210 Mr. A. R. Forsyth. On the Theta-Functions. [Dec. 22, 



function of the sum of two pairs of variables may be combined in a 

 product with, any one function of the difference of the same pairs, 

 256 equations are necessary to give the complete expression of the 

 theorem. These are written down in sixteen sets of sixteen each, that 

 which is common to each set being the function of the difference of 

 the pairs of variables. Denoting by 



. . . Hx + £,y + V ), 



G' . . . $(x-f,y- V ), 



S . . . $(x, y), 



e... v ), 



one such equation is 



c 2 e e ' =e<? V + W + W + °A 2 



where c is the value of # when x, y are both zero. The obvious 

 analogy with the case of the single theta- functions 



e 2 (0) e O + v) e o - v ) = e 2 («*) e 2 (?) - H 2 0)H 2 (?) 



(using the ordinary notation) need hardly be pointed out. 



In Section IV many of the properties already proved for the double 

 theta-functions are generalised for the "r" tuple theta-f unctions. 

 Among these are : — 



(a.) The periodicity ; 



(/3.) The product theorem, which gives the product of four func- 

 tions as the sum of 4 } ' products of four functions ; and from it several 

 general equations are deduced ; 



(7.) The analogue of the main theorem of Section II, which is for 

 the " r " tuple functions 



fa A 2 , . . . , XA 9 'A=e'^%% K ^^ t ^^S^' 



v & • • • > v r) 1/5 j < =1 



and this is used, as before, to obtain 



(£.) The r differential equations of the form 



— 2xA h r *—=r ) — + 2flvV 2 — = 0, 



dx r \ Jl,/ dx r dk r 

 and the -|r(r--l) of the form 



Ps,t- + o— -7-5 ' 



dp s , t v dx s dx t 



all satisfied by <I> ; and to indicate a method of obtaining the constants 

 in the expansions of the <I>'s in powers of the a>'s. 



