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HISTORY OF PHYSICAL ASTRONOMY.

bodies, he computed the derangements caused by the planets Jupiter andSaturn throughout three successive revolutions. In the immense calcula-tions to which these researches gave rise, he received efficient aid fromLalande, who was then just entering upon his long career. He finally ar-rived at the conclusion that the comet would be retarded 100 days by theaction of Saturn, and 518 days bv the action of Jupiter . Hence the wholeperiod of revolution would extend to 70 years, 211 days, and as the comethad passed its perihelion on the 14th of September, 1082, it ought to returnto it again on the 13th of April, 1759. Clairaut announced the result ofhis labours to the Academy of Sciences on the 14th of November. 1758.He took the precaution, however, of stating that the omission of manysmall quantities, which was rendered necessary by the method of approxi-mation he employed, might cause the actual time of the comet’s arrival inperihelion to differ as much as a month from the computed time. He alsoadded “that a body which passes into regions so remote, and which ishidden from our view during such long periods, might be exposed to theinfluences of forces totally unknown, such as the action of other comets, oreven, of some planet too far removed from the sun to he ever perceived.”

All the astronomers of Europe were now looking forward with anxiousexpectation to an event which was destined to exercise so important aninfluence on the fate of the Newtonian theory. At length the comet wasseen for the first time on the 25th of December, 1758, by George Palitsch,a native of Saxony , and an amateur astronomer. It reached its perihelionon the 13th of March, just one month earlier than the time fixed by Clair-aut , but still within the limit assigned by that illustrious geometer.

Having revised his labours, Clairaut reduced the error in the computedtime of the comet to 22 days, and at a subsequent period he reduced it to19 days. He would have approached still nearer the truth if he had beenin possession of more accurate values of the masses of Jupiter and Saturn .

The return of this comet so near the predicted time was one of themost brilliant triumphs which the Newtonian theory had yet achieved. Itestablished beyond all doubt that comets, while mainly controlled by thepreponderating influence of the sun, were also liable to be deranged intheir motions by the other bodies of the system, and the theory of their per-turbations henceforth formed a subject of deep importance to geometers.Clairaut employed in his researches the method of approximation he hadpreviously devised for the lunar theory, hut he introduced into it severalingenious modifications, in order to adapt it to the peculiar difficulties ofcometary perturbation. It will readily be perceived that the same facilitiesdo not exist in this case for calculating the influence of a disturbing bodyas when the question relates to a planet. In the latter case the eccen-tricity and inclination of the disturbed body being small fractions, it is al-ways possible to expand the perturbing function into a rapidly convergingseries, arranged according to ascending powers of these quantities; but, asthe orbits of comets are very eccentric, and inclined at all angles to theelliptic, the series into which the perturbing function is developed con-verges with such slowness as to render the usual method of integration quiteimpracticable. Little progress was made in these researches until the year1780, when the Academy of Sciences of Paris having proposed to geometersthe theory of cometary perturbation, Lagrange composed an admirablememoir on the subject, w'hich obtained for him the prize of the Academy .The method which that illustrious geometer invented for the purpose ofobviating the peculiar difficulties of the problem is at once ingenious and