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.

200 PRECESSION AND NUTATION.
And the moment of inertia
S r d(c5) 87r d(c')
- fIp = a(c), if (C))=fcP dc
1 5 d c 1{ de
Hence, the Earth being heterogeneous,
a. sin 0. cos 0. --.
r3 - (c)
27. In the investigation of the value of a, we have
supposed that the only force which tends to give the Earth
a rotatory motion about AC, (fig. 3 and 4), is the action
of a distant body. This, however, is not strictly true;
for, since AD, the axis about which the Earth is at any
instant revolving, does not coincide with AE the axis of
the figure, the centrifugal force will diminish the effect
produced by the distant body. With an ellipticity, however, so small as that of the Earth, this diminution is not
sensible.
28. PROP. 11. To investigate the quantity of solar
precession for any given time.
Suppose EC, (fig. 7), to be the projection of the ecliptic on the surface of a sphere described about the Earth's
center; S, P, Q, the projections of the Sun's place, the
pole of the Earth, and the pole of the ecliptic; join SP
by an arc of a great circle. By Prop. 9 and 10, the
value of a is
SB.S.
— B sin. cos 0,
r3
a2 _ (c)
B being = a2 + if the Earth be homogeneous, and = -(c)
if the Earth be heterogeneous. Now, 0 is the angle made
by the Earth's axis with the line joining the centers of
the Sun and Earth, and is, therefore, in (fig. 7), represented by SP: hence
3B. co SP;
ta = - sin SP. cos SP;
r3