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.

134 FIGURE OF THE EARTH.
Consequently, the attraction on E, in the direction perpen.
dicular to the axis of the spheroid.^.^. - e( )
= CW. 2kr. /(e) sin-' e e }
where e is the eccentricity of the smaller spheroid, which
is the same as that of the larger, as the spheroids are
similar. In the same spheroid, it will be observed, this
attraction is proportional to CW.
15. PROP. 10. If from any point E, (fig. 7) on the
surface of a spheroid, a line EX be drawn perpendicular
to the axis, and a spheroid XY be described concentric,
similar, and similarly situated, to the given spheroid, touching that line at its pole X; then the attraction of the
given spheroid on the point E, in a direction parallel to
its axis, is equal to the attraction of the smaller spheroid
on a point at its pole X.
The demonstration of this is, in all respects, similar to
the demonstration of Prop. 8. The spheroids must be divided into wedges, by planes passing through the line EX;
and the sections of the spheroids, made by one plane, will
be similar and concentric ellipses.
16. PROP. 11. To find the attraction of E, in a
direction parallel to the axis of the spheroid.
By Prop. 10, this is equal to the attraction of the spheroid XY, on a particle at X. But, by Prop. 6., the attraction of the spheroid XY, on a particle at X
47rk.. W e- sin-' e;
therefore the attraction of the larger spheroid on E, in a
direction parallel to its axis, is
4,k. EC. 1 - (- e2) sin-'e.
In the same spheroid this is proportional to EC.