Prof. Challis on the Eccentricity of the Moon's Orbit. 133 



Let 



. a(l-<?) 1 



COSiIr = — - , 



1 er e 



so that 



sin <£ _ simjr 



Hence 



= cos(^-^!L^JE?sinf) nearly j 

 \ • 2/u. / 



.-. g=f + ^ a3i/l - e8 (30- e sm^), 



6 being supposed to commence with yfr and <£. 



It is not necessary for my purpose to pursue the reasoning 

 further. The equation last obtained, which is identical with the 

 principal result of Mr. Thacker's solution, sufficiently proves 

 that the two methods are the same in principle, and lead to the 

 same inferences respecting the motion of the apses. It is, how- 

 ever, important to remark, that my method shows clearly that 

 the analytical reasoning may legitimately commence from a fixed 

 ellipse. Let us now consider the bearing this inference has on 

 the lunar theory. 



It is well known that in the lunary theory the analysis fails to 

 give the true development of the moon's radius-vector and true 

 longitude, if the approximation commences from a fixed ellipse. 

 What is the reason that the process fails in this instance, while it 

 succeeds in that just considered ? I reply, that the different con- 

 ditions of the lunar problem entirely account for this difference. 

 In the first place, the force in the lunar theory is not wholly central, 

 and the equation (C) is consequently only approximate. Next, 

 there is introduced into the lunar theory a limitation, according 

 to which the moon's true longitude and radius-vector can never 

 differ much from mean values*. In consequence of this limi- 

 tation, which is essential to the subsequent treatment of the 

 problem, the eccentricity of the moon's orbit is always small. 

 Lastly, in the lunar theory it is necessary to conduct the ap- 

 proximation, not only according to the disturbing force, but ac- 

 cording to the eccentricity also. The failure just spoken of 



* In Mr. Airy's Lunar Theory (Mathematical Tracts, 3rd edition) this 

 limitation is introduced where it is said, in pajije 29, that " for 6' we shall 



Eut the value which it would have if the motions of the sun and moon were 

 r>th uniform." 



