344 



REPORT 1865. 



Any infinite product of cither of the types occurring in these equations he 

 calls an elliptic product ; and every infinite product which can be formed in 

 more ways than one by the multiplication of two elliptic products, leads 

 directly to one of his special formula?. The five following elliptic products 

 are of great importance in the theory ; they correspond to the suppositions 



co 



n( i-f^y n( 1 - f -)= K - i)'Y' 

 1 



CO 



n(i + 2 2 '"- 1 ) 2 n(i - ff m )=-z<f 

 1 



co 



11(1- r/'- 1 ) n(l - 2*") = I( - 1)"Y/'" N ' 

 1 



>' 



(B) 



CO 



n(i + r/"'-') rr(i -q* m )=i<f m -+• 

 1 



co 



11(1- q"<) = 2( — l)"V/ l3 "'-l "" ; 

 1 



the first two are the equations (19) and (20) of art. 124; the last is a cele- 

 brated formula due to Eider. 



The infinite products in the numerators and denominators of the fractions 

 equal to u and u' (equations 27 and 28, art. 125) are all elliptic products of 

 one or other of these five types, in some cases with q 2 , or q a , or — q substi- 

 tuted for q, Hence a comparison of any two of the fractions equal to u or 

 to u' gives immediately one of Jacobi's special formulas. The demonstration 

 of the formula (2) in art. 128 will serve as an example of this process. 



Again, Jacobi has shown that the Eulcrian product (art. 124, E.) 



CO 



co 



n(i + q'")=n T= - ( 



1 1 " l 



which Euler had himself represented by the fraction 



00 1— q- m 2,( — \y"q 3 ">"-+"> 



l—q'" ~ W _ l \m ( A(3mH»>)' 



(C) 



can be represented by six other fractions of which both the numerators and 

 denominators are elliptic products ; either the numerator or denominator, or 

 both, being of one of the types (B). Thus, for example, 



co 



n(l + r/"-=)(l-fr/"-)(l- 5 ^) 



11(1 + 2"')= * 

 1 



2q 



i(3m 8 -fm) 



\«i/v3m* 



(D) 



co 2(-l)'"2 



n(l- 2 c » | - 3 ) 2 (l-r/'") 



1 



Here again a comparison of any two of the seven equal fractions gives one 

 of the special formula; : thus writing q 2i for q in the two fractions (C) and (D), 

 wc find 



