of homogeneous Ellipsoids. 
S 6 5 
whose length is qx and its base dy ,d%\ and hence %x .dy. 
dz, taken within the limits prescribed, is no other than the 
mass of the sphere = — . k 3 . Therefore 
4<E7 
x a. 
The same reasoning, it is evident, will apply to the remain- 
ing attractions B and C : and hence the attractions of a sphere 
upon a point within the surface, acting perpendicularly to the 
planes of any three great circles that intersect at right angles, 
are thus expressed. 
A = a 
4®r 
B = 6 x — 
3 
C=CXy, 
These three forces compose a force, directed to the centre of 
the sphere, and equal to ~ x v/V -f- b 2 -j- c : it is therefore 
directly proportional to the distance from the center. 
For a point without the surface of a sphere, we have h = 
h' = h" = s/ c? -j- b 2 + c 2 : hence it is easy to infer, that the 
formulas (6) will become, 
A = a x 
B = h x 
— . k 3 
3 
a x M 
(«* + b* + c*lf 
j 
(a 1 + b 1 -f- c 2 )| 
4'Sr 
C = c x 
where M = — . k 3 = 
3 
3 
. k 3 
(a z + b z 4- C a )4- 
b x M 
(a 2 + o z + c 1 )! 
c x M 
forces compose a force = 
(a z -f b z + c 2 )-| (a 1 -f- b x + c 2 )-| * 
: the mass of the sphere. These three 
M 
a* + b z + c 
3 B 
•, directed to the center: 
MDCCCIX. 
