the Attractions of Spheroids of every Description. 
11 
the sphere being positive, and those on the inside negative. 
Therefore, relatively to the whole spheroid, we shall have 
V 
4' T 
f J rJ^Y' ( A ) 
We must next compute the value of -r- in the same cir- 
dV 
dr 
cumstances as before. Relatively to the sphere, it is — y . L 
for the point without the surface : and, by making r = p, it is 
— y . P for the point in the surface. In order to find the 
other part of the quantity in question we may suppose, with 
Laplace,* the attracted point to be raised up, in the prolonga- 
tion of p, the distance Jr above the surfaces of the spheroid 
and sphere; then, if f denote the distance of the molecule dm 
from the attracted point in its new position J'-j" and ,j^y~ will 
be two consecutive values of the same function which cor- 
respond to the values r and r -{- Jr; therefore, supposing r 
to vary, the fiuxional coefficient will, by the principles of the 
T d m f*dm 
differential calculus, be — ^ y 
fdm r-dm 
™ when Jr = 0. There* 
ir 
fore, by adding together the two parts of (yyj, we shall get 
Sr S' 
dV 
dr 
/ 
dm 
7 
(B) 
3 1 
observing that the second term on the right-hand side is to be 
valued on the supposition of Jr = o. 
Let y denote the cosine of the angle contained by p and an- 
other radius of the sphere drawn to the molecule dm ; then/, 
the distance of the molecule from the attracted point in the 
* Liv, 3e, No. io. 
C a 
