relating to the Moon*s Orbit. iO& 



to be arbitrary and independent. For considerations quite sepa- 

 rate from the process of integration by the variation of parame* 

 ters, and arising out of the hypothesis of the approximation, may 

 show that those constants may have certain analytical expressions 

 or values, and certain relations to each other. Thus the process 

 of integration by the method in question proves that there is a 

 constant part of the radius-vector ; the process of approximation 



(m^ e^\ 

 1 — -^ + -^ ) . So by the 



variation of parameters the eccentricity is proved to have a con- 

 stant part ; but that process is incapable of arriving at such an 



equation as e^ = 1 ^ + ^s^ns' much less of deciding whether 



h and C have a relation to each other. All that is proved by the 

 method of the variation of parameters must be true ; but other 

 results not inconsistent with its indications, and not deducible 

 by it, may also be true. I contend, therefore, that it affords no 

 handle for an argument against my views. 



I have now met, I think successfully, the whole congeries of 

 Mr. Adams's arguments, the number of which does not make up 

 for their want of force. By attacking an equation, the evidence 

 for which is irresistible, Mr. Adams took up a false position 

 which it was impossible to maintain. Nothing is now wanting 

 to establish fully the solution of the lunar problem, of which I 

 have indicated the initial steps, than to show, by carrying the 

 approximation further, that it explains the variation and evec- 

 tion, and gives more approximate values of the eccentricity and 

 the mean motion of the apse. I have not yet had time to do 

 this, but I feel confident that the solution, if rightly conducted, 

 will bear this test. 



In concluding this long letter, I propose to give the method 

 of determining, by a direct process to the first approximation, 

 the mean motion of the moon^s node, which, as before intimated, 

 was contained in my paper. The differential equations of the 

 motion are, 



d'^z fjLZ m'z _ 



The last term in the third equation is a small quantity of the 

 third order on account of the small ratio of z to r. If, therefore, 

 we neglect this term, and suppose, as Newton does, the orbit to 

 be a circle, we have for determining the period of the moon's 



