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sibly coincide, through a small arc on each side of the extremity of their axis, it is difficult to ascertain in which of these curves the comets move, from observations made, as they necessarily must be, at their perihelia; but probably they all move in extremely excentric ellipses, although in most cases the parabolic curve coincides most nearly with their observed motions. Some few seem to describe hyperbolas; such being once visible to us, would vanish for ever, to wander through boundless space, to the remote systems of the universe. If a planet be supposed to revolve in a circular orbit, whose radius is equal to the perihelion distance of a comet moving in a parabola, the areas described by these two bodies in the same time will be as unity to the square root of two, which forms such a connexion between the motion of comets and planets, that, by Kepler's law, the ratio of the areas described during the same time by the comet and the earth may be found; so that the place of a comet at any time in its parabolic orbit, estimated from the instant of its passage at the perihelion, may be computed. But it is a problem of very great difficulty to determine all the other elements of parabolic motion-namely, the comet's perihelion distance, or shortest distance from the sun, estimated in parts of the mean distance of
the earth from the sun; the longitude of the perihelion; the inclination of the orbit on the plane of the ecliptic; and the longitude of the ascending node. Three observed longitudes and latitudes of a comet are sufficient for computing the approximate values of these quantities; but an accurate estimation of them can only be obtained by successive corrections from a number of observations, distant from one another. When the motion of a comet is retrograde, the place of the ascending node is exactly opposite to what it is when the motion is direct; hence the place of the ascending node, together with the direction of the comet's motion, show whether the inclination of the orbit is on the north or south side of the plane of the ecliptic. If the motion be direct, the inclination is on the north side; if retrograde, it is on the south side.
The identity of the elements is the only proof of the return of a comet to our system. Should the elements of a new comet be the same, or nearly the same, with those of any one previously known, the probability of the identity of the two bodies is very great, since the similarity extends to no less than four elements, every one of which is capable of an infinity of variations. But even if the orbit be determined with all the accuracy the case admits of, it may be difficult, or even impossible, to
recognise a comet on its return, because its orbit would be very much changed if it passed near any of the large planets of this, or of any other system, in consequence of their disturbing energy, which would be very great on bodies of so rare a nature. Halley computed the elements of the orbit of a comet that appeared in the year 1682, which agreed so nearly with those of the comets of 1607 and 1531, that he concluded it to be the same body returning to the sun, at intervals of about seventy-five years. He consequently predicted its re-appearance in the year 1758, or in the beginning of 1759. Science was not sufficiently advanced in the time of Halley, to enable him to determine the perturbations this comet might experience; but Clairaut computed that it would be retarded in its motion a hundred days by the attraction of Saturn, and 518 by that of Jupiter, and consequently, that it would pass its perihelion about the middle of April, 1759, requiring 618 days more to arrive at that point than in its preceding revolution. This, however, he considered only to be an approximation, and that it might be thirty days more or less: the return of the comet on the 12th of March, 1759, proved the truth of the prediction. MM. Damoiseau and Pontecoulant have ascertained that this comet will return either on the 4th or the 7th of No
vember, 1835; the difference of three days in their computations arises from their having employed different values for the masses of the planets. This is the first comet whose periodicity has been established; it is also the first whose elements have been determined from observations made in Europe, for although the comets which appeared in the years 240, 539, 565, and 837, are the most ancient whose orbits have been traced, their elements were computed from Chinese observations.
By far the most curious and interesting instance of the disturbing action of the great bodies of our system is found in the comet of 1770. The elements of its orbit, determined by Messier, did not agree with those of any comet that had hitherto been computed, yet Lexel ascertained that it described an ellipse about the sun, whose major axis was only equal to three times the length of the diameter of the terrestrial orbit, and consequently that it must return to the sun at intervals of five years and a half. This result was confirmed by numerous observations, as the comet was visible through an arc of 170°; yet this comet had never been observed before the year 1770, nor has it ever again been seen, though very brilliant The disturbing action of the larger planets afford a solution of this anomaly, as Lexel ascertained that in
1767 the comet must have passed Jupiter at a distance less than the fifty-eighth part of its distance from the sun, and that in 1779 it would be 500 times nearer Jupiter than the sun; consequently the action of the sun on the comet would not be the fiftieth part of what it would experience from Jupiter, so that Jupiter became the primum mobile. Assuming the orbit to be such as Lexel had determined in 1770, La Place found that the action of Jupiter, previous to the year 1770, had so completely changed the form of it, that the comet which had been invisible to us before 1770, was then brought into view, and that the action of the same planet producing a contrary effect, has, subsequently to that year, removed it, probably for ever, from our system. This comet might have been seen from the earth in 1776, had its light not been eclipsed by that of the sun.
Besides Halley's comet, two others are now proved to form part of our system; that is to say, they return to the sun at intervals, one of 1207 days, and the other of 63 years, nearly. The first, generally called Encke's comet, or the comet of the short period, was first seen by MM. Messier and Mechain in 1786, again by Miss Herschel in 1795, and its returns in the years 1805 and 1819 were observed by other astronomers, under the impression that all four were different