OF THE PERTURBATIONS OF THE PLANETS. 
213 
d ./(» - AT, (?) 
d -57 
and, by integrating, 
= -(l -e*y 
■A 
2 e sin (v — ot) . d v 
1 + e cos (v — 
d .f(v — ot, e) sin (v — ot) a/ 1 — e 2 sin (v — ot) s/ 1 — e 2 
1 + <?cos(v-«r) (l + ecos(y-OT)) 3 _ 
^2 + e cos (v — sin (v — tv) \/ l — e 2 
0 + <?cos (v—^y 
(1-0" 
d .f(y — ot, v) 
(^1 -(-£ cos (v — ot)^)' 
These values being substituted in the foregoing formula, we shall find this re¬ 
sult, after dividing all the terms by the coefficient of dvs, 
( 1 + ecos (v — rs)y 
(1 — e 2 f 
or, more concisely, 
. (d s — e?cr) = — 
^2 + e cos (v — sin (v — ct) 
1 — e 1 
de — dvs. 
a 3 \/ 1 — e 3 
. (ds — d-ft) = — 
^2 + e cos (v — sin (v — ot) 
. de — dvs. . . . ( 9 ) 
r* - v - / 1 — e 2 
From the equation between the mean and the true anomalies we deduce, 
V = i + £ — O, 
d> representing a function of the mean anomaly £ -f- £ — w ' and as the differ¬ 
entials of £ and v are independent of the differentials of g, e , and ®r, we shall 
have, 
dv 7 , dv i , dv , ' , 
Te di + d~e de + d^ d ™ — ° .( 10 ) 
Now, 
dv _ d . <P dv _ d. <P 
ds de ’ dvs dvr 
and, because, d> is a function of g — vs, 
d. <P d. <P dv , dv 
-77 = ~-d ^ : consequently, ^ = 1 • 
The equation may therefore be thus written, 
