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.

38 LUNAR THEORY.
or -2 a. +I 3e cos o +,3 - - 3e coscO - a}
x {cos (2 - 2m) 0 - 23 - 2e' cos (2 - m) 0 - -
+ 2e'. cos (2 - 3m)0 - 3/3 + }.
Multiplying together these series, and setting down all the
terms of the third order, we have
~3e~e'
co 2- cos - 2 + - cos (2 - m) 0 - f -
Se
_ 3 -cos+ - cos (2 - 2m) - 3/3 +-
~2a~ ~2
3, J — cos (2-2m +c) 0-2/3-a
— m- a 2
3e
- - cos (2 - 2m - c) 0 - 23 + a
-2e' cos (2 - m) 0 - -
+ 2e' cos (2 - 3sn) 0 - 33 +.
Of this the only part which must be preserved, is
3 3e
-- 2a Icos (2 - 2m) 0- 23 - - cos (2 -2m - c) 0 -23 + a,
2 2
because the coefficients of 0 in the other terms (all which
are of the third order), namely 2 - m, 2 - 3m, 2 - 2m. + c,
are neither small nor nearly equal to 1.
P
The remaining terms of - are of the fourth order;
hence, our value of h- is
3 k2 3 k2
a (1- +- cos 2g0 - 2y)
4 4
m2 a
--- il+3e'cosmO+/F —3e cosc-a+ 3cos(2-2m) 0-2/3
9e (2 ) 0 -3 + a
- - cos (2 - 2rn - c) O - 2/3 + a}.
2