Prof. Challis on the Theory of the Moon's Motion. 525 



case unless the trinomial h^—^fjb7'-\-Cr^ be equivalent to a single 

 term of the second order of small quantities. To satisfy this 

 condition, the constants ^ and C must be related to each other. 

 The required relation may be found as follows. Put for r in the 

 above expression w-\-v, and assume w to be much larger than v. 

 Then the expression becomes 



h^-2lM{w + v)^-C{w^-vY, 



and the appropriate condition is satisfied if 



and 



Hence it follows that 



— 2/i- + 2Cm;=0. 



w=j^ and h^=. ^. 



The reasoning by which the above relation between h and C 

 is deduced, appears to be perfectly cogent. I cannot, after the 

 fullest consideration, perceive that any step can be called in 

 question. It may be observed that the constant C was intro- 

 duced by an integration performed anterior to any limitation of 

 the question, and that the relation between h and C results from 

 the limitation given to the problem by assuming the moon's 

 true motion in longitude to differ little from a mean motion. 

 It is not my intention to introduce the equation h^C = ^^ into 

 the investigation at present, because I wish to prove, first, that 

 if h and C be regarded as independent of each other, the solu- 

 tion of the lunar problem deduced from the equation (C) is iden- 

 tical with the ordinary solution. It must, however, be borne in 

 mind, that if the foregoing reasoning be good, neither method 

 of solution ought, in strict logic, to be proceeded with until the 

 above relation between the constants has been deduced*. 



* As in this part of the reasoninp; my method is distinguished from every 

 other that has been applied to the lunar theory, I will endeavour to put the 

 argument in as succinct a form as possible, that it may be the more readily 

 seized. 



Suppose the expression —Cr^-~}i?-\-'2,}ir-{- —— to contain terms higher 



in value than the last term. 



Then the only legitimate process of approximation is to integrate, ne- 

 glecting the small term, so as to obtain an approximate value of r, to sub- 

 stitute this value in the small term and integrate again, and so on. 



By this process the approximation commences with a fixed ellipse of 

 arbitrary eccentricity, and is found in succeeding steps to introduce terms 

 which may increase indefinitely with the time, and which are therefore in- 

 compatible with the hypothesis that the true values of the radius-vector and 

 the longitude dififer little from mean values. 



Consequently on that hypothesis the above expression cannot contain 



