138 ON PROFESSOR CHALLTS'S NEW THEOREMS [20 



satisfy the first of those equations, since the term of the second order 



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

 Sun, which does not occur at all in the relations in question. 



The contradiction to Professor Challis's theory, which is presented 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 offered that 

 the taking it into account would tend to reconcile the discrepancy. 



At the bottom of page 282, Professor Challis attempts to shew, 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 circumstances of this fluc- 

 tuation 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 in- 

 versely 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 

 velocity of the orbit being always proportional to that of the body at the 

 same instant ; and the eccentricity of the orbit might be any whatever, 

 and would not at all depend on the disturbing force. 



