ON THE THEORY OF NUMBERS. 367 



For, eliminating c and d", we find that (a) has as many solutions as 

 2 a ~ 1 m=vd' + pS" has solutions in which (?'<2ju ; and (a') has as many solu- 

 tions as the same equation has solutions in which <S" <2v : i.e. (a) has as many 

 solutions as (a'); inasmuch as to eYery solution of 2 a ~ l m = vd'+ftB" in 

 which d'<2fi, hut 2">2»<, there corresponds a solution, in which d'>2fi, 

 hut $"<2v, and vice versa ; for example, if d'<2fi, but 1">2 V , let 2hv be the 

 multiple of 2v next inferior to I" , then 



2 a m = v(d' + 21-fi) + / u(S'-2i>) 

 is a solution of the equation, in which d' -\-2l-^i>2^, but o — 2A.x2e. 

 Similarly it will be seen that the solutions of the systems 



d'&+d"Z"=2 a m 

 d' + d" =2u 

 V-B" =-2 v 

 d!l' +d"l" =2*m~ 

 d'-d" =2/x 

 ^ + 3" =2v - 

 are eqtial in number. 



Also the number of solutions of either of the systems 



d'V+d"h"=2 a mT 



d'+d" =2Hl . . . 



i'=r =i _ 



d'l' + d"l"=2«m- 

 d' = d" =3 

 3' + 3" =2«(7_ 

 in each of which d, $ are two given conjugate divisors of m, is 2 a ~ 1 d. 



Let us now attribute to p, v, d, I in the systems (a), (b), (c), (a'), (&'), (c'), 

 all values, in succession, for which those systems are resoluble. We shall 

 evidently obtain the sum If(d' + d"), which occurs in the equation to be proved, 

 by extending the summation, first, to all solutions of the various systems 

 («), secondly, to all the solutions of the various systems (b), and lastly, to all 

 solutions of the various systems (c). Similarly we shall obtain the sum 

 2f(d'— d") by extending the summation to all solutions of the systems (a'), 

 (b), (V). But the terms f( d' + d") arising from any one of the systems (a) 

 or (6), are cancelled in the difference 



2f(d'-d")-2f(d' + d") 

 by the terms f(d'—d") arising from the corresponding system (a') or (b'). 

 That difference is, therefore, equal to the excess of 2/(d'— <f '), extended to all 

 solutions of the systems (c'), above 2f(d' + d") extended to all solutions of the 

 systems (c) ; so that, finally, 



S [/(cZ' -d") -f(d' + d")-] = 2*- ' 2d [/(0 ) -f(2 a d)-\. 



II. Let m be an uneven number, and f(x) an uneven function ; we have 

 the equation 



(b) 

 (c) 



2f(d' + 2m')=2f(i^y 



the summations in the left- and right-hand members extending respectively to 

 all solutions of the equations 



m=2m' 2 + d'h', 

 2m=m\+d 1 c 1 , 

 the indeterminates d' I' d x ^ m x being positive and uneven, but m' being even 

 or uneven, positive, negative, or zero. 



