154 PROBLEMS IN THREE DIMENSIONS. [CHAP. V 



These functions possess the property 



S f S ,,, S/ _/ V+l 1 /K\ 



- -- -T ......... m- 



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

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



(6), 



with A determined by the condition 



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



^k/V? . 2 ^ -** Si--** ( 7 &amp;gt;- 



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, 



= Afi(l - n*)*(p + 1)* 3f cot- 1 ?- 3 + ^ sin a) ...... (8), 



with the surface condition 



(9). 



For the circular disk (f = 0) this gives 



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



At the two surfaces of the disk we have 



/i 2 )^ sin w, = + kq (1 - yu, 2 )i sin &&amp;gt;, 



and substituting in the formula 



we obtain 2T= Jf^.q 8 ..................... (11). 



