On the Attraction of Ellipsoids. 369 



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

 of the meridian, we have 



fl e* sin X cos X . . 



cot u = -, ; 5-r- = 2e sm A cos A, q. #., 



1 - e 2 cos- X 



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



Substituting in (16) this value of cot 6, and the values of R 

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



T + ? 7 (1-3 sin 2 X) -(u 2 rcos 2 X| 2* sinX cosX 

 r 2 2 r 4 



- + o> 2 ; sin X cos X, 



\ r* J 



or, approximately, 



{M a C 1 A C 1 A 



+ ^-^=(l-3sin 2 X)-<o 2 acos 2 X 2e = 3~^W. 



If we neglect quantities of the second order, this equation 

 becomes 



"^ O "^ i 2 . /I T\ 



=6 + d)*a. (17) 



a 2 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 3C-A 



tie T + r-. 



a 2 ' 2 a 



4 



but 



If 



T + ; 

 2 2 



2B 



