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.

136 FIGURE OF THE EARTH
The attraction on any particle of a spheroid, perpendicular to the axis, equals its distance from the axis x Q,
where
1
2k-{r sin-' e')
The attraction perpendicular to the plane of the equator equals its distance from the plane of the equator x P,
where
P=.4k7r e 1 _/(1 - e2) sin-' e}.
[e2 \
If the spheroid differ little from a sphere,
Q 1 —l - e P= — 1+-~.
4k7r 4kr 4)
3 5 3 5
These expressions may be found by expanding the values
just given: or by making the attractions at the pole and
equator coincide with those found in (11) and (9).
APPLICATION OF THESE THEOREMS TO THE FIGURE
OF THE EARTH.
20. In investigating the figure of the Earth, we shall
suppose that the Earth was originally a homogeneous fluid
mass, every particle attracting every other particle with an
accelerating force, proportional to the mass of the attracting
particle directly, and the square of the distance of the attracted particle inversely. This mass we suppose to revolve
about an axis in 23h. 56m. 48.
21. Now if the Earth had no motion of rotation, it
would evidently assume a spherical figure. For the mutual attraction of the particles would collect the whole into
one mass; and if any one part were then protuberant above
the rest, the direction of gravity would not be perpendicular to its surface, and it would not remain in that
form, but would run down. The form then must be such
as would leave no part protuberant above the rest; that
is, it must be spherical?