not remain permanent, since by a remarkable property of perturbations of this class, which geometers have demonstrated, but the reasons of which we cannot stop to explain, any change produced on the axis of the disturbed planet's orbit is necessarily accompanied by a change in the contrary direction in that of the disturbing, so that the periods would recede from commensurability by the mere effect of their mutual action. Cases are not wanting in the planetary system of a certain approach to commensurability, and in one very remarkable case (that of Uranus and Neptune) of a considerably near one, not near enough, however, in the smallest degree to affect the validity of the argument, but only to give rise to inequalities of very long periods, of which more presently.*

(670.) The variation of the length of the axis of the disturbed orbit is due solely to the action of the tangential disturbing force. It is otherwise with that of its excentricity and of the position of its axis, or, which is the same thing, the longitude of its perihelion. Both the normal and tangential components of the disturbing force affect these elements. We shall, however, consider separately the influence of each,

and, commencing, as the simplest case, with that of the tangential force; let P be the place of the disturbed planet in its elliptic orbit A P B, whose axis at the moment is A S B and focus S. Suppose Y P Z to be a tangent to this orbit at

* 41 revolutions of Neptune are nearly equal to 81 of Uranus, giving rise to an inequality, having 6805 years for its period.

P. Then, if we suppose A B=2 a, the other focus of the ellipse, H, will be found by making the angle Z PH=YPS or YP H=180°-Y P Z, or SP H=180°-2 Y P S, and taking PH 2 a-S P. This is evident from the nature of the ellipse, in which lines drawn from any point to the two foci make equal angles with the tangent, and have their sum equal to the major axis. Suppose, now, the tangential force to act on P and to increase its velocity. It will therefore increase the axis, so that the new value assumed by a (viz. a') will be greater than a. But the tangential force does not alter the angle of tangency, so that to find the new position (H') of the upper focus, we must measure off along the same line PH, a distance P H' (=2 a' SP) greater than PH. Do this then, and join SH' and produce it. Then will A'B' be the new position of the axis, and SH' the new excentricity. Hence we conclude, 1st, that the new position of the perihelion A' will deviate from the old one A towards the same side of the axis A B on which P is when the tangential force acts to increase the velocity, whether P be moving from perihelion to aphelion, or the contrary. 2dly, That on the same supposition as to the action of the tangential force, the excentricity increases when P is between the perihelion and the perpendicular to the axis F HG drawn through the upper focus, and diminishes when between the aphelion and the same perpendicular. 3dly, That for a given change of velocity, i. e. for a given value of the tangential force, the momentary variation in the place of the perihelion is a maximum when P is at F or G, from which situation of P to the perihelion or aphelion, it decreases to nothing, the perihelion being stationary when P is at A or B. 4thly, That the variation of the excentricity due to this cause is complementary in its law of increase and decrease to that of the perihelion, being a maximum for a given tangential force when P is at A or B, and vanishing when at G or F. And lastly, that where the tangential force acts to diminish the velocity, all these results are reversed. If the orbit be very nearly circular* the points F, G, will be So nearly that the cube of the excentricity may be neglected.

so situated that, although not at opposite extremities of a diameter, the times of describing AF, F B, B G, and G A will be all equal, and each of course one quarter of the whole periodic time of P.

(671.) Let us now consider the effects of the normal component of the disturbing force upon the same elements. The direct effect of this force is to increase or diminish the curvature of the orbit at the point P of its action, without producing any change on the velocity, so that the length of the

axis remains unaltered by its action. Now, an increase of curvature at P is synonymous with a decrease in the angle of tangency SPY when P is approaching towards S, and with an increase in that angle when receding from S. Suppose the former case, and while P approaches S (or is moving from aphelion to perihelion), let the normal force act inwards or towards the concavity of the ellipse. Then will the tangent PY by the action of that force have taken up the position PY. To find the corresponding position H' taken up by the focus of the orbit so disturbed, we must make the angle SPH' 180°-2 SP Y', or, which comes to the same, draw PH' on the side of P H opposite to S, making the angle HP H' twice the angle of deflection Y P Y' and in PH' take PHP H. Joining, then, SH' and producing it, A'S H'M' will be the new position of the axis, A' the new perihelion, and SH' the new excentricity. Hence we conclude, 1st, that the normal force acting inwards, and P moving towards the perihelion, the new direction S A' of the perihelion is in advance (with reference to the direction of P's

=

revolution) of the old-or the apsides advance when P is anywhere situated between F and A (since when at F the point H' falls upon H M between H and M). When P is at F the apsides are stationary, but when P is anywhere between M and F the apsides retrograde, H' in this case lying on the opposite side of the axis. 2dly, That the same directions of the normal force and of P's motion being supposed, the excentricity increases while P moves through the whole semiellipse from aphelion to perihelion - the rate of its increase being a maximum when P is at F, and nothing at the aphelion and perihelion. 3dly, That these effects are reversed in the opposite half of the orbit, A G M, in which P passes from perihelion to aphelion or recedes from S. 4thly, That they are also reversed by a reversal of the direction of the normal force, outwards, in place of inwards. 5thly, That here also the variations of the excentricity and perihelion are complementary to each other; the one variation being most rapid when the other vanishes, and vice versâ. 6thly, And lastly, that the changes in the situation of the focus H produced by the actions of the tangential and normal components of the disturbing force are at right angles to each other in every situation of P, and therefore where the tangential force is most efficacious (in proportion to its intensity) in varying either the one or the other of the elements in question, the normal is least so, and vice versâ.

(672.) To determine the momentary effect of the whole disturbing force then, we have only to resolve it into its tangential and normal components, and estimating by these principles separately the effects of either constituent on both elements, add or subtract the results according as they conspire or oppose each other. Or we may at once make the angle HP H" equal to twice the angle of deflection produced by the normal force, and lay off PH"=PH+twice the variation of a produced in the same moment of time by the tangential force, and H" will be the new focus. The momentary velocity generated by the tangential force is calculable from a knowledge of that force by the ordinary principles of dynamics; and from this, the variation of the axis is

EFFECTS ON THE APSIDES AND EXCENTRICITIES.

431

easily derived. The momentary velocity generated by the normal force in its own direction is in like manner calculable from a knowledge of that force, and dividing this by the linear velocity of P at that instant, we deduce the angular velocity of the tangent about P, or the momentary variation of the angle of tangency SPY, corresponding.

(673.) The following résumé of these several results in a tabular form includes every variety of case according as P is approaching to or receding from S; as it is situated in the arc F A G of its orbit about the perihelion or in the remoter arc G M F about the aphelion, as the tangential force accelerates or retards the disturbed body, or as the normal acts inwards or outwards with reference to the concavity of the orbit.

from which it appears that the variation of the axis arising from a given variation of velocity is independent of r, or is the same at whatever distance from S the change takes place, and that cæteris paribus it is greater for a given change of velocity (or for a given tangential force) in the direct ratio of the velocity itself.