524 Pi*of. Challis on the Theory of the Moon's Motion, 



dr 

 have iT=0, and consequently 



As this equation contains possible but not unequal roots, it is 

 satisfied simultaneously with the derived equation 



— ^ + Cr-nV=0. 

 Hence 



Lb Tl Uj 



= 5 + -^ nearly.. 



This will afterwards be found to be the value of the mean distance 

 in the first approximation to the orbit. 



As the equation (C) gives the value of -^ to the second order 



dr 

 of small quantities, we may deduce from it the value of -j, and 



therefore that of r, to the first order of small quantities ; and 

 these values, it may be observed, will be independent of the 

 sun's longitude. I proceed next to the integration of that 

 equation. 



The course which first presents itself is, to substitute for r in 

 the term involving w'^ a mean value a, thus omitting, as hereto- 

 fore, small quantities of the third order. By this process the 

 approximate orbit is shown to be a fixed ellipse described by the 



action of the central force -^ . This result does not recognize 



the existence of a disturbing force, although it has been fully 

 taken into account in the investigation. I find also, by carrying 

 the approximation further on the same principle, that terms in- 

 consistent with the hypothesis of the approximation arise, of the 

 same kind as those met with in the usual method of treating the 

 lunar theory when the approximation commences with a fixed 

 ellipse. Since, however, no rule by which approximations are 

 conducted has been violated, some step incompatible with the 

 particular conditions of the question must have been taken. But 

 the only step that has been taken peculiar to the case, is that of 



substituting a for r in the term -77-, on the assumption that 



some other term of the equation is of a higher order of value 

 than this term. It must therefore be concluded that the equa- 

 tion (C) contains no term of a higher order of value than that 



of the term -5—. It is, however, certain that this is not the 



