336 
PROFESSOR DONKIN ON THE 
The definitions of all the elements (relative to the moving axes) in terms of the six 
new variables x, y, z, u, v, w, have the same form as those of the corresponding 
elements (relative to the fixed axes) in terms of x, y, z, n, v, w. The two relative 
elements a, e are the same as the corresponding absolute elements ; / is the inclina- 
tion of the plane of the ellipse to the moving plane of xy, and v the longitude of the 
node reckoned from the axis of and since the place of the body in the ellipse is 
evidently the same, the relations between the remaining elements ts- and (s) (or s) and 
the corresponding absolute elements are purely geometrical. 
Comparing these results with those of art. 79? we see that the independence of the 
formulae for the mechanical and geometrical variations of the elements of the true 
osculating ellipse is completely established. 
83. In all that precedes, the three variables (the angular velocities of the 
system of moving axes about the axes themselves) are entirely arbitrary ; they may 
be either explicit functions of t, involving only determinate constants, or they may 
depend in any way upon the relative or absolute elements of the orbits of any or all 
the planets, and their differential coefficients with respect to t. In the case in which 
the expressions for 00 ^, 00 ^, involve only the relative elements, when these expressions 
are introduced in the formulae (95.), art. 79, and these formulae completed by the 
addition of the terms in art. 55, and when the corresponding sets of equations are 
formed for each planet, we obtain a set of simultaneous differential equations in- 
volving all the elements of all the orbits, and their differential coefficients with 
respect to t. The integration of these equations would determine all the elements 
as functions of C and thus the motion of all the planets, relatively to the axes of 
coordinates, would be known. Lastly, the motion of the whole system, relatively to 
fixed space, would be found by integrating the system of equations 
cos X— 1^1 sin X 
N' sin L=(Wo sin X-f-^yi cos X r 
X':=<y,-N'cosL 
(96.) 
where are now given functions of t, and L is the inclination of the plane of 
xy to that of xy, N is the longitude of the ascending * node of the plane of xy, 
reckoned from the axis of x, and X is the longitude of the axis of x, reckoned upon 
the plane of xy,from the node, in the direction of positive rotation. 
In the case in which co^,, cannot be expressed in terms of the relative elements, 
the integrations which determine the relative motion of the system cannot be sepa- 
rated from those which determine the position of the axes in fixed space ; but the 
ecjuations (96.) must be considered simultaneously with the other differential equa- 
tions of the problem. 
* Ascending relatively to a positive rotation, i. e. from x to y. 
