AND THE FIGURE OF THE EARTH. 
87 
d\ 
d h 
dY 
d r 
h , 
+ 2a-f — + 
+ 2 ^ + t 
3 ir g h 
is( 1 - e 2 “) 
7 Tg (SyU, 3 ~ 1) 
o (1 — e 2 ) 
11. When the particle attracted is external, the series in (3.) does not give a finite 
expression. Instead of taking it separately, I make it a case of a general theorem 
which follows. 
12. To find the attraction of a spheroid on a point within it, the density being any 
function of the distance from the centre, and the eccentricity being small. 
Let p p r' represent the law of density ; then the value of V, for the portion com- 
prised between the surface and a sphere of radius r, is 
2 * tf? f - 1 P ^ • r ' ( p o + p i TT + • • • + P* pa + • •) d r' d yJ. 
Integrating by parts we have 
/I — n 
fpr'.d 1 n .dr' = + {n— 1) -ft- + (n — 1) n + (n—l)n(n+l) + &c. 
Therefore the ( n + 1 ) th term of V is 
2%zr f ^n (^r-i+(»“l)iy + ( w -l)»|Kn + ( w “l) w (»+l) 
<p. R 
MV 
R w + 2 ' 
•■)dy 
NowR={^ + (^-^,' 2 } =a( 1+e 2 yJ 2 ) 2 = a (l — y p/ 2 ), rejecting e 4 
« . — 
to the same degree of 
and higher powers of e ; and p (m) R = p (m) a - p {m _ x) 
accuracy. The last member of the expression for V need only be calculated when 
n — 0 ; for all the rest of the terms (involving f _ l P re d y! where n > o) vanish. 
The first member need only be calculated when n = 0 , and when n — 2 ; for when 
n is odd, it vanishes as before ; and also when n is even and greater than 2 : for the 
functions of R involve no higher powers of yJ than the square ; and consequently 
they vanish, when multiplied by P 4 , P 6 > • . &c., and integrated with respect to yJ , from 
- 1 to + 1 *. 
When n = 0 , the term is 
2 * § S- 1 ( R <P, R ~ R ) d p' ~ 2 v § f_ j (r p, r - p u r) d y/, 
/ t / a 2 e 9 w/ 2 \ 
j [ap^ - (pa. — - - p n a J d yJ - 4 k g» (r p, ,r - p n r), 
= 4 7T § {a p t a — p u a — a 2 p a — (r <p t r — (p n r)) . 
* See Art. 9. 
