OF THE PERTURBATIONS OF THE PLANETS. 
197 
plane passing through the origin of the coordinates placed in the sun’s centre ; 
and z, z' being perpendicular to the same plane. 
Further, let M, m, m' denote the respective masses of S, P, P' ; r and r' the 
distance of P and P' from S, and § the distance between the two planets ; then, 
putting (a = M 4- m, the direct attraction between S and P will be and 
the resolved parts of this force, acting in the respective directions of x, y , 2 , and 
tending to diminish these lines, will be 
[J.X [X.1J pz 
^ 5 ^3 ^ * 
H)f} 
The planet P' attracts S with a force = -p§, of which the resolved parts are, 
ml x / m! y m! z' 
r !3 ) j/3 > r l3 • 
VI' 
The same planet P' attracts P with the force of which the partial forces 
m' ( x ' — x) m! {y — y) wl ( z — z) 
f 5 f 5 f 
Were S and P attracted by V in like directions with equal intensity, the rela- 
tive situation of the two bodies would not be changed, and the action of P' 
might be neglected : but the attractions parallel to the coordinates being un- 
equal, the differences of these attractions, viz. 
m ( x' — x) m' x 1 w! {y' — y) m' y' m (z' — z) m! z 1 
yti ) gi jJi 5 fl 3 f'3 1 
are exerted in altering the place of P relatively to S. These last forces increase 
the coordinates x, y, z ; and, therefore, they must be subtracted from the 
former forces which have opposite directions, in order to obtain the total 
forces acting in the directions of the coordinates and affecting the motion of 
P relatively to S, viz. 
