336 REPORT— 1861. 



eventually be constant ; i. e. there will exist an infinite series of determinants, 

 all composed of the same primes, and all having the same number of pro- 

 perly primitive classes. As it is possible to find determinants contained in a 

 series of this kind, and having only one class in each genus, it appears that 

 the number of the positive determinants, which have only one class in each 

 genus, is infinite. This result, which was anticipated by Gauss (Disq. Arith. 

 art. SOi), is remarkable, because it is probable, from the result of a very- 

 extensive induction, that there are but 65 negative determinants, of which 

 the greatest is — 1848, having the same property. 



lOl. Summation of the series expressing the number of Properly Primitive 

 Classes. — It appears from the last article that, to obtain expressions in a finite 

 form for the number of classes, we may confine our attention to the order 

 of properly primitive forms, and may suppose that the determinant is not 



divisible by any square. To sum the series 2;|— )- upon this supposition, 



Dirichlet employs the formulae given by Gauss in his memoir, " Summatio 

 Serierum quarundam siugularium," to which we have already referred in this 

 Report (art. 20). The ninth section is occupied with the demonstration of 

 these formulae ; in the tenth they are applied to the summation of the series 



S[- ) -. Two different methods are given by Dirichlet, by either of which 

 this summation can be effected. 



(i.) If k be the index of periodicity of /-\ so that /-^^=/^-Y and 



^ /D\ ! ^' 



2( — I =0, the summation indicated by the symbol S extending to all values 



of « prime to 2D from 1 to k, we have, writing V for S | — )-, 



Jo 0*— 1 "*' 

 wherey(a?)=2/ — jo;", sothat/(l)=0. Integrating by the ordinary method 

 of decomposition into partial fractions, we find 



m=k-l , , ri 



m=l ^ ^•^o 



2mir. 



m=k~\ 



=j;/(.^)h(...=),.|(:-^)]. 



To simplify this complicated expression, it is requisite to transform the 

 symbol/ — j by the law of reciprocity, and to consider separately the eight 



cases which arise from every possible combination of the hypotheses (a) D 

 positive or negative, (/3) D even or uneven, (y) D, or |^D, =sl, mod 4-, 

 or_^ 3, mod 4. As an example of the process, we shall take the two cases 



