APPENDIX.

597

(46.) The more nearly the mean motions of the two planets are com-mensurable, the less will be the angle through which the line of apsideswill have to advance during each synodic revolution of the two planets,and consequently the greater will be the eccentricity depending uponthe disturbing force.

(47.) The progressive movement of the line of apsides is mainly dueto the part of the disturbing force which acts in the direction of theradius vector. For, in the first place, the tangential force vanishes alto-gether at the point of conjunction, where the disturbing force is mostintense; and, in the second place, its influence in altering the position ofthe apsides is very feeble near conjunction (where alone the disturbingforce is sensible), since, by (31), its effect vanishes at the aphelion, which,in this case, always coincides with the point of conjunction. On the otherhand, the radial force is not only most intense at conjunction, but is alsomost effective (24) at the same point*.

(48.) If we had considered the perturbation of Neptune by Uranus ,it would readily appear, by (22), that it is the perihelion of the disturbedorbit which, in this case, would require to be turned towards the point ofconjunction, in order that the line of apsides might progress; and it ismanifest that, by a due adjustment of the eccentricity, the amount of ad-vance during each synodic revolution might be rendered exactly equal tothe advance of the line of conjunction. Thus the eccentricity would con-stantly retain the same value, while the line of apsides would regularlyprogress, coinciding always with the line of conjunction.

(49.) With a view to illustrate the foregoing remarks, we insert theprincipal terms of the mutual perturbations of both planets, exclusive ofthose arising from independent eccentricity, the influence of which weare not at present considering. In calculating these terms, a mean dis-tance equal to 19.18239 has been assigned to Uranus . The mean dis-

* If we exclude from consideration the tangential force, as incapable of exercising anyinfluence upon the character of the perturbative effect, it is easy to see that the disturb-ing force, by acting outwards in the direction of the radius vector, tends continually totwist round the line of apsides in the direction of the planet’s motion. First, let theplanet be advancing towards conjunction. In this case the tangential angle is obtuse,and the planet is naturally receding from the position in which the radius vector is per-pendicular to the tangent; but the deflective influence of the central force is so powerful,that the tangential angle is actually contracting, and is again rapidly approaching to aright angle. Hence the disturbing force, by weakening the influence of the central body,diminishes the deflection of the orbit, and thereby retards the perpendicularity of thetangent with respect to the radius vector. It is manifest, therefore, that the planet willrequire to revolve through an angle of somewhat more than 180° from the perihelion beforeit arrives at the aphelion, or, in other words, the line of apsides will have progressed. "Withrespect to the influence of the disturbing force after the passage of the aphelion, it is to beremarked, that if the central force were diminished in due proportion throughout the wholesemi-ellipse, extending from the aphelion to the perihelion, the position of the line ofapsides would not undergo any change. It is evident, therefore, that the character ofthe effect produced after the passage of the aphelion (whether progression or regression)will be the same as if the central force was increased a little before the arrival of the planetin the perihelion. Now when the planet is approaching the perihelion, the tangentialangle is rapidly opening out, from the natural tendency of the body to persevere in thesame direction ; and the efficacy of the central force consists in opposing its enlargement,and thereby retarding the'arrival of the planet at the apse. It is evident, therefore, thatthe effect of an increase of the central force will be to retard, in a still greater degree, thearrival of the planet in such a position. It appears, then, that the action of Neptune uponUranus causes the line of apsides to progress both before and after conjunction. It maybe shewn, in a similar way, that the action of Uranus upon Neptune will cause the line ofapsides of the disturbed orbit to progress at each conjunction.