loo Pi'of. Cballis on certain Questions 



sophical Magazine, I arrived at an equation which will be very 

 frequently referred to in the course of this discussion, viz. 



5? + ;5---2^3+C-0 (C) 



I saw at once that this equation, which, as I believe, had not 

 been previously noticed by any writer on the lunar theory, might 

 be employed in obtaining directly the true form of an approxi- 

 mate value of the radius-vector. The process for this purpose 

 will be considered in the course of the discussion. 



I have a few remarks to make on statements contained in 

 page 28 of Mr. Adams's letter, which refer to views expressed in 

 the introductory part of my paper. My objections to the logic 

 of the process usually followed in the lunar theory are there 

 stated in these words : — "As it is certain that the mean motions 

 of the apse and node are consequences of the sun's disturbing 

 force, there must be some direct means of deducing them from 

 the dynamical equations of the motion. In fact, the method of 

 calculating the motion of the moon's node in the third book of 

 Newton's Principia shows, step by step, that the motion results 

 from the dynamical conditions of the problem, and in this respect 

 is logically more exact than the analytical method, which only 

 shows, on the hypothesis of such motion as Newton deduced, 

 that the dynamical equations are satisfied. There must be some 

 hiatus in the analytical method which requires to be filled up in 

 order that the logic of the lunar theory may be free from 

 reproach." My paper supplied this defect in the case of the 

 mean motion of the node as well as in that of the apse, and by 

 processes analogous to each other, as I shall take occasion to 

 show hereafter. 



Again, with reference to the principle of introducing the quan- 

 tities c and ff, I said that they were introduced hypothetically j 

 the apse and node being thereby supposed to have certain mean 

 motions; and that these hypotheses, suggested probably by 

 observation, are the real basis of the approximation in the usual 

 treatment of the lunar theory. The symbolical ^oXwiiow obtained 

 on these suppositions is, as Mr. Adams urges, proved to be cor- 

 rect by its satisfying the differential equations of motion ; but 

 its applicability to a particular instance (as that of the moon's 

 motion) is not proved till the hypotheses on which the solution 

 was based are shown, by direct or indirect comparison with 

 obseiTation, to hold good in that instance. This is the confir- 

 mation by observation that I spoke of. The solution which I 

 have proposed, being based on the single supposition that there is 

 a mean motion in longitude, requires :only that we establish by 

 observation that the moon's motion satisfies this condition. I 



