oj homogeneous Ellipsoids. 355 
the directions parallel to k' and k " ; it may, in like manner, be 
shewn, that the attractions B and B' have the same proportion 
as kk" has to hh" ; and the attractions C and C', the same pro- 
portion as kk' to hli'. 
The points in the surfaces of the two ellipsoids, which are 
determined by the co-ordinates, h cos. m, h! sin. m cos. n, h a 
sin. in sin. n, or a , b, c , and k cos. m, k' sin. m cos. n, k" sin. m 
sin. n , or a', b\ c ', may not improperly be called correspond- 
ing points of the surfaces : they are such points as are situated 
on the same sides of the planes of the principal sections, and 
have their co-ordinates respectively proportional to the axes 
to which they are parallel. This being premised, the result 
of the foregoing investigation may be enunciated, as in the 
following theorem : 
“ If two ellipsoids of the same homogeneous matter have 
“ the same excentricities, and their principal sections in the 
“ same planes ; the attractions which one of the ellipsoids ex- 
“ erts upon a point in the surface of the other, perpendicularly 
“ to the planes of the principal sections, will be to the attrac- 
“ tions which the second ellipsoid exerts upon the correspond- 
“ ing point in the surface of the first, perpendicularly to the 
“ same planes, in the direct proportion of the surfaces, or 
“ areas, of the principal sections to w r hich the attractions are 
“ perpendicular.” 
For the principal sections, being ellipses, their areas are 
proportional to the products of the semi-axes. 
When the attracted point, of which the co-ordinates are a, 
b, c, is placed without the ellipsoid having k, k', k" for its semi- 
axes ; then the point, of which a', b', c' are the co-ordinates, is 
necessarily within the other ellipsoid : and, on account of the 
