154 PROBLEMS IN THREE DIMENSIONS. [CHAP. V 



These functions possess the property 



......... (5). 



For the motion of a planetary ellipsoid (f = ) parallel to the 

 axis of y we have n = 1, s = 1, as before, and thence 



(6), 



with A determined by the condition 



<ty = _<ty 

 d d' 



for f = , v denoting the velocity of the solid. This gives 



In the case of the disk (f = 0), we have A = 0, as we should 

 expect. 



Again, for a planetary ellipsoid rotating about the axis of y 

 with angular velocity q, we have, putting n = 2, s= 1, 



ma) ...... (8), 



with the surface condition 



d<f) _ f dx dz 



~ ~~ 



^sina, ..................... (9). 



For the circular disk ( = 0) this gives 



|7r^. = -A; 2 q ..................... (10). 



At the two surfaces of the disk we have 



yu 2 )* sin a), = + kq (1 - /a 9 )* sin <w, 



and substituting in the formula 



we obtain 2r=if / oc 5 .q 2 ..................... (11). 



