21 G ON THE PHYSICAL STRUCTURE OF THE EARTH. 



may briDg about equality in the two principal moments of inertia of the 

 shell. The most favorable case would be for a homogeneous shell. 

 Hence we have only to solve the very simple problem: Given the thick- 

 ness 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 d and Ai in (7), 



a^ J, {a'-W)=a^ />, {(h'-h^), or a,'-a^ b^=^' ^^ ^f-^'-^' 

 which gives 



This may be written 



^i-V2V-^+V 5^5 + 1. 



If we take e=2c73 for the outer ellipticity of the shell, and ei=T7^o 

 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 semi- 

 axis, 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 77^,7, cor- 

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

 cluded that, whether the shell is thin or whether the Earth has become 

 almost altogether solid, the moment of the inertia of the shell with 

 respect to its polar axis must be always greater than the moment of 

 inertia for its equatorial axis. 



The tendency of the fluid nucleus to increase in ellipticity might 

 produce a result worthy of examination by volcanologists, namely, a 

 possible increase in the development of volcanic phenomena in equa- 

 torial as compared to polar regions with the progressive solidification 

 of the Earth up to a certain point. Until the thickness of the shell 

 has become very great, recent periods should exhibit a greater devel- 

 opment of volcanic energy towards the equator than toward the poles 

 as compared to remote epochs. 



NOTE. 



On the annual precession calculated on the hypothesis of the Earth's solidity. 



In discussing the influence of the internal structure of the Earth 

 upon precession it has been frequently assumed that with the ellipticity 

 3^0 the annual precession of a homogeneous solid shell or completely 



