(107.) Secondly. When the line of apses is directed toward the sun, the whole effect of the force is to make it progress, that is, to move in the same direction as the sun the sun passes through about 27° in one revolution of the moon, and therefore departs only 16° from the line of apses; and therefore the apse continues a long time near the sun. When at right angles to the line joining the earth and sun, the whole effect of the force is to make it regress, and therefore, moving in the direction opposite to the sun's motion, the angle between the sun and the line of apses is increased by 36° in each revolution, and the line of apses soon escapes from this position. The effect of the former force is therefore increased, while that of the latter is diminished: and the preponderance of the former is much increased. It is in increasing the rapidity of progress at one time, and the rapidity of regress at another, that the force perpendicular to the radius vector indirectly increases the effect of the former in the manner just described.

(108.) From the combined effect of these two causes the actual progression of the line of apses is nearly double of what it would have been if, in different revolutions of the moon, different parts of its orbit had been equally subjected to the disturbing force of the sun.

(109.) The line of apses upon the whole, therefore, progresses; and (as calculation and observation agree in showing) with an angular velocity that makes it (on the average) describe 3° in each revolution of the moon, and

that carries it completely round in nearly nine years. But as it sometimes progresses and sometimes regresses for several months together, its motion is extremely irregular. The general motion of the line of apses has been known from the earliest ages of astronomy.

(110.) V. For the alteration of the excentricity of the moon's orbit: first, let us consider the orbit in the position in which the line of apses passes through the sun, fig. 31. While the moon moves from B1 (the perigee), to B, (the apogee), the force in the direction of the radius vector is sometimes directed to the earth, and sometimes from the earth, and therefore, by (57.) and (59.), it sometimes diminishes the excentricity and sometimes increases it. But while the moon moves from B, to B1, there are exactly equal forces acting in the same manner at corresponding parts of the half-orbit, and these, by (58.), will produce effects exactly opposite. On the whole, therefore, the disturbing force in the direction of the radius vector produces no effect on the excentricity. The force perpendicular to the radius vector increases the moon's velocity when moving from B, to B1, and diminishes it when moving from B, to B2; in moving, therefore, from B, to B1, the excentricity is increased (65.), and in moving from B1 to B2, it is as much diminished (66.). Similarly in moving from B2 to B, the excentricity is diminished, and in moving from B3 to B, it is as much increased. This force, therefore, produces no effect on the excentricity.

G

On the whole, therefore, while the line of apses passes through the sun, the disturbing forces produce no effect on the excentricity of the moon's orbit.

(111.) When the line of apses is perpendicular to the line joining the earth and sun, the same thing is true. Though the forces near perigee and near apogee are not now the same as in the last case, their effects on dif ferent sides of perigee and apogee balance each other in the same way,

(112.) But if the line of apses is inclined to the line joining the earth and sun, as in fig. 32, the effects of the

forces do not balance.

While the moon is near B, and

near B2, the disturbing force in the radius vector is directed to the earth; at B, therefore (58.), as the moon is moving towards perigee, the excentricity is increased; and at B2, as the moon is moving from perigee, the excentricity is diminished. From the slowness of the motion at B2 (which gives the disturbing force more time to produce its effects), and the greatness of the force, the effect at B2 will preponderate, and the combined effects at B2 and B will diminish the excentricity. This will appear from reasoning of the same kind as that in (98.). At B, and B, the force in the

radius vector is directed from the earth: at B1, therefore, by (59.), as the moon is moving from perigee, the excentricity is increased, and at B, it is diminished: but from the slowness of the motion at B, and the magnitude of the force, the effect at B, will preponderate, and the combined effects at B, and B, will diminish the excentricity. On the whole, therefore, the force in the direction of the radius vector diminishes the excentricity. The force perpendicular to the radius vector retards the moon from B1 to B2, but the first part of this motion may be considered near perigee, and the second near apogee, and, therefore, in the first part, it diminishes the excentricity, and in the second increases it; and the whole effect from B1 to B2 is very small. Similarly the whole effect from B, to В is very small. But from B to B1, the force accelerates the moon, and therefore, by (68.) (the moon being near perigee), increases the excentricity; and from B2 to B, the force also accelerates the moon, and by (68.) (the moon being near apogee), diminishes the excentricity; and the effect is much greater (from the slowness of the moon and

*

* To the reader who is acquainted with Newton's 3rd section, the following demonstration of this point will be sufficient. Four times the reciprocal of the latus rectum is equal to the sum of the reciprocals of the apogeal and perigeal distances. The effect of an increase of velocity at perigee in a given proportion is to alter the area described in a given time in the same proportion, and therefore, to alter the latus rectum in a corresponding proportion. Consequently an increase of velocity at perigee in a given proportion alters the reciprocal of the apogeal distance by a given quantity, and, therefore, alters the apogeɛl distance by a quantity nearly proportional to the square of the apogeal distance; and, therefore, the ratio of the alteration of apogeal distance to apogeal distance (on which the alteration of excentricity depends).

4

the greatness of the force) between B2 and B, than between B, and B1, and therefore the combined effect of the forces in these two quadrants is to diminish the excentricity.

On the whole, therefore, when the line of apses is inclined to the line joining the earth and sun, in such a manner that the moon passes the line of apses before passing the line joining the earth and sun, the excentricity is diminished at every revolution of the moon.

(113.) In the same manner it will appear that if the line of apses is so inclined that the moon passes the line of apses after passing the line joining the earth and sun, the excentricity is increased at every revolution of the moon. Here the force in the radius vector is directed to the earth, as the moon moves from perigee

is nearly proportional to the apogeal distance. Similarly, if the velocity at apogee is increased in a given proportion, the ratio of the alteration of perigeal distance to perigeal distance (on which the alteration of excentricity depends) is nearly proportional to the perigeal distance. Thus if the velocity were increased in the same proportion at perigee and at apogee, the increase of excentricity at the former would be greater than the diminution at the latter, in the proportion of apogeal distance to perigeal distance. But in the case before us, the proportion of increase of velocity is much greater at apogee than at perigee. First, because the force is greater (being in the same proportion as the distance). Second, because the time in which the moon describes a given angle is greater (being in the same proportion as the square of the distance), so that the increase of velocity is in the proportion of the cube of the distance. Third, because the actual velocity is less (being inversely as the distance), so that the ratio of the increase to the actual velocity is proportional to the fourth power of the distance. Combining this proportion with that above, the alterations of excentricity in the case before us, produced by the forces acting at apogee and at perigee, are in the proportion of the cubes of the apogeal and perigeal distances respectively.