formation of lunar tables, founded solely upon the theory of universal gravitation, a number of attempts have been made, which, though not altogether successful, can scarcely fail to be of service in ultimately bringing to perfection a subject of so great importance to navigation. In the Annales de Chimie, (XIII. 250.) M. de Laplace has shewn, with great clearness and precision, the advantages which the lunar theory may derive from the concurrent labours of astronomers, as well as the points in which it is incomplete, and to which their labours should be directed. By the labours of geometers, the lunar theory had already made such advancement, that, in the seventh book of the Mécanique Celeste, the greatest difference between the coefficients of the inequalities of the analysis there given, and those of the tables of M. Burg, was reduced to 8.5". Hence it was natural to conclude, that, by means of approximations carried still farther, the theory would represent observations within the limits of the errors of which they are susceptible. The two papers to which the Academy adjudged a reward in 1820, fulfil this condition, and are the result of immense labour; leaving no doubt, that, on a comparison with our present lunar tables, the formule they contain, when reduced to tables, will agree with observation within the limits already indicated. This is directly established by the author of the first paper, M. Damoiseau, who, according to his theory, has formed new tables, which, compared with sixty observations of Bradley, and sixty observations made since the year 1802, only produce slight errors of the same order with those of the tables of Burg and Burckhardt. We may therefore hope, that, by the examination of a great number of observations, the author will improve still farther the arbitrary elements of the theory, and at

length give to his tables all the accuracy which can be desired.

The authors of both these Memoirs have set out from differential equations of the celebrated problem of the three bodies, in which the differential of the true movement of the moon, referred to the ecliptic, is supposed constant; and they have determined the mean longitude, the latitude, and parallax of that body, in series of sines and co-sines of the angles, increasing proportionally to its true movement. This is the method employed by Laplace, in the seventh book of the Mecanique Celeste already referred to, and appears to give the most converging approximations. Indeed, the disturbing forces present themselves under that form, or are easily reducible to it. To reduce them to another form,-for example, that of the series of sines and co-sines of the angles, increasing proportionally to the time,-the approximations would require to be carried very far, by reason of the considerable inequalities of the lunar orbit; which would render the analysis more complicated, and the approximations less convergent. Other forms of series have been tried, and it would be easy to imagine a great number; but none appears better calcu lated to give the coefficients of the lunar inequalities. Nevertheless, some very small inequalities, of which the argument increases with great slowness, may be better determined by other methods. In the preceding, these inequalities, in virtue of repeated integrations, acquire, as divisors, the squares of the very small coefficients of the true longitude of their arguments. In the final result, these square divisors disappear, and are reduced to the first power; so that this result, being the difference of quantities very great in relation to itself, becomes inexact, unless we are careful to preserve, in the course of the computation, all the

quantities of its order. By neglecting this circumstance, several geometers have failed in determining the inequality depending on the longitude of the node of the lunar orbit. Uniformity of method certainly gives elegance to analysis; but when it is proposed to approximate, as nearly as possible, anaÎysis to observation, the methods employed must be varied according to the nature of the inequalities; for it is in the selection of these methods, and in foreseeing the quantities that may become sensible by successive integrations, that the art of approximation consists, an art no less useful to the progress of science, than the discovery of analytical methods.

Laplace having discovered, by theory, the cause of the inequalities in the secular motion of the moon, the two papers above referred to have verified and confirmed the results to which that eminent philosopher was conducted by his profound analysis, particularly that relative to the motion of the perigee in proportion to its magnitude. The form

of the analytical expressions of the first, being the same which he had adopted in the seventh book of the Mécanique Celeste already referred to, he was enabled to compare these expressions with his own; and he found, that they agreed in the degrees of approximation which are common to both, but that the authors of the papers having carried these approximations farther, the new terms introduced by them have produced differences, inconsiderable, indeed, in regard to the secular equations of the mean motion, and of the perigee, but sensible in relation to the motion of the nodes. The following table exhibits the numerical coeffi cients, by which, in order to find the secular equations, we must multiply the integral of the product of the differential of the time by the excess of the square of the eccentricity of the terrestrial orbit above the same square at any arbitrary epoch of time, which, in this case, was fixed at the commencement of 1801 :

The authors of the Second Memoir, MM. Plana and Carlini, in the expression of the secular inequality of the mean motion, have not attended to the terms depending on the square of the eccentricity of the lunar orbit; and which, rendered sensible by the small divisors which they acquire in the course of the integrations, produce the difference of results observable in the two communications. Laplace thinks that the difference, in regard to the secular inequality of the perigee, proceeds from the nature of the approximations employed, by the authors' having reduced their expressions to series, disposed according to the ascending powers of the relation of the motion

2d Memoir. 0.00760102. -0.0311110 0.0053877

of the sun to that of the moon, a relation less than a twelfth. MM. Plana and Carlini find, in the mean lunar motion, a secular inequality equal to the product of -0.1398", by the cube of the number of periods elapsed since 1801. This inequality, which would increase the longitude of the moon at the moment of its eclipses, in the years 719 and 720 before our era, about 37, depends, according to them, on supposing the true ecliptic transposed to a fixed ecliptic, for example, that of 1801; but they have not attended to the secular transposition of the lunar orbit to the same ecliptic, which would have destroyed the result at which they have arrived. Laplace has shewn,

that the part of the secular equation relative to the inclinations, depends only on the inclination of the lunar orbit to the true ecliptic, and that the rapidity of the motion of the nodes of the moon, renders insensible the secular variation of that inclination.

M. Damoiseau having, at the special request of M. de Laplace, carefully re-examined his analytical and numerical calculations, upon the subject of the lunar inequality called parallactic because it depends on the parallax of the sun, found that, supposing this parallax a four hundredth part of that of the moon, the inequality in question would be 121.15". Proceeding on the same hypothesis, Laplace himself found it 122.01", and MM. Plana and Carlini 122.90". According to the tables of Burg, it is 122.378"; according to those of Burckhardt, 122.97"; which gives respectively 8.6303", and 8.6721" for the mean parallax of the sun, upon the parallel, whose terrestrial radius is that of a sphere of the same mass as the earth, and of a density equal to its mean density. The mean 8.65" appears to be the probable value of the solar parallax.

The small inequalities which astronomers have imagined they have detected in the mean motion of the moon, is the only point of the lunar theory which still remains to be explained. Future observations, in proving its reality, will determine its value. Fortunately, in the interval of half a century, this inequality may be safely confounded with the mean motion; for as long as it shall remain unknown, it will be sufficient for the purposes of navigation to rectify, from half century to half century, the mean lunar motion. But when its existence shall be fully established, the investigation of its cause will then become an object of importance in Physical As

tronomy.

The remarkable comet which ap. peared in July 1819, gave occasion to some important astronomical investigations, relative to the orbits described by these eccentric bodies. Signor Nicolas Cacciatore, Director of the Royal Observatory at Palermo, made his observations, which comprise the period between the 3d of July and the 11th of August, with an entire circle of Ramsden. The parabolic elements which result from these observations, differ but little from those obtained by M. Bouvard of Paris, of which an account will be found in the Journal de Physique, xc. 11. The same astronomer states, that he observed phases in the nucleus of the comet, which led him to conclude, that comets are not of themselves luminous, and that their nucleus, coma, and tail, shine by reflected light; but, from the remarks of M. Arago, in the Annales de Chinie, XIV. 217, it is evident that the appearances which misled Signor Cacciatore, can only be regarded as irregularities; that, in the course of even a few days, comets undergo a sensible change of form; but that these changes and irregularities have yet furnished no data for enabling astronomers to determine the nature of the light, whether inherent or reflected, which comets emit. M. Pictet adds, that no explanation can be given of the phases observed by the astronomer of Palermo, without supposing the comet to revolve round its axis, and to possess a surface of opposite powers, one part reflecting, and the other absorbing light.

Dr Brinkley, of Trinity College, Dublin, has published, in the Journal of the Royal Institution, his observations on this comet, and the elements of its orbit; the instruments he employed were an astronomical circle, eight feet in diameter, and a transit instrument. His computation was founded on observations made on the

4th, 5th, and 6th of July, and the elements so obtained were further corrected by observations made on the 13th

and 20th of the same month. The result was as follows:

Longitude of node Inclination

Place of perihelion

In correcting the first approximations, Dr Binkley employed a method different, it is believed, from any that had been formerly used. Instead of changing the approximate perihelion distance, and the approximate time of passage through the perihelion, by small quantities, as in M. de Laplace's method, he obtained two equations, in which the unknown quantities were the corrections of the perihelion distance, and of the time of passage through the perihelion. This was done by investigating the fluxions of the anomalies, heliocentric longitudes, and latitudes, computed by help of the approximate perihelion distance, and approximate time of perihelion, and of three observations. The operations by this method, which, at first sight, might be supposed to lead to intricate formula, were found considerably shorter than by Laplace's method, when great exactness is required; and it has the additional advantage of being particularly applicable in cases where it is necessary to investigate the elliptic orbit.

The comet of 1819 performs its eccentric revolution in the space of about three years, and consequently would appear again in 1822. The celebrated Dr Ölbers of Bremen has given

some details as to the nature of the path it will describe till that event take place; and Professor Enke, of Berlin, having considered the effects of Saturn, Jupiter, Mars, the Earth, Venus, and Mercury, on this erratic body, throughout the whole interval from 1786 till 1819, has found that the attraction of Jupiter alone will have any material effect on the time of the next perihelion, which, as the distance from Jupiter will only be 1.136, will be retarded nine days from this cause.

This return of comets at periods which, by the great improvement of astronomical science, and the accurate methods of a refined calculus, can be predicted beforehand, has led the author of an article which appeared in a London periodical, remarkable, certainly, for any quality rather than profound science, to endeavour to prove that the phoenix of the ancient Egyptians-the symbol, as some had ima gined, of a particular celestial revolution, or, according to others, of that principle of incessant decay, and reproduction, which guarantees the permanence and indestructibility of Nature, even in her changes-was nothing more or less than a hieroglyphical painting of the celebrated comet of 1680.* It is astonishing how Dr