of homogeneous Ellipsoids. 353 
— dh"k" sin. m sin. n sin. <p sin. if k z e 2 sin. z m cos. 'n 
4- e'* sin. 2 m sin. % n 4- e 2 sin. 4 cos. 3 tf + e 11 s ^ n - s ^ n> } \- 
In these values of A and A', it is plain that the quantities- 
h, h', h" are alike concerned with the quantities k, k', and k " : 
and hence, by interchanging the semi-axes of the two ellip- 
soids, we may represent each of the expressions for A and A' 
in two forms, which, when expanded, are identical : thus 
A — | {h cos. m — k cos. <py 4 " {h' sin. m cos. n — k' sin. a? cos. 4 )* 
[h" sin. m sin. n—k" sin. cp sin. 4) 2 |f = { {k cos. m — h cos. 
<pY-\- s ^ n - m cos » n —h' sin. cp cos. i f )* 4 ~ s ^ n - m s * n - n—h lr 
sin. (p sin. if ) 2 
A'= {{hcos.m^k cos. (h' sin. m cos. n — k' sin. cp cos. if ) 2 
+ (*' sin. m sin. n — k" sin. c p sin. if ) 2 {4 = j cos. m -j- h cos. 
<p) 2 4 - f sin. m cos. n-~h' sin. cp cos. if ) 2 4 ~ (&" sin. m sin. n — h" 
sin. (p sin. if) 2 j-f. 
In the formula (3) 
A = [ k'k n fj sin. (p cos. <p .do? . dty { — — 
the symbols A and A 1 express the distances of the attracted 
point, situated in the surface of the ellipsoid whose semi-axes 
are h, h', h", and determined by the co-ordinates a , b, c, or h 
cos. m, h' sin. m cos. h" sin. m sin. n, from the extremities 
of a prism of the matter of the ellipsoid first considered, paral- 
lel to the axis k, and having k' k" sin. q> cos. <p . dp . dfy for its 
base, and its length equal to 2k cos. <p : and, if we take a point 
in the surface of the last mentioned ellipsoid, that shall have 
k cos. m, k' sin. m cos. n , k" sin. m sin. n (which we may de- 
note by a', b', d) for its co-ordinates ; and conceive a prism 
of the matter of the other ellipsoid, parallel to k and h, that 
