212 
MR. IVORY ON THE THEORY 
/{V-V, 
( 1 — <?*)■*• . d v 
1 + ecos (u — zr)^~ 
the integral being taken on the conditions that it vanishes when v — zs — 0 , 
and that e and zs are constant. If now we make e and zs variable, we shall 
have, 
d.f(v — zr, e) d , . f (v — w, e) 7 d . flv — zr, e) 1 in, ^ 
— ~Tv dv + rfe de + ~ d« - d* = d.f(v-v,e). 
But the partial differential relatively to v, is no other than the expression of 
d £ : wherefore. 
+ d ■ f(v - g) de + dzg 
d zr 
d .f(v — zs,e). 
By introducing a new symbol this equation may be separated into the two 
which follow, 
dX^-\- dz — dzs — d ,f{v — zs, e ), 
7 7 d •/(» ~ e ) j _ , d -A v - «•> e) 7 
dz-dzs — j- £ de-] ^ d 
zr. 
In the integral 
£-fs — 7Z=f(v- zr,e), 
£ -f g is the mean motion of the planet reckoned from a given epoch, s however 
representing a quantity that varies incessantly by the action of the disturbing 
forces, the amount of the variation being determined by the second formula 
in which the value of s alone has not been previously ascertained. The mean 
anomaly of the planet is £ + z — zs ; and the integral shows that there is the 
same finite equation between the mean and the true anomalies in the disturbed 
orbit as when there is no disturbing force. It follows therefore that, in both 
the cases, the true anomaly, the true motion of the planet, and the radius 
vector, are deducible from the mean anomaly by the same rules and by the 
solution of Kepler’s problem. 
In order to find the value of the new variable z, it is necessary to eliminate 
the differential coefficients from its expression. Differentiating relatively to e 
and nr, we shall get, 
d . f(y — zr , e) 
d e 
cos (v— zr) + 3 e + e 4 cos (v— zr) 7 
— */» - jy — dv ’ 
