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.

204 PRECESSION AND NUTATION.
32. If we call the two parts of Solar Nutation, mentioned in (28) and (31), x and y, we shall easily perceive
that they are connected by this equation,
+ Y~ '$ w
37r.B s.inI. cos ) ' 3 T B. sin I
This is the equation to an ellipse, whose axes are in the
ratio of cos I 1. Thus is explained the construction in
Woodhouse's Astronomy, new edition, p. 367.
33. PROP. 13. To investigate the motion of the pole
produced by the Moon in one sidereal revolution.
By (9), it appears that, instead of considering at once
the effect of the Sun and Moon upon the Earth, we may
first investigate the effect produced by one, and then add
to it the effect produced by the other. For the effect
produced by the Moon, the investigation is exactly similar
to those of Prop. 11 and 12, and the same figure may be
used; observing that, as EC (fig. 7) is the great circle
apparently described by the Moon, Q is not now the pole
of the ecliptic, but the pole of the Moon's orbit. There
is only one difference: putting E for the Earth's absolute
force, M for the Moon's, T' for the time of a sidereal
revolution of the Moon, 1' for the inclination of the Earth's
axis to the axis of the Moon's orbit, a' for the mean distance,
dt r2 2 7r. a'
- will be, and T' will be --
dl /a'(1-e2) ((E+M) (E+M)
37rB 3rrB M
and, therefore, instead of T' we must put T7 - M
If the Moon's mass be - th of the Earth's, this = -7 )
n T' (n + l)
Thus we find for the motion of the pole parallel to the
Moon's orbit,
37r. Bsin'.cos I' — sin 21
T'w (+)n 2 1)