relating to the Moo'n/s Orbit. 85 



tegration, so that it is not surprising* that Professor Challis 

 should have met with such difficulties in performing the inte- 

 gration. 



The relations between r, 6, and t, given in page 281 (which 

 profess to include all small quantities of the second order),, are 

 said to be derived from the equations (B) and (C). It is easy 

 to seCj however, that they do not satisfy the first of those equa- 

 tions, since the term of the second order 



in the right-hand member of that equation involves the longi- 

 tude of the sun, which does not occur at all in the relations in 

 question. 



The contradiction to Professor Challis's theory, which is pre- 

 sented by the eccentricity of the orbit of Titan, is supposed by 

 him to be occasioned by the large inclination of that orbit to the 

 plane of the orbit of Saturn. But in page 280 it is remarked 

 that the inclination of the orbit is taken into account ; and even 

 if this were not the case, no proof is ofi'ered that the taking it into 

 account would tend to reconcile the discrepancy. 



At the bottom of page 282, Professor Challis attempts to show, 

 a priori, that the eccentricity of the moon^s orbit must be a 

 function of the disturbing force in the following manner. 



If there were no disturbing force, the value of the radius vector 

 drawn from the earth's centre in a given direction, would be 

 constantly the same in different revolutions. But if a disturbing 

 force act in such a manner as to cause the apsidal line to make 

 complete revolutions, the value of the above-mentioned radius- 

 vector would fluctuate in different revolutions, between the two 

 apsidal distances. Hence it is argued that, since if there were no 

 disturbing force there would be no such fluctuation of distance, 

 therefore the total amount of such fluctuation, and consequently 

 the eccentricity, must be a function of the disturbing force. 



But, on consideration, it will appear that this argument is 

 fallacious. No doubt it may be inferred that some of the cir- 

 cumstances of this fluctuation of distance will depend on the 

 disturbing force which causes it, but it cannot be asserted, 

 without investigation, that the total amount of such fluctuation 

 must necessarily depend on the disturbing force. 



As a simple example, we will suppose the principal force 

 to vary inversely as the square of the distance, and a central 

 disturbing force to be introduced which varies inversely as the 

 cube of that distance. In this case we know, by Newton's 9th 

 section, that the motion would be accurately represented by 

 supposing it to take place in a revolving ellipse, the angular 



D2 



