293-294J SOLUTION IN SPHERICAL HARMONICS. 529 



Hence the equations (1) are satisfied, provided 



Also, substituting in (2), we find 



2nA - (n + l)(2w + 3) B= 0, 



Hence the general solution of the system (1) and (2) is 



1 g f ^ dp n _ nr^ +s _ d_ 

 ~> 12(2n + l) das + (n+l)(2n+l)(2fc + 3) ^r 2w+1 J ~* Uf 



= lyf 

 fj, (2 ( 



, 



- , 



(n '+ 1) (2n + 1) (2w + 3) dy 



/* (2 (2/1 + 1) dz (TO + 1) (2/i + 1) (2/1 + 3) 



where w 7 , t/, w' have the forms given in (9) of the preceding Art. 



The formulae (8) make 



1 nr 2 



xu + yv + zw = - 2 2 ( p n + ^n<t> n .......... (9). 



Also, if we denote by f, ij, the components of the angular 

 velocity of the fluid (Art. 31), we find 



These make 2 (a?f + yiy + ^rj) = Sw (TO+ I)%n ............... ( n )- 



* This investigation is derived, with some modifications, from various sources. 

 Cf. Thomson and Tait, Natural Philosophy, Art. 736 ; Borchardt, 1. c. ; Oberbeck, 

 "Ueber stationare Fliissigkeitsbewegungen mit Beriicksichtigung der inneren 

 Reibung," Crelle, t. Ixxxi., p. 62 (1876). 



