CHAP. XI. MOTION OF THE APSIDES. 363 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 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 CH. XI. PROGRESSION OF THE LUNAR APOGEE. 365 of the sun's attraction is to weaken the gravity of the earth by a force, whose law of variation, instead of the inverse square, follows the direct proportional relation of the distance; while, in the quadratures, the reverse takes place, the whole effect of the radial disturbing force here conspiring with the earth's gravity, but the portion added being still, as in the former case, in the direct ratio of the distance. Therefore the motion of the moon, in and near the first of these situations, will be performed in an ellipse, whose apsides are in a state of advance; and in and near the latter, in a state of recess. But, as we have already seen (art. 556.), the average effect arising from the mutual counteraction of these temporary values of the disturbing force gives the preponderance to the ablatitious or enfeebling power. On the average, then, of a whole revolution, the lunar apogee will advance. (573.) The above reasoning renders a satisfactory enough general account of the advance of the lunar apogee; but it is not without considerable difficulty that it can be applied to determine numerically the rapidity of such advance: nor, when so applied, does it account for the whole amount of the movement in question, as assigned by observation—not more, indeed, than about one half of it; the remaining part is produced by the tangential force. It is evident, that an increase of velocity in the moon will have the same effect in diminishing the curvature of its orbit as the decrease of central force, and vice versa. Now, the direct effect of the tangential force is to cause a fluctuation of the moon's velocity above and below its elliptic value, and therefore an alternate progress and recess of the apogee. This would compensate itself in each synodic revolution, were the apogee invariable. But this is not the case; the apogee is kept rapidly advancing by the action of the radial force, as above explained. An uncompensated portion of the action of the tangential force, therefore, remains outstanding (according to the reasoning already so often employed in this chapter), and this portion is so dis tributed over the orbit as to conspire with the former cause, and, in fact, nearly to double its effect. This is what is meant by geometers, when they say that this part of the motion of the apogee is due to the square of the disturbing force. The effect of the tangential force in disturbing the apogee would compensate itself, were it not for the motion which the apogee has already had impressed upon it by the radial force; and we have here, therefore, disturbance re-acting on disturb ance. (574.) The curious and complicated effect of perturbation, described in the last article, has given more trouble to geometers than any other part of the lunar theory. Newton himself had succeeded in tracing that part of the motion of the apogee which is due to the direct action of the radial force; but finding the amount only half what observation assigns, he appears to have abandoned the subject in despair. Nor, when resumed by his successors, did the enquiry, for a very long period, assume a more promising aspect. On the contrary, Newton's result appeared to be even minutely verified, and the elaborate investigations which were lavished upon the subject without success began to excite strong doubts whether this feature of the lunar motions could be explained at all by the Newtonian law of gravitation. The doubt was removed, however, almost in the instant of its origin, by the same geometer, Clairaut, who first gave it currency, and who gloriously repaired the error of his momentary hesitation, by demonstrating the exact coincidence between theory and observation, when the effect of the tangential force is properly taken into the account. The lunar apogee circulates, as already stated (art. 360.), in about nine years. (575.) The same cause which gives rise to the displacement of the line of apsides of the disturbed orbit produces a corresponding change in its excentricity. This is evident on a glance at our figures 1. and 2. of art. 571. Thus, in fig. 1., since the disturbed body, proceeding from its lower to its upper apsis, is acted on by CHAP. XI. EXCENTRICITIES AND PERIHELIA. 367 a force greater than would retain it in an elliptic orbit, and too much curved, its whole course (as far as it is so affected) will lie within the ellipse, as shown by the dotted line; and when it arrives at the upper apsis, its distance will be less than in the undisturbed ellipse; that is to say, the excentricity of its orbit, as estimated by the comparative distances of the two apsides from the focus, will be diminished, or the orbit rendered more nearly circular. The contrary effect will take place in the case of fig. 2. There exists, therefore, between the momentary shifting of the perihelion of the disturbed orbit, and the momentary variation of its excentricity, a relation much of the same kind with that which connects the change of inclination with the motion of the nodes; and, in fact, the strict geometrical theories of the two cases present a close analogy, and lead to final results of the very same nature. What the variation of excentricity is to the motion of the perihelion, the change of inclination is to the motion of the node. In either case, the period of the one is also the period of the other; and while the perihelia describe considerable angles by an oscillatory motion to and fro, or circulate in immense periods of time round the entire circle, the excentricities increase and decrease by comparatively small changes, and are at length restored to their original magnitudes. In the lunar orbit, as the rapid rotation of the nodes prevents the change of inclination from accumulating to any material amount, so the still more rapid revolution of its apogee effects a speedy compensation in the fluctuations of its excentricity, and never suffers them to go to any material extent; while the same causes, by presenting in quick succession the lunar orbit in every possible situation to all the disturbing forces, whether of the sun, the planets, or the protuberant matter at the earth's equator, prevent any secular accumulation of small changes, by which, in the lapse of ages, its ellipticity might be materially increased or diminished. Accordingly, observation shows the mean |