208 Mr. A. K. Forsyth. On the Theta-Functions, [Dec. 22, 



and similarly for the y's. Since the assumption has been made that 

 the sums of the four similarly situated numbers in the characteristics 

 of the functions are all even, the equation comprises 4,096 cases 

 (=16 3 ). 



This general result seems to be new, but numerous particular cases 

 occur in Rosenhain's paper. All the important parts of his theory 

 are deduced, viz., the quadratic relations between the constant terms 

 in the ten even functions ; the nine ratios of all but one of these by 

 that one are expressed in terms of three independent constants & l3 & 2 , 

 Jc s ; and it is proved that the fifteen quotients of all the functions but 

 one by that one can be expressed in terms of two new variables 

 a,^, # 2 , the expressions being given. The connexion between a^, x 2 , and 

 .x, y is 



J Vz J 



J J J -/Z 



where Z=s(l -z) (1 -Jcft) (l-h*z) (1 - k s *z), 



and A, B, A', B' are perfectly determinate constants. 



The Quadruple periodicity is investigated at the beginning of the 

 section ; and afterwards definite-integral expressions for the periods 

 are obtained, as, for example, 



Jo Vx 



and it is proved that K satisfies a linear differential equation of the 

 fourth order in each of the quantities \, & 2 , & 3 . 



It may be mentioned that in dealing with the particular functions a 

 current-number notation # , # l5 . . . , # 15 is used in preference 



to the cumbrous <£ 



Section II gives the expansions of all the functions 



(i) in trigonometrical series, 



(ii) in ascending powers of x and y. 



Much use is throughout made of a theorem 



f A \ 1 2KAlogr d* 



a 0*0 j p G/) being single theta -functions) proved by means of 

 the known values of the single theta-functions. From this many 

 properties are deduced : — 



