AND THE FIGURE OF THE EARTH. 
85 
8. Returning to the theory of attractions, we have, when the particle is internal, 
V 
=f f-\f* ? r '( P o + P i7+' 
= 2 J § ( p 0 >•’ + Pi r + P 2 
••+P £+..)dr'd l >'df 
^•2 \ 
7 + • • + P* r <n- l + • •) dr ' 
for that portion of the body which is comprised between a sphere of radius r, and the 
surface of the body, supposed to be a surface of revolution round the axis of z ; for, 
in that case R, the value of r' at the surface, is independent of <p’ . 
9. Suppose, for example, we wish to find the attraction of a homogeneous spheroid, 
on a point within it. In this case is constant, and 
a being the semi-major, and c the semi-minor axis. 
First, all the even terms vanish ; for the general even term is 
y.2 n -(- 1 
P V 
2 ra -j- 1 yJ 2 n 
dr d (d 
f' p ( l-p» 
2m — 1 J-\ r 2n + l\ a 2 "^c 2 / 
d (Jj 1 
/•> 
“P 2«— lj ~ 1 *2»+l 
/\ _ uj* >2\ 2 »-t 
Now P 2n + 1 consists of odd powers of (d ; and j 2 can be expanded 
in even powers of (d ; therefore the integral of the product (which is an odd function), 
taken from [d = — 1 to fd — 1, is 0. Also x V 2n + l d(d = 0 *. All the odd terms 
above the third vanish ; for the (2 n + 1 )th term is 
1 Yi | />*. r 2 d [d, for it may be shown that^/^ P e Pr d(d d <p' = 0, if i and / 
/ 1 u/ 2 U.'-\ U ~ L 
be different integers. Now when n is greater than 1, l — ^ ^ ) is a rational 
and entire function of fd, and, therefore, capable of being expressed in a series of 
Laplace’s coefficients'!', the highest of which will be of the (2 n — 2)th order; and 
therefore no term of this expansion can be of the same order as P 2 n ; and the inte- 
gral of the product of any two of different orders, between these limits, vanishes. So 
the second member of this vanishes. 
* See Pratt. Mec. Phil., § 180. 
t See Pratt. Mec. Phil., § 176. Poisson, Theorie Math, de la Chal., chap. viii. Laplace, Mec. Cel. liv. iii. 
chap. ii. 
