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.

144 FIGURE OF TIIE EARTII
Taking only the first power of c, we have
7r 2 q
- c. q —, or -= c -.
2 a 7r
1 1
34. In the Earth it is found that m =, or =-
289 434,
hence, supposing the ellipticity small, e=: or, supposing
230
the eccentricity nearly =1, C=-61. That is, the Earth
would be an oblate spheroid, with axes either in the proportion
of 230: 231, or in the proportion of: 681. It is found by
measurementt, that the ratio of the axes is nearly 300 301:
hence, the Earth is not homogeneous.
35. In the spheroid of small ellipticity, the proportion
of gravity at the pole to that at the equator is the same
as the ratio of the axes, or is the ratio 31: 230, supposing the Earth homogeneous. By observation, it is found
to be about 188: 187.
36. Resuming the consideration of the general equation
of (29), and the construction of (30), it is easily seen that,
upon giving to q a certain value, the curve will touch the
line of abscissoe: and upon increasing q' the curve will not
meet the line of abscissae at all. In the former case, then,
there is but one form of equilibrium, and, in the latter,
equilibrium is not possible. To find e', the value of e,
which gives but one form, we may observe, that two roots
of the equation have become coincident, or are equal: if
then we take the differential coefficient of the first side of
the equation, it must have one of the equal roots: or the
same value of e will make it = 0. This gives
9 2 9 8 sin-e'
e 7 e ée14 e V/(1-e2 0'
solving this equation by approximation, e'=,92995, whence
b' a
-=,36769, — 2,7197. Substituting this value of e in the
a b'