Mr. Ivory on the Attractions 
35 s 
namely, h. It is plain that one value of h, and only one, may, 
in all cases, be determined from this equation. For, by taking 
h small enough, the function on the left hand side will become 
greater than any positive quantity how great soever ; and by 
taking h great enough, the same function will become less 
than any positive quantity how small soever : and while h in- 
creases from o, ad infinitum y the function continually decreases 
from being infinitely great to be infinitely little. Therefore 
there is only one ellipsoid, having the required conditions, 
whose surface wall pass through the attracted point.* When 
h is determined, then h' = s/h* -J- e % h" =. v/ hf -f- e ' 3 : and in 
consequence of the equation, 
we may suppose, a — h cos. m, h == h' sin. m cos. n> c = h" 
sin. m sin. n. 
Let these values of a , b, c be substituted in the last expres- 
sions for A and A' : then 
A = | (hcos. m — k cos. <?>) 2 + {h* sin. m cos. n-—k' sin. cp cos. 40 2 
-f- (//' sin. rn sin. n — k" sin. <p sin. 40 2 } 2 
A'= | ( h cos. m -\-k cos. <£>)*-{- [h' sin. m cos. w — k' sin. cp cos. 4')“ 
-f- [h" sin. m sin. n — k" sin. cp sin. 4 )* j \ : 
and because A ' 2 = hr -f- e*, h"* = If -J- e'% k'* = k 2 -f- e 2 , k" 2 = 
k* -J- e> \ we shall readily obtain 
A — \ lf — Aik cos. m cos. cp — ck! k' sin. m cos. n sin. cp cos. 4/ 
■ — %h"k" sin. m sin. n sin. cp sin. 4* + ^ “f* e * s ^ n * * m cos * * n 
-f- e'* sin. sin. 2 n -f- e* sin. 2 cp cos. *4 + sin. *<p sin. 2 4 } 2 
A'= | h* -j- 2 hk cos. m cos. cp — 2 h'k' sin. m cos. n sin. cp cos. 4* 
* Mecan. Celeste, Tom. II. p. 50. 
