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.

ATTRACTION OF OBLATE SPHEROID. 135
17. PROP. 12. If a point E, (fig. 8) be placed at
the interior surface of a shell, bounded by similar and
concentric spheroidical surfaces, the attraction of the whole
shell will be 0.
For suppose the small pyramids EF, EG, to be formed
by the same planes passing through E; let a plane pass
through the axis of the pyramids, and through the common center of the spheroids: through H the point of bisection of EM, draw WHKL. Since EM is bisected in
H, the tangent at K is parallel to EM; and since the
ellipses are similar and concentric, the tangent at L is
parallel to that at K; it is therefore parallel to FG; and FG
therefore is bisected in H, or FH=HG. But EH=HM;.. EF= MG. Now the attractions of the pyramid FE,
and the frustrum MG, upon a point at E, are proportional
to their lengths EF and MG, by Prop. 4.; they are,
therefore, equal; and they are in opposite directions; therefore they destroy each other. Now the whole shell may
be divided into pairs of pyramids, in each of which it
may be shewn that the attraction is 0; therefore the attraction of the whole shell = 0.
18. PROP. 13. To find the attraction of a spheroid
on a point within it.
From E the point, (fig. 8), draw EC perpendicular to
the plane of the equator. By Prop. 12, the attraction of
the shell external to the spheroid EKM is o; and by
Prop. 9., the attraction of the spheroid EKM on E, in
the direction perpendicular to the axis of the spheroid, is
CW. krr.\./( - e) sin-' e - 1.
e e '
By Prop. 11, the attraction in direction parallel to the
axis, is
EC. 4kw /(1- e) s ein1.
[e, ea
19. The results of all these propositions may be thus
stated.