196 PROCEEDINGS OF THE AMERICAN ACADEMY, 



so that, by (80), 



(98) C.,,^ = — =: ^^ '^^— for ^ m ^ a (excepting m = 1 ) 



\/2 TT m ! 



and 



(99) C„.^i = —= ^_+JV_ for 1 ^ ;„ ^ ^. 



V ^ TT »J ! 



It will be observed that C'a,^ is less and C2,„+i is greater tlian — =^ ; 

 for very large values of ?n we have, by Stirling's formula, 



(100) c.. = ^. c,„„ = ^^', 



so that Cg,,, diminishes and C2,„4_j increases without limit as m increases 

 without limit; this fact will, perhaps, to a certain extent offset the con- 

 clusion that might be drawn from (61) that the values of the P's with 

 odd suffixes are likely to be less than those of the P's with nearly the 

 same even suffixes (on account of the presence of the odd functions S 

 under the integral sign in the former and of the even* functions *S' in the 

 latter) in estimating the relative importance of the terms of y"(.s) as given 

 by (78) that contain it's with odd suffixes and R's with even suffixes. 

 For the C's with the smaller suffixes we have 



(101) 



Cq — — ; : ) C4 — . ) Cg — ) etc., 



\/2 TT 8 V 2 TT 1 6 V2 TT 



C3 = — ^7^= ' C\ = — ^^ , C^ = p= , etc., 



2 V2 TT 8 \/2 TT 1 6 \/2 TT 



so that, by (27), (78), (G8), (51), (63), and (101), 

 (102) 



^(0 = -f{s) = -4== [1 - 1(1 - hp.) (1 - 2s-^ + U^ 

 H- !' V 2 TT 



- I (1 - ^P4 + ri'oPe) (\-3s-+s'- ^3/; + etc. 



iPsi^- ^- ^'') - 4 (P3 - A Ps) (5 - 3 s' + tV «') + etc.]e- ?. 



XI. Application to any Particular Case. 



It seems reasonable to suppose that, in any case (whatever the values 

 of s and s)f(s) will have small values for values of s beyond the limits 

 of possible values, so that the extension of the limits to — go and + x 

 will not make much difFerence in the values of the integrals by which 



