meters have demonstrated (it is to Lagrange that we owe this most important discovery) that, besides these, there exists no third class of effects, whether arising from the radial or tangential disturbing forces, or from their combination, such as can go on for ever increasing in one direction without self-compensation; and, in particular, that the major axes of the planetary ellipses are not liable even to those slow secular changes by which the inclinations, nodes, and all the other elements of the system, are affected, and which, it is true, are periodic, but in a different sense from those long inequalities which depend on the mutual configurations of the planets inter se. Now, the periodic time of a planet in its orbit about the sun depends only on the masses of the sun and planet, and on the major axis of the orbit it describes, without regard to its degree of excentricity, or to any other eleThe mean sidereal periods of the planets, therefore, such as result from an average of a sufficient number of revolutions to allow of the compensation of the last-mentioned inequalities, are unalterable by lapse of time. The length of the sidereal year, for example, if concluded at this present time from observations embracing a thousand revolutions of the earth round the sun, (such, in short, as we now possess it,) is the same with that which (if we can stretch our imagination so far) must result from a similar comparison of observations made a million of years hence.

ment.

(564.) This theorem is justly regarded as the most important, as a single result, of any which have hitherto rewarded the researches of mathematicians. We shall, therefore, endeavour to make clear to our readers, at least the principle on which its demonstration rests; and although the complete application of that principle cannot be satisfactorily made without entering into details of calculation incompatible with our objects, we shall have no difficulty in leading them up to that point where those details must be entered on, and in giving such an insight into their general nature as will render it evident what must be their result when gone through.

CH. XI.

PERMANENCE OF THE MAJOR AXES.

359

(565.) It is a property of elliptic motion performed under the influence of gravity, and in conformity with Kepler's laws, that if the velocity with which a planet moves at any point of its orbit be given, and also the distance of that point from the sun, the major axis of the orbit is thereby also given. It is no matter in what direction the planet may be moving at that moment. This will influence the excentricity and the position of its ellipse, but not its length. This property of elliptic motion has been demonstrated by Newton, and is one of the most obvious and elementary conclusions from his theory. Let us now consider a planet describing an indefinitely small arc of its orbit about the sun, under the joint influence of its attraction, and the disturbing power of another planet. This arc will have some certain curvature and direction, and, therefore, may be considered as an arc of a certain ellipse described about the sun as a focus, for this plain reason, that whatever be the curvature and direction of the arc in question, an ellipse may always be assigned, whose focus shall be in the sun, and which shall coincide with it throughout the whole interval (supposed indefinitely small) between its extreme points. This is a matter of pure geometry. It does not follow, however, that the ellipse thus instantaneously determined will have the same elements as that similarly determined from the arc described in either the previous or the subsequent instant. If the disturbing force did not exist, this would be the case; but, by its action, a variation of the elements from instant to instant is produced, and the ellipse so determined is in a continual state of change. Now, when the planet has reached the end of the small arc under consideration, the question whether it will in the next instant describe an arc of an ellipse having the same or a varied axis will depend, not on the new direction impressed upon it by the acting forces, — for the axis, as we have seen, is independent of that direction, not on its change of distance from the sun, while describing the former arc, - for the elements of that arc

are accommodated to it, so that one and the same axis must belong to its beginning and its end. The question, in short, whether in the next arc it shall take up a new major axis, or go on with the old one, will depend solely on this, whether the velocity has undergone, by the action of the disturbing force, a change incompatible with the continuance of the same axis. We say by the action of the disturbing force, because the central force residing in the focus can impress on it no such change of velocity as to be incompatible with the permanence of any ellipse in which it may at any instant be freely moving about that focus.

(566.) Thus we see that the momentary variation of the major axis depends on nothing but the momentary deviation from the law of elliptic velocity produced by the disturbing force, without the least regard to the direction in which that extraneous velocity is impressed, or the distance from the sun at which the planet may be situated in consequence of the variation of the other elements of its orbit. And as this is the case at every instant of its motion, it will follow that, after the lapse of any time however great, the amount of change which the axis may have undergone will be determined by the total deviation from the original elliptic velocity produced by the disturbing force; without any regard to alterations which the action of that force may have produced in the other elements, except in so far as the velocity may be thereby modified. This is the point at

which the exact estimation of the effect must be intrusted to the calculations of the geometer. We shall be at no loss, however, to perceive that these calculations can only terminate in demonstrating the periodic nature and ultimate compensation of all the variations of the axis which can thus arise, when we consider that the circulation of two planets about the sun, in the same direction and in incommensurable periods, cannot fail to ensure their presentation to each other in every state of approach and recess, and under every variety as to their mutual distance and the consequent intensity of their mutual action.

Whatever velocity, then, may be gene

CHAP. XI.

MOTION OF THE APSIDES.

361

rated in one by the disturbing action of the other, in one situation, will infallibly be destroyed by it in another, by the mere effect of change of configuration.

(567.) It appears, then, that the variations in the major axes of the planetary orbits depend entirely on cycles of configuration, like the great inequality of Jupiter and Saturn, or the long inequality of the Earth and Venus above explained, which, indeed, may be regarded as due to such periodic variations of their axes. In fact, the mode in which we have seen those inequalities arise, from the accumulation of imperfectly compensated actions of the tangential force, brings them directly under the above reasoning: since the efficacy of this force falls almost wholly upon the velocity of the disturbed planet, whose motion is always nearly coincident with or opposite to its direction.

(568.) Let us now consider the effect of perturbation in altering the excentricity and the situation of the axis of the disturbed orbit in its own plane. Such a change of position (as we have observed in art. 318.) actually takes place, although very slowly, in the axis of the earth's orbit, and much more rapidly in that of the moon's (art. 360.); and these movements we are now

to account for.

(569.) The motion of the apsides of the lunar and planetary orbits may be illustrated by a very pretty mechanical experiment, which is otherwise instructive in giving an idea of the mode in which orbitual motion is carried on under the action of central forces variable according to the situation of the revolving body. Let a leaden weight be suspended by a brass or iron wire to a hook in the under side of a firm beam, so as to allow of its free motion on all sides of the vertical, and so that when in a state of rest it shall just clear the floor of the room, or a table placed ten or twelve feet beneath the hook. The point of support should be well secured from wagging to and fro by the oscillation of the weight, which should be sufficient to keep the wire as tightly stretched as it will bear, with the certainty of

not breaking. Now, let a very small motion be communicated to the weight, not by merely withdrawing it from the vertical and letting it fall, but by giving it a slight impulse sideways. It will be seen to describe a regular ellipse about the point of rest as its center. If the weight be heavy, and carry attached to it a pencil, whose point lies exactly in the direction of the string, the ellipse may be transferred to paper lightly stretched and gently pressed against it. In these circumstances, the situation of the major and minor axes of the ellipse will remain for a long time very nearly the same, though the resistance of the air and the stiffness of the wire will gradually diminish its dimensions and excentricity. But if the impulse communicated to the weight be considerable, so as to carry it out to a great angle (15° or 20° from the vertical), this permanence of situation of the ellipse will no longer subsist. Its axis will be seen to shift its position at every revolution of the weight, advancing in the same direction with the weight's motion, by an uniform and regular progression, which at length will entirely reverse its situation, bringing the direction of the longest excursions to coincide with that in which the shortest were previously made; and so on, round the whole circle; and, in a word, imitating to the eye, very completely, the motion of the apsides of the moon's orbit.

(570.) Now, if we enquire into the cause of this progression of the apsides, it will not be difficult of detection. When a weight is suspended by a wire, and drawn aside from the vertical, it is urged to the lowest point (or rather in a direction at every instant perpendicular to the wire) by force which varies as the sine of the deviation of the wire from the perpendicular. Now, the sines of very small arcs are nearly in the proportion of the arcs themselves; and the more nearly, as the arcs are smaller. If, therefore, the deviations from the vertical are so small that we may neglect the curvature of the spherical surface in which the weight moves, and regard the curve described as coincident with its pro