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



dr* 

 Consequently the value of -7-5 must include all small quantities 



of the second order. I have shown that by merely making use 

 of the equation 6 = nt -f e, the following equation may be obtained, 



g + £-^ + c=* . . . . (C 



2w' 

 in which C is put for C, - f — 2n'h, C, and h being the arbi- 

 trary constants introduced by the integration. It may here be 

 remarked, that as this equation was obtained by proceeding to 

 small quantities of the second order, it is needless to retain the 



general value of r in the term =— . Putting a for r in that 



n' 2 a 2 

 term, and substituting C for C =— , we have 



at* r* r 

 This equation gives by integration, 



r = a(l— ecos-^r), — (/+T)=^r— esiw^r, 



a ~ c" e ~ p* ' 



Hence by substituting for C its value, and putting m for the 

 ratio of n' to n, it will be found that 



a /, m 2 \ n C/ 



C/* 2 m 2 



But as the approximation gives the values of r only to the first 

 order of small quantities, the above value of a is not true to 

 small quantities of the second order, and in fact is not confirmed 



by the next approximation. Hence we have simply «= ^. 



\j 



It is evident that the orbit, as determined by this first ap- 

 proximation, is a fixed ellipse of arbitrary eccentricity e, and that 

 if ct be the longitude of the perihelion, the values of the radius- 

 vector and true longitude are the following : 



r = a(\ — e eos(nt + e— ■or) ^) 



6 = nt + e + 2e sin (nt + e — *r-). 



These values are to be substituted in the small terms of the 

 equations (A) and (B) to proceed to the second approximation. 

 In the Supplement to the Philosophical Magazine for December 



