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.

208 PRECESSION AND NUTATION.
The velocity, therefore, with which the inclination is increased,
is
6r2. B
T'2 w (n +.) sin qP.cos q P. sin QPq.
T'2 W (n + l)
Now, sin qP. sin QPq = sin i. sin Q;
and cos q P = cos I. cos i + sin I. sin i. cos Q.
Their product
= cos I. sin i. cos i. sin Q + sin I. sin2i. sin Q. cos Q;
or, the velocity of increase of inclination
6 r. B 2,.t.. 4 7r
=TO(+)u l(cos I.sin i.sin- +1sinI. sini.sin —
T 2W(n + 1) or 2
Integrating this with respect to t, the inclination
6r2. B (r. 2rrt
-= I T,~ ---. - cos I. sin 2i cos --
T'2w (n+l) 47r T
+. sin I.sin".cos-.
The terms subtracted froin I are periodical, depending upon
cos -, or cos (180~- long. Moon's ascending node,) and
r
upon the cosine of twice that angle; the first is a part of
Lunar Nutation; the second is usually neglected.
40. If we call x and y the first terms of nutation
in (36) and (89), it will easily be seen that they are connected by this equation,
$l f 2TW'2 (n+l1) 2 2J 2T'w (n+l) 12
327r Br. cos 2l. sin 2 iJ rB. T. cos I. sin2iJ
This is the equation to an ellipse, in which the axes have
the ratio of cos21: cos. This explains the construction
in Woodhouse's Astronomy, page 357.