318 Lord Rayleigh on Conduction of Heat in a Spherical 



where h ly h^ . . . are the values of 7i corresponding according 



to (9) with the various values of m, and ft, ft . . . are of the 



form 



1 = cosm l x— cos m x a ~\ 



>>•••• (24) 

 s 1 = — a(cos m±x — sin m^a/w^a) J 



It only remains to determine the arbitrary constants Af, 

 A 2 , . t . to suit prescribed initial conditions. We will limit 

 ourselves to the simpler case of q = 0, so that the values 

 of m are given by (22). With use of this relation and 

 putting for brevity a=l, we find from (24) 



ftft dx= 3— cos nil cos m 2 , 



a P 



1 j a ^ + l 

 5 X 5 2 ax = 32 — cos m x cos m 2 ; 



'0 P 



so that 



( 1 2 dx + /3/a.\ sfrdv^O, .... (25) 



ft, ft being any (different) functions of the form (24). Also 



'^^^^{U.^}. . (26) 



There is now no difficulty in finding A 1? A 2 , . . . to suit 

 arbitrary initial values of and its associated s, i. e. so that 



e=A 1 ft + A,0 2 + .. 



(27) 



t 



Jo 



f 



Jo 



S 

 Thus to determine A 1? 



=A 1 ft + A a ft+ ... I 



= A 1 5 1 +A 2 5 2 + ... 



f 1 (® ft + pl'a . Ssj)^ = A 1 C (ft 2 + £/« . Sl 2 )dx 



+ A 2 (0 1 2 +l3/*.s 1 s 2 )da!+.. 



Jo 



in which the coefficients of A 2 , A 3 . . . vanish by (25) ; so that 

 by (26) 



M i+ ^1 = i^Jo 1(0<?i+/3a *- s ^- • (28) 



An important particular case is that in which © is constant, 

 and accordingly S = 0. Since 



f 1 sin 7^ l + a/3 



1 V 1 dX= C0SWi = — ■ tt^-COSWi, 



Jo ™i «P 



