ON THE PHYSICAL STRUCTURE OF THE EARTH. 213 



by Mr. Hopkins mul Archdeacon Pratt; and, as we have seen, such an 

 assumption is not only unwarranted, but is absolutely contradicted 

 by the established laws of hydrodynamics. My conclusions are not 

 only in harmony with those laws, but necessarily require them to be 

 kept constantly in view throughout the whole investigation. 



(7) The result obtained in section (3) allows of an immediate 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 difterence of the moments of inertia of any 

 spheroidal stratum to its greatest moments of inertia. It will readily 

 appear that the ditJ'erence 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 

 elipticities are equal. 



If p be the density of any spheroidal stratum of equal density, then 

 for that stratum 



Ci—Ax J p (^^+?/") d.v (ly dz— J^p {z-^y'^) dx dy dz 

 ^'i J'p {X"-^y'^) dx dy dz 



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

 both from numerator and denominator. As we shall see presently, 



0, 



'=^ (-:;:> 



C,-.4, 1 / . h 



where &i and a^ are the semi-axes of the stratum; and for all other 

 strata of equal density we would have 



C^-A, 1 



-A,_ 1 /■ h' \ 



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



«i ~ <f> tta * ' ' «.„ ' 



and hence 



~ Or ~ (J, - A, (7„ -2A/ aV' 



