Physical Structure of the Earth. 247 



(7) The result obtained in section (3) allows of an imme- 

 diate and easy application to the inquiry before us, if we admit 

 that the strata of equal density in the shell have all equal 

 ellipticities — an admission which has been already shown to 

 be a particular case of a rigorous and exact deduction from 

 hydrodynamical principles. In this case let us consider the 

 ratio of the difference of the moments of inertia of any sphe- 

 roidal stratum to its greatest moment of inertia. It will 

 readily appear that the difference of the greatest and least 

 moments of inertia, of all the strata divided by the sum of the 

 greatest moments of inertia will be the same as that for a 

 homogeneous shell whose inner and outer ellipticities are 

 equal. 



If p be the density of any spheroidal stratum of. equal den- 

 sity, then for that stratum 



C?! — Ai _ j H^ 2 + y 2 )dx dy dz — jp (a? +f)dx dy dz 



Qi j]o {a? + y 2 )dx dy dz 



and as p may be placed outside the sign of integration, it 

 disappears both from numerator and denominator. As we 

 shall presently see, 



where & x and a x are the semiaxes of the stratum ; and for all 

 other strata of equal density we would have 



0,-a, l/.-yi 



C s -A 3 1/, W\ C„-A„ 1, 



2V 1 <W'--- 0. ~a a*y 



C 3 2^ 

 Now if these strata are all similar, and have equal ellipticities, 



and hence ai a * °* " a »' 



d-Ai^-A, C3-A3 0.-A, 



Ci C 2 A 3 C, 



K-5). 



where b and a are the outer semiaxes of the shell composed 

 of all the strata of equal density. But 



Vi ._^ 2 \_C-A_ C 1 + C 2 + ... + q-(A 1 + A 2 + ... + A ?t ) 

 2\ W C ~ CH-ft+.-. + Q. 



This is the symbolical form of the proposition just stated. 



In a homogeneous solid of revolution the general expres- 

 sion for the moment of inertia is 



ir\y l xdx ; 



