250 On the Physical St ructure of the Earth. 



its rotation to become unstable, so as to bring about conditions 

 which might result in a change of the axis of rotation. It is 

 easy to show, on the most favourable suppositions, that this 

 could not occur. The increasing ellipticity of the inner sur- 

 face of the shell would be due to the increasing oblateness of 

 the surface of the fluid nucleus, and this would be at its maxi- 

 mum if the nucleus approached a state of homogeneity. But 

 the fluid cannot approach this state unless the radius of the 

 nucleus is so small that the variation in density due to pres- 

 sure becomes insensible, whence all its strata would possess 

 the same density. This condition, with a certain thickness of 

 the solid shell, may bring about equality in the two principal 

 moments of inertia of the shell. The most favourable case 

 would be for a homogeneous shell ; hence we have only to 

 solve the very simple problem : — Given the thickness of a 

 homogeneous spheroidal shell at its pole, required its thick- 

 ness at the equator so as to make its principal moments of 

 inertia equal. We have from the expressions for C^ and Aj 

 in (7), 



a 2£ (a 2_ 6 2 )=ai 2£ i(ai 2_ V)) 



or 



«1 — «l 01 — £ ; 



which gives 



This may be written 



If we take e=^x for the outer ellipticity of the shell, and 

 e 1 =-^-r for its maximum inner ellipticity, we can easily find 



the values of j- and ^ ; from whence it appears that in order 



to have equal moments of inertia the thickness of the shell 

 should be *047 of its equatorial semiaxis, and the mean radius 

 of the nucleus would thus be reduced from the original value 

 when the whole mass was fluid by a fraction less than one 



twentieth. Under these conditions the ellipticity of ^— ~ ; cor- 



responding to homogeneity, could not exist ; and hence it may 

 be concluded that, whether the shell is thin, or whether the 



