OF THE PERTURBATIONS OF THE PLANETS. 
205 
from the perihelion if e be positive, and from the aphelion if the same quan- 
tity be negative. 
It must however be observed that the preceding determination rests entirely 
]? ] t 2 
on the assumption that, in the equation e 2 = 1 — — , the quantity — is posi- 
tive and less than unit. Without entering upon any detail, which our present 
purpose does not require, all the possible cases of the problem will be suc- 
cinctly distinguished by writing the equation in this form, 
~=(l~e) X a. 
The quantity on the left side being essentially positive, the two factors on the 
other side must both have the same sign. If they are positive, the orbit will 
be an ellipse ; if they are negative, and consequently e greater than unit, the 
curve described by the body will be a hyperbola ; and it will be a parabola, 
when e = 1, and a and l —e pass from being positive to be negative, at which 
limit the equation will assume this form, 
A 2 
Y=0X~. 
h°- 
In all the cases Y+~ e is the perihelion distance. 
The nature of the orbit being found, we have next to determine the relation 
between the time and the angular motion of the planet. For this purpose we 
have the equation, r 2 dv = li dt */(*, from which, by substituting the values 
of r and h, we deduce 
dt \/ 1 
aJ 
(1 - e*fdv 
0 + e cos (u — 
Let 
\/ y. 
a? 
n ; then, by integrating, 
n t 
+s ~*-frx 
(1 -e*)±dv 
(' + ecos (u — ■37) 
the quantity under the sign of integration being taken so as to vanish when 
v — 7S = 0, and s being a constant quantity. The mean motion of the planet 
reckoned from a given epoch, is equal to ^ ^ H- s ; and the mean anomaly, to 
