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When this comet was discovered at Marseilles by M. Pons, on the 26th of November 1819, it was in the neck of the constellation Pegasus. It appeared tolerably bright, and a nucleus was several times perceived, but no appearance of a tail. It was not observed in Germany till December 22, 1818, and continued visible till January 12, 1819, describing in its course an apparent arch of about twenty three degrees.

In calculating the parabolic orbit of this comet, Professor Encke used only the German observations. After various trials he found that he could not obtain any parabola that would represent them within three minutes, and the error was much greater in the early observations made at Marseilles, being above thirty minutes in right ascension, and six minutes in declination; a sure indication that the orbit varied much from a parabola, and that it was necessary to notice its elliptical form.

After some preliminary calculations, Prof. Encke assumed the following elliptical elements of the orbit in 1819, using always the mean time for the meridian of Seeberg, and the mean equinox for the year in which the comet was observed.

d.

Time of passing the perihelion 1819 Jan. 27.275

Longitude of the perihelion

Longitude of the ascending node

Inclination of the orbit to the ecliptic

157° 5' 53"

334 43 37

13 38 42

Eccentricity (comet's dist. from sun = 1) 0.849
Mean distance from the sun

Periodical revolution, about 1203 days.
Motion direct.

2.2131

These elements represent the place of the comet, during its appearance, to a great degree of accuracy. For, out of forty two observations of the right ascension and declination, thirty five are given to less than half a minute, and no error exceeds 1' 33" so that there can be no doubt that the true figure of the orbit in 1819 is very nearly given by the above elements.

When this result had been obtained, it became an interesting object of inquiry to find whether the comet had been before observed by astronomers. Upon looking over the general table of the orbits, Professor Encke selected the first comet of 1805 (marked 107 in Delambre's table) as a former appearance, with an interval of four revolutions. It was near the constellation Ursa Major, when first discovered by Mr Pons, Oct.

20, 1805, and appeared as a star of the fourth magnitude, with a nucleus and a very faint tail, 21° in length. It continued visible the 15th of November. The parabolic orbit had been computed by several persons, with considerable difference in the results, which had been imputed, in great measure, to the incompleteness of the observations. In these different calculations the time of passing the perihelion varied from November 17th to 18th; place of the perihelion 147° to 149°; of the node 340° to 345°; inclination 151° to 171°; perihelion distance 0.346 to 0.379. The parabolic elements, published by the accurate and indefatigable Professor Bessel, (Monatliche Correspendenz, b. xiii. xiv.) did not represent the right ascensions and declinations without errors of 26', 11', 8', 7', 6', ,&c. Instead of which, Professor Encke found that the following elliptical orbit would satisfy all the observations made in 1805.

d.

Time of passing the perihelion 1805, Nov. 21.529. Longitude of the perihelion

Longitude of the ascending node

Inclination of the orbit

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156°47' 24"

334 20 10

13 33 30

Eccentricity

Mean distance from the sun

0.8461

2.2131

In fifty observations of the right ascension and declination, thirty two are given, by these elements, to less than half a minute, and the greatest error is 1' 39".

The near agreement of these elements with those of the comet of 1819 made Professor Encke feel confident that it was the same comet. This discovery having been communicated to Dr Olbers, led him to a farther investigation of the subject, and he, with his usual sagacity and promptness, immediately perceived and gave the information to Professor Bode, that it had also appeared in 1786 and in 1795.

During its appearance in 1795, it was about three minutes. in diameter, not well defined, without any tail or nucleus, and was first discovered near the constellation Cygnus. It was found extremely difficult to make the observations at that time correspond with a parabolic orbit, and the results of different astronomers were very discordant; making the time of passing the perihelion December 14th or 15th; longitude of the perihelion 157° to 171°; the node 353° to 361°; inclination 10° to 24°; perihelion distance 0.212 to 0.258. After a new reduction and computation of the observations, Professor

Encke found that the following elliptical elements would represent very well the observations made at that time.

d.

Time of passing the perihelion 1795, Dec. 21.47.

Longitude of the perihelion

Longitude of the ascending node
Inclination of the orbit

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156° 41' 20"

334 39 22

13 42 30

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In 1795 the comet was frequently observed in a part of the heavens where there were no bright stars, favourably situated to compare with it, and, on this account, the observations of different astronomers sometimes varied considerably from each other. However, out of thirty eight observations of the right ascension and declination, twenty three were within one minute, and but very few of the observed places varied more than two minutes from the calculations.

On the 17th and 19th of July 1786, this comet was observed in the constellation Aquarius by Messrs Messier and Mechain. These two observations are not sufficient to determine the elements, three being the least number that can possibly be used. No notice is therefore taken of it in Delambre's table. It appears however from the above elements of Encke's comet, that it must have been near the perihelion about the 31st of January 1786, and the calculations of Professor Encke confirm those of Dr Olbers, proving that the following elements will satisfy these two observations.

d.

Time of passing the perihelion 1786, Jan. 30.88
Longitude of the perihelion

Longitude of the ascending node

156°38'

334 08

Inclination of the orbit

Eccentricity

13 36

0.848

Mean distance

2.208

It may be observed that other elements may be found, which will also satisfy these two observations, but as these elements vary but little from those found for Encke's comet in 1795, 1805 and 1819, there can be no doubt that it was the same.

To compare these results together, we must allow for the precession of the equinoxes during the intervals of the successive appearances. Reducing them all therefore to the mean equinox of 1806, the elements deduced from the observations of the four different appearances of the comet will be respectively,

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These elements agree remarkably well with each other, especially when we take into consideration that the disturbing forces of the planets have been wholly neglected in making the calculations. Three revolutions of the comet have been completed in each of the intervals between 1786, 1795 and 1805; and four revolutions between 1805 and 1819. Hence it follows that the periodical revolution is about one thousand two hundred and five days, which is rather less than that of the newly discovered planets. In the perihelion it passes within the orbit of Mercury, and at the aphelion is about mid-way between the orbits of these small planets and that of Jupiter.

From these elements it would follow that if the comet pass the perihelion between the months of October and February, it would appear, upon its approach towards the earth, so much to the north of the ecliptic, as to render it visible in our northern climates. But, if it should happen at an opposite season of the year, it would not, in general, be visible, except in places south of the equator; unfortunately, this will in some measure be the case at its next appearance in May 1822.

To determine with precision the time and place when the comet will next appear, would require a complete calculation of the disturbing forces of all the planets upon the comet from the year 1795 to 1822. The methods of doing this (though much abridged and simplified from what they were, when Clairaut and his associates, with immense labour, calculated the return of Halley's comet) are yet extremely laborious; and the difficulty in the present instance may sometimes be very much increased, from the circumstance that the comet may pass very near to Mercury, and the attraction of this planet might be so much augmented by its proximity, that its disturbing force, notwithstanding the smallness of its mass, might exceed that of Jupiter, the greatest planet in the system. The calculations made by Professor Encke, to ascertain nearly the effect of the disturbing forces of the planet upon the

comet, are given in the third of the papers mentioned at the beginning of this article. It is unnecessary to go into any particular detail relative to this subject. We shall merely observe, that the result of his inquiries leaves an uncertainty of about one day in the time of passing the perihelion. He has therefore used two sets of elements, in computing the places of the comet. The first set, which he supposes to be the most accurate, is as follows: counting the mean time from the meridian of Seeberg, and the longitudes from the mean equinox of May 24, 1822.

d. h.

Time of passing the perihelion, 1822, May 24, 0. Log. of the mean distance

0.34722

Long. of the perihelion

157° 12' 7"

Long. of the ascending node

334 23 24

Inclination of the orbit to the ecliptic

13 20 36

Eccentricity

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In the second set, the time of passing the perihelion is May 25d. Oh.; the log. of the mean distance 0.34746; the rest of the elements are not altered.

Professor Encke has calculated, in both these hypotheses, the places of the comet, at intervals of four and of two days, from Feb. 25, 1822, to July 27. From this table we have -made the following extracts.

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It appears, from this table, that at the time of sunset at Seeberg, in the latitude of 50° 56′ N. the comet on the 25th Feb. 1822, will be 261° above the horizon of that place; March 13th it will be 20°; March 29th, 143°; and April

19 2

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9.8559.607

26 34

9.966 9.440

46 35

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