and Mathematics to Seismology. 197 



Thus we have finally 



= 3 sin d cos 0(M/E)(>/R) 3 -i-(l + 2gpa/19n). (68) 

 This result can also be got by noticing that 

 5^=tan- 1 (T/ff), 



= T/g, to a first approximation, 

 where 



T eeE sin 8^1—6 oostyi 



= 3g sin 6 cos 0(M/E) (aR) 3 + (1 + 2gpafl9n) . (69) 



is the tangential component of the surface-force. 

 Comparing (68) with (41) we have 



ty : top : : 1 : 1 + 2gpa/19n, . . . (70) 



or the self-gravitational forces reduce the apparent change 

 of level, as calculated in Thomson and Tait's ' Natural Phi- 

 losophy,' in precisely the same ratio as they reduce the 

 ellipticity of the surface. 



This last result might probably be deduced at once from 

 the fact that v a and v a are reduced in the same proportion, 

 but 1 have preferred an explicit mathematical proof. 



Comparing (66) and (68), we have 



$*/8f = l + ^gpa/lVn), .... (71) 



showing that the apparent change in star's altitude — the star 

 being, it will be remembered, in the same vertical plane with 

 the moon — is always in excess of the apparent change of 

 level. 



Numerical Estimates. 

 § 31. As before, we shall take 



a= 64xl0 7 , 

 gpa = 35 x 10 8 grammes wt. per sq. cm., 



(M/E)(a/R) 3 = 1/(182 x.10 5 ). 



We shall consider only the greatest and least values of n 

 specified in § 7, exhibiting the results side by side ; 0, it will 

 be remembered, is measured from the line joining the centres 

 of the earth and moon. 



