IN THE MOTIONS OF THE EARTH AND VENUS. 
69 
Earth); the variation of the elements of the second planet’s orbit will be given 
by the following equations: 
d a' 
dt 
d n' 
~dt ~ + 
2 n a 
r 
d R 
[i/ ’ de' 
3 vJ~ a! d R 
de' _ v! a! ,,, d R n’ a! (1 — e n ) a Id R ( d R\ 
dt u! e' ' C ' de' ‘ u! e \d7 d'tz) 
dix' 
dt 
dj_ 
d t 
n' a! 
(i - 
3 n ' 2 a! d R 
t + 
dR 
de 1 
2 v! a'* d R 
I v. as ju.' 
?n (cd x + y.z/) 
where R or — - ^7 — 
( * + y + 2T ✓{(*--*)• + far'-jr + (*■-«)■} iS expanded in 
terms depending on the mean motions of the two planets. These expressions 
are true only on the supposition that the actual orbit of rd is in the plane of 
xy , or is so little inclined that the square of the inclination may be neglected. 
The values of a', e', &c. on the right-hand side of the equations ought in strict¬ 
ness to be the true variable values. But it will in general be sufficiently accu¬ 
rate to put for e' the value E which it had near the time for which the investi¬ 
gation is made, and to consider it as constant: or at any rate the expression 
E -f- F*, where F is the mean value of its increase when t = 0 : and similarly 
da! de r 
for the others. Determining thus the values of-yy, yy, &c. and from them 
those of d, c\ &c., they are to be substituted in the expressions 
d — d ^ 1 + 77 e' 2 + (^— e' + ■— e' 3 — &c.^ cos (n't J r <■' — ■&') 
+ -y e' 2 + &c. ^ cos (2 d t + 2 s' — 2 to-') + &c. j- 
v' = n't -f- s' + ^ 2 e' — ■ye' 3 -f- &c.) sin (d t + s' — w') 
+ (y- e' 2 — e' 4 + &c.^ sin (2 n't + 2 s' — 2 w') J r &c. 
and the true values of the radius vector and longitude are obtained. 
