lOd Prof. Challis on certain Questions 



disturbances of which may countervail the disturbing force of 

 the sun. 



I conic now to an argument in which I admit that Mr. Adams 

 has taken a right view. I attempted by reasoning conducted 

 verbally, to confirm the results of an analysis which stood in need 

 of no such confirmation. If that reasoning had been good, it 

 would have shown that the eccentricity is proportional to the 

 square of the ratio of the periodic times, and not, as stated in 

 Theorem II., simply proportional to that ratio. I am therefore 

 thankful to Mr. Adams for proving that this verbal reasoning 

 was in fault. 



The assertion in page 36, that " if the disturbing force were 

 increased the total fluctuation in the value of the radius-vector 

 would be the same as before," begs the question at issue. So 

 also does the " fatal objection " which follows it. To determine 

 generally the orbit described under given initial circumstances, 

 it would be necessary to commence the approximation on the 

 single supposition that the ratio of the moon's radius-vector to 

 that of the sun is small, a problem which has never been 

 attempted. To this supposition is always added that of a mean 

 motion of the radius-vector differing little from the true motion. 

 I have shown that this latter supposition conducts to a special 

 value of the eccentricity depending on the ratio of the mean 

 motions of the moon and sun, and have thus given at least a 

 negative proof that no part of the eccentricity would be constant 

 if that particular relation did not exist. To prove the same thing 

 positively would require the solution of the general problem 

 above mentioned. It would be waste of time to say more in 

 defence of what I only stated to be a probability. 



As Mr. Adams appeals to the method of the variation of para- 

 meters in support of the last argument, I shall take the occasion 

 to say a few words on the principle of this method. In the first 

 place, I remark that it has been applied in the lunar theory only 

 on the hypothesis of a mean motion of the radius-vector, and 

 has decided on that hypothesis that the non-periodic part of the 

 eccentricity is constant; but not having been applied to the 

 more general problem, it has not decided that there would be 

 a constant part of the eccentricity under all initial circum- 

 stances. Again, the method of the variation of parameters is 

 simply a process of integration applicable to differential equa- 

 tions of a cei-tain form, and requiring the same rules of treat- 

 ment whether the differential equations be approximate or exact. 

 Consequently in the former case as well as in the other, the con- 

 stants introduced by the integration necessarily present them- 

 selves as independent of each other and arbitrary in value ; but 

 in the case of an approximate solution, they are not thus proved 



