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.

42 LUNAR THEORY.
55. PROP. 20. To integrate accurately to the second
order the differential equation for u.
$ 3k~ mAssume u = a 1 - -- -+ e. cos (cO - a)
4 2
+ A cos (2g0 - 2 y) + B cos (2 - 2m) - 213
+ C cos (2 - 2m - c) - 2/3 + a + Dcos (m0 +/3 -,
according to the direction in (6).
Substituting this value in the differential equation, and
making = 0 the coefficient of each cosine,
S m" ae
ae (1-c) - 2 =0;
2
3m2 m2.. c2 = 1 - -- c= - —, nearly.
2 4
(1 -g2) _3k2 a
aA (1- 4g')- =0;
4
3k2 k2.' = —, nearly, since g nearly = 1.
4 (1 - 4go) 4
aB (1 - 2 - 2 ) + 3m2a = 0;
3m2.. B = = m", nearly.
(2 -2m)2 - 1
15
aC(1 - 2-2m-c 2) - m2ae = 0;
15 m~e 15 m2e
2 1 - (2 - 2m - c)2 = I - (1 - 2m)2 nearly
(since c nearly 1) = me, nearly.
8
aD (1- m) + - m2ae =0;
2
3 me' 3.'. D = - ~ = - m2 e, nearly.
2 1 -m2 2