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from the earth. And for a short time he perceived them advancing off it at each end, when turning round in their orbits. The eclipses of the exterior satellites only take place when the ring is in this position. Mr. Lassell, with a powerful telescope, made by himself, has seen Iapetus, the nearest of the two, on several occasions, even when the opening of the ring was very wide, which made it extremely difficult to see so minute an object. Of the situation of the equator of Uranus we know nothing, nor of his compression; but the orbits of his satellites are nearly perpendicular to the plane of the ecliptic; and, by analogy, they ought to be in the plane of his equator. Uranus is so remote that he has more the appearance of a planetary nebula than a planet, which renders it extremely difficult to distinguish the satellites at all; and quite hopeless without such a telescope as is rarely to be met with even in observatories. Sir William Herschel discovered the two that are farthest from the planet, and ascertained their approximate periods, which his son afterwards determined to be 13d 11h 7m 12-6 and 8d 16h 56m 288.6 respectively. The orbits of both seem to have an inclination of about 101.2 to the plane of the ecliptic. The two interior satellites are so faint and small, and so near the edge of the planet, that they can with difficulty be seen even under the most favourable circumstances: however, Mr. Lassell has ascertained that the more distant of the two revolves about Uranus in 4 days, and that nearest to the planet in 2 days, and from a long and minute examination he is convinced that the system only consists of four satellites. Soon after Neptune was seen Mr. Lassell discovered the only satellite known to belong to that planet. The satellites of Uranus and Neptune, the two planets on the remotest verge of the solar system, offer the singular and only instance of a revolution from east to west, while all the planets and all the other satellites revolve from west to east. Retrograde motion is occasionally met with in the comets and double stars.
Lunar Theory-Periodic Perturbations of the Moon- - Equation of Centre Evection Variation Annual Equation Direct and Indirect Action of Planets The Moon's Action on the Earth disturbs her own Motion - Excentricity and Inclination of Lunar Orbit invariable Acceleration Secular Variation in Nodes and Perigee - Motion of Nodes and Perigee inseparably connected with the Acceleration Nutation of Lunar Orbit Form and Internal Structure of the Earth determined from it Lunar, Solar, and Planetary Eclipses · Occultations and Lunar Distances Mean Distance of the Sun from the Earth obtained from Lunar Theory Absolute Distances of the Planets, how found.
OUR constant companion, the moon, next claims our attention. Several circumstances concur to render her motions the most interesting, and at the same time the most difficult to investigate, of all the bodies of our system. In the solar system, planet troubles planet; but, in the lunar theory, the sun is the great disturbing cause, his vast distance being compensated by his enormous magnitude, so that the motions of the moon are more irregular than those of the planets; and, on account of the great ellipticity of her orbit, and the size of the sun, the approximations to her motions are tedious and difficult, beyond what those unaccustomed to such investigations could imagine. The average distance of the moon from the centre of the earth is only 238,793 miles, so that her motion among the stars is perceptible in a few hours. She completes a circuit of the heavens in 27d 7h 43m 11*5, moving in an orbit whose excentricity is about 12,985 miles. The moon is about four hundred times nearer to the earth than the sun. The proximity of the moon to the earth keeps them together. For so great is the attraction of the sun, that, if the moon were farther from the earth, she would leave it altogether, and would revolve as an independent planet about the sun.
The disturbing action (N. 101) of the sun on the moon is equivalent to three forces. The first, acting in the direction of the line joining the moon and earth, increases or diminishes her gravity to the earth. The second, acting in the direction of a
tangent to her orbit, disturbs her motion in longitude. And the third, acting perpendicularly to the plane of her orbit, disturbs her motion in latitude; that is, it brings her nearer to, or removes her farther from, the plane of the ecliptic than she would otherwise be. The periodic perturbations in the moon, arising from these forces, are perfectly similar to the periodic perturbations of the planets. But they are much greater and more numerous; because the sun is so large, that many inequalities which are quite insensible in the motions of the planets, are of great magnitude in those of the moon. Among the innumerable periodic inequalities to which the moon's motion in longitude is liable, the most remarkable are, the Equation of the Centre, which is the difference between the moon's mean and true longitude, the Evection, the Variation, and the Annual Equation. The disturbing force which acts in the line joining the moon and earth produces the Evection: it diminishes the excentricity of the lunar orbit in conjunction and opposition, thereby making it more circular, and augments it in quadrature, which consequently renders it more elliptical. The period of this inequality is less than thirty-two days. Were the increase and diminution always the same, the Evection would only depend upon the distance of the moon from the sun; but its absolute value also varies with her distance from the perigee (N. 102) of her orbit. Ancient astronomers, who observed the moon solely with a view to the prediction of eclipses, which can only happen in conjunction and opposition, where the excentricity is diminished by the Evection, assigned too small a value to the ellipticity of her orbit (N. 103). The Evection was discovered by Ptolemy from observation, about A.D. 140. The Variation produced by the tangential disturbing force, which is at its maximum when the moon is 450 distant from the sun, vanishes when that distance amounts to a quadrant, and also when the moon is in conjunction and opposition; consequently, that inequality never could have been discovered from the eclipses its period is half a lunar month (N. 104). The Annual Equation depends upon the sun's distance from the earth it arises from the moon's motion being accelerated when that of the earth is retarded, and vice versá―for, when the earth is in its perihelion, the lunar orbit is enlarged by the action of the sun; therefore, the moon requires more time to perform her revolution. But, as the earth approaches its aphelion, the
moon's orbit contracts, and less time is necessary to accomplish her motion-its period, consequently, depends upon the time of the year. In the eclipses the Annual Equation combines with the Equation of the Centre of the terrestrial orbit, so that ancient astronomers imagined the earth's orbit to have a greater excentricity than modern astronomers assign to it.
The planets disturb the motion of the moon both directly and indirectly; their action on the earth alters its relative position with regard to the sun and moon, and occasions inequalities in the moon's motion, which are more considerable than those arising from their direct action; for the same reason the moon, by disturbing the earth, indirectly disturbs her own motion. Neither the excentricity of the lunar orbit, nor its mean inclination to the plane of the ecliptic, have experienced any changes from secular inequalities; for, although the mean action of the sun on the moon depends upon the inclination of the lunar orbit to the ecliptic, and the position of the ecliptic is subject to a secular inequality, yet analysis shows that it does not occasion a secular variation in the inclination of the lunar orbit, because the action of the sun constantly brings the moon's orbit to the same inclination to the ecliptic. The mean motion, the nodes, and the perigee, however, are subject to very remarkable variations.
From the eclipse observed at Babylon, on the 19th of March, seven hundred and twenty-one years before the Christian era, the place of the moon is known from that of the sun at the instant of opposition (N. 83), whence her mean longitude may be found. But the comparison of this mean longitude with another mean longitude, computed back for the instant of the eclipse from modern observations, shows that the moon performs her revolution round the earth more rapidly and in a shorter time now than she did formerly, and that the acceleration in her mean motion has been increasing from age to age as the square of the time (N. 105). All ancient and intermediate eclipses confirm this result. As the mean motions of the planets have no secular inequalities, this seemed to be an unaccountable anomaly. It was at one time attributed to the resistance of an ethereal medium pervading space, and at another to the successive transmission of the gravitating force. But, as La Place proved that neither of these causes, even if they exist, have any influence on the motions of the lunar perigee (N. 102) or nodes,
they could not affect the mean motion; a variation in the mean motion from such causes being inseparably connected with variations in the motions of the perigee and nodes. That great mathematician, in studying the theory of Jupiter's satellites, perceived that the secular variation in the elements of Jupiter's orbit, from the action of the planets, occasions corresponding changes in the motions of the satellites, which led him to suspect that the acceleration in the mean motion of the moon might be connected with the secular variation in the excentricity of the terrestrial orbit. Analysis has shown that he assigned the true cause of the acceleration.
It is proved that the greater the excentricity of the terrestrial orbit, the greater is the disturbing action of the sun on the moon. Now, as the excentricity has been decreasing for ages, the effect of the sun in disturbing the moon has been diminishing during that time. Consequently the attraction of the earth has had a more and more powerful effect on the moon, and has been continually diminishing the size of the lunar orbit. So that the moon's velocity has been gradually augmenting for many centuries to balance the increase of the earth's attraction. This secular increase in the moon's velocity is called the Acceleration, a name peculiarly appropriate at present, and which will continue to be so for a vast number of ages; because, as long as the earth's excentricity diminishes, the moon's mean motion will be accelerated; but when the excentricity has passed its minimum, and begins to increase, the mean motion will be retarded from age to age. The secular acceleration is now about 11"-9, but its effect on the moon's place increases as the square of the time (N. 106). It is remarkable that the action of the planets, thus reflected by the sun to the moon, is much more sensible than their direct action either on the earth or moon. The secular diminution in the excentricity, which has not altered the equation of the centre of the sun by eight minutes since the earliest recorded eclipses, has produced a variation of about 1° 48′ in the moon's longitude, and of 7° 12' in her mean anomaly (N. 107).
The action of the sun occasions a rapid but variable motion in the nodes and perigee of the lunar orbit. Though the nodes recede during the greater part of the moon's revolution, and advance during the smaller, they perform their sidereal revolution in 6793d 9h 23m 9.3, or about 18 6-10 years; and the