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.

170 FIGURE OF THE EARTH,
For the ellipticity of the surface we must make c = c: thus
we obtain. l -- -.-( -
5 ( qc \ ( - 3 3
5m tan q q2c qc. tan qc
e=.
-qc2 qc q2C"
2 _ q~ c2 _
tan qc tan2qc
3z
1 -22
qc 5m q2c
If 1 - - e =.
tan qc 2 q-c
3-z5 wr 57 -66. Suppose q=. -. Then qc= -: =5,53452;
OC c
5m
and e =.0,37703. And, since n the increase of gravity
5m 5m
at the pole = - e, by (62), n = -.,62297. In the
2 2
1 5m 1
Earth m =-: - = - therefore on these assumptions,
289 2 115'
i 1
e = —, n = These are very nearly the same as the
305 184,6
observed values of e and n.
67. PROP. 30. To compare the mean density witli the
density at the surface.
If the whole spheroid consisted of matter whose density = mean density, its mass would be the same as it
mass
actually is. Hence, the mean density = lume. Now,
volume
neglecting the ellipticity, the mass may be found by considering it as a series of spherical shells of different density: and since the surface of one of these, whose radius
is c, is 47r. c~, the mass of the shell whose thickness is ic
jnAD APdjçst In js ijlt;Jj3 apJr 4A.î<CLj. a2 ^ai^fmVjp ju J
the mass,
du
d= 4-pe2, u = 4rfpc2 = 47r.p,
de