Mr. Ivory on the Attractions 
3 66 
this force is, therefore, directly as the mass, and inversely as 
the square of the distance from the center of the sphere. 
For an ellipsoid in general, we have x — k cos. tp,y = k' sin. 
op cos. i | >,% — k" sin. tp sin. 4 : in order to transform the for- 
mulas (5), we must first compute the values of dy . dz, dx . dz, 
dx .dy. For this purpose, let the fluxion of y be taken, making 
<p the only variable, so that dy = k' cos. tp cos. 4 x dtp : then, be- 
cause y must be constant in the expression of the force A, when 
% varies, we must make 
dz — Jc" cos. <p sin. 4* • da? -f- k" sin. <p cos. 4 • dpi 
0 = k' cos. tp cos. pi . dp — k' sin. <p sin. 4 . dp/, 
and, by exterminating dtp, we get dz == k" x dp/ : there- 
fore 
dy .dz = k'k” cos. (p sin. p . d<p . d \ {/. 
Again, because the value of x depends only on the angle <p, 
we have dx — — k sin. <p . dtp: and, by taking the fluxions of 
y and % relatively to the variable 4> we have dy ==. — k' sin. p 
sin. 4 . dp>, dz = k" sin. <p cos. 4 . dpi : therefore, 
dx .dy — kk' sin. z p sin. pi . dtp . dpi 
dx . dz — kk" sin. *p cos. pi . dp . dpi : 
in these expressions the sign — , which stands before the 
values of dx and dy, has been neglected : for that sign marks 
only that x and y decrease when the angles tp and 4 increase, 
and does not affect the absolute magnitudes of the fluents, 
which are alone the subjects of our research. Observing that 
= t -f 6% and k" z — t -f- e"\ the formulas (5) will now 
become, by substitution, 
