346 KEPORT — 1869. 



the form 



n(l-f/"'+^)(l_5«^'+^)(l_5«3'+^), . . . =s^(-i)'+V''^*^''^"^'^ 



where / in the products has the values 0, 2, 3, &c., and in the summations i 

 and h have the vahies 0, +1, +2, +3, . . . . 



Jacobi has given a Table of these equations. I select three of them for 

 demonstration, which -Rill give an idea of the whole sufficient for our present 

 purpose. 



(1) n{{l-qf+\l-q''+y(l-rf'+')}=^(-n+''-f/''+^'''+\ 



(2) n{(l-</+')(l-,/+2)(l_./+^y(l-,/'+«)=} 



(3) n{(l-*/'+^y(l-.r/'+^)^} = v(- l)'+^/''+2*^+'". 

 By the formula (I. ), 



vc_2y+i- 3i-'+3/.-+ /_ 



n{(i-.^)«'+^'(i-,/'>^)(i_./+«)n(i-r/+^)(i-^«'+^)a-5''+')}, 



consequently formula (I.) reduces itself to 

 or 



n(i-5^'+^)=n(i-/"+^)(i -</'+'), 



which is at once seen to be true. 



To prove the second formula, we observe that 



by formula (I.) is equivalent to 



n(l-j3'+iXl-52'+-)(l-r/'+-^)n(l-5«'+3Xl-/''+3)(l-/'+''); 

 and the equation reduces itself to 



the same as before. 

 "We also have 



-.(-1) q =n(i — J XI — J XI — !/ ). 



a formula not noticed by Jacobi. 

 To prove formula (3)' we have 



= ri(i-,/+^Xi-/'^'Xi-'/^')n(i-r/'+^)(i_./'--^Xi-'/''-'-'); 



and the equation reduces itself to 



n(i-,/+2xi_,/.--f4)^i_,/+G)=n(i-./'+^Xi-5^'+'')> 



each member of wliich is immediately seen (o be equivalent to U(l — q-'+-). 

 In this manner the follo\ving expressions for tlie modulus are deduced 

 wbicli will b(> found dciucustrated ou imp's 7C> and 77. 



