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XXV. On the Problem of Three Bodies. By the Rev. J. Challis, M.A.^ F.R.S., 
F.R.A.S., Plumian Professor of Astronomy and Experimental Philosophy in the 
University of Cambridge. 
Received May 15, — Read May 22, 1856. 
The determination of the motions of three bodies mutually attracting according 
to the law of gravity being a problem too complicated for exact solution, mathe- 
maticians 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 pro- 
blems have taken. In the theory of the moon’s motion, Laplace, Pontecoulant, 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 prin- 
ciple 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. 
3 z 
MDCCCLVI. 
