satellite in the first case, and to retard it in the second. The same reasoning will also show that both the first and third satellites exert forces on the second, tending to retard its angular motion in the first case, and to accelerate it in the second. All these actions tend to preserve the law; in the first case by making the line of conjunctions of the first and second satellites regress, and that of the second and third progress, till they coincide; and in the second case, by altering them in the opposite way, till they coincide.

(143.) Perhaps there is no theoretical permanence of elements on which we can depend with so great certainty as on the continuance of this law. The greatest and most irregular perturbations of Jupiter or of his satellites, provided they come on gradually, will not alter the relation between their motions; the effect of a resisting medium will not alter it; though each of these causes would alter the motions of all the satellites; and though similar causes would wholly destroy the conclusions which mathematicians have drawn as to the stability of the solar system, with regard to the elements of the planetary orbits. The physical explanation of this law was first given by Laplace, in A.D. 1784.

(144.) We have terminated now the most remarkable part of the theory of these satellites. There are, however, some other points which are worth attending to, partly for their own sake, and partly as an introduction to the theory of the planets.

(145.) The orbit of the third satellite, as we have mentioned, has a small excentricity independent of per

turbation. Consequently, when the conjunction with the second takes place near the independent perijove of the third, the effect of the disturbance on the second is rather greater than at any other time; and this produces an irregularity in the excentricity of the second, and in the motion of its apses, depending on the distance of the line of conjunction from the independent perijove of the third. The departure from uniformity in the angular motion of the third, also produces a departure from uniformity in the regression of the line of conjunction, and this contributes to the same irregularity.

(146.) The disturbing force in the direction of the radius vector, produced by an inner satellite, is sometimes directed to the central body and sometimes from it, but, on the whole, the former exceeds the latter (86.). Now the principal part of the effect really takes place when the satellites are near conjunction; consequently, when the line of conjunction passes near the independent perijove of the third satellite, the force by which the third satellite is urged to the planet is greater than at any other time; and as the line of conjunction revolves, the force alternately increases and diminishes. This produces an irregularity in the major axis, and consequently in the motion of the third satellite (47.), depending on the distance of the line of conjunction from the perijove of the third.

(147.) The disturbing force in the direction of the radius vector produced by an outer satellite is sometimes directed to the central body, and sometimes from

it, but, on the whole, the latter exceeds the former (80.). For the reasons, therefore, in the last article, there is in the motion of the second satellite an irregularity depending on the distance of the line of conjunction from the independent perijove of the third, but opposite in its nature to that of the third satellite.

(148.) Each of these irregularities in the motion of one of these satellites produces an irregularity in the motion of the others; and thus the whole theory becomes very complicated when we attempt to take the minute irregularities into account.

(149.) The motion of the fourth satellite is not related to the others in the same way in which they are related among themselves. Its periodic time is to the periodic time of the third nearly in the proportion of 7:3. Some of the irregularities then which it experiences and which it occasions are nearly similar to those in the motions of the planets. These, however, are small; the most important are those depending on the changes in the elements which require many revolutions of the satellites to go through all their various states, but which, nevertheless, have been observed since the satellites were discovered. We shall proceed with these.

(150.) First, let us suppose that the third satellite has no excentricity independent of perturbation, and that the fourth satellite has a sensible excentricity, its line of apses progressing very slowly, in conseqneuce principally of the shape of Jupiter (so slowly as not to have gone completely round in eleven thousand revo

lutions of the satellite). When each of the satellites has revolved a few hundred times round Jupiter, their conjunctions will have taken place almost indifferently in every part of their orbits. If the orbit of the fourth as well as that of the third had no independent ellipticity, there would be no remarkable change of shape produced by perturbation, as the action of one satellite upon the other would be the same when in conjunction in all the different parts of the orbit. But the orbit of the fourth being excentric, the action of each satellite on the other is greatest when the conjunction happens near the perijove of the fourth satellite. We may consider then that the preponderating force takes place at this part of the orbits; and we have to inquire what form the orbit of the third satellite must have, to preserve the same excentricity at every revolution. It must be remembered here that the effect of Jupiter's shape is to cause a more rapid progress of the line of apses of the third satellite, if its orbit be excentric, than of the line of apses of the fourth.

(151.) Considering, then, that the preponderating force on the third satellite in the direction of the radius vector is directed from the central body towards the perijove of the fourth, and that the preponderating force perpendicular to the radius vector accelerates it as it approaches that part, and retards it afterwards, it is plain from (51.), (65.), and (66.), that, if the perijove of the third satellite were in that position, the forces would cause the line of apses to regress; and this regression, if the excentricity of the third be small, may be con

siderable (though the preponderance of force which causes it is extremely small), and may overcome so much of the progression caused by Jupiter's shape, as to make the real motion of the line of apses as nearly equal as we please to the motion of the line of apses of the fourth. But the motion of the line of apses of the fourth will itself be affected (though very little) by the greater action of the third satellite on it at the same. place; and the part in the radius vector being directed at its perijove to the central body, and the part perpendicular to the radius vector retarding it before it reaches the perijove, and accelerating it afterwards, will cause a small increase of progression of its apse. The state of things will be permanent, so far as depends on these forces, when the increased progression of the apse of the fourth satellite is equal to the diminished progression of the apse of the third; and thus the progression of the apse of the fourth will be somewhat increased, and the third satellite's orbit will have a compression corresponding in direction to the perijove of the fourth, and an elongation in the same direction as the apojove of the fourth. This would be the case if the third satellite had no excentricity independent of perturbation; but we may, as in other cases, consider that the same kind of distortion will be produced in the orbit if it has an independent excentricity.

(152.) Now let us suppose the fourth satellite to have no excentricity independent of perturbation, and the third satellite to have an independent excentricity. The greatest action will now be at the apojove of the