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However, the values of the elements determined separately can only be regarded as approximate, because they are so connected that the estimation cof any one independently will induce errors in the others, for the excentricity depends upon the longitude of the perihelion, the s mean motion depends upon the major axis, the · longitude of the node upon the inclination of the orbit, and vice versâ, consequently the place of a planet computed with the approximate data, will differ from its observed place : then the difficulty is to ascertain what elements are most in fault, since the difference in question is the error of all, but that is obviated by finding the errors of some thousands of observations, and combining them so as to correct the elements simultaneously, and to make the sum of the squares of the errors
a minimum with regard to each element. The method of accomplishing this depends upon the Theory of Probabilities, a subject fertile in most "important results in the various departments of science and of civil life, and quite indispensable in the determination of astronomical data. A series of observations continued for some years will give approximate values of the secular and periodic inequalities, which must be corrected from time to time till theory and observation agree; and when all these quantities are determined in numbers, the longitude, latitude, and distances of the planet
from the sun are computed for stated intervals, and formed into tables, arranged according to the time estimated from a given epoch, so that the place of the body may be determined from them by inspection alone, at any instant, for perhaps a thousand years before and after that epoch. By this tedious process, tables have been computed for eleven planets, besides the moon and the satellites of Jupiter. Those of the four new planets are astonishingly perfect, considering that these bodies have not been discovered more than thirty years, and a much longer time is requisite to develop their inequalities.
The oblate form of several of the planets indicates rotatory motion; this has been confirmed, in most cases, by tracing spots on their surface, by which their poles and times of rotation have been determined. The rotation of Mercury is unknown, on account of his proximity to the sun; and that of the new planets has not yet been ascertained. The sun revolves in twenty-five days and ten hours about an axis which is directed towards a point half-way between the pole-star and Lyra, the plane of rotation being inclined by 7° 20', or a little more than seven degrees, to the plane of the ecliptic. From the rotation of the sun, there is every reason to
believe that he has a progressive motion in space, although the direction to which he tends is unknown: but in consequence of the reaction of the planets, he describes a small irregular orbit about the centre of inertia of the system, never deviating from his position by more than twice his own diameter, or a little more than seven times the distance of the moon from the earth. The sun and all his attendants rotate from west to east, on axes that remain nearly parallel to themselves in every point of their orbit, and with angular velocities that are sensibly uniform. Although the uniformity in the direction of their rotation is a circumstance hitherto unaccounted for in the economy of nature, yet from the design and adaptation of every other part to the perfection of the whole, a coincidence so remarkable cannot be accidental; and as the revolutions of the planets and satellites are also from west to east, it is evident that both must have arisen fron the primitive cause which has determined the planetary motions. Indeed, La Place has computed the probability to be as four millions to one, that all the motions of the planets, both of rotation and revolution, were at once imparted by an original common cause, but of which we know neither the nature nor the epoch.
The larger planets rotate in shorter periods than the smaller planets and the earth, their compression is consequently greater, and the action
of the sun and of their satellites occasions a nutation in their axes, and a precession of their equinoxes similar to that which obtains in the terrestrial spheroid, from the attraction of the sun and moon on the prominent matter at the equator. It is an evident consequence of Kepler's law of the squares of the periodic times of the planets being as the cubes of the major axes of their orbits, that the heavenly bodies move slower the farther they are from the sun. In comparing the periods of the revolutions of Jupiter and Saturn with the times of their rotation, it appears that a year of Jupiter contains nearly ten thousand of his days, and that of Saturn about thirty thousand Saturnian days.
The appearance of Saturn is unparalleled in the system of the world; he is a spheroid about 900 times larger than the earth, surrounded by a ring even brighter than himself, which always remains suspended in the plane of his equator, and viewed with a very good telescope, it is found to consist of two concentric rings, divided by a dark band. The mean distance of the interior part of this double ring from the surface of the planet is about 22240 miles, it is no less than 33360. miles, broad, but, by estimation, its thickness does not much exceed 274 miles, so that it appears like a plane. By
the laws of mechanics, it is impossible that this body can retain its position by the adhesion of its particles alone; it must necessarily revolve with a velocity that will generate a "centrifugal force sufficient to balance the attraction of Saturn. Observation confirms the truth of these principles, showing that the rings rotate about the planet in ten hours and a half, which is considerably less than the time a satellite would take to revolve about Saturn at the same distance. Their plane is inclined to the ecliptic, at an angle of 28° 39' 45"; and, in consequence of this obliquity of position, they always appear elliptical to us, but with an excentricity 80 variable as even to be occasionally like a straight line drawn across the planet. In the beginning of October, 1832, the plane of the rings passed through the centre of the earth; in that position they are only visible with very superior instruments, and appear like a fine line across the disc of Saturn. About the middle of December, in the same year, the rings became invisible, with ordinary instruments, on account of their plane i passing through the sun. In the end of April, 1833, the rings vanished a second time, and reappeared in June of that year.
. It is a singular result of theory, that the rings could not maintain their stability of rotation if they were every where of uniform thickness;