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.

156 FIGURE 0F THE EARTH
de
regard to c. Let fp. -d Xd (c), the integral being made to
vanish when c = 0. If we suppose the polar semi-axis of the
interior surface to be c, that of the exterior to be c, then
de
fp d for the shell, between these limits, = X (c) - X (c). Hence
it will easily be seen, that the force in the direction of i
47r 2
_ = { x (c) xX (c)}.f,
that in the direction of y
4wr 2
-, -.-{(c) )} -
that in the direction of z
-- {x (c) - (c)}. h.
$ 5
51. PROPI. 24. To find the attraction of a heterogeneous
spheroid of the same kind as that in Prop. 22, upon any point
within it.
Let y be the polar semi-axis of that spheroidal surface
which passes through the given point, and through all other
points at which the density is the same. And let c be the
semi-axis of the exterior surface. Then the given point is
external to all the spheroids whose semi-axes are less than y,
and to these, therefore, the integration in Prop. 22. must be
applied. Taking these integrals, then, from c = o to c = y, we
find for the forces in direction of x, y and z,
{ 1 +h2 - 3f2 - 3gs 4
3 + h )5 (f2 + g2 + h2)
47r f 1 12h2- 3f2- 3g 1
3 (f +g+ 7) h 5 (f2 + g2 + ) ()
4ir~ 6h - 9f - 9g2 i
3 (f2 + g5 + h )g + 2) 7If (,) h.
Again, the given point is interior to all the spheroidal surfaces
whose polar semi-axes are greater than y, and less than c;