IN THE MOTIONS OF THE EARTH AND VENUS. 
93 
whence 
„(°) 4 
C r = — s 
tv TT 
7 r {a ,s + 2 a' a. cos 2w + a 2 } 
or, putting a for y, 
n (°) 4 
t • S 
Now let sin&/ = 
i it a! ’ ^{1 +2* cos 2 w + a 9 } 
sin 2 a) 
y'l 1 + 2 a cos 2w+ a*} 
tion it is found that 
or ; and a' = - ^C - 1 — : after substitu- 
l + \/ 1 - « 9 
n ( 0 ) -±-nj_ q l 
H — TT ad 1 ■+■ a ; • ^ • ^/{ 1 + 2 «' cos 2 «/ + u 1 -} 
sin 2 w' 
In the same manner, making sin J' = ^ { , - + = «" = = 
do) 
and so on, we get for C 4 the expression 
— , ( 1 + a!) ( 1 + a”) ( 1 + o' . S*, (») — — y y Try 
r 'V n > v ' a/{1 +2 «Wcos2 ®W + } 
The values of a", &c. decrease very rapidly ; and when a is insensible, 
S«(”) 
V{l + 2 « (7l) cos 2 + 
,(«) 
y, becomes S»« .1 or y . Consequently 
cr ; } 
cf = I (1 + «) (1 + «") (1 + «”) • &c. 
the factors being continued till becomes insensible. The calculation is very 
a ai (o) 
easy; for, if we make sin/3 — a, sin/3' = tan 2 77 , sin/3" = tan 2 77 , &c. then 
= y sec 2 y . sec 2 y- . sec 2 ^ 77 . &c. For Venus and the Earth (M 6 c. C61. liv. VI.) 
a . ( 0 ) 1 
a or y = 0,7233323: using this number in the calculation, C, =-^x 2,386375. 
. . COS 2 CO 1 _(°) ^ 1 _.(!) . 
34 - A g ain > ^/ { a» + 2 a ' C .co S 2«, + a »} =T C i •COs2,»-2-C j (l+COS4») 
1 ( 2 ) 
+ 7 C 1 (cos 2 + cos 6 <w) — & c. ; integrating between the same limits as 
before. 
