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.

SUPPOSED HOMOGENEOUS. 137
22. But, in consequence of the Earth's rotation, every
particle has a centrifugal force, or a tendency to recede
from the axis of rotation. The effect of this, it is plain,
will be to enlarge the Earth at its equator, and to flatten
it at the poles. It is our object now to shew, that the
figure which the Earth would assume is accurately that
of an oblate spheroid.
23. To prove this we shall shew that, upon giving a
proper value to the ellipticity, the whole force which acts
upon any point at the surface is perpendicular to the surface; and that, if two canals of any form be made in the
fluid, terminated at any points in the surface, and leading
to the same point in the interior, the pressure on this point
is the same from the fluid in both canals.
24. Let T be the time of revolution: the centrifugal
force on the particle E, fig. 6, 7, and 8, is
4 -2 4,,79
2X, or -- CW
and is in the direction XE. Adding this to the forces
mentioned in (19), we have, the whole force acting upon
the point E in direction EX= (Q - T-). CW: that in direction EC = P. EC. The forces therefore upon any point,
estimated in directions perpendicular to the axis, and perpendicular to the equator, are still proportional to the distances from the axis, and from the plane of the equator.
25. PROP. 14. In order that the fluid in a canal from
the equator to the center, and the fluid in a canal from
the pole to the center, may produce the same pressure on
a particle at the center, the whole force at the pole must
be to that at the equator as the radius of the equator to
the radius of the pole.
Let two sections of the polar canal be made by planes
at the distances p and p + 3p from the center. The pressure which is produced by the fluid included between
these, estimated by its action upon a given surface, is pro