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HISTORY OF PHYSICAL ASTRONOMY.

by a small quantity termed the perturbing function. lie then demon-strated, by a process similar to that -which he employed in the less gene-ral case of a planet revolving round the sun, that the expression for theinstantaneous variation of each constant contained only terms involvingpartial differentials of the perturbing function relative to the constants,each differential being multiplied by a function only of the constants.

In this memoir Lagrange remarked that the same principles wereapplicable to the determination of the motions of the planets aroundtheir centres of gravity, taking into account the action of the disturbingplanets upon the equatorial matter of the disturbed planet. Havingexhibited a statement of the general conditions of the problem and pointedout its connexion with the theory of the variation of elements, he con-cluded by announcing his intention of making it the subject of a futurememoir. At no subsequent period of his life did he carry into effect thisresolution; but, indeed, Poisson a few months afterwards rendered such astep unnecessary, by communicating to the Institute an excellent memoiron the subject. Euler first gave the differential equations relative to themotion of a body about its centre of gravity when it has received an initialimpulse and is not afterwards exposed to the action of any forces. Theseequations being three in number, and of the second order, their completeintegrals will contain six arbitrary constants. Their integration wouldtherefore lead to the equations of a planet’s motion about its centre ofgravity if it were not continually disturbed by the action of the sun andthe other planets of the system. By considering the latter, however, asso many perturbing bodies which tend to produce a continual change inthe elements of rotation, the variations of these elements may be investi-gated by a process similar to that employed in determining the variationsof the elements of the orbit. The formul® for these variations being thenintegrated and substituted in the primitive equations depending solely onthe initial impulse will conduct to three complete equations, by means ofwhich the positions of the planet’s axis and the velocity of rotation cor-responding to any assignable time may be readily ascertained. By ajudicious selection of constants, Poisson obtained formulas for the variationsof the elements of rotation exactly analogous to those which Lagrange andLaplace had already arrived at relative to the orbit of the planet. Thisremarkable result constituted the crowning triumph of the theory of thevariation of elements, for by means of it the two great problems of thesystem of the world, although differing essentially in conception, were mostunexpectedly exhibited in a relation of close affinity, and the geometerwas thus enabled to contemplate all the effects of planetary perturbationthrough the medium of the same analysis*.

* Memoire sur la variation des constanles arbitraires dans les questions de Mecanique,Journal de I'Ecole Polytechnique, tome viii. p. 266, et seq. In this memoir Poisson in-vestigates the expressions for the variations of the arbitrary constants in the generalproblem of Mechanics by a more direct analysis than that which Lagrange had employedin his original demonstration of the same results. It is right, however, to state that thelast-mentioned geometer was conducted about the same time to a similar improvement ofliis own method, (Memoires de 1’lnstitut, 1809, p. 343, et seq). A complete account ofthe theory of the variation of arbitrary constants by means of Poisson 's analysis will befound in Pontecoulant’s Thcorie Analytique du Systeme du Monde , tome i. liv. ii.chap. iii. See also on this subject the seventeenth chapter of Mr. De Morgan 'sTreatise on the Differential and Integral Calculus, Library of Useful Knowledge.This chapter contains a concise, but singularly able exposition of the most comprehensivetheories of mechanical science. The reader is conducted by means of them to the verythreshold of the great problems of the system of the world.