ON THK THEORY OP NUMBERS. 333 



signs of summation have the same signification as in the similar equation (a) 

 of the last article ; and, as in that equation, the right-hand member may be 

 expressed in the simpler form 



\n/n' \n /\n/ n^ 

 If we now write 1 +p for s, and, multiplying by p, allow p to converge to 

 zero, the limit of the left-hand number is (H — H') ^'^^" / . The series 



2( — 11-1 • converges to a finite limit; forf — ) and (i)are each of them 



expressions of the form 2 2^8 |_V g and e denoting positive or negative 



units, and Q an uneven number composed of unequal primes dividing D ; 

 their product is therefore another expression of the same form, in which I, e 

 and Q are not simultaneously equal to 1, because we have expressly excluded 



that combination of the particular characters which causes ( - | to coincide 



with I —|. It can therefore be shown, by reasoning as in the last article, 

 that the second Lemma of art. 99 is applicable to the series, and that it con- 

 verges to the finite limit S|— j|-|-. Similarly, it may be shown that 



S|* I converges to a finite limit. The limit of the right-hand member 



of the equation (rf) is consequently zero on account of the evanescent factor 

 p ; from which it follows that H = H'. Let G^, G^, . . be the different possible 



genera ; A,,, h^, . . the number of classes they severally contain ; ( ^Jthevalue 



of (j) for the genus G. The equation H — H'=0 comprises 2^—2 equations 

 of the type 



{iM&'' 



+ . . =0, 



corresponding to the 2^—2 different expressions symbolized by <p. If we mul- 

 tiply each of these equations by the coefficient of kk in it, and add the pro- 

 ducts to the equation 



2^1 -f 2/<2 + 2^3 -f- . . . = 2h, 



we arrive at the conclusion 2Vji=2/i. For the coefficient of A, in the re- 

 sulting equation is the product 11 l-j-|X YX.j ; and this product is 2\ 



if G, and G* are identical, but is zero in every other case, as one at least of 

 the factors will be zero. 



103. The seventh section (Crelle, vol. xxi. p. 1) commences with the proof 

 of the theorem that the number of sets of representations of any number M 

 prime to 2D by quadratic forms of determinant D, is equal to the excess of 

 the number of those divisors cf of M which satisfy the equation 



a 2 e « \p)=^' 

 above the number of those divisors which satisfy the equation 



2 2 e 2 (^pj=:-l, 



