THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 73 
The effect of the disturbing action of a body m’ on the motion of m around the 
Sun is given by the expressions 
2 (uv —x ce qe2(y—y y ey ae 2! 
tee Eg 2x) Ee — a): 
Introducing these into the equations given above we have in the case of dis- 
turbed motion 
ix 2 NBEO tat Shae aoe =) 
aa t+ (1 4 m) —; = TV Hi a 
dy 2 ( yo > yefy—y _ y (1) 
ae + k#(1+m) = =m k ee ¥) 
a bs 
1 9 / > Zz i) 9 z'—2z tf 
oi + kh (1+ m) 2m ae —— =) 
The second members of equations (1) show the difference between the action of 
the body m’ on m and on the Sun. The action of any member of bodies m’, m’’, m’” 
ete., can be included in the second members of these equations, since the action of all 
will be similar to that of m’. 
b) 
The second members can be put in more convenient form if we make use of the 
function 
(Ove nv’ (' kei eee) 
tee WA Pe 
Differentiating relative to « 
CO an ( 1 dd =) 
dx AB ale Fas” 
But since 
we have 
Gia x —x 
dz” Ame’ 
A. P. S.—VOL. XIX. J. 
