﻿222 Prof. M. P. Rudski on 



Here we have made the supposition that the body is incom- 

 pressible. [R means the mean radius of the Earth.] 



We must now calculate the products and the moments of 

 inertia about old axes after deformation. 



As this calculation implies the knowledge of the displace- 

 ments, we shall take the expressions of the displacements 

 given by Thomson and Tait * for the case of a sphere 



?=i[< 8R2 - 5 ^ 2+ ^ w ']- • ™ 



Before calculating the products of inertia we remark that 

 the deformation is evidently symmetrical with respect to the 

 plane passing through the axis of z and the axis of rotation. 



Hence, taking this plane for the plane XZ t, we obtain 

 first, if D, E, F are the products of inertia, 



D=F=0; 



secondly, 



<j>=^ \a\z*-sF)-%xzac\ . . . (IV.) 



Now, the calculation of 



E=2m(* +£)(*+{), 



with the help of the formulas IT., III., and IV. is very easy, 

 and gives 



» 



B =- a - c -i9^2pr MR2 - M2 ' 



where M is the mass of the earth, 



u the equatorial velocity (w = g)R). 



We shall also need the difference C — A. This difference, 

 being very little changed by the deformation, may be 

 directly calculated from the known values 



C= 0-3321 . MR 2 , 



A = 0-3310 . MR 2 , 

 so that 



C-A=0-0011MR 2 . 



* Treat, on Nat. Phil. arts. 837, 838._ 

 t In this manner we introduce coordinates moving respectively to the 

 body, but evidently it does not change anything in the results. 



