(31) 



342 Sir William Thomson on the Motion of 



Now let 



t = ^g) 2 +...+^g)" H +&c. 



be the spherical harmonic developments of P and yjr relatively 

 to the centre of the rigid globe as origin — the former necessarily- 

 convergent throughout the largest spherical space which can be 

 described from this point as centre without enclosing any part 

 of the core, the latter necessarily convergent throughout space 

 external to the sphere. By (33) we have 



Hence (32) gives 



which, by 

 becomes 



%*lh*i (85) 



j-f^p^o, 



Now, remarking that a solid spherical harmonic of the first de- 

 gree may be any linear function of x } y, z, put 



P,-=Aa? + By + C^ ..... (37) 



which gives 



? 2 =A 2 -fB 2 + C 2 , 

 and 



i f(Wpj= (A 2 + B 2 + C 2 ) .| • f(Vcr = ^ x vol. of globe = |^r/. 



Hence, by (36), 



w= \^ + IJjM¥^+ ¥^ + ••■)'•• (38) 



and therefore, by comparison with (31), 



' w -lJH^ + V n+ -} ■ • • (39) 



13. When the radius of the globe is infinitely small, 



W=i M *, ...."... (40) 

 where yu denotes once and a half the volume of the globule, 

 and q the undisturbed velocity of the fluid in its neighbourhood. 



