dZ dY 



y j z j 

 dn an 



102.] MOTION OF AN ELLIPSOID. 115 



Next let us consider the function 



U = yZ-zY. ..................... ..... (10). 



With the same notation as before we find 



Vdin 

 7- = mZ n 

 \_dn J 



= ny(ir -H+G}- mz(^TT -G + 

 by formulae of the same type as (6) and (8). Since 



the ratio of the expression last written to ny mz is 

 6 2 (4-7r -H+G)- c 2 (47r 



a constant. Hence all the conditions of the problem are satisfied 

 by making 



where 



the lower limit being the same as in (9). 



The values of % 2 , ^ 3 may be written down from symmetry. 



The student may, as an exercise, prove the equivalence of the 

 above formulae, in the case where one of the axes of the ellipsoid 

 is infinite, to those of Arts. 88 (d) and 89 (c). 



Example 4. In the particular case of a sphere we have 

 a = b = c. We then find 



(where r* = # 2 + J/ 2 + -z 2 ), and therefore 



x a 8 # 



& = - 2 7- , 



with similar formulae for &amp;lt;^ 2 , &amp;lt;^&amp;gt; 3 . The values of XD Xz&amp;gt; Xs are zero &amp;gt; 

 as is obvious a priori. 



82 



