HISTORY OF PHYSICAL ASTRONOMY.

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effective, and has formed the basis of all subsequent researches relative tothe same object. It is founded on the theory of the variation ofarbitrary constants ; but in integrating the differential expressions of thecomet’s elements the usual mode of procedure is not adhered to. Inthe lunar and planetary theories the integrations are so effected, that theyconduct to formulas by means of which the place of the disturbed body maybe readily calculated for any assignable instant whether past or future. Asimilar course not being practicable in the theory of comets, on account ofthe peculiarities to which we have already referred, Lagrange proposedto substitute instead of it what has been termed the method of quadratures.This consists in dividing the orbit of the comet into a number of distinctarcs, and then summing up the effect of perturbation for each arc. By thisprocess a fresh set of elements is obtained at the end of each summation,and these form the basis of computation for the following arc. The appli-cation of this process to the whole of the orbit would demand an enormousamount of calculation, but Lagrange shewed that such a course was not ne-cessary. When the comet is near the perihelion, the method of quadraturesis indispensable; but, when it is traversing the superior part of its orbit,the ordinary method of integration may be practised on account of the greatdistance of the comet compared with the distance of the disturbing body.

In 1770 a comet appeared, tlie circumstances connected with which led tovery interesting results. The observations on it seemed inexplicable byany parabolic orbit that could he devised, until Lexell finally shewed thatthey might be all reconciled with the supposition of the comet revolvingin an ellipse in a period of nearly six years. The opinion of Lexell wasconfirmed in the most satisfactory manner by Burchardt, who for this pur-pose undertook a careful discussion of all the observations. Strange tosay, the comet never afterwards appeared, notwithstanding the shortness ofits period. In order to account for this fact, Lexell remarked that it hadbeen always invisible until the year 1770, but that in 1767 it passed sonear Jupiter that it was thrown by the powerful disturbance of that planetinto a new orbit and thereby rendered visible; and that in 1779 it againpassed so near the planet as to he thrown into another orbit and renderedinvisible. Laplace undertook an analytical investigation of this interestingquestion, and he found that the disturbing action of Jupiter would becapable of producing the singular effects ascribed to it by Lexell *. Nodoubt, therefore, can exist on the cause either of the appearance or thesubsequent disappearance of the comet.

As this comet approached very near the earth, it offered a favourableexample for ascertaining to what extent comets in general affect the otherbodies of the system. The result of its action on the earth would he todiminish the force of that body towards the sun, and thereby to lengthenthe sidereal year. No such change has been detected by astronomers,whence we may conclude that the masses of comets are very small.Laplace found that, if the sidereal year had increased I s since the year1770, the mass of the comet would not have exceeded -^o’o^th part of theearth’s mass. It is evident, however, that the mass is much less thanthis, for, although the comet passed through the system of Jupiter ’s satel-lites both in 1707 and in 1779, it did not produce the slightest perceptiblederangements in the motions of any of those bodies.

Laplace, after having investigated with the most brilliant success everysubject connected with the system of the world, finally conceived the design* Mec. Cel, liv. ix. chap. if.