305-306] SOLUTION IN SPHERICAL HARMONICS. 



559 



306. The equations of small motion of an incompressible 

 fluid are, in the absence of extraneous forces, 



du I dp _ 9 ^ 

 -TT = - + vV 2 u, 

 dt pdx 



dv I dp _ 9 



Tt = - p f + vV * v ' 

 to* + v ^ 



dt pdz 



with *! + *! + *? = o. ..(2). 



dx dy dz 



If we assume that u, v, w all vary as e M , the equations (1) may 

 be written 



1 ,, . ^ . 



( 



where h* = -\/v ........................... (4). 



From (2) and (3) we deduce 



V 2 p = .............................. (5). 



Hence a particular solution of (3) and (2) is 



1 dp I dp I dp 



U = > V = > W= ............ (6) ' 



and therefore the general solution is 



1 dp , 1 dp . 1 dp 



w== i^-;7 + w > v== irj+ v > W = T^J 



n?ji dx * 



where u' t v f , w' are determined by the conditions of the preceding 

 Art. 



Hence the solutions in spherical harmonics, subject to the 

 condition of finiteness at the origin, fall into two classes. 



