Mathematical tracts on the lunar and planetary theories, the figure of the earth, precession and nutation, the calculus of variations, and the undulatory theory of optics.

di4
Hence A will be changed to A + - c + &c.; and theredc
fore the attraction of the spheroid E"F"
dÂ
=p (A + d c + &c.).
The difference of the attraction of the two spheroids, or the
attraction of the shell included between them, is therefore
dA
ultimately = p.- -c. If, then, u be the attraction of the
de
heterogeneous spheroid, whose polar semi-axis = c, we find
du u dA
= ultimate value of =. -;
dc cc dc
dA
~'. U=J;P dc;
where in the differentiation e must be considered as a
function of c; and in the integration, p and e must both
be considered as functions of c.
49. Now,
A= -i - r c()8 + 3( 2e) - c(f,^ - _ r
3 l(f 2 i-g 2 +h25 (f (+ g~ + hî)2 J
and when we differentiate this with respect to c, since f,
g, and h, are perfectly independent of c, the only variable
ternis will be c' (1 +2e) and c5e. Let
d(c3. 1 + 2e) d (c' e)
p * dc =:(c); ~p. d! =J(c);
both integrals being made to vanish when c = o. Then by
the expression found in the last article, the attraction in
the direction of x
4r (f2 + g2h+ h")_' (3) - - (c)2 f
3 l(/ + g+ h)32 5 (f2 + g2 + )
Similarly from the values in (47), we find the attraction in the
direction of y