1881.] 



'particularly those of Two Variables. 



207 



prize given by the Paris Academy of Sciences in 1846, and was- 

 published in the " Memoires des Savans Etrangers," torn. xi. Taking 

 as the definition of the general donble theta- function 



•{'GcM 



m —oo n— oo \-r x ; ny 



= 2 2 (_l) m X+ M Ppit 2m +^) 2 ^4(^+v)V^ 2m+ MH2n+v) e (2m+^)^+(2r.+v)^ 



and denoting the product of four functions, in which the characteristic 

 numbers and the variables have the subscript indices 1, 2, 3, 4 re- 

 spectively, by 



no 



there is investigated, by the guidance of Professor H. J. S. Smith's 

 paper on the Single Theta- Functions in the first volume of the 

 " Proceedings of the London Mathematical Society," the theorem 



4n *{C;'M= 



+»*{(\:> p m 



+ n*{(^t>x} 



+™{( A+ # +1 M 



+ (_1)P' into 



™{(„ A / + iM 



-»{( A S5r>.x} 



+ (_1)A< 



-W( A #*M 



4- (-l)AM-r' into 



-»{U p #iM 



in which 



2(A 1 +X 1 )=2(A 2 +X 2 )==2(A3 + \3)==2(A 4 ^-X 4 )=\ 1 + \o + A3 + \ 4 ==2A^ 

 and similarly for fi, p, v; and 



2(X l + x 1 )=2(X 2 + x 2 )=2(X s ^x s ) = 2(X 4 + x i )=x 1 ^x 2 + x 3 + x 4 , 



