334 REPORT — 1861. 



the symbols o and c having the same signification as in art. 101. Of this 

 theorem, which coincides with tiiat of art. 87) since 





two demonstrations are given, one purely arithmetical, the other derived from 

 the equation (5) of art. 101, the proof of which in Dirichlet's memoir does 

 not involve the theorem of art. 87, but is deduced from the arithmetical 

 principles on which that theorem itself depends. We have already referred 

 (art. 95) to some of the particular results which can be deduced from the 

 general theorem. 



It is evident from the mode of formation of the equation (6), or of the cor- 

 responding equation for a positive determinant, that it may be generalized 

 by taking instead of the power (^ax^ + 2bxy -\-cy)-^, any function of 

 ax'^-\-2bxy-\-cy^ which renders the two members of the equation convergent ; 

 i. e. we may write, in the case of a negative determinant. 



Si . ^(a.a-^ + 2 J,^y + Ci.y*) + S^ • K«2^' + 2*2^ + c,y *) + . . . 



=2s(?)K««'). 



Dirichlet illustrates this observation by giving to ^ the exponential form q", 

 which satisfies the condition of convergence, if the analytical modulus ol q 

 be inferior to unity. Each double sum, such as Sgr^J^+sioy+cy' jn the left- 

 hand member of the equation 



=2.(5y.' 



can then be replaced by 2oA(|'(2A) (or sometimes by fewer) products of 

 the form 



W= — X «= — 00 



in which each simple series such as 



■ (2aAt!+Po)' 



V=-^ I Cj,.A^^^\i 



a. 



S 2' 

 v^ — 00 



can be expressed by means of the elliptic function ; the right-hand mem- 

 ber can also be expressed by means of elliptic series. If, for example, 

 D= — 3, we have the equation 



2 qi^o+yfx S g3.(20=+ 2 g(6f+2)'x S g3(2»+i)» 



1}= — 00 V= — 00 V^ — 00 t)=— 00 



~k = 1— 2«<''*+l> k = Q l_26(6*+5) • 



It does not appear that this remarkable transformation, which is only 

 very briefly noticed by Dirichlet, has been further examined. (See a note by 

 Mr. Cayley in the Cambridge and Dublin Mathematical Journal, vol. ix. 

 p. 163.) 



In the eighth section Dirichlet assigns the relation between the numbers 

 of properly and improperly primitive classes. When the determinant is ne- 



i 



