On the Attraction of Ellipsoids. 369 



But, from the property of the elliptic section made by the plane 

 of the meridian, we have 



, Q e* sin X cos A rt . x 

 cot = - = 2e sm A cos A, q. p.. 



1 - e 2 cos 2 A 



where e is the excentricity and f the ellipticity of this ellipse. 



Substituting in (16) this value of cot 9, and the values of R 

 and P from (13) and (14), the equation of equilibrium becomes 



\M 3C-A 



+ ~ 



(1-3 sin 2 A) - w z r cos 2 A J 2e sin A cos A 



C- A 



3 - + w 2 r ) sin A cos A, 



r* 2 r 4 



or, approximately, 



j + o ^r^ (1-3 sin 2 A) -rfa cos 2 AJ 2 = 3^^+^M. 



\ Ct> & (I (i 



If we neglect quantities of the second order, this equation 

 becomes 



&M 0-A , 



r- = o + or#. (17) 



a 2 a 4 



We have thus arrived at a relation which enables us to ex- 

 press the unknown quantity C-A, in terms of quantities which 

 are all known, and, therefore, to eliminate the former from any 

 other equation in which it may occur. 



Now let R e and R p denote the equatorial and polar attrac- 

 tions respectively ; we have from the general value of R (13), 



M 3 C-A 



but 



M 



"' 



_M ~ 



~~*~~' 



