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.

EXPLANATION OF TERMS IN LATITUDE. 53
67. PROP. 28. To explain the effect of the terms in
the expression for s.
The first of these is k.sing (0 -. If this depended on 0 -, it would shew that the Moon moved in a
g __
plane (39). But, as it depends on gO-y, or 0-(y+g-i1),
the Moon's motion in latitude may be represented by supposing her to move in a plane, the tangent of whose inclination to the ecliptic is k, and supposing the intersection
of this plane with the ecliptic to move with a retrograde
motion which is to the whole motion of the Moon as
g -: 1, and which therefore is nearly uniform. This is
exactly analagous to the notion of the perigee in (61), with
the single difference, that c being< 1, and g> 1, the motion
in one case is direct, and in the other, retrograde.
68. The second term
3m
k.. sin (2 - 2m -g) - 2/3 + y,
8
has precisely the same relation to the first, which the evection has to the elliptic inequality; and the alteration which
it produces in the place of the node and the inclination
of the orbit, may be found in the same manner. Thus,
y is the longitude of the node, if the second term did
not exist, and /3 the longitude of the Sun, when 0 = o.
Now, during the description of a portion only of the orbit,
we nlay, without material error, suppose in our expressions,
that the Sun and node are stationary: then, for gO -y,
the Moon's distance from the node, we must put 0 -y;
for (2 - 2 m) 0-2, double the excess of the Moon's longitude above the Sun's, we must put 20 - 23; and for
(2 - 2 -g) - 2 3 +y, we must put
(20-23 -0+ y), or (0-y)+2.(y-3).
Om
Hence, s = k {sin ( - y) +- sin 0 - + 2 (y - /3)
8
Sm Sm.
= k Il + — cos.sy-3. sinO-y+k- sin 2.y-/.cos0-y.
8 8