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

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 a 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

jection on a horizontal plane, it will be then moving under the same circumstances as if it were a revolving body attracted to a center by a force varying directly as the distance; and, in this case, the curve described would be an ellipse, having its center of attraction not in the focus, but in the center*, and the apsides of this ellipse would remain fixed. But if the excursions of the weight from the vertical be considerable, the force urging it towards the center will deviate in its law from the simple ratio of the distances; being as the sine, while the distances are as the arc. Now the sine, though it continues to increase as the arc increases, yet does not increase so fast. So soon as the arc has any sensible extent, the sine begins to fall somewhat short of the magnitude which an exact numerical proportionality would require; and therefore the force urging the weight towards its center or point of rest, at great distances falls, in like proportion, somewhat short of that which would keep the body in its precise elliptic orbit. It will no longer, therefore, have, at those greater distances, the same command over the weight, in proportion to its speed, which would enable it to deflect it from its rectilinear tangential course into an ellipse. The true path which it describes will be less curved in the remoter parts than is consistent with the elliptic figure, as in the annexed cut; and, therefore, it will not so soon have its

B

motion brought to be again at right angles to the radius. It will require a longer continued action of the central

Newton, Princip. i. 47.

force to do this; and before it is accomplished, more than a quadrant of its revolution must be passed over in angular motion round the center. But this is only stating at length, and in a more circuitous manner, that fact which is more briefly and summarily expressed by saying that the apsides of its orbit are progressive.

(571.) Now, this is what takes place, mutatis mutandis, with the lunar and planetary motions. The action of the sun on the moon, for example, as we have seen, besides the tangential force, whose effects we are not now considering, produces a force in the direction of the radius vector, whose law is not that of the earth's direct gravity. When compounded, therefore, with the earth's attraction, it will deflect the moon into an orbit deviating from the elliptic figure, being either too much curved, or too little, in its recess from the perigee, to bring it to an apogee at exactly 180° from the perigee; -too much, if the compound force thus produced decrease at a slower rate than the inverse square of the distance (i. e. be too strong in the remoter distances), too little, if the joint force decrease faster than gravity; or more rapidly than the inverse square, and be therefore too weak at the greater distance. In the former case, the curvature, being excessive, will bring the moon to its apogee sooner than would be the case in an elliptic orbit; in the latter, the curvature is insufficient,

and will therefore bring it later to an apogee. In the former case, then, the line of apsides will retrograde; in the latter, advance. (See fig. 1. and fig. 2.)

(572.) Both these cases obtain in different configurations of the sun and moon. In the syzigies, the effect