Hy per geometric Function with n parameters. 185 



which gives the corresponding result in the case when n — 2 

 to the one already pointed out for I»(u), viz. 



and useful results such as 



uni+«, (9^)2^, 



/ . 7T 7rs w \ 

 x cos! usin ■— 1, 



for r=l, 2, 3, 4, 5 ; and for r=6, 



x W6) 6 



1 ! II l-f a, 



1 



nil + a * / f/.\- S„ r ^COS 71-/6 / . , n lf/S? . K ZO\\ 



^ (u/6) b le cos (m sm 7r/6 — 7r/b(Za + o/2) j 



(27r) v 6 



+ cos(w-7r/6(2a+5/2))]; ■ 



and so on. 



The other (?i — 1) solutions of the equation 



satisfied by 



are the series 



«"* r oF n _ 1 (l + «! — a r , ,..Uvr a r' 1 ~ a ,o 



with /' = !, 2,...w — 1. Calling these solutions ?/j, t y 2 ,.. •//„_], 

 since the sum of the parameters for y r is %oL — na, r% it is 

 plain that the asymptotic expansion of y r is of the form 



r ' 2 



%fl{n)zT*r. z - <,«(*) 





4/« 



