226 
MR. IVORY ON THE THEORY 
If it can be proved that these elements, after an indefinite lapse of time, will 
remain of inconsiderable magnitude as they are at present, the convergency of 
the series will be established, and the form of the terms of which R consists, 
will enable us to compute the changes in all the elliptic elements, and to de- 
cide the great question of the stability of the system. But we cannot enter 
upon any extended discussion of these points, and shall conclude this paper 
with some remarks in illustration of the problem we have solved, and of the 
manner in which we have solved it. 
6. If we suppose that there is no disturbing force, or that R = 0, we shall 
have by the equation (I), 
1 2 dx 2 + dy* + dz 2 
a r dt* . [x 9 
and if V represent the velocity of the planet at the extremity of r, then, 
Tro d x 2 -f dy 2 + d z* 1 2 
V 2 = t-t- , and — = V 2 . 
d t~ . 9 a r 
This last equation shows that the mean distance a of an elliptic orbit depends 
only upon the radius vector drawn to any point, and upon the velocity at that 
point. Conceive that the straight line r extends from the sun in a given 
direction and to a given length, and from its extremity suppose that a planet 
is launched into space with the velocity V, the foregoing equation will deter- 
mine the mean distance a of the immoveable ellipse in which the planet will 
revolve. The point from which the planet is projected, and consequently r, 
remaining the same, ~ and V 2 will vary together ; and if we suppose that a 
becomes equal to a, at the same time that V 2 is changed into V 2 + d . V 2 by 
forces which act continually but insensibly, we shall have these equations, 
d. — = - (Z.V 2 , and - = - - f'd . V 2 . 
a 9 a a J 
It has been shown that the disturbing forces acting in the directions of x, y, z, 
and tending to increase these lines, are respectively and, by the 
principles of dynamics, double the sum of the products of these forces, each 
being multiplied by the element of its direction, is equal to the change effected 
on the square of the velocity : wherefore, 
dy + 
d R 
d z 
d z 
)= 
2d'R=d.V 2 ; 
