whether the progression produced by the force perpendicular to the radius vector near B, will or will not exceed the regression produced near B1? To answer this we must observe, that the rate of this progress or regress depends entirely upon the proportion * which the velocity produced by the disturbing force bears to the

* Suppose, for facility of conception, that the force, perpendicular to the radius vector, acts in only one place in each quadrant between syzygies and quadratures. The portions of the orbit which are bisected by the line of syzygies will be described with greater velocity in consequence of this disturbance (abstracting all other causes) than the other portions. Now the curvature of any part of an orbit does not depend on the central force simply, or on the velocity, but on the relation between them; so that the same curve may be described either by leaving the central force unaltered and increasing the velocity in a given proportion, or by diminishing the central force in a corresponding proportion, and leaving the velocity unaltered. Consequently, in the case before us, the same curve will be described as if, without alteration of velocity, the central force were diminished, while the moon passed through the portions bisected by the line of syzygies. If now the imaginary diminution of central force were in the same proportion (that is, if the real increase of velocity were in the same proportion) at both syzygies, which here coincide with the apses, the regression of the line of apses produced at perigee, would be equal to the progression produced at apogee. But the increase of velocity produced by the force perpendicular to the radius vector near apogee, is much greater than that near perigee. First, because the force is greater, in proportion to the distance. Second, because the time of describing a given small angle is greater in proportion to the square of the distance; so that the acceleration produced while the moon passes through a given angle, is proportional to the cube of the distance. Third, because the velocity, which is increased by this acceleration, is inversely proportional to the distance; so that the ratio in which the velocity is increased is proportional to the fourth power of the distance. The effect at the greater distance, therefore, predominates over that at the smaller distance; and therefore, on the whole, the force perpendicular to the radius vector produces an effect similar to its apogeal effect; that is, it causes the line of apses to progress.

velocity of the moon; and since from B2 to B3, and from B to B, the disturbing force is greater than that from B to B1, and from B1 to B2, and acts for a longer time (as by the law of equable description of areas, the moon is longer moving from B2 to B, and B1, than from B1 to B1 and B2), and since the moon's velocity in passing through B2, B3, B4, is less than her velocity in passing through B, B1, B2, it follows, that the effect in passing through B2, B3, B4, is much greater than that in passing through B, B1, and B2. Consequently, the effect of this force also is to make the line of apses progress.

(100.) On the whole, therefore, when the perigee is turned towards the sun, the line of apses progresses rapidly. And the same reasoning applies in every respect when the perigee is turned from the sun.

(101.) In the second place, suppose that the line of apses is perpendicular to the line joining the earth and sun. The disturbing force at both apses is now directed to the earth, and consequently, by (50.) and (53.), while the moon is near perigee, the disturbing force causes the line of apses to progress, and while the moon is near apogee the disturbing force causes the line of apses to regress. Here, as in the last article, the effects at perigee and at apogee would balance if the disturbing force were inversely proportional to the square of the distance from the earth. But the disturbing force is really proportional to the distance from the earth: and, therefore, as in (98.), the effect of the disturbing force while the moon is at apogee preponderates over the other; and

therefore the force directed to the centre causes the line

of apses to regress.

(102.) We must also consider the force perpendicular to the radius vector. In this instance, that force retards the moon while she is approaching to each apse, and accelerates her as she recedes from it. The effect is, that when the moon is near perigee the force causes the line of apses to progress, and when near apogee it causes the line of apses to regress (65.) and (66.). The latter is found to preponderate, by the same reasoning as that in (99.). From the effect, then, of both causes the line of apses regresses rapidly in this position of the line of apses.

(103.) It is important to observe here, that the motion of the line of apses would not, as in (56.), be greater if the excentricity of the orbit were smaller. For though the motion of the line of apses is greater in proportion to the force which causes it when the excentricity is smaller; yet, in the present instance, the force which causes it is itself proportional to the excentricity (inasmuch as it is the difference of the forces at perigee and apogee, which would be equal if there were no excentricity): so that if the excentricity were made less, the force which causes the motion of the line of apses would also be made less, and the motion of the line of apses would be nearly the same as before.

(104.) It appears then, that when the line of apses passes through the sun, the disturbing force causes that line to progress; when the earth has moved round the

sun, or the sun has appeared to move round the earth, so far that the line of apses is perpendicular to the line joining the sun and the earth, the line of apses regresses from the effect of the disturbing force; and at some intermediate position, it may easily be imagined that the force produces no effect on it. It becomes now a matter of great interest to inquire, whether upon the whole the progression exceeds the regression. Now the force perpendicular to the radius vector, considered in (99.), is almost exactly equal to that considered in (102.); so that the progression produced by that force when the line of apses passes through the sun is almost exactly equal to the regression which it produces when the line of apses is perpendicular to the line joining the earth and sun; and this force may, therefore, be considered as producing no effect (except indirectly, as will be hereafter mentioned). But the force in the direction of the radius vector, tending from the earth in (98.), is, as we have mentioned in (80.), almost exactly double of that tending to the earth in (101.), and, therefore, its effect predominates: and, therefore, on the whole, the line of apses progresses. In fact, the progress, when the line of apses passes through the sun, is about 11° in each revolution of the moon; the regress, when the line of apses is perpendicular to the line joining the earth and sun, is about 9° in each revolution of the moon.

(105.) The progression of the line of apses of the moon is considerably greater than the first consideration would lead us to think, for the following reasons.

(106.) Firstly. The earth is revolving round the sun, or the sun appears to move round the earth, in the same direction in which the moon is going. This lengthens the time for which the sun acts in any one manner upon the moon, but it lengthens it more for the time in which the moon is moving slowly, than for that in which it is moving quickly. Thus; suppose that the moon's angular motion when she is near perigee is fourteen times the sun's angular motion: and when near apogee, only ten times the sun's motion. Then she passes the sun at the former time, (as seen from the earth), with ths of her whole motion, but at the latter with only ths; consequently, when near perigee, the time in which the moon passes through a given angle from the moving line of syzygies (or the time in which the angle between the sun and moon increases by a given quantity), is 14ths of the time in which it would have passed through the same angle had the sun been stationary; when near apogee, the number expressing the proportion is ths. The latter number is greater than the former; and, therefore, the effect of the forces acting near apogee is increased in a greater proportion than that of the forces acting near perigee. And as the effective motion of the line of apses is produced by the excess of the apogeal effect above the perigeal effect, a very small addition to the former will bear a considerable proportion to the effective motion previously found; and thus the effective motion will be sensibly increased.