222 



PIERCE. 



Substituting these various integrations (67), (68), (69), and (70) 

 in (66), we have 



P 



A: 



4c 



- 4I(- 1)^- \- + 2 sin^ BZi- 1)^ -^, 

 nln n -\- W 



/(-I 



+ 2co^BY.i- 1)2 : 



^•" 



2 2 VttT 



n 



22 42...^j2 



ra-l 



Jn-l (B) 



B 2 



n+l 



" k^ 52'^"M2 + ^ 

 + 2 sin 51 (- 1)2 ^-^-^^--^ ^^^ ^ J„+i (5). 



71 = 2, 4, 6,- • • oo 





n 27rfe . , ,A 1 TT 



£> = . IS between () and -• 

 A 2 



(71) 



This result may be expessed in a power series by expanding the 

 Bessel's Functions by equation (6), Art. 120 of Byerly's Fourier's 

 Series, giving 



n-l 



B 2 



J.-1 (5) = -„ZT . ^ ^ 



1 - 



B^ 



o2 h±i 



+ 



B' 



B' 



n+ 1\ /». + 3 



2!2^ 



3!2« 



n -\~ 1\ fn -}- 3\ /w + 5 



+ 



(72) 



Jn+l (B) 



"+1 



B 2 



n+l 



2 - r 



n + 3 



£2 



+ 



5^ 



02 ('L±l 



iJ« 



?i + 3\ //t + 5 



9 ! 94 



312*^ 



A? + 3\ //?. + 5 



/? + 7 

 9 



+ ■ 



r3) 



