Central Forces. 133 
It is also evident that the particle in the moveable ellipse will be 
at an apsis of the curves, represented by (19) when it is at the 
perihelion or aphelion of the ellipse. Again, supposing (as above,) 
that t= then m=V 0; if “.n=1, or P= v—mv=v—v=0, 
and the. particle revolves in a quiescent ellipse; but if Wn is <1 
and >0, then v—~mv is positive, and the ellipse revolves in the same 
direction as the particle, or in consequentia; but if Wn is >1, 
v—mv is negative and the ellipse revolves in a contrary direction to 
the motion of the particle, or in antecedentia; in the former case the 
apsides are said to go forwards, or in consequentia, but in the latter 
backwards, or in antecedentia. Universally, whatever may be the 
law of force, provided that m is a positive number <1 and >0, the 
apsides progress; but if m is positive and >1, they regress; because 
the apsides of the moveable ellipse progress or regress with respect to 
the motion of the particle in these cases. See Prin. prop. 43, sec. 
9, B. I, also cor. 7. prop. 66, sec. 11. 
I will now suppose that a particle of matter is describing a curve 
around a given centre of force, and is at the same time attracted’ 
towards another centre of force, situated in the plane of the descri- 
bed curve; to determine the equations of its motion. 
It is evident that the attraction towards the second centre of force 
will alter or disturb the motion of the particle around the first centre ; 
I shall therefore call it (according to usage,) the disturbing force. I 
shall (as heretofore,) denote the distance of the particle from the first 
center of force at any time (¢) by r, and shall suppose that » denotes 
the angle described by r around the centre in any time, v being reck- 
oned from some fixed line in the plane of the motion. 
Let ; denote the area described by r around the first centre of 
force in a unit of time; then by (G), (Vol. XVI, p. 286,) c’*?= 
ridy? r2.dv? 
qa or e==r* Xa (1). 
If there was no disturbing force, e’ would be constant ; but it evi- 
dently becomes variable by its action. Let the disturbing force be 
resolved into two; one in the direction of r, the other perpendicular 
to r, which let be denoted by T’; it is evident (by what was shown 
in Vols. XVI and XVII,) that the first of these cannot affect ¢’, but 
T will render it variable. 
