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.

126 FIGURE OF THE EARTH.
then the attraction of one stratum ultimately= p, whence
kbx kb
u =-~, or for the whole pyramid=
ON THE ATTRACTION OF AN OBLATE
SPHEROID.
8. PROP. 6. To find the attraction of an oblate spheroid
on a particle placed at its pole.
Let B, (fig. 4), be the pole of the spheroid, BD the
axis; let the spheroid be divided into wedges, by planes
passing through BD, two of which are BPD, BQD, making
with each other the very small angle w; in these planes
draw BP, BQ making with BD the angle 0, and Bp, Bq,
making with BP, BQ the very small angle S0; and suppose the wedge divided into pyramids similar to BPq. Let
x be the abscissa of P, measured along the axis of the
spheroid; y its ordinate; let BP= r. If through qp a
section pqts be drawn perpendicular to the axis of the pyramid; since qp ultimately = yw, and qt = rO, the area of
this section = rywQ0; therefore by (7), the attraction of the
krymc8
pyramid - r = kywrO. This is in the direction BP:
but as the whole attraction of the spheroid will evidently
be in the direction BD, we must resolve the attraction of
the pyramid into one parallel to BD, and one perpendicular
to BD: the former will be effective, but the latter will be
counteracted by forces in the opposite direction. The effective part
=kywo. - = kr. sin. cos. w. 0.
r
Let a be the equatoreal radius of the spheroid, b the semiaxis;
then y2 - (2bx - y 2)=