"The terms A and B include the secular terms, and also those of long period as well as those which acquire large coefficients by the small divisors, which depend upon the near approach to commensurability in the mean motions of Neptune and Uranus. These coefficients will vary very sensibly by a change in the value of the mean motion of Neptune arising from a more accurate determination of its orbit. But the principal effect of these terms can for a limited period, such as a century, for instance, be included in the ordinary forms of elliptic motion, and the residual portion will assume a secular form, which is no more liable to change, from a new correction of the mean motion of Neptune, than the other small coefficients of the equations of perturbation. The elliptic portions of A and B may therefore be neglected until longer observation has given a more precise value of Neptune's mean motion, and the residual portion is contained in the following equation.

"The following particular values of 8v and 8r, derived from the preceding formulæ, will be useful in computing the orbit of Neptune from observation.

Professor Peirce also communicated the following elements of the orbit of the satellite of Neptune, computed from the combination of all of Lassell's and Mr. Bond's observations; and he also communicated the corresponding mass of the primary.

"Time of sidereal revolution, 5 days 21 hours 12.4 minutes. "Inclination to ecliptic, 29°.9.

66

Longitude of ascending node (the motion being supposed direct), 119°.8.

"Time of greatest northern elongation, November 26.53, Greenwich mean solar time.

"Greatest elongation, 16′′.5.

"Distance of satellite from Neptune, 230,000 miles.

"Corresponding mass of Neptune, 1880, the mass of sun being

unity.

"The time of sidereal revolution is not liable to an error of more than a few minutes, and the greatest elongation cannot be less than 16.3, or more than 17".0. The mass of Neptune, therefore, cannot be less than 100, or greater than 1700."

Three Hundred and second Meeting.

January 4, 1848.-MONTHLY MEETING.

The PRESIDENT in the chair.

Mr. Everett read a letter from M. Leverrier, acknowledging his election as a Corresponding Member of the Academy.

Mr. Everett also submitted to the Academy a paper received from M. Leverrier, containing a succinct abstract of the first of two memoirs lately read by him to the Academy of Sciences at Paris, on the subject of periodical comets. It was the intention of M. Leverrier that this communication should reach the Academy in advance of the publication of the Compte Rendu for the 25th of October, in which the abstract of the first memoir appears in extenso. Owing to the great length of the passage of the vessel by which M. Leverrier's communication was transmitted, this expectation was disappointed. As the Compte Rendu, however, of course possesses but a limited circulation in this country, a translation of this interesting paper was read by Mr. Everett to the Academy.

After alluding to the stability of the orbits of the planets, caused by their moderate eccentricity, small inclination, and the great preponderance of the central force, M. Leverrier observes, that

"It is not so with respect to the comets. Those of them, which move in planes but little inclined to the ecliptic, cut very near the orbits of one or more of the planets. It may accordingly happen, that they will pass in the neighbourhood of the planets themselves, and that the disturbing force, thus rendered preponderant, may turn them from their course. Thus the comet, which, left to itself, would have continued to move in a parabola, may by the action of Jupiter be brought for ever, or only for a limited period, into an ellipse of moderate extent. The same cause which shall have compelled the comet to describe this ellipse may be able hereafter again to control its movement, and to force it for ever from our planetary system, by throwing it into a hyperbolic curve."

M. Leverrier then adverts to the discovery of a comet by Messier in 1770, which was afterwards known as Lexell's, in

consequence of its being discovered by that astronomer to move in an elliptical orbit of five years and a half period. To the objection made against this theory, that it had not before been seen, Lexell replied, that it might be a new comet, drawn into an elliptical orbit by the action of Jupiter, and that it would approach that planet again in 1779, which might then, perhaps, throw it off from our system, to return no In point of fact, astronomers have looked in vain for

more.

the return of Lexell's comet !

curve.

In the month of November, 1843, M. Faye saw a comet, whose observed movement could not be reduced to a parabolic Dr. Goldschmidt discovered that it described an ellipse of a period of seven years and a half. The objection to this theory, that it ought to have been seen before, was answered, as in the case of Lexell's comet, by reference to the action of Jupiter.

As the region of the heavens where this approach to Jupiter took place was nearly the same for both comets, M. Leverrier was led to admit the possibility, that the comets of 1770 and 1843 might be the same, although their orbits were altogether different.

In 1844, M. de Vico, at Rome, discovered a comet, which was shown by M. Faye to move in an orbit of five years and a half. The possibility that this was Lexell's comet of course conflicted with M. Leverrier's first impressions, just mentioned, but increased the probability that Lexell's comet might be recovered in one or the other of the recent discoveries.

"The only difficulty," says M. Leverrier, "was, that the calculations became immensely laborious, and I was obliged to devote to them several years, including the last (1846). Although my researches are brought to a close, however great my desire to submit them to the Academy, the necessity of passing some days in collecting the documents relative to the comet of De Vico will oblige me to confine myself at present to that of M. Faye."

The elements of the comet of 1770, being different from

those of the comets of 1843 and 1844, M. Leverrier first undertook to follow the former into the neighbourhood of Jupiter and the other regions which it would have traversed up to the years 1843 and 1844, and to ascertain, in this way, if the comet of 1770 might not place itself upon the orbits of one or the other of those discovered by M. Faye or M. Vico.

On approaching the subject more nearly, M. Leverrier found that the calculations of Laplace, in the Mécanique Céleste, as to the direction given by Jupiter to the comet of 1770, could not be depended on. Slight changes in the elements of the orbit give routes so different to the aphelion, that it remains uncertain whether it passed within or beyond the orbit of Jupiter, through the system of the satellites or outside of them. M. Leverrier was accordingly obliged to commence by studying the movement of the comet of 1770, leaving to it all the latitude which resulted from the observations made at the time. In pursuing this course,

"I established," says M. Leverrier," the following points:

"1. That it was impossible that the comet should have been arrested within the system of Jupiter, without falling into the planet itself; an event very improbable, it is true, without being absolutely inadmissible. "2. I showed that Jupiter might have forced the comet to pass off in an hyperbola round the sun. In this case, we could not expect to see it again, as it would continually move on to a greater distance from our system, to enter into other spheres of attraction.

"3. It is possible that the comet, after having escaped the action of Jupiter, might have pursued its course in ellipses of very long period. But it is much more probable that it continued to move in ellipses whose moderate period must permit us often to witness its return. I have formed a complete table of all the possible ellipses, which will serve henceforth as the basis of our further inquiries."

The first inquiry will, then, be, whether the elements of the new comet (that of Faye), as calculated from the observations, present themselves among the systems of this table. If so, the problem is solved.

Should this not be the case, it will be necessary to inquire