and from apogee: and is directed from the earth as the moon moves to perigee and to apogee; which directions are just opposite to those in the case already considered. Also the force perpendicular to the radius

FIG. 33.

vector retards the moon both near perigee and near apogee; and this is opposite to the direction in the case already considered. On the whole, therefore, the excentricity is increased at every revolution of the

moon.

(114.) In every one of these cases the effect is exactly the same if the sun be supposed on the side of the moon's orbit, opposite to that represented in the figure.

(115.) Now the earth moves round the sun, and the sun therefore appears to move round the earth, in the order successively represented by the figs. 31, 32, and 33. Hence then; when the sun is in the line of the moon's apses, the excentricity does not alter (110.); after this it diminishes till the sun is seen at right angles to the line of apses (112.); then it does not alter (111.): and after this it increases till the sun reaches the line of apses on the other side. Consequently, the excentricity is greatest when the line of apses passes through the sun; and is least when the line

of apses is perpendicular to the line joining the earth and sun.

The amount of this alteration in the excentricity of the moon's orbit is more than 4th of the mean value of the excentricity; the excentricity being sometimes increased by this part, and sometimes as much diminished; so that the greatest and least excentricities are nearly in the proportion of 6:4 or 3:2.

(116.) The principal inequalities in the moon's motion may therefore be stated thus:

1st. The elliptic inequality, or equation of the centre (31.), which would exist if it were not disturbed. 2nd. The annual equation (90.), depending on the position of the earth in the earth's orbit.

3rd. The variation (93.), and parallactic inequality (94.), depending on the position of the moon with respect to the sun.

4th. The general progression of the moon's perigee (104.). 5th. The irregularity in the motion of the perigee, depending on the position of the perigee with respect to the sun (109.).

6th. The alternate increase and diminution of the excentricity, depending on the position of the perigee with respect to the sun (115.).

These inequalities were first explained (some imperfectly) by Newton, about A.D. 1680.

(117.) The effects of the two last are combined into one called the evection. This is by far the largest

of the inequalities affecting the moon's place: the moon's longitude is sometimes increased 1° 15′, and sometimes diminished as much by this inequality. It was discovered by Ptolemy, from observation, about A.D. 140.

(118.) It will easily be imagined that we have here taken only the principal inequalities. There are many others, arising chiefly from small errors in the suppositions that we have made. Some of these, it may easily be seen, will arise from variations of force which we have already explained. Thus the difference of disturbing forces at conjunction and at opposition, whose principal effect was discussed in (94.), will also produce a sensible inequality in the rate of progression of the line of apses, and in the dimensions of the moon's orbit. The alteration of disturbing force depending on the excentricity of the earth's orbit will cause an alteration in the magnitude of the variation and the evection. The alteration of that part mentioned in (94.) produces a sensible effect depending on the angle made by the moon's radius vector with the earth's line of apses. All these, however, are very small: yet not so small but that, for astronomical purposes, it is necessary to take account of thirty or forty.

(119.) There is, however, one inequality of great historical interest, affecting the moon's motion, of which we may be able to give the reader a general idea. We have stated in (89.) that the effect of the disturbing force is, upon the whole, to diminish the moon's gravity

to the earth and in (90.) we have mentioned that this effect is greater when the earth is near perihelion, than when the earth is near aphelion. It is found, upon accurate investigation, that half the sum of the effects at perihelion and at aphelion is greater than the effect at mean distance, by a small quantity depending on the excentricity of the earth's orbit: and, consequently, the greater the excentricity (the mean distance being unaltered) the greater is the effect of the sun's disturbing force. Now, in the lapse of ages, the earth's mean distance is not sensibly altered by the disturbances which the planets produce in its motion; but the excentricity of the earth's orbit is sensibly diminished, and has been diminishing for thousands of years. Consequently the effect of the sun in disturbing the moon has been gradually diminishing, and the gravity to the earth has therefore, on the whole, been gradually increasing. The size of the moon's orbit has therefore, gradually (but insensibly), diminished (47.): but the moon's place in its orbit has sensibly altered (49.), and the moon's angular motion has appeared to be perpetually quickened. This phenomenon was known to astronomers by the name of the acceleration of the moon's mean motion, before it was theoretically explained in 1787, by Laplace: on taking it into account, the oldest and the newest observations are equally well represented by theory. The rate of progress of the moon's line of apses has, from the same cause, been somewhat diminished.

SECTION VI.-Theory of Jupiter's Satellites.

(120.) JUPITER has four satellites revolving round him in the same manner in which the moon revolves round the earth; and it might seem, therefore, that the theory of the irregularities in the motion of these satellites is similar to the theory of the irregularities in the moon's motion. But the fact is, that they are entirely different. The fourth satellite (or that revolving in the largest orbit) has a small irregularity analogous to the moon's variation, a small one similar to the evection, and one similar to the annual equation: but the last of these amounts only to about two minutes, and the other two are very much less. The corresponding inequalities in the motion of the other satellites are still smaller. But these satellites disturb each other's motions, to an amount and in a manner of which there is no other example in the solar system; and (as we shall afterwards mention) their motions are affected in a most remarkable degree by the shape of Jupiter.

(121.) The theory, however, of these satellites is much simplified by the following circumstances:First, that the disturbances produced by the sun may, except for the most accurate computations, be wholly neglected. Secondly, that the orbits of the two inner satellites have no excentricity independent of perturbation. Thirdly, that a very remarkable relation exists (and, as we shall show, necessarily exists)

between the motions of the first three satellites.

Before proceeding with the theory of the first three