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.

166 FIGURE OF THE EARTH
at the equator, neglecting the small terms, is 4- (c). the
3 C2
4 7r2 4-r2
centrifugal force is -.a = — c, nearly. Hence,
4 2 47r (c) 3sr m (c)
C = M. - -= M
3 2 CT2 C4
Hence, the whole force at the surface
4__r 0 (c) 6 \P() (P (C) 9 hS(c) (f +g2)}
3 C 5 c4 c4 5 c
=-{Jc ) _ _*,(c ) - ((2e+m). *(c) *Ii) (f2+g24)
61. The equation of Prop. 26, becomes, at the surface,
e.0(c) _3 Nf(c) 37r_
c3 5' c5 2c 0
3sr m 0(c)
or, since -_ = -.
2 T. 2 c3
qSn' 0 (c) 3x1dc)
e - } 5' c' = 0.
( 2 C3 5 * C5
Substituting from this the value of xk (c), we find, for the
force at any point of the surface, the following simple
expression
4w *c )f 1-(2e-r - f2 ( ). ( g)
3 ) 2 ) \ c2 ] \
C
62. For the force at the pole we must make f= o, g= 0,
and the force, therefore,
47r C (c)
=- '. - (1 -2e-m).
3 C2
For the force at the equator we must makef2 + gl = a2 = C2
nearly, and, therefore, the equatoreal gravity
t (1 - (2e ) - - e)
3 C2,