564 PROCEEDINGS OF THE AMERICAN ACADEMY. 



provided that 



'' MttV 1 



x=i log? 



This is of course under the assumption that ■pijif) may be represented 

 in the form (62). But this fact may he proved at once. P^or let \}/(i) 

 be any function which satisfies (60) and (61), and 4>{t) the particuhir 

 one above obtained. The function \f/ {t)/^ {t) is doubly periodic, 

 analytic save for poles, and can therefore be expressed as a quotient 

 of products of sigma functions 



'^cr(^-ft)... (7(^-/3,)' ^^' ^^'• 



But this quotient when multiplied by (t), which is expressed as a 

 product of sigma functions, must yield i/' (t), an entire function. This 

 necessitates that for each zero of a {t — jj) in the numerator 

 (i = 1, . . .jH) there must be a congruent zero x = ^j of a (.r — ^j) in 

 the denominator. Such pairs of corresponding factors may be com- 

 bined leaving only an exponential factor c'^'+^. Thus \j/ (t) appears in 

 the same form as cj) (0 • 



Our result is therefore that the element -pij (x) of P^x) is of the form 

 (66) (t = \ogx/\ogq), where pi, . . .,p„, (Ti, ■ . .fCrn are the constants 

 that appear in the series representations (58), and where the condi- 

 tions (67) are fulfilled. 



It is interesting to apply these results to the equation (56) in which 

 n = 1, iJL = 1, Pi = 0,(Xi = TT V— l/log^, and Yq (x) and Y^ (.r) reduce 

 respectively to yo (x) and y^ (x) defined in (57). In this case we have 

 therefore 



(68) yo(x) = yy,{x)p{.v) 



where 



— n/logi\- , — log a; /I 



log qy 

 The constant c may be determined Ijy writing (68) in the form 



y,(x) = [{x-l)y.^A-^)][_^~] 



and allowing x to approach 1. Since (r(0) = 0,a'(0) = 1, this gives 

 (see (57)) 



