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.

44 LUNAR THEORY.
The only terms to be preserved are
r2k 2 sin (g - y) - sin (2 - 2m - g)0 - 2/ +;
because g is very nearly = 1, and 2 - 2m - g = 1 - 2m, nearly,
and therefore differs little from 1: and therefore, by (45) and
(47), both these terms, on integration, will be important.
ds
Third, d- = kg. cos (gO - y) = k. cos (gO - y) nearly,
wh;ih is of the first order; taking therefore the first term
T
only of the expression for h-U3S that is, (51)
- m2. sin (2 - 2m)0 - 2(3,
we have
T ds S.T -.. Mn2k. cos (gO - y). sin (2 - 2 m)O - 2P
= - -m2 k {sin(2-2m+g)O- 213-y+sin(2-2m-g)O-2/3+y}.
4
The only term to be preserved is
- m2k. sin(2 - 2m - g) - 23+ + y.
Collecting these parts, the equation of (41) becomes
d2s 3 3
= + + - m$ k. sin (gO-y) - - mksin (2-2m-g) 0-23+y.
dOu 2 2
57. PItOP. 22. To integrate the differential equation
for s.
Assume s = k {singO - y + Asin(2 m -g) - 28 + y},
and substitute in the equation above: then, making =0 the
coefficient of each sine,
k(l-g)+O-, +
k(i -g~) +c 7mnk= =, go=p+ 2;