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.

INFLUENCE OF A DISTANT BODY CALCULATED. 199
24. In the same manner, it would be found that the
sum of the y23m for the spheroid = - kac x c'. Hence,
15
the moment of all the impressed forces
aS 4,r
—. - ka2cc (a2 - c2) sin 0. cos 0.
r3 15
25. Now, to find the angular velocity which this would
generate in 1" about the axis whose projection is A, we
must (Whewell On the Motion of Points constrained, Sc.
Art. 206, Earnshaw's Dynamics, Art. 144) divide this by
the moment of inertia, or the sum of (x2 +y2)êm for the
whole spheroid. This sum is found in the same manner
to be
4,7
- ka2c (a2 + c2).
15
Hence, the angular velocity generated in 1", or a,
SS a2-c2
= 3. 22.sin 0. cos 0.
r3 a2 + c2
26. PROP. 10. To calculate a, supposing the Earth
heterogeneous.
Suppose the Earth composed of strata of different densities, bounded by spheroidal surfaces of different ellipticities,
as in the Treatise on the Figure of the Earth. Let c be
the semi-axis of any one of these spheroids, e its ellipticity;
c the semi-axis of the external surface, e its ellipticity; p
the density at any point; p and e being functions of c.
Then, (sincea2-c2=2c2e nearly, and a +c2= 2C2 and a2c =c
nearly) as in Prop. 22. of the Figure of the Earth, we
shall have for the moment of the impressed forces,
3S 8 r d(c5e)
sin 0. cosO 1-. p. dr3 snc.15 de
$S 8ir
7sin 0. cos 0. - (c).
r3 15