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.

142 FIGURE 0F THE EARTH
The force at the pole, by Prop. 6,
4-,r. kb -. - sin-, e
{e2 e3
The attraction at the equator, by Prop. 7,
=2-r.kb {sin-le-;
the centrifugal force there
4 7r2 4 7r2 b
- a = _ e
T" T",/(1- eé)
hence, the whole force at the equator
= 2r. kb J sin-' e - /( e)} 4w er,, fl. V/(1 -ee-)l 4w2 b
These must be in the proportion of a: b, or 1: i /(1 - e);
that is,
-Â2k 1 - e__' )sin-le' k - sin-e /(- e2 -w
2k{ /(e' e es e- T'_ 2 /(1 -e:):: 1: v/(1-e2).
Let q =: substituting and reducing,
ki T d d
3(1 - e2) (3 - 2e2) (1 - e2) sin.... e- - -- -— ~~sin-le + q = 0.
2e
e" e3
The solution of this equation will give e, when q is known.
30. As this is a transcendental equation, it can be
solved only by approximation; but some properties of its
roots may be found thus. The left side of the equation is
positive, when e = 0, and when e = 1, which are the extreme
values of e that can be admitted; and it has therefore no
roots, or an even number. And by constructing a curve,
as fig. 10, in which the abscissa is e, and the ordinate is
proportionate to the value of the first side of the equation,