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.

168 FIGURE 0F THE EARTH,
6a. Priop. 29. To find the ellipticity of the Earth on
any assunied law of density of the strata.
Differentiating the equation
e.(>(c ) (c)
e - 5' c Y, X(c)- (c)} 2T-...(I),
we find
de
c3p(c). d- + 3 (c) - 3e.c2 (c) = o... (II),
and differentiating this,
d2e 2pce de 2pc 6. I)
dâ~ * + fpe ' d-5 + e= 0... (lII).
d 2- de 2f
Now, when p is given in terms of c, we must substitute it in
this last equation, and by integration find e. The expression
will contain two arbitrary constants: one of these may, in general, be conveniently determined by substituting in equation
(II), and the other by substituting in equation (I). Equation
(III) may be transformed into one of a simpler form, thus.
v
Let fpc2 = p, and let pe=v, or e = -: then, upon substituting in (III), we get
d2v 6v c dp
- v -U -= 0~...0....(IV).
de2 c2 p de
sin q c
65. Example. Suppose p =A. -, A and q being
c
constant. As this gives a density diminishing from the center
to the surface, it is probable that it will pretty nearly represent that of the Earth. On substituting in (IV), we get
d2v 6v
2 + qv = o:
dc2 C2
the complete solution of which is
3 3
v = C {sin (qc + C') + - cos (qc + C') -, sin (qc + C')},
qC q~? C21 (q+qc