AND THE FIGURE OF THE EARTH. 89 



where C is the value of (A) when r' = 0. As before, the last term need be calculated 

 only when w = 1 ; and the first when w = 1, and w = 3. 

 When w = 1, it is 



When w = 3, it is 



• +24<^R)=-|-(^2_l)^%a.e^. 



The whole value of V is 



'fpf' «*-^. i/'o l\«* „■) 



4 



And 



14. Instead of the eccentricity e = \/ ( 1 — ^) it will be more convenient to em- 

 ploy the ellipticity g = 1 — —*. These give « = -g * ^^^ ^^^ values of V become for 

 an internal point, 



for an external point, 



15. To find the attraction on the supposition that the body is composed of sphe- 

 roidal layers, homogeneous in themselves, but differing from one another in density 

 and ellipticity. 



First, on an internal point. 



Let r', as before, be the radius vector of any layer ; a' its equatorial radius ; § cp a' 

 its density, and s' its ellipticity, being some function of a' as % «'• Then 



a' == r' (1 + g' fJ^) and <|) «' = <|) r' + r' </>' r' >^ / jW»'2 = <^ r' + F r' . (jiJ^, suppose. 



Consequently to the term before produced in (12.) by ^ r* we must add a term 

 similarly produced by F r' . (jb'"^. Also, instead of taking, in the first instance, the por- 

 tion comprised between the surface and a sphere of radius r, we must take the por- 



* See Puissant, vol, i. p. 259, where the word ellipticity is used in this sense. 

 MDCCCXLI. , N 



