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



rejected (Phil. Mag. for August 1854 p. 108), because it ap- 

 peared to lead to the inference that the eccentricity is simply 

 proportional to the disturbing force. But by taking account, as 

 is done above, of the rate of angular motion of the apse, this 

 inference by no means follows, and the eccentricity is still a 

 function of the disturbing force, although the form of it may be 

 discoverable only by analytical investigation. 



It is particularly to be remarked, that the above general rea- 

 soning only proves that, whatever be the law of the disturbing 

 force, and whether it be central or not, for a certain value of the 

 eccentricity the motion of the apse will be uniform. But it does 

 not prove, that, under particular conditions, for instance, when 

 the disturbing force is wholly central, and the law of it is given, 

 uniform motion of the apse may not be consistent with any value 

 of the eccentricity. In a similar manner, the general theorem, 

 that every mass, whatever be its form, has three axes of perma- 

 nent rotation, is not invalidated by finding that masses of par- 

 ticular forms have an unlimited number of axes of permanent 



rotation. Hence the fact, that when the central force is ^ + %, 



the motion of the apse is uniform whatever be the eccentricity, 

 is no argument against the general proposition, unless it should 

 appear that in this instance the eccentricity can under no cir- 

 cumstances be a function of the disturbing force. This, how- 

 ever, is not the case ; for the equation which gives the apsidal 

 distances for that law of force is 



from which it follows that 



C — — J. 2 ' 2 > 



and consequently that e is a function of /x' if /« 2 C=/i. 2 . 



I think that I have now established the truth, in all essential 

 particulars, of the new views respecting the moon's orbit, which 

 were first published in the Philosophical Magazine for April 

 1854; and that I have pointed out a step towards a great sim- 

 plification of the lunar theory. I have not yet had time to carry 

 the method through the third approximation. 



Cambridge Observatory, 

 January 11, 1855. 



P.S. In the reasoning of the foregoing article it has been 

 assumed, in accordance with the view generally taken by mathe- 

 maticians, that analysis fails to give the true developments of 

 the moon's radius-vector and longitude when the approximation 



