The determination of the motions of three bodies mutually attracting according to the law of gravity being a problem too complicated for exact solution, mathematicians have employed various methods of solving it approximately. It is well known that of these methods the one which appears to be the most obvious and direct, introduces terms which may increase indefinitely with the time, and render the solution inapplicable to any observed case of motion. This difficulty occurs whether the problem be to find the perturbation of the moon’s motion by the sun, or the perturbation of the motion of one planet by another, and the necessity of meeting or evading it has very much determined the courses which the solutions of these problems have taken. In the theory of the moon’s motion, Laplace, Pontécoulant, and others, have appealed to the results of observations of the motions of the moon’s perigee and node, to justify the assumption of a form of solution which is not attended with the above-mentioned difficulty. Although this way of proceeding may lead to correct results, there can be no doubt that it is an abandonment of the principle of determining by analysis alone the form of development which is appropriate to the conditions of the problem. Again, in the theory of the motions of the planets, recourse is had on the same account to the method of the variation of parameters, more especially for determining the secular inequalities. Now it will perhaps be admitted that that method, elegant and exact though it be, is yet not indispensable, and that when it succeeds, there must be some direct method which would be equally successful and conduct to the same results. The discovery of such a method I have long considered to be a desideratum in the theory of gravitation, and having after much labour found out one by which the forms of the expressions for the radius-vector, longitude and latitude, and both the secular and the periodic inequalities, are evolved by the analysis alone, and which is applicable as well to the lunar as the planetary motions, I thought it might deserve the attention of the Royal Society. I propose in this communication to enter at length into the details of the method, and then to add a few remarks on its general principle, and to explain why, in common with the method of the variation of parameters, it succeeds in determining the motion of the apses of a disturbed orbit.
XII. Spectroscopic observations of the sun.—No. II
