OF THE PERTURBATIONS OF THE PLANETS. 
211 
dr __ e sin (v — ot) 
r d v 1 + e cos (v — -or) 
^_ a (1 — c 3 ) _ h? 
1 ~~ 1 + e cos (v — tb) 1 + e cos ( v — et)’ 
e d zs sin (v — w) + d e . cos (v — zs) = 
Qdh 
~1T 
(l -{- ecos (v — sr)). ... (7) 
It is obvious that this last formula is tantamount to the equating to zero of 
the differential of r relatively to the variables, h, e, zs, or a, e, zs ; it may there¬ 
fore be thus written. 
dr dr dr 
— d a + "t— d c —f~ — d zs — 0. 
d a 1 de 1 
d e 
( 8 ) 
The equations that have been investigated, enable us to deduce from the 
disturbing forces the variable elements of the ellipse that coincides momen¬ 
tarily with the real path of the planet; a being the mean distance, e the eccen¬ 
tricity ; m the place of the perihelion, and A 2 the semi-parameter. We have 
next to find the relation between the time and the angular motion in the 
variable orbit. This will be accomplished by means of the equation r 2 dv 
— hdt,J(A ; from which we obtain, by substituting the values of r and A, 
dt vV (1 — e-)^. dv 
aj (T + ecos(v — 7s)y 
The integral J' -L / supposed to commence with the time, is the mean 
motion of the planet: when there is no disturbing force, a being constant, the 
mean motion is proportional to the time and equal to n x t ; but the action of 
the disturbing forces, by making a variable, alters the case, and requires the 
introducing of a new symbol £ to represent the mean motion. Thus we have 
i=r^xdt-, d^ = 
(1 - . d v 
^1 + ecos (i> — et)^ 
The value of £ cannot be obtained directly by integration on account of the 
variability of e and zs. Let f (v — zs, e ) express that function of the true ano¬ 
maly which is equal to the mean anomaly in the undisturbed orbit; that is, 
suppose, 
2 e 2 
