Prof. Challis on the Eccentricity of the Moon's Orbit. 131 



to the remarkable result, that the motion of the apses is uniform 

 whatever be the eccentricity of the orbit. I have no grounds for 

 concluding that the author is of opinion that any argument can 

 be deduced from this result against the theorem I have advanced 

 respecting the moon's orbit, viz. that the uniformity of the pro- 

 gression of the lunar apses depends on a certain value of the 

 eccentricity; but as such an argument appears prima facie to be 

 deducible, I have undertaken to show that this is not really the 

 case. For this purpose I propose to solve the problem by a 

 method somewhat different from that employed by Mr. Thacker. 



The force is wholly central, and equal to — 2 —fi'r, the second 



term is supposed to be always small compared to the other, and 

 only the first power of /i' is retained. From the usual differ- 

 ential equations of the motion, viz. 



may be derived, after substituting r cos 6 for x, and r sin 6 for y, 

 the following : 



d?r dff 2 fi . n 



dO _ h 



dJ-7*' (2) 



d*r A 2 a , 



y-pr+Jr~#*«0 ( 3 ) 



Multiplying (3) by 2dr and integrating, 



^! + i 2 _^_^, 2 +c-o (o 



dt *+ r 2 — -/*» +b_0 (O) 



Hence, eliminating dt by (2), we have 



dd= Mr 



r v/-Cr 2 + 2fir-h 9 + fi'r 4 ' 



As the integral of this equation can only be obtained approxi- 

 mately, and the approximation is to proceed according to the 

 powers of ft}, we get by expanding to the first power of this 

 quantity, 



jo hdr p/h ?«*dr 



r(-Cr s +2^r-A 2 )* 2 ' (-Cr 8 + 2/*r- A 9 )*' 

 This equation being integrated by the usual rules, gives 



a . , It 1 — fir Sfifi'h , O 

 + y= cos -1 — ~ . , H . cos~' 



_ lllL [ r * ZitfC- Z fi^jlP-fir) 3/<V(/*-Cr)1 

 2 V R ' \ C + " CV 2 - h*C) + C'V - A 8 C) )' 

 K2 



