Professor Mac Cullagh's Lectures on the Attraction of Ellipsoids. 395 



{M 3 C—A 1 C—A 

 — + n — J— (l-3sin^\)-a)Vcos'^\ ?2£siiiXcos\=(3 — j |-ft'V)sin\cos\, 



or, approximately, 



f^+|^^(l-3sin^\)-«,^acos^x|2e=3^^ + e«^«. 



{a' 2 a* ^ ' J a* 



If we neglect quantities of the second order, this equation becomes 



2eJf ^C-A , , , 



— ;-=3 — + w-a. (17 



a^ a^ ^ ' 



We have thus arrived at a relation which enables us to express the un- 

 known quantity C~A, in terms of quantities which are all known, and, there- 

 fore, to eliminate tlie former from any other equation in which it may occur. 



Now let Re and Ep denote the equatorial and polar attractions respectively ; 

 we have from the general value ofi2(13), 



P 31 3 C-A 



„ M ,C-A 



but 



But. 



c = a(l-e), .-. i = i(l + 2e)andl = l(l + 4e), 

 „ _M 2Me ^C-A 



Gf = Rp and Ge = R,- w^a ; 

 ^ ^ 2eM 9 C-A 



C — A 

 Eliminating j— by means of equation (17), we get 



Gp — Ge _ 5 w^a 



<?. ~~"^2~G/' 

 or, 



G„ — Ge 5 , . „ , 



-^^^+e=-,. (18) 



VOL. XXII. 3 F 



