164 PROBLEMS IN THREE DIMENSIONS. [CHAP. V 



where % is a function of X only. This requires 



V . x + ?& + ?fK = .......... (2). 



* ydy z dz 



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



= 2 , 

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





(X - p) (\-v) 

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



(3), 



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>/d\ = pzdy/dh pydz/d\ ............... (4), 



for X = 0. Assuming 



< s >* 



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

 G 



1 l \ 

 ~ #) ' 





* This expression differs only by a factor from 



rfO dQ 

 If'**' 



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

 point (x, y, z). Since v 2 i2 = it easily follows that the above is also a solution of 

 the equation y 2 = 0. 



