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.

50 LUNAR THEORY.
The first two depend on the eccentricity e and the mean
distance from the perigee cpt-a: their sum constitutes the
elliptic inequality in longitude. The next term, which is
called the variation, is proportional to sin 2. (pt - mpt + 3),
or sin2 (Moon's mean longitude- Sun's mean longitude).
It is therefore = 0 when the Moon is in conjunction; it is
greatest and positive when the difference of longitudes = 45~;
it is again = 0 when the difference = 90~, or when the Moon is
at quadrature: it has its greatest negative value when that
difference = 1350; and is again = 0 when the difference of longitudes = 180~, or when the Moon is in opposition. The Moon's
true place therefore is before the mean place from syzygy
to quadrature, and behind it from quadrature to syzygy.
(Newton, Lib. III. Prop. 29.) The last term, depending on
sin (2 - 2m - c) pt - 23 + a, is called the evection: it appears to increase the elliptic inequality when the axis of
the Moon's orbit is in syzygies, and to diminish it when
that axis is in quadratures: the reasoning of the last article
applies to it in every respect.
There are, besides, (Prop. 24.) these terms,
k2
- (sin 2gpt - 2 y), and - me' sin (mpt + - t ).
4
The former of these depends upon the Moon's distance from
the mean place of her node, and is nearly the difference
between her longitude, measured onr her orbit, and her
longitude, measured on the ecliptie: it is called the reduction. The latter depends on the Sun's mean anomaly: it
appears that, while the Sun (apparently) goes from perigee
to apogee, the Moon's true place is behind her mean place:
while the Sun goes from apogee to perigee, the Moon's true
place is before her mean place. (Newton, Prop. 66. Cor. 6.)
This is called the annual equation. The alteration in the
parallax, from this cause, is very small, being of the third
order (see the last term in the expression for u, Art. 55.).
65. In respect of magnitude, the evection is far the
most important of the inequalities which are produced by
the disturbing force of the Sun. And its effect on the