APPENDIX.

589

of mean distance. Hence it follows that in the new, as well as in theundisturbed ellipse, the planet is at the mean distance. Let p denote theplace of the planet, p t the tangent of the undisturbed orbit, and p t' thetangent of the new orbit. Now, since in an ellipse the tangent at eitherextremity of the minor axis is parallel to the major axis, the position ofthe latter will be determined by drawing a 's b' parallel to p t'. The lineof apsides has therefore progressed through an angle equal to the amountof deflection occasioned by the disturbing force.

(18.) The eccentricity of the new ellipse will be determined by theangle spi', representing the maximum value of the tangential angle.The greater this angle is the more eccentric is the ellipse. In the presentcase this angle is less than it was in the undisturbed orbit. Hence itappears that the eccentricity is diminished by the action of the disturbingforce.

(19.) Let the planet now be moving towards the aphelion, and, when ithas arrived at r, let it be disturbed by a small force tending to increasethe central force at s. Since the tangential angle is now in the course of

contracting, the immediate effect of thedisturbing force will be to make it equalto the tangential angle at q, a point moreadvanced in the orbit ; and since the dis-tance, velocity,and force maybe supposedto be equal at p and Q, ( in consequence of thevicinity of the planet to the apse, the cir-cumstances which determine its path maybe regarded as identical at those points.

Hence, the new orbit of the planet maybe represented by supposing the major axis of the original ellipse tohave revolved through an angle equal to p s q, in a direction contrary tothat of the planet’s motion; in other words, the line of apsides liasregressed.

(20.) In the foregoing case it has been assumed that the circumstancesof motion at p and a are identical. Let us suppose, liow r ever, that whilethe planet is moving towards the aphelion, its distance from the apse atthe instant when the disturbing force acts, is sufficiently great to occasiona sensible difference in the circumstances of motion at p and q. Sincethe tangential angle is closing more slowly at p than at q (11) the planetwill be somewhat longer of arriving at the apse in the new orbit than if ithad started in the original orbit from q. Hence it is obvious that theline of apsides will have revolved through an angle bsj' (see the abovefigure), somewhat less than the angle psq. In this case, then, the regres-sion of the apsides is less considerable than that which results when theplanet is in the immediate vicinity of the apse. Again, since the velocityat p is presumed to be sensibly greater than that at q, while at the sametime the motion of the planet in either case is almost perpendicular tothe radius vector, it is clear that the velocity of the planet at the aphelionwill now be quicker than it was in the original orbit. The eccentricityhas therefore been diminished by the disturbing force. This, indeed, isreadily perceived from the mode in which the two ellipses intersect eachother; for the aphelion distance of the new orbit is manifestly less thanthat of the original orbit.

(21.) It will be found, by reasoning precisely similar to that employedabove, that when the planet is revolving from the aphelion to the perihe-