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 n = 1 ; and the first when n — 1, and n = 3. 
When n — 1, it is 
(R 2 <t>, R - 2 R </>„ R + 2 r - K) d yJ = 4 r d y! 
When w = 3, it is 
(f 4 ' 2 -!) (R^,R-4R^„R+12R^„,R-24R^R 
+ 24 <J>, r) = — 4 - ((t 2 — y) <p a . e 2 . 
The whole value of V is 
4 
And 
T~ a ~ If a * $ a ~ To ~ t) a * e2 } * 
d V 4 7r p l" e 2 o 3 / „ 1 \ « 5 , .>7 
- 57 = 4 « - g- « 3 <J> « - Jo l^ 2 - 3 1 r” 4> 0 •« 7 • 
14. Instead of the eccentricity e = x/(> — Jr) it will be more convenient to em- 
ploy the ellipticity g = 1 
e 2 
These give g = And the values of V become for 
an internal point, 
- (<P„ 
for an external point, 
4 * gyi'P 
~ | ^ r + y) -ja 2 (j>a-j(y, 2 -j-)e<paJ; 
- y a 3 $ a - -j p( - y) « 5 </> a J . 
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 ; § (p a 1 
its density, and g' its ellipticity, being some function of a! as % a!. Then 
a 1 — r' (1 -f- s' yJ 2 ) and (pa 1 — (p r [ -{- r' <p' r' % r' yJ 2 — <p r l + F r ' . y! 2 , suppose. 
Consequently to the term before produced in (12.) by <p r' we must add a term 
similarly produced by F r' . yl 2 . 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 
