76 
[176 
176. 
NOTE ON JACOBI’S CANONICAL FOBMULiE FOB DISTUBBED 
MOTION IN AN ELLIPTIC OBBIT. 
[From the Quarterly Mathematical Journal, vol. I. (1857), pp. 355—356.] 
Consider a body (afterwards called the disturbed body) revolving about a central 
body under the influence of their mutual attraction and of any disturbing forces. 
Then referring the disturbed body to axes through the central body and parallel to 
fixed lines in space, write 
x, y, z, the coordinates of the disturbed body, 
r, , the radius vector, = J(x 2 + y 2 + z 2 ), 
M , the mass of the disturbed body, 
M" , the mass of the central body. 
Write also 
Д the disturbing function, taken negatively, i.e. the sign of R is taken as in 
the Mécanique Celeste. 
The equations of motion then are 
d?x 
(M + M")x 
dR 
dt 2 
ry*3 
dx ’ 
d 2 y 
(M+M")y 
dR 
dt 2 
гу*Ъ 
dy ’ 
d 2 z _ 
(M+ M") z 
dR 
dt 2 
ry> 3 
dz ’ 
and the motion may be represented by supposing that the body moves in an ellipse 
with variable elements, and such that the direction and velocity of motion are always 
the same as in the actual orbit.