



TT \n/n ^ f 



Secondly, let the determinant D be positive ; and let us retain the same 

 notation as in the former CEise. If in the series 



^ ^^\aKx'^<ihT,xy^cj,yy 



(in which it is convenient to suppose that the forms (oj., h^^ c!c)> representing 

 the properly primitive classes of determinant D, have their first coefficients 

 positive, and their last coefficients negative) we suppose the sign of double 

 summation S;. to extend only to those integral values of x and y which 

 render the value of the form (a^, b,^, c^) prime to 2D, and which further 

 satisfy the inequalities 



we obtain, by a comparison of arts. 87 and 100, the equation 



S = i:lEf5U; ; (e') 



in which n denotes any positive number prime to 2D, and which corresponds 

 to equation (c). 



If—-— be the nth term of the series S^ — — — -^ ottt-j « « equal to 



Jc^^f (ax^+2bxy+cy^y+p ^ 



the number of points having coordinates of any one of the forms 



