APPENDIX.

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b through g, and will diminish from b to a through f; and on the otherhand, the apsides will regress from f to g through a, and will progressfrom g to f through b. The points where the variations of both elementsattain their maximum values, and also those at which they severallyvanish, will be the same as in the former case.

(25.) Let us now consider the effect of a small disturbing force actingin a direction perpendicular to the radius vector. In all such cases theeccentricity of the orbit is supposed to be so inconsiderable, that the dis-turbing force may be regarded as acting in the direction of the tangent,and consequently as tending wholly either to accelerate or retard themotion of the planet. Let us suppose, then, that it tends to increase thevelocity of the planet, and first let it act when the planet is at theperihelion. The velocity being now increased, the central force will haveless control over the planet, and the latter in consequence taking a widersweep, will now recede, farther at the mean distance. It is manifest, also,since the tangential angle continually enlarges from the perihelion to themean distance, that it will now open out to a greater extent than it for-merly did. Now, the greater the maximum value of the tangential angle,the more eccentric is the orbit. Hence the effect of the disturbing forceis to increase the mean distance, and also the eccentricity. It is manifestthat the position of the line of apsides cannot suffer any alteration fromthe action of such a force.

(26.) Let us now suppose the disturbing force to act at the aphelion.Since the velocity is increased, the central force will be less effectivein deflecting the motion of the planet, and the latter in consequencetaking a wider circuit, will not approach so near the centre of force atthe mean distance as it formerly did. Moreover it is manifest, since thediminution of the tangential angle continues from the aphelion to themean distance, that when it attains its minimum value, it will be lessacute than it formerly was. Hence the effect of the disturbing force inthis case is, to increase the mean distance, and to diminish the eccen-tricity.

(27.) It is manifest that the conclusions above deduced are equallyapplicable if we suppose the disturbing force to act at a little distance oneach side of the apse, whether the latter refer to the perihelion or theaphelion, for the circumstances which determine the path of the planetare then almost the same as if the disturbing force had acted exactly atthe apse.

(28.) Let us suppose the planet to be revolving from the lower to theupper apse, and let the disturbing force act when it has arrived at the

mean distance e. At this point thedeflection of motion in the undis-turbed orbit exactly compensates forthe angular displacement of the radiusvector - , and the tangential angle inconsequence remains for an instantinvariable. The velocity of the planet,however, being now increased by theaction of the disturbing force, themomentary deflection of motion willbe diminished, and therefore the tan-gential angle will still continue to open out. Let e' be the point at whichthe momentary deflection of motion in the new orbit is equal to the cor-