593

APPENDIX.

responding change of direction of the radius vector, and let e' t' be atangent to the orbit at that point. Then, since the angle s e't' is greaterthan s e t, it follows that the eccentricity is increased by the disturbingforce. It is evident, however, that the increase of that element is verysmall, for the planet is in the most favourable position for the retardationof its motion by the central force, and consequently the deflection ofmotion is speedily brought to an equality with the angular displacementof the radius vector. The position of the major axis of the new orbit willbe determined by drawing a' b' parallel to e't', whence it is evident thatthe line of apsides has progressed. It may be shewn, in a similar manner,that when the planet is revolving from the upper to the lower apse, theeffect of the disturbing force at the mean distance is, to diminish theeccentricity, and to make the line of apsides 7-egress. In every point ofthe orbit, the mean distance is obviously increased by the action of thedisturbing force.

(29.) Since the eccentricity is increased at e and diminished at b by theaction of the disturbing force, it is evident that there is some intermediatepoint at which the disturbing force produces no effect upon that element.For a similar reason, there must be some point between b and the follow-ing point of mean distance at which the eccentricity does not undergo anychange from the action of the disturbing force. These neutral points arefound by strict investigation to be f g, the extremities of the ordinate pass-ing through the upper focus of the ellipse. (See the figure at page 590).

(30.) Generally it may be shewn, that from f to g through a, the eccen-tricity is everywhere increased by the action of the disturbing force, thevariation increasing from nothing at f until it attains its maximum at a,and subsequently diminishing by equal degrees until it vanishes at a.On the other hand, from g to f through b, the eccentricity is every-where diminished by the action of the disturbing force, the variationbeing a maximum at b, from which point it diminishes towards f and g.

(31.) With respect to the line of apsides, it has been found to progressif the planet, while revolving from the perihelion to the aphelion, shouldbe at the mean distance when the disturbing force acts, and to regress,if the planet, in the course of revolving from aphelion to perihelion,should have arrived at the corresponding point of the orbit; while, again,at either apse it does not undergo any change of position. Generally itmay be shewn, that from a to b through g, the line of apsides progresses,and that from b to a through f, it regresses, the variation attaining itsmaximum values at f and g, and vanishing at a and b. Thus it appears,that in passing from a disturbing force acting in the direction of theradius vector, to one acting at right angles to that direction, an inter-change takes place between the points of maxima and minima of thevariations of the eccentricity and the apse.

(32.) If the disturbing force act in a direction contrary to that of theplanet’s motion so as to diminish the velocity, the effects produced uponthe eccentricity and the line of apsides will be precisely the reverse ofthose abovementioned. The eccentricity will increase from a to bthrough g, and will diminish from b to a through f; while again the lineof apsides will regress from f to g through a, and will progress from g tof through b. Moreover the points at which the variations attain theirmaximum values, and also those at which they vanish, will be the sameas in the case wherein the disturbing force tends to increase the velocityof the planet.