HISTORY OF PHYSICAL ASTRONOMY.

113

The complete establishment of this sublime theory was the last greateffort of Lagrange’s genius. We have already remarked that Euler ’sresearches contain the earliest traces of the use of the variation ofelements in computing the perturbations of the planetary motions. Al-though that geometer, however, has the undoubted merit of first employingthis fertile method of investigation, the magnificent expansion which itsubsequently received, and to which it owes all its practical value, is whollydue to the illustrious mathematician above mentioned*. In PhysicalAstronomy, it is indeed very naturally suggested by observation; for theelements of the planetary orbits had been already found by astronomersto be in a state of slow variation, although their elliptic forms did notappear to be undergoing any change. It is in questions of this kind, wherethe perturbing forces are small compared with the principal forces whichanimate the system, that the variation of elements can be most advan-tageously employed. Lagrange resolved to embody his final researches onthis subject in a new edition of the Mecanique Analytique; and with thisview he undertook a complete revisal of that immortal work; but thefatigue he incurred in consequence brought on a fever which carried himoff before the termination of his labours, on the 10th April, 1813.

Lagrange deserves to be ranked among the greatest mathematicalgeniuses of ancient or_ modern times. In this respect he is worthy of aplace with Archimedes or Newton, although he was far from possessing thesagacity in physical enquiries which distinguished these illustrious sages.From the very outset of his career he assumed a commanding positionamong the mathematicians of the age, and during the course of nearly halfa century previous to his death, he continued to divide with Laplace thehomage due to pre-eminence in the empire of the exact sciences. Hisgreat rival survived him fourteen years, during which he reigned aloneas the prince of mathematicians and theoretical astronomers.

In recent times the general theory of planetary perturbation has derivedmuch improvement from the profound researches of Poisson , Plana, Ivory,Lubbock , and other geometers. The most remarkable innovation is due,however, to M. Hansen. That distinguished geometer conceives theelliptic elements of the planet to be invariable, and assumes the time aloneto be subject to perturbation f. The method of investigation founded on thisrefined notion possesses some important advantages over the ordinarymethods employed for the same purpose. M. Hansen has applied it withsuccess to the theory of Jupiter and Saturn , and also in researches on thelunar perturbations.

The derangements in the elliptic motion of a planet, occasioned by theaction of another planet on it, depend upon an algebraic expression, in-volving the mutual distance of the two planets, and termed the perturbing

* Euler explained the method of the variation of elements in his memoir on theperturbations of the planets, which he transmitted to the Academy of Sciences in 1756;but his investigation was imperfect, because he omitted to take into account the variationof the epoch. The same defect characterized the researches of Lagrange, which appearedin the volume of the Turin Academy for 1763. It was not until he published his famousmemoirs on the planetary perturbations in the volumes of the Berlin Academy for1782-3-4, that the last-mentioned geometer gave a complete theory of the subject byinvestigating the expressions for the variations of the six elements of the undisturbedorbit.

+ Pontecoulant has expounded this method in the Connaissance des Temps for 1837.

I