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.

52 LUNAR THEORY.
From these equations
15
- m sin 2 (a - 3)
15
1 +- m n cos 2 (a - 3)
15
=- m sin 2 (a - 3) nearly,
8
E = e + cos2(a- ) + m sin 2 (a -,)}
l 8 J l8
15
=e ll+i - m cos 2 (a -)} nearly.
8
And the united inequalities are represented by
15 15
e { 1 +-m. cos 2 (a - F)} oos { - a + - m. sin 2 (a -/3)}.
8 8
This is the same as the expression for the elliptic inequality
in an orbit whose eccentricity
15
=eil +- m.cos2(a-3)},
8
and in which the longitude of the perigee
15
= a - - mL. sin 2 (a - 3).
8
Hence, to find the Moon's place, when we have found the
longitude of the perigee on the supposition of its uniform
progression, we must subtract from that longitude
15
- m. sin 2 (long. perigee - long. Sun),
8
and apply the equation due to an elliptic orbit, whose eccentricity
15
= e 1 + - m. cos 2 (long. perigee - long. Sun).
(Newton, Lib. III. Scholium to the Lunar Theory.)