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



It would lead me into too complicated calculations, to add to 

 the equation (C") the terms of the fifth order which receive small 

 denominators by integration, and to obtain the complete values 

 of r and 6 to the third order of small quantities. I have indi- 

 cated above a process proper for this purpose ; and having made 

 trial of this process to some extent, I found terms of the third 

 order identical with those given in Pontecoulant's Lunar Theory. 

 In fact, I have no reason to conclude that my solution differs in 

 ultimate analytical expression from previous solutions. The 

 order of the reasoning differs from that of any former method, 

 and is, I think, strictly logical. There is also one important 

 particular in which the reasoning itself is unlike that of any 

 other method. I have commenced the process of approximation 

 after obtaining an exact integral, which had not been previously 

 discovered. On that integration, and on results to which it leads, 

 depends the proof of the new theorem I have published respect- 

 ing the relation of the eccentricity of the moon's orbit to the 

 disturbing force. I propose to conclude the present communi- 

 cation with some further consideration of this point. 



In the Philosophical Magazine for February (p. 135), I have 

 adduced the following argument to show that the eccentricity of 

 the moon's orbit has a special value determined by the disturbing 

 force. The fluctuations of value of the radius-vector, as the 

 moon returns in successive revolutions to the same longitude, 

 depend on the disturbing force in such a manner that the change 

 in each complete revolution would vanish if the disturbing force 

 vanished. The total fluctuation, in the case of a uniform apsidal 

 motion, is the same in all directions after a large number of 

 revolutions of the moon, and depends both on the partial fluc- 

 tuations and on the interval during which they are additive or 

 subtractive, that is, on the rate of motion of the apses. Hence 

 as the motion of the apses, as well as the partial fluctuations, 

 would vanish with the disturbing force, it follows that the total 

 fluctuation in any given direction, and consequently the eccen- 

 tricity of the orbit, is a quantity which contains the disturbing 

 force as a factor. This conclusion does not hold good when the 

 force is wholly central, as the orbit in that case may be a circle, 

 and the eccentricity is consequently not restricted to one value. 



If the foregoing general argument be true, the same conclu- 

 sion must be obtainable in a special manner in the course of the 

 investigation of the moon's motion, and I am ready to admit 

 that the new theorem is not established unless it can be so ob- 

 tained. It will, therefore, be worth while to inquire how far the 

 theorem receives support from the results of the third approxima- 

 tion. First, it may be remarked, that as it has been proved that 

 the expression for e 2 contains no term involving m 3 , if the arbi- 



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