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HISTOKY OF PHYSICAL ASTBONOMY.

tlie moon would ensue, in virtue of which the elongated axis would oscil-late perpetually on each side of its mean place.

D’Alembert w-as the first of Newton’s successors who undertook to in-vestigate the subject of the moon’s physical libration. In 1754 thisgreat geometer, encouraged by his researches on the Precession of theEquinoxes, applied the same method of investigation to the problem ofthe moon’s motion about her centre of gravity. He did not, however,pay sufficient attention to the modification rendered necessary by theslow rotatory motion of the moon, and the near commensurability ofthe motions of revolution and rotation. For these reasons the resultsobtained by him did not accord so well with observation as those to whichhe was conducted by his previous researches of a similar kind relative tothe motion of the earth. The Academy of Sciences of Paris havingoffered their prize of 1764 for a complete theory of the moon’s libration,Lagrange composed an admirable memoir on the subject, which obtainedfor him the prize. It was in this investigation that he first employed theprinciple of virtual velocities in combination with the dynamical principlerecently discovered by D’Alembert . By this step he reduced every ques-tion relating to the motion of a system of bodies to the integration of aseries of differential equations of the second order, whence the only dif-ficulties that remained to be overcome were those of a purely analyticalnature. This refined conception forms the basis of his celebrated work ,the Mecanique Analytique , which he published at a subsequent period of hislife, and in which all the great problems of mechanical science are inves-tigated by a process divested of every trace of geometrical reasoning.

Lagrange, in the memoir above mentioned, proceeded first to considerthe figure which the moon would assume in consequence of the variousforces exerted upon the particles composing her mass, which he supposed,with Newton, to have been originally in a fluid state. It does not appearto have occurred to the latter, that the centrifugal force generated by therotatory motion of the moon would affect her figure to an extent com-parable with the effect occasioned by the action of the earth. Lagrange,however, found that both effects were of the same order, and that the moonwould in reality acquire the form of an ellipsoid, the greatest axis beingdirected towards the earth, and the least being perpendicular to the planeof the equator. The greatest axis, and the mean axis, will both lie in thelast-mentioned plane; the mean position of which, as we have already stated,is parallel to the plane of the ecliptic. Lagrange also found that the excessof the axis turned towards the earth over the least axis was four timesgreater than the excess of the axis at right angles to it over the same axis.

Considering next the effect of the earth’s attraction upon the rotatorymotion of the moon, Lagrange found that the mean motion would beaffected by a series of inequalities corresponding to those of the meanmotion in longitude. The velocity of rotation is on this account some-times accelerated beyond its mean state, and at other times retarded behindit, whence there ensues a real libration similar to that remarked by New-ton. Lagrange shewed that it was not necessary to suppose that at theorigin the motions of rotation and revolution were exactly equal. If theydiffered by an arbitrary quantity confined within certain narrow limits,the effect of this difference would be merely to occasion a slight inequality,in the motion of rotation, in virtue of which the axis directed towards theearth would librate continually on each side of the line joining the earthand moon. The most careful observations of the moon’s surface have not