OF THE PERTURBATIONS OF THE PLANETS. 
213 
d.f(v — zr,e) /i p 2 e sin (v — ot) . d v 
d -cr 
and, by integrating, 
= “(1 - e 2 )' 
/6 'Zt i 
•y ( T 
+ e cos (t; — vt)) 3 
d ,f[v — t>s, e) sin (v — zr) V 1 — e 2 sin (v — zs) \/ 1 — e 2 
Te 1 + ecos(t ;-«r) + e cos {v - CT )) 3 _ 
(^2 + e cos (v — zr)^) sin (u — ar) */ I — e 2 
0 + e cos (u — ay)^ 2 
d ./(p — ot, g) ( 1 — g 3 ) 1- 
( ^' Er ^1 + e cos (v — 
These values being substituted in the foregoing formula, we shall find this re- 
sult, after dividing all the terms by the coefficient of dzr, 
^ . (dt-dz?) = - ^ ^ / rfe - dzr, 
(1-e 2 ) 4 
or, more concisely, 
^2 + e cos (v — ay)) sin (u — ay) 
a 3 */ 1 — e 2 (J j N 
^ . (de-dvr) 
l _ 8 . d e — dvr. . . . (9) 
From the equation between the mean and the true anomalies we deduce, 
v = i + £ — <£>, 
O representing a function of the mean anomaly £ -f- s — to- : and as the differ- 
entials of £ and v are independent of the differentials of g, e, and ay, we shall 
have. 
dv , , dv , , dv 7 
d s 
Now, 
du 
d £ C?£ 
iZ £ ' dzr 
d . <P dv 
( 10 ) 
<Z. 0 
d ay 
and, because, is a function of g — ar, 
cZ . 0 d . 0 dv . dv 
- T7 = -~d consequently, ^ + j- 
W 
1. 
The equation may therefore be thus written, 
