OF THE PERTURBATIONS OF THE PLANETS. 
227 
and consequently, 
d.^ = - 2d'R, and ^- = ~-2fd / R. 
These results agree with the investigation in the fourth section of this paper ; 
and they coincide with the remarkable equation first discovered by Lagrange, 
from which he inferred the invariability of the mean distances and the periodic 
times of the planets, when the approximation is extended to the first power 
only of the disturbing force. 
It has already been observed that ^ ^ 1 + s is the disturbing force per- 
pendicular to the plane passing through the sun and the coordinate z 3 that is, 
cZ R 1 | ^ 
to the planet’s circle of latitude ; and likewise that . — — is the disturb- 
ing force in the same plane perpendicular to r the radius vector. The elements 
of the direction of these forces are respectively and : wherefore, 
-v/l +s 2 
^ (d R _ . d K , \ 
2 Tl ds ) 
d R 
is the variation produced in the square of the velocity in the direction perpen- 
dicular to r. But dv being the small angle described round the sun in the 
time d t, the space described by the planet perpendicular to r, is r d v ; and 
v d v _ __ 
consequently — - --= is the planet’s velocity in that direction. Wherefore, 
CL Z y 
using the symbol £ to denote a variation caused by the disturbing forces per- 
pendicular to the radius vector, and observing that these forces produce no 
momentary increase or decrease of that line, we get, 
o a \ d ^ j \ _ it ^ dv 1 _ 9 v dv 2 
2 Ux + ds ds ) — ^ * dt\y. ~ r “ * * dt 2 .p : 
consequently, 
„/c?R d R \ dv 2 v. r* dv 
2 ^ W d * + in ds ) = 14 * • art = s • dwr, 
y 2 
r 3 dv 
and, as ^ -= and its square vary by no other cause but the action of the 
forces perpendicular to r, we have 
Now this is the same differential equation that has already been obtained by 
2 g 2 
