IN THE MOTIONS OF THE EARTH AND VENUS. 
93 
whence 
„(°) 4 
a = — s • 
4 -7T ^ 
■k ^<o ' ^/ { a ,s + 2 da. cos 2 co + a 9 } 
or, putting - a for 
(°) 4 
Cl = - 7 • S 
£ 57 a' 
a/ / 1 + 2 a cos 2 co + a 2 }* 
Now let sin*;' = -rrr , a 
v'l 1 + 2 a cos 2 co + « } 
tion it is found that 
sin 2 co j , i _ v' l — cc 9 /v , . 
tt ; and a =- ; : after substitu- 
l + Vi — 
( °)_^_ _l_ 
W ~ 5r ad 1 ~i~ a J 1 * ^// 1 + 2 «' cos 2 co' + cc' 1 } 
sin 2 co' 
In the same manner, making- sin a" = >- f -r~~77~ VlL"o ~-rv • a " = -— ^ --. - a . : 
5 & v { 1 + 2 a' cos 2 j _j_ y' j_ a /2 
( 0 ) 
and so on, we get for C 4 the expression 
4,(1 +“') (1+4").(1 + «'”’) • S,» l/{l+2t ,(.) C0 1 s2<( ,W + .(.)>} 
The values of a!, a", &c. decrease very rapidly; and when c^ ni is insensible, 
- n ) 
V{l + 2«Wcos2 J n) + cc^ 2 } 
becomes S*(») .1 or. Consequently 
Cr — —f (1 + °0 (1 + a ") (1 + a ") • & c . 
the factors being continued till u ( ' 7l/ becomes insensible. The calculation is very 
3 Ql ( 0 ) 
easy; for, if we make shi |8 =a, sin /3' = tan 2 77 , sin/3''= tan 2 —, &c. then C 4 
= sec 2 • sec 2 . sec 2 ^-. &c. For Venus and the Earth (Mec. Cel. liv. VI.) 
a • ( 0 ) 1 
a or = 0,7233323: using this number in the calculation, C + = X 2,3863/5. 
* . cos 2 co l _,(o) 1 (i) 
34 - A S am > + 2 - a'a. ' w T g» ~ + F } = 2 ~ C t -COS 2*,-(1 + COS 4 «) 
1 ( 2 ) 
+ — Cr (cos 2 cy + cos 6 cy) — &c.; integrating between the same limits as 
before. 
