164 PROBLEMS IN THREE DIMENSIONS. [CHAP. V 



where % is a function of X only. This requires 



V . X + I&+?^ = 0.. .(2). 



* ydy z dz 



Now, from Art. 109 (4), (6), 



= ---- 



y dy z dz l \y d\ z d\) d\ 



X)/ 1 1 \d x 



a 



On substitution in (2) we find, by Art. 109 (7), 

 log {(* + &amp;lt;* + X). 



Whence = 



the second constant of integration being chosen as before. 



For a rigid ellipsoid rotating about the axis of x with angular 

 velocity p, the surface-condition is 



d(j&amp;gt;/d\ = ipzdyldX. ^ydz\d\ ............... (4), 



for X 0. Assuming 



we find that the surface-condition (4) is satisfied, provided 

 /! , 1\ 7o -/8. 





This expression dififers only by a factor from 



where is the gravitation-potential of a uniform solid ellipsoid at an external 

 point (a-, y, z). Since v 2 fi = it easily follows that the above is also a solution of 

 the equation ^&amp;lt;p = Q. 



