322 Lord Rayleigh on Conduction of Heat in a Spherical 

 Also 

 2V C l , a9 , ot » *j i sin 2wii , 2 sin 2 m x ( „. 



To determine the arbitrary constants B 1 . . . . from the given 

 initial values of 6 and 5, say © and S, we proceed as usual. 

 AVe limit ourselves to the term of zero order in spherical 

 harmonics, i. e. to the supposition that 6, s are functions of r 

 only. The terms of higher order in spherical harmonics, if 

 present, are treated more easily, exactly as in the ordinary 

 theory of the conduction of heat. By (43) 



6 = Bi#i + B 2 2 + i 



S=B l5l +B 2 5 2 + ... 



and thus 



P (e^ + ZS/a . $ Sl )r*dr= B x ( * (0* + 0/a . s^Alr 



Jo Jo 



+ B 2 \ l [6 A + PI a . S&) t*dr + ...., 



Jo 



in which the coefficients of B 2 , B 3 , vanish by (46). The 



coefficient of Bx is given by (47) . Thus 



9rn 2 C 1 



I sin 2m a 2 sin 2 m l 



^ l \ 1 -~¥inT+~~^afi~ 



(49) 

 by which B} is determined. v ' 



An important particular case is that where © is constant 



and accordingly 8 vanishes. Now with use of (42) 



C l fl <2rf , _ Sm 1Ul ~~ ??l ! C0S 7Ul Sm 7??1 _ (1 "*" a fi) Sm ??l ! 



J l . " mj 3 3w l 3a^m l 



so that 



t, f . sin 2 m! , 2 sin 2 wit] 2j»i sin m. . @ ._. 



B '( 1 -^r + -3^-} : 3^ — •• ^ 



Bj, B 2 , .... being thus known, 6 and 5 are given as functions 

 of the time and of the space coordinates by (43), (44). 

 To determine the pressure in this case we have from (45) 



+ s/a _l + afiy^ sm 2 m.e- ht ,-.,. 



© 



a$ jU &*£/ sin2m\' ' 



