»Jl 



00 



ON THE THEORY OF NUMBERS. 



oo ^i- 1 "* 



? H0 J=4 Q+l)g ' rin^+l)^ 



1 9 2 n l + <f" +1 



CO 



H=2S ( — l)"2i (2 " +lr sin(2>i+ l).r, 

 



361 



(") 



J 



00 



a.i+1 



" 10l i =4 or^ lSiu(2H+1>v ' 



0! 



H0!_ 



co 



2zQ, l ^ ,2 " +lr sin(2n + ]>r 



e o 



where Q„=l + 2?- 1 + 2gH + ... + 2rf 



(iii) 



f "" H 2 e 



and designating the definite integral r)fi\ I — — ,— dx by ttJ, we obtain 



Jo ©" 



oo 



oo 



>h 



0j = 



J = 8S-2iL-8S^ 



U-+H 



1 W 



» (2n + l)^»+»< 2 "^ 



5 C 1; ■ "T+7^ ' 







co 



Let A(n) represent the sum of those divisors of n whose conjugates are 

 uneven, and ri(«) the sum of those divisors of n which do not surpass Vn, 

 and which are even or uneven, according as their conjugate divisors are un- 

 even or even ; we find immediately 

 co 

 9l J=8S[A(»)-ri(»)]2"j 

 1 

 also 



CO 4»+3 



eJ=2S(-l)"[«l>(4H-r-3)-*(4n + 3)]2 4 , 

 



co -f- n co 

 J = 4S„ ^ Sg^iKsn+iXSn+^+s)-^] 



— »0 



CO 4»+3 



=4ZF(4n + 3)g 4 » 

 

 if F' (4n + 3) represent the number of solutions of the equation An + 3 = ac— b 2 , 

 in which a and c are positive and uneven, a is less than c, b is even, 

 and less in absolute magnitude than a. But M. Hermite has shown that 

 F(4>i + 3) = F(4n -1-3). For F' {An -f 3) is evidently the number of quadratic 

 forms {a, b, c) of determinant — (4n+3), in wbich the second coefficient is 

 even, and less than either extreme coefficient, and in which also the first co- 

 efficient is less than the third. But each uneven reduced form is equiva- 

 lent to one, and only to one, of the forms (a, b, c). For the reducing trans- 



