214 
MR. IVORY ON THE THEORY 
p t (ds-dT*) + ^ e de + dn^O. 
But, v being a function £ + s, it follows that. 
d v dv a 2 •/ 1 — e 3 
de d$ f* 
and thus it appears that the equation we are considering is identical with the 
formula (9) : from which we learn that. 
d e 
1 -<? 3 
It remains now to say a word about the longitudes and latitudes of the 
planet reckoned on the immoveable plane of xy. The variable quantities N 
and i denote the longitude of the ascending node, and the inclination of the 
orbit, in respect to the fixed plane: let P represent the place of the same node 
on the moveable plane of the planet, this arc being reckoned from the same 
origin as the true motion v : then, because the momentary plane in which the 
planet moves, in taking a new position, turns about a radius vector, it is ob¬ 
vious that, if rfN be the motion of the node in the fixed plane of xy, cos i X d N 
will be its motion in the variable plane of the orbit. Wherefore we have. 
d P = cos i X d N, and P = f cos i . e?N, 
a constant being supposed to accompany the integral. This being observed, 
it is obvious that the same equations as in the case of the undisturbed orbit, 
will obtain between the quantities under consideration, viz. 
tan (X — N) = cos i tan (v — P), 
^ = tan i sin (X — N). 
The foregoing investigations prove that the motion of a disturbed planet 
may be accurately represented by a variable ellipse coinciding momentarily 
with the real path of the planet. The variations, in the magnitude, the form, 
and the position of the ellipse, have been expressed by equations that depend 
upon the disturbing forces. A new inquiry presents itself: to exhibit the 
differentials of the elements of the variable orbit in the forms best adapted 
for use. 
