188 The Rotation of an Elastic Spheroid. [Jan. 16, 



fact that in the oscillation with which we are concerned u u v iy W\, yjr 

 will all be small quantities of the order of the ellipticity of the 

 spheroid, while X/a; will be small of the order to 3 or e 3 / 2 . Hence, if we 

 neglect small quantities of the order of the square of the ellipticity 

 the equations (4) reduce to 



Ox Oy Oz 



l... (6), 



cZaj dy dz ~ J 



while to the same order of approximation the boundary equations (5) 

 may be supposed to hold good at the surface of a sphere r = a. 



4. The equations (6), with the boundary conditions (5) at the 

 surface of the sphere r = a, are the equations obtained in consider- 

 ing the equilibrium of a sphere distorted by forces throughout its 

 mass derivable from a potential function ^{O-iXz—O^jz). By using 

 the well-known solution of this problem we therefore obtain 



I \5 ttVl + e'/eJ 



1 + e'/e 



a/ii/ 8 r \ 6 1,4 6 y(e 2 xz-e x yz) 



V = Vo + ^i = zGA 1+ - \ +f— — ^ — ^ 



L V 5 ayl + e'/ej l + e /e <T 



z(6 % xz—0 x yz) 



*l + e'/e a 2 



15 iv 2 , 5w 2 a 2 



where e = , e = . 



1Qtt P 38w 



We have thus expressed u, v, w by means of two arbitrary constants 

 0i, 6> 2 . If with these values of u, v, w we form expressions for the 

 angular momenta of the body about the axes Ox, Oy, and then express 

 that the rates of change of these angular momenta are zero, we obtain 

 two equations giving the ratios of the quantities X , 2 and involving 

 X. Eliminating h 6 % from these equations we are led to an equation 

 for X, which is found to reduce to 



Xlw — — . 



5. If we had neglected the elastic distortions we should have 

 obtained to the same order of approximation 



