32 Mr Hicks, On tiuo pulsating spheres in a fluid. [Nov. 22, 



Also Q = 2 T W/»*fc 



Now P 4 2 remains unchanged when — /a is written for \i. Hence 



j +1 p>d/*=o, 



and Q = 4SC M a n £p„P n/ xc^, 



the summation extending to all positive integral values of m, n, 

 except m = n. 



Let now (P m P„^ = <l>. 



Then f*P m P n M>dfJ. = - f_P m PJl* +f_* d P = jj^" 



Now (Ferrers' s Spherical Harmonics, § 24), 



<j> = 1 Mn + 1) _ 



(m-w)(m + n + l) \ 2w+l mV " +1 •* 



to(to + 1) p , p _ p ) 

 2m + 1 nV m+1 ^'J' 



Hence I <3>c^ vanishes unless n = to + 1 or to — 1. Its values in 

 the two cases are 



2(m + 1) and 



(2m + 1) (2to + 3) (2m - 1) (2m + 1) " 



Since ^u, = P 1 the value of the integral may also be found by sub- 

 stitution in Prof. Adams' formula for I P m P n P p dfi. (Proc. Roy. Soc. 

 xxvil. p. 71.) 



Substituting the above values, 



n as ( 2m n a , 2(to+1) 



V [(2m - 1) (2m + 1) - 1 "' T (2m + 1) (2m + 3) ^»' u "' +1 



_ -1 ftv m r< n 4-s.n n 



~ 2 (2m - 1) (2m + 1) - 1 '» + 3 °^" 



