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.

100 PLANETARY THEORY.
x {1 + - + +____ - 3+ '+ &C.}
92a 2..4a 2.4.6a3
_-1 {1 + (1 + + &c.l
1i fla2 1/l' $a'4 /1..5\ 2 a'6
a ~2a \a2.4a \2a.4.UJ
1 at' 1 1.3 a' 1.3 1.3.5 a'5
aa 2 a.4 2a 2.4 2.4.6 a5 J&
(e~'-1 + e-0o~-1)
+ (&c.) ((2w~ + E- V-) + &c.
1 2/12 a'e 1.S\2 a'4 l.'.5 2 a'6
a 2 \24) 4 2.4.6) a6
2 1f a' 1 1.3 a 1'3 1.3.3.5 a',.
+- -.- -.. + &c.
a a2 a 2 2.4 as 2.4 2.4.6 a' J
As - is small, these series converge very rapidly.
122. Case 2. Suppose that a' is not very inuch less
a
than a. Put a for -- and 7r- 2\ for ra): then
a,+,+ 1 = C(' ) - C(') cos 2 / + C() cos 4, -&c.
a V/(1+-2acos2\a+ &a)
Integrating both sides with respect to i from, =0 to = 2,
we get 7r C =1
4 a a(1 +, a cos2 + a2)
and the problem is now reduced to finding the value of this
definite integral. Now let
sin 2i '
/(l + 2a cos 22J +- a'), cos 2 + - a, sin 2
cos /= tan,\ =
/(1 + 2a cos 2 \4 + a2) cos 2 \ + a