of homogeneous 'Ellipsoids. 3S3 
very successfully cultivated, and is fertile in resources and 
methods that are applicable to every particular instance. 
The fluents in the formulas ( 6 ) for a point without the 
surface, are derived from the ellipsoid whose semi-axes are 
h,h',h", in the same manner as the fluents already considered 
kk'k ' 1 
are derived from the given ellipsoid : and, because is equal 
to the mass of the latter solid, divided by the mass of the for- 
mer one, it is easy to infer that we have only to substitute It 
for k in the fluents of the formulas (7), to obtain the expres- 
sions of the attractions of the given ellipsoid upon a point 
without the surface. Thus the two cases, when the attracted 
point is within the solid or in the surface, and when it is with- 
out the solid, differ only in the limits of the fluents: in the 
former case, the fluents, beginning when the variable quan- 
tity is infinitely great, are to be extended till it has decreased, 
so as to be equal to the least of the semi-axes of the given 
ellipsoid ; and, in the latter case, the fluents are to be ex- 
tended only till the variable quantity has decreased, so as to 
be equal to h, the least of the semi-axes of the ellipsoid, whose 
surface passes through the attracted point. In the former 
case, the values of the fluents are the same for all points 
within the ellipsoid, and in its surface ; in the latter case, these 
values depend upon the position of the attracted point. 
The preceding formulas, being founded on the most gene- 
ral hypothesis, are applicable to all figures bounded by finite 
surfaces of the second order. The case of the sphere, which 
corresponds to the supposition that the excentricities e* and e ,% 
are both evanescent, has already been considered, and, as it 
is attended with no difficulty, it needs not be again discussed 
