190 PEOCEEDINGS OF THE AMERICAN ACADEMY. 



On applying (72) and (73) to the first and second of equations (75), 

 respectively, we obtain, by (54) and (36), 



. a a 











Substituting these values of A.^^ and A2:+i in (6G) vpe have 

 A; 



(76). 



(20! 







+^/Wi ^2*+i -^^^Z- 1)' (27TT7! 







or, replacing (2^)! and (2 ^ + 1)! by their values in (37), observing the 

 last line of (36), and employing the notation of (62), 



(77) 6 (s) =2,^2* A. '%. +5,^2.+i Pu-.x ^2i-+i ; 







again, by (64) and (68), 



a ^ 



(78) fis) =2^^02, P2,R., +2,^2^-+i ^^*+i ^2^1- 







Substituting this expression of f(s) in (61) we obtain 



a s" p «" 



P.,„ =^^. C,, P,, • r*S„„7?,,, • c?5 + X C,,^,P,,^, 'Js,^ R,,^, • ds, 



s " « 



P2.+i=Xc,,p.sJs,.,.+,Ru-ds^2^^a,^,p,,^,-J^^^^^^ 



(79) 



IX. The Coefficients of /(«). Conditions for the Li.mits. 



In all that contributed to the determination of the form of f (s) as 

 given by (78) no assumption has been made with regard to the rallies of 

 the P's with even suffixes not greater than 2 u and the P's with odd 

 suffixes not greater than 2/3+1 [excepting that, by (51), P^ — 1, 



