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.

130 FIGURE OF THE EARTH.
Hence, the effective attraction of the pyramid
2kbh cos%2. cos30. q. ÎO
a (1 - e). cos'0 + sin0'
Let w be the effective attraction of the wedge;
dw 2kb2 cos30
dO = a * (1 - e). cos2 + sin2 0
To integrate this, let sin 0 v;
rcw 2kb2 1 - VI
_-, ---cos0q.ô0.;
dv a s (1 - e2) + e
2kb62 I ev.W. w =- cos P. q.{ — + 3.) tan-1l e
a e- e,/(1 -e) V/1-)
Taking this from, to 0= +, or from = - 1,
2 2
to v = + 1,
2 2 02 1 e
w -- cos. _.. tan-1 e
a cos {/(1-e) -/(1-e) e2)
=4kbcos'p. -.{ sin - ( - e}
e3 6 2
Hence, if u be the attraction of the spheroid,
du - -e1)
d = 4kb cos2p { sin 'e -;
{1e J
integrating from = —, to - + -
u 2kb r b sin-le - e
11. If the spheroid differ little from a sphere, putting,
as before, a = b (1 + e), we find
e2=, - ()b2
a