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.

ATTRACTION OF OBLATE SPHEROID. 127
Putting for x and y their values rcos and rsin0, this
becomes
r' sin2 0 = (2br cos 0 - r cos2 0),
2b cos 0 2b cos
whence r=
aw -hcr ' 1 - e esin2 0
1- " sin2 0
aputting e for the eccentricity of the generating ellipse.
Hence, the attraction of the pyramid ultimately
2b cos2 0. sin 0. 0
= l- eOsin 0
and if w be the attraction of the wedge,
dw cos 0. sin 0.- = 2kbw.
dO 1 - e sinr0
Let cos = z; then
dw z2
-== -2kbw.
dz 1-e +ez'z
~dx~ I~ - e^ + e 2e:
integrating
2kb ---- v ( - e) tan- z
=- e e-e-e
Taking th to or from = 1, to 0,
w = f1k b(w-2)
w = 2kbw { /('e e') tan-' e
4ca C/ — ^ 7 e?)^
This is the attraction of a wedge whose angle is w; and
since the attraction of every wedge with an equal angle
must be the same, the attraction of the whole spheroid will
be found by putting *2 7 in tle place of w; that is, the
attraction
- kb tan-' ~-}
4 4.kb { ( eS2)e t e.
4. kb /(1 -e e) sinl e
e2 e 3