[Report on] Researches in physical astronomy

The method of the variation of parameters as applied to the investigation of the perturbations of the solar system has been successively developed in modern times. This method gives the variations of the elements of the elliptical orbit in terms of the differentials of a certain functionR of these elements, and of the disturbing forces. Euler, Lagrange (1783), Lagrange and Laplace (1808) obtained the formulae fordα, de, dϖ, dp, dq wherep = tan φ sin θ,q = tan φ sin θ. Poisson first gave the expression fordε . Pontecoulant, p. 330, has introduceddt anddv instead ofdp anddq ; but those developments gave expressions neglecting the square of the disturbing force. Mr. Lubbock has published (in a Paper in the Phil. Trans. April 1830,) expressions which include the effect of any power of the disturbing force. This method has been principally applied to the secular inequalities; but it is susceptible of being applied with no less strictness to periodical inequalities, all of which may be represented by certain changes in the elements of the elliptical orbit. But the same problems may also be approximately solved directly; for we obtain a differential equation involving the radius vector and the time. In this equation there occurs the same functionR of which we have already spoken; and this function is expanded according to terms involving cosines of the mean motions of the disturbing and disturbed planet, and cosines of the difference of certain multiples of these motions. This expression has been treated of by various authors, and among others Mr. Lubbock has himself (in memoirs read May 19 and June 9, 1831,) given the expansion ofR in a form suited to his present object.