570 Lord Bayleigh on the Calculation of the Frequency of 



integration may be limited to the plane part. Of this the 

 two halves contribute equally. Now when = ^tt, 



cj,/c = t(-lYA 2n+1 (r/cy-^\ (14) 



d$ldn = d<f>lrdO= -2q 2 (r/c) + 4? 4 (r/c) 3 - . . . (15) 

 Thus 



T-a(-ljVA fcW {- i ^ r+s ^ 5 -.;.}, • (16) 



where A 2ft +i is given by (10) and (12) ; it is of course a 

 quadratic function of q 2 , q^ &c 



The summation with respect to n is easily effected in par- 

 ticular cases by decomposition into partial fractions according 



to the general formula 



( 



1 i = _^X I 1 —\ 



2»:+2s+l)(2w+2s'+l) 2 («—*') L2n + 2.s-' + l 2n + 2s + lJ 



1 



(17) 

 If s'= —s, we have 



1 = A/ i L-Jj 



(2n + 2s + l)(2m— 2s + l) 4s\2n-2s+l 2n + 2s+l/ 



If/ 1 1 , , , . 1 



= ^{(-2^I-2^3--- 1 + 1 + i + --- + 2^I 



+ 2^I + -)-(2^I + 2^ + ---)}=°- - (18) 

 If s' = s 9 (17) fails, but we have by a known formula 



- 1 - «2 K2 ' • • fO a 1\2' \ LV ) 



(2n + 2s + \) 2 8 3 2 5 2 "•• (2*-l) 



Thus for the term in q 2 \ we have in (16) 



4 «v_L/ _ _J_ + _J LJl (20) 



7T ^2ti + oI 2/1 + 3^2*1 + 1 2n-lj ? " ^ ; 

 in which by (18) 



S(2n + 3)- 1 (2n-l)" 1 = 0, 



by (17) 



2(27z + 3)- 1 (2n + l)- 1 = i^(2^ + l)- 1 -P(272 + 3)- 1 



1 /i , 1 , l , \ 1/1 1 \ 1 



■ Ba iV 1+ 3 i+ S» + "7 - 5V3» + 5 r2+ -") = 2' 

 and by (19) 



2(27i + 3)- 2 =1tt 2 --1. 



The complete term (20) in q 2 is accordingly 



^(2-K) (21) 



