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 1 dp 



-ji = -7^4- 

 dt pdx 



dv I dp , n x 



-jI = --^-+vV*V t ..................... (1), 



at pay 



dw I dp 



-JT = -f + 

 dt pdz 



. , du dv dw /cn 



with 3- + j- + ^j~ = ..................... V 2 )- 



dx dy dz 



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

 be written 



(v^^) 



( V2 ^&amp;gt; 



(V+A) w 



where h? = \/v ........................... (4). 



From (2) and (3) we deduce 



V 2 j9 = .............................. (5). 



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



and therefore the general solution is 



1 dp , 1 dp , I dp 



U = TTJ+ U &amp;gt; V = T^^T + V &amp;gt; w = T - 

 * 



dz 



/t _. 

 ... (7), 



where u t v , 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. 



