OF THE PERTURBATIONS OF THE PLANETS. 
211 
dr e sin (v — -sr) 
r d v 1 + e cos (u — ■or) 
a ( 1 — e 2 ) Td 
y% i 1 — ^ 
1 + e cos (u — ct) 1 + e cos ( v — ot) 
<2 d Ti ( \ 
e dzs sin (v— -sr) + . cos (v — zs) = (l + ecos {v— vs)). ... (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, is, or a, e, sr ; it may there- 
fore be thus written. 
^da+^de+ ~d^=0 (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 ; cr the place of the perihelion, and h 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 [h ; from which we obtain, by substituting the values of r and h, 
dt vV ( l — e 2 )^ . d v 
aj ^1 + ecos(u — 
The integral ^ 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 
„ /■»! u, 7 , 7 „ (1 — . dv 
£ = / 3 X dt ; dY — t — “y 
J a 7 ^1 + ecos (v — 
The value of Y, cannot be obtained directly by integration on account of the 
variability of e and is. Let f(v — is, 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 
