620 Prof. Challis on the Tlieory of the Mom^s Motion. 



completely establish the six laws laid down by me ; but I could 

 have wished, before communicating them to your valuable 

 Journal, to have had an opportunity of repeating them with dif- 

 ferent substances, particularly with double refracting crystals ; 

 but the short, rainy days of November, and the return of the col- 

 lege duties of Michaelmas term, compel me to postpone further 

 experiments to a i)eriod of brighter sunshine and greater leisure. 

 In the meantime, I think the facts I have already obtained will 

 prove of some interest to such of your readers as are engaged in 

 optical researches. ^i 



Trinity College, Dublin, . 

 November 4, 1854. 



LXII. The Theory of the Moon's Motion. — Second Approximation. 

 By Professor Challis*. 



MR. ADAMS has made no reply to the arguments in the 

 Philosophical Magazine for August, by which I met his 

 objections to my new solution of the problem of the moon's 

 motion. As those objections are all completely answered, I 

 consider myself entitled to assert that the judgement passed 

 on the paper which contained the first approximation of the 

 solution remains unsupported. I have distinctly proved, that, 

 in forming his opinion of it, Mr. Adams relied on false rea- 

 soning. I feel, therefore, no hesitation in extending the 

 method to higher approximations, being well persuaded that 

 such extension will more fully demonstrate its logical accuracy 

 and the soundness of the deductions drawn from it. The object 

 of the present communication is to obtain expressions for the 

 radius-vec1;or and true longitude of the moon to the second ap- 

 proximation, after recapitulating, for the sake of clearness, the 

 reasoning of the first approximation. [1 



It will be proper to begin with stating the limitations of the 

 problem. Three bodies being supposed to attract each other 

 according to the law of gravity, and the velocity and direction 

 of the velocity of one of them at a given point of space being 

 given at a given instant, it might be proposed to determine by 

 successive approximations its subsequent motion on the suppo- 

 sition that the motion is principally due to the action of one of 

 the other bodies. This, however, is not the problem relating to 

 the moon's orbit which I propose to solve. I make the addi- 

 tional limitation, suggested by observation, that the moon's 

 motion is such that it always differs to a small amount from 

 uniform motion in a circular orbit of given radius. The problem 



* Communicated by the Author. 



