

Prof. Challis on the Theory of the Moon's Motion. 381 



trary constants C and h satisfy the equation C/i 9 =/z, 2 , the con- 

 sequent value of the eccentricity is the same as that given by 

 the second approximation. Thus the third approximation is not 

 contradictory to the second. At the same time it is incapable of 

 establishing the truth of the theorem, as appears from the fol- 

 lowing considerations. 



In the method of treating the lunar theory which I have pro- 

 posed, no use is made, as in previous methods, of the radius- 

 vector and longitude of an assumed revolving ellipse ; but instead 

 of this process, the terms containing the sun's longitude are 

 separated from those which do not contain it explicitly, and the 



former being omitted in the values of JJ and ~, equations are 



formed which define the orbit that I have called the mean orbit. 

 These equations, as given by the third approximation, are the 

 following : 



^ + #! V, r , n M h 



dt* + r* r +b -^- d J = ^- 



dO 

 The development of ^- was found to contain an additional term 



independent of the sun's longitude, but as it is of the fourth 

 order, it is omitted in the third approximation. By the above 

 two equations the mean orbit is proved to be a revolving ellipse 



for which c= j, and the value of h' 2 already obtained shows that 



to the second order of small quantities ~=l-—, ft thus 



h 4 



appears that the apsidal motion is deduced in the third approxi- 

 mation from equations which are the same in form as if the force 

 were wholly central, and consequently, so far as this approxima- 

 tion shows, the eccentricity is arbitrary. But on proceeding to 

 the fourth approximation, the term above mentioned must be 



taken into account, in consequence of which r- — is no longer 



dt G 



constant for the mean orbit, and the force cannot be regarded as 

 wholly central. The mean motion of the apse and the form of 

 the mean orbit are now dependent in part on a tangential force ; 

 and it is perhaps not too much to assert, that under these cir- 

 cumstances the mean orbit cannot be a circle, and consequently 

 that the eccentricity is not arbitrary. Again, the process by 

 which I obtain the value of c appears to indicate that in suc- 

 ceeding approximations, this quantity will be a function of e 

 and m. If so, it cannot be a function of m exclusively, unless 

 the arbitrary constant C satisfy the condition C/i 2 =yu, 2 . It would 

 be necessary, however, in order to settle these points, to complete 



