its perihelion may be determined, and its position in its parabolic orbit fixed. For this purpose a parabola must be conceived to pass through several observed places of the comet; and then having ascertained that it is still found on this curve at other epochs of its visible route, the elements of its motion may be ascertained. These are its perihelion distance, the position of its perihelion, the instant of its passage through that point, the inclination of its orbit to the plane of the ecliptic, and the place of its nodes. When these five elements are known, the place of the comet may be assigned at any given epoch of its revolution, and that place compared with actual observation when it is visible. But the accurate determination of these elements requires difficult and delicate calculations; for which we must refer such as wish to be acquainted with them to the Mécanique Céleste, where he may find the most accurate and general method that has yet been given.

But as the difficulty of making accurate observations on these bodies exceeds both the means and the practice of most of those whose curiosity may be excited by their appearance, several easy methods of approximating to their elements have been adopted. For this purpose, the apparent diameter of the comet being taken as frequently as possible, a judgment may be formed of its relative distance at different times. Its degree of motion and its brightness also deserve attention; for when it moves with the greatest velocity, or appears brightest, it may fairly be presumed that it is near its perihelion. If the place of the comet can be observed when it has no latitude, the position of its node and the time of passing it would be accurately determined: but as this can seldom be done, these elements must be approximated to by other means. The proper course of a comet may be found by observing its distance, on several successive nights, from two fixed stars, the latitudes and longitudes of which are known; U

or by finding its altitude when it is in the same azimuth with two known stars; as from these observations the place of the comet for each night is easily computed by spherical trigonometry; and thus the course of the comet becomes determined. Longomontanus also pointed out an easy method of tracing out the course of a comet mechanically, which is to find out two stars in the same line with the comet, by stretching a fine thread or a hair before the eye, so as to pass over the three bodies; and then to do the same for other two stars and the comet. Then, upon a celestial globe or a planisphere, draw a fine line connecting each of these pairs of stars, and the intersection of these lines will give the situation of the comet at that time. This repeated for several successive nights, and all the points joined, will indicate the path of the comet in the heavens; and consequently a great circle drawn through three distant places, will show, by its intersection with the ecliptic, nearly the place of the node, and the inclination of its orbit to the plane of the ecliptic; which being thus found by a good number of observations, independent of each other, the mean of all the results may be taken; and, if the observations have been made with care, this mean may be regarded as a tolerable approximation to the truth.

The great eccentricity of the orbits of these bodies causes a small error in the observations, to produce a great effect in the results of their calculated elements, and renders it difficult to ascertain them with accuracy; besides the thickness of the atmosphere with which a comet is generally surrounded, and the variations to which it is subject, also render it almost impossible to ascertain, with any degree of precision, when either the limb or the centre passes the wire of the telescope: much uncertainty must therefore accompany the result thus deduced. The only safe method, therefore, to determine the periods of comets, is to compare the elements of all those that have been

computed; and where any remarkable coincidence is perceived, an identity may be inferred; it being extremely improbable that two different comets should have the same inclination, the same perihelion distance, and the places of the perihelion and the node the same. By this means, the periodic time being determined, the major axis of the orbit becomes known from the laws of Kepler, and, the perihelion distance being likewise obtained from observation, will also give the minor axis of its orbit.

The perfect identity of the observed elements of a comet, with those of one that had previously appeared, is not absolutely necessary for concluding that they are the same; for the elements of the orbit may, like those of the planets, experience perturbations by which they are in some measure changed. And, consequently, if the elements of the one afford only a near approximation to those of the other, the doctrine of probabilities furnishes the means of judging what conclusion is to be drawn from these resemblances. It was in this way that Dr. Halley was enabled to foretel the appearance of the comet in 1759; which he thus ascertained to have appeared in 1456, 1531, 1607, and 1682, and which therefore occupies between 75 and 76 years in completing one revolution; and consequently ought to appear again in 1835. According to the third law of Kepler, therefore, if the semitransverse axis of the earth's orbit be taken for unity, the mean distance of that comet from the Sun will be equal to the cube root of the square of 76, or to 17.95. For if d denote the mean distance of the earth from the Sun, and D that of the comet, t and T the times of their sidereal revolutions, the laws of Kepler give,

In the present case we have T = 76t; if, there

fore, the mean distance of the earth be supposed to be unity, or d 1, we shall have

The transverse axis of the orbit will therefore be 35.9; and as its perihelion distance was observed to be 0.58, it follows that its aphelion distance was 35.32. The greatest distance of this comet from the. Sun, therefore, exceeded thirty-five times that of the earth, while its least distance was only about half that of the earth; hence the orbit it describes is extremely eccentric.

The intervals between the returns of the same comet to its perihelion are not equal to each other. That from 1531 to 1607 was longer by three months than that from 1607 to 1682; and this last was eighteen months shorter than that from 1682 to 1759. It therefore appears that the movements of comets are subject to variations, like those of the planets, but in a greater degree.

The comet of 1770 presents a striking example of these inequalities. The observations on that comet, calculated first by Lexell, and afterwards by Burkardt, assigned for the time of its revolution a period of five years and seven months; but it has not since reappeared; which has induced astronomers to conclude that the nature of its orbit must have been changed since its appearance in 1770. See Biot's Astronomie Physique, tome troisième.

The orbits of about one hundred comets have been calculated with sufficient accuracy for ascertaining their identity on any future appearance. Many of these orbits are inclined to the plane of the ecliptic in large angles, and many of them approach much nearer the Sun than the earth does. Their motions are also different from those of the planets, some of them being direct and others retrograde, nearly half the number moving each way. The different motions

of the comets, and the various inclinations of their orbits to the plane of the ecliptic, must not be regarded as the work of chance, but as calculated to answer beneficial purposes, or avoid baneful consequences; for if these orbits had been nearly coincident with that of the earth, both bodies might have arrived at the common point of intersection of their orbits at the same time; in which case a derangement of both motions must, at least, have been the necessary result. But, according to all the observations that have been made respecting their present distribution and direction, there is not the least reason to apprehend any such consequence.

The following table contains a list of the last twenty-one of the principal comets that have been observed, with the time of passing their perihelia, and their nearest approach to the Sun.