relating to the Moon's Orbit, 99 



that the publication of it by an order of the Council of that 

 body would bring credit neither to myself nor to the Society. 

 If that opinion of the paper can be maintained, I shall admit 

 that Mr. Adams had good reason for advising the Council not 

 to print it. But I am well persuaded that the decision was rash_, 

 and made on mistaken grounds, and that my paper contained 

 important additions to the lunar theory. I am therefore glad, 

 as well for the sake of the interests of science as on my own 

 account, that Mr. Adams has consented to publish his objections 

 in the Philosophical Magazine. The article in the July Number 

 gives me the opportunity I desired of vindicating my views. 

 I shall discuss the objections seriatim, and in the order in which 

 they occur, after premising some general remarks on the ques- 

 tions at issue. 



It is well known that the theoretical determination of the 

 motion of the moon^s apse has been attended with difficulties. 

 Newton gave, in the ninth section of the First Book of the Prin- 

 cipia, methods of determining apsidal motion when the force is 

 wholly central, but left nothing applicable to the circumstances 

 of the moon's motion. The successors of Newton, who applied 

 analysis to the lunar theory, finding that the process of approxi- 

 mation failed on starting from a fixed ellipse, altered the hypo- 

 thesis of the approximation by assuming the apse to have a mean 

 motion. This was done by " introducing " (to use Mr. Adams's 

 expression) the quantity usually denoted by c. Laplace intro- 

 duces this factor hypothetically, and refers to a subsequent veri- 

 fication. (Theorie de la Lune, vol. vii. sect. 4.) Plana, on the 

 contrary, is led to it by the method of the variation of parame- 

 ters. {Theorie de Mouvement de la Lune, Chap. II. § 3.) On 

 the principle of the latter method I shall say a few words in the 

 sequel, at present I am concerned with the introducing process. 

 Now I think I may assert that mathematicians have felt that 

 there is something unsatisfactory in this process ; that it is a pis 

 alter, to which recourse was had because no better method was 

 discovered. Though it is undoubtedly legitimate and leads to 

 true results, a method which would conduct directly to the form 

 of the expression for the radius-vector to the first order of small 

 quantities would seem to be more logical and more complete. I 

 long since directed my attention to the discovery of such a 

 method; and having at length remarked that the approximate 

 solution of the problem of the moon's motion, as usually treated, 

 rests on two distinct hypotheses, viz. a mean motion of the 

 radius-vector and a mean motion of the apse, it occurred to me 

 to attempt the investigation on the single hypothesis of a mean 

 motion of the radius-vector. On doing so in the manner indi- 

 cated in my communication to the April Number of the Philo- 



H2 



