ON THE THEORY OF NUMBERS. 343 



From the nature of the identity (2) it is evident that we may substitute 

 any function whatever (which renders the two series convergent) for the 

 exponential of q. Thus, for example, we find 



(-ir =s (- i )" i+ " _ i: ri±ii x(F) 



=n 1 ,. n_J_- n_l 



V p^J 



i+ 



where p 1 , p a p 3 are primes of the forms 8u + 1 ; 8n + 3 ; 8n + 5 or 7, respect- 

 ively ; and 17 is a positive or negative unit determined by the congruence 



P1—1 ft— J 

 (-1) 8 2 4 =,, mod Pl . 



It would seem that the method of Dirichlet which we have here described 

 maybe employed to prove all the theorems of Jacobi's memoir in which the 

 two forms compared have different determinants. Those in which the two 

 forms compared have the same determinant, or determinants differing only 

 by a square factor, are of a more elementary character, and are capable of 

 immediate verification. But Dirichlet's method may also be extended to cases 

 in which one or both of the forms compared bas a positive determinant. One 

 example will suffice. If P=(2a + l) 2 + 86 2 =(2a + l) 2 -8/3 2 , we have 

 S (_l)'Y[(2m + l) 2 + 8^] = S(-l) m +>[(2m + l) 2 -8H 2 ], 



representing any function whatever which renders the series convergent, 

 and the limits of m and n in the first sum being 0, 00 , and — co , + 00 ; in 

 the second sum 0, 00 , and 1, j(2»i + l). 



129. We proceed to indicate very briefly the origin of the principal formula? 

 in Jacobi's memoir. Three of them are distinguished from the rest as general, 

 being deduced from the equation (7) of art. 124, without any specialization. 

 If in that formula we write successively +z and — 2 for v, and multiply the 

 results together, the left-hand member becomes 2( — l)»y< 2 +»V"+"; the right- 

 hand member may be written in the form 



GO CO 



n(i-2 4m - 2 ) 2 (i-2 4 "') x n(i-2 4 "0(i-2 4 "'-v-)(i- 2 4m --v 2 ), 

 1 1 



where the second infinite product, by the equation (J), is equal to 

 2(— l) m q 2m *z 2 ">, and the first to 2(— l) m q-"'\ Hence 



S (— l)'»2»' 2 +"V ,! +"=2(— l)"' + "2 2( '" 2+ " 2, ~ 2 " i , .... (A) 



which is one of Jacobi's general formulae. The other two general formulas, 

 and most of the special ones, are obtained in like manner by considering infi- 

 nite products which are capable of being expressed in more ways than one as 

 the product of two Theta functions. To arrive at his special formulas, Jacobi 

 transforms the equation (7) by writing q a , where a is positive, for q, and +q b 

 for v. He thus obtains the equations 



co 



T\(\ -y2»m -«-iyi Q2ma—a+b\( 1 n' ima \ = ^Y 1 y«(y<»» 5 4-»'* 



1 



CO 



TI(1 + 2 S»«-«-»)(l + 2 2m -«+6) ( l_ 2 2m«.- ) = S5 <».. , +m6 # 



1 



