OF THE PERTURBATIONS OF THE PLANETS. 
203 
the plane of the orbit is inclined to the immoveable plane of x y, and the posi¬ 
tion of the line in which the two planes intersect one another. 
3. We begin with the more simple case of the problem, when the planet is 
urged solely by the central force of the sun. On this supposition, there being 
no disturbing forces, we must make R = 0 in the equations of the last §. 
By the formulas (3) and (4), we have, 
r 2 dv = h dt (Jj, 
1 + s- 
■ d\=ti 
T 
and h, h', are constant quantities. Now^j== is the projection of r upon 
the plane of xy and the area l + -- 3 . dX is the projection of the area r 2 dv 
upon the same plane ; wherefore, if i denote the angle of inclination which the 
plane containing the radii vectores r and r + dr, has to the plane of xy , we 
shall have 
cos i = 
l + s 2 
dX 
r 2 dv 
which proves that a plane passing through the sun’s centre and any two places 
of the planet infinitely near one another, has constantly the same inclination 
to the immoveable plane of xy. And it further proves that the planet moves 
in one invariable plane; for, unless this were the case, the areas described 
round the sun in any consecutive small portions of time, could not constantly 
have the same proportion to their projections upon the plane of xy. 
The orbit in its proper plane will be determined by the equations (2) and 
(3), viz. 
dr 2 r 2 d v 3 2 1 
]Tdl 2 + ^d¥ ~T + ~ °> 
r 2 dv — hdt 
a and h being arbitrary quantities. By exterminating d t ,J[x from the first 
equation, 
Id. 
dr 
Id 
2 
r 4 d 
*j-H-—■ 0; 
v 1 r r 1 a 
2 D 2 
