All the periodic and secular inequalities deduced from the law of gravitation are so perfectly confirmed by observation, that analysis has become one of the most certain means of discovering the planetary irregularities, either when they are too small, or too long in their periods, to be detected by other methods. Jupiter and Saturn, however, exhibit inequalities which for a long time seemed discordant with that law. All observations, from those of the Chinese and Arabs down to the present day, prove that for ages the mean motions of Jupiter and Saturn have been affected by a great inequality of a very long period, forming an apparent anomaly in the theory of the planets. It was long known by observation that five times the mean motion of Saturn is nearly equal to twice that of Jupiter: a relation which the sagacity of La Place perceived to be the cause of a periodic irregularity in the mean motion of each of these planets, which completes its period in nearly 918 years, the one being retarded while the other is accelerated; but both the magnitude and period of these quantities vary in consequence of the secular variations in the elements of the orbits. Suppose the two planets to bu on the same side of the sun, and all three in the same straight line, they are then said to be in conjunction (N. 83). Now, if they begin to move at the same time, one making exactly five revolutions in its orbit, while the other only accomplishes two, it is clear that Saturn, the slow-moving body, will only have got through a part of its orbit during the time that Jupiter has made one whole revolution and part of another, before they be again in conjunction. It is found that during this time their mutual action is such as to produce a great many perturbations which compensate each other, but that there still remains a portion outstanding, owing to the length of time during which the forces act in the same manner; and if the conjunction always happened in the same point of the orbit, this uncompensated inequality in the mean motion would go on increasing till the periodic times and forms of the orbits were completely and permanently changed: a case that would actually take place if Jupiter accomplished exactly five revolutions in the time Saturn performed two. These revolutions

are, however, not exactly commensurable; the points in which the conjunctions take place are in advance each time as much as 8°.37; so that the conjunctions do not happen exactly in the same points of the orbits till after a period of 850 years; and, in consequence of this small advance, the planets are brought into such relative positions that the inequality which seemed to threaten the stability of the system is completely compensated, and the bodies, having returned to the same relative positions with regard to one another and the sun, begin a new course. The secular variations in the elements of the orbit increase the period of the inequality to 918 years (N. 84). As any perturbation which affects the mean motion affects also the major axis, the disturbing forces tend to diminish the major axis of Jupiter's orbit and increase that of Saturn's during one half of the period, and the contrary during the other half. This inequality is strictly periodical, since it depends upon the configuration (N. 85) of the two planets; and theory is confirmed by observation, which shows that, in the course of twenty centuries, Jupiter's mean motion has been accelerated by about.3° 23', and Saturn's retarded by 5° 13'. Several instances of perturbations of this kind occur in the solar system. One, in the mean motions of the Earth and Venus, only amounting to a few seconds, has been recently worked out with immense labor by Professor Airy. It accomplishes its changes in 240 years, and arises from the circumstance of thirteen times the periodic time of Venus being nearly equal to eight times that of the Earth. Small as it is, it is sensible in the motions of the Earth.

It might be imagined that the reciprocal action of such planets as have satellites would be different from the influence of those that have none. But the distances of the satellites from their primaries are incomparably less than the distances of the planets from the sun, and from one another; so that the system of a planet and its satellites moves nearly as if all these bodies were united in their common center of gravity. The action of the sun, however, in some degree disturbs tho motion of the satellites about their prinjary?

C

COAS ARTILLERY

SECTION IV.

Theory of Jupiter's Satellites-Effects of the Figure of Jupiter upon his
Satellites Position of their Orbits-Singular Laws among the Motions
of the first three Satellites-Eclipses of the Satellites-Velocity of Light
-Aberration-Ethereal Medium-Satellites of Saturn and Uranus.

THE changes which take place in the planetary sys-
'tem are exhibited on a smaller scale by Jupiter and his
satellites; and, as the period requisite for the develop-
ment of the inequalities of these moons only extends to
a few centuries, it may be regarded as an epitome of
that grand cycle which will not be accomplished by the
planets in myriads of ages. The revolutions of the
satellites about Jupiter are precisely similar to those of
the planets about the sun: it is true they are disturbed
by the sun, but his distance is so great, that their
motions are nearly the same as if they were not under
his influence. The satellites, like the planets, were
probably projected in elliptical orbits: but, as the masses
of the satellites are nearly 100,000 times less than that
of Jupiter; and as the compression of Jupiter's sphe-
roid is so great, in consequence of his rapid rotation,
that his equatorial diameter exceeds his polar diameter
by no less than 6000 miles; the immense quantity of
prominent matter at his equator must soon have given
the circular form observed in the orbits of the first and
second satellites, which its superior attraction will al-
ways maintain.
The third and fourth satellites, being
farther removed from its influence, revolve in orbits
with a very small eccentricity. And although the first
two sensibly move in circles, their orbits acquire a
small ellipticity, from the disturbances they experience
(N. 86).

It has been stated, that the attraction of a sphere on an exterior body is the same as if its mass were united in one particle in its center of gravity, and therefore inversely as the square of the distance. In a spheroid, however, there is an additional force arising from the bulging mass at its equator, which, not following the exact law of gravity; acts as a disturbing force. One

VOLUITRA TRACJ

effect of this disturbing force in the spheroid of Jupiter is, to occasion a direct motion in the greater axes of the orbits of all his satellites, which is more rapid the nearer the satellite is to the planet, and very much greater than that part of their motion which arises from the disturbing action of the sun. The same cause occasions the orbits of the satellites to remain nearly in the plane of Jupiter's equator (N. 87), on account of which the satellites are always seen nearly in the same line (N. 88); and the powerful action of that quantity of prominent matter is the reason why the motions of the nodes of these small bodies are so much more rapid than those of the planet. The nodes of the fourth satellite accomplish a tropical revolution in 531 years; while those of Jupiter's orbit require no less than 36,261 years;-a proof of the reciprocal attraction between each particle of Jupiter's equator and of the satellites. In fact, if the satellites moved exactly in the plane of Jupiter's equator, they would not be pulled out of that plane, because his attraction would be equal on both sides of it. But, as their orbits have a small inclination to the plane of the planet's equator, there is a want of symmetry, and the action of the protuberant matter tends to make the nodes regress by pulling the satellites above or below the planes of their orbits; an action which is so great on the interior satellites, that the motions of their nodes are nearly the same as if no other disturbing force existed.

The orbits of the satellites do not retain a permanent inclination, either to the plane of Jupiter's equator, or to that of his orbit, but to certain planes passing between the two, and through their intersection. These have a

greater inclination to his equator the farther the satellite is removed, owing to the influence of Jupiter's compression; and they have a slow motion corresponding to secular variations in the planes of Jupiter's orbit and equator.

The satellites are not only subject to periodic and secular inequalities from their mutual attraction, similar to those which affect the motions and orbits of the planets, but also to others peculiar to themselves. Of the periodic inequalities arising from their mutual at

traction, the most remarkable take place in the angular motions (N. 89) of the three nearest to Jupiter, the second of which receives from the first a perturbation similar to that which it produces in the third; and it experiences from the third a perturbation similar to that which it communicates to the first. In the eclipses these two inequalities are combined into one, whose period is 437.659 days. The variations peculiar to the satellites arise from the secular inequalities occasioned by the action of the planets in the form and position of Jupiter's orbit, and from the displacement of his equator. It is obvious that whatever alters the relative positions of the sun, Jupiter, and his satellites, must occasion a change in the directions and intensities of the forces, which will affect the motions and orbits of the satellites. For this reason the secular variations in the eccentricity of Jupiter's orbit occasion secular inequalities in the mean motions of the satellites, and in the motiong of the nodes and apsides of their orbits. The displacement of the orbit of Jupiter, and the variation in the position of his equator, also affect these small bodies (N. 90). The plane of Jupiter's equator is inclined to the plane of his orbit at an angle of 3° 5′ 30′′, so that the action of the sun and of the satellites themselves produces a nutation and precession (N. 91) in his equator, precisely similar to that which takes place in the rotation of the earth, from the action of the sun and moon. Hence the protuberant matter at Jupiter's equator is continually changing its position with regard to the satellites, and produces corresponding mutations in their motions. And, as the cause must be proportional to the effect, these inequalities afford the means, not only of ascertaining the compression of Jupiter's spheroid, but they prove that his mass is not homogeneous. Although the apparent diameters of the satellites are too small to be measured, yet their perturbations give the values of their masses with considerable accuracy— a striking proof of the power of analysis.

A singular law obtains among the mean motions and mean longitudes of the first three satellites. It appears from observation that the mean motion of the first satellite, plus twice that of the third, is equal to three