102 Prof. Challis on certain Questions 



the approximative process of integration made use of to obtain 

 the values of r and 6, does not decide whether or not the con- 

 stants are mutually related, because it is conducted in an irregular 

 manner. After assuming that there is a mean motion of the 

 radius-vector, it makes the additional assumption of a mean 

 motion of the apse ; whereas my method of integrating the equa- 

 tions (1) and (^) to the first approximation deduces the mean 

 motion of the apse from an assumed mean motion of the radius- 

 vector ; and in the course of making this deduction, a certain 

 relation between the constants h and C (which may replace a and 

 e) is found which gives for e a special value. The reasoning 

 which leads to this result will be defended in a subsequent part 

 of the discussion. 



The assertions in the next paragraph (p. 30) respecting the 

 " variation '^ and " evection,^' are simply not true, the previous 

 values of r and 6 belonging to my method of integration for the 

 same reason that they belong to the ordinary method. After 

 obtaining the value of r to the first order of small quantities by 

 the process given in my communication to the April Number 

 (p. 281), I may use it in conjunction with the equations (1) and 

 (2) to obtain the values of r and 6 to the second order of small 

 quantities precisely according to the usual method, the only dif- 

 ference being, that having deduced instead of assuming the form 

 of the first value of r, I am entitled to ascribe to e a special value. 



In the latter part of the same paragraph Mr. Adams has 

 fallen into a misconception respecting Theorem I., which may 

 have arisen from the terms in which the theorem is enunciated. In 

 my article in the April Number (p. 280), after obtaining the equa- 

 tion (C), I say, " this equation proves Theorem I.'^ I could not, 

 therefore, mean any other relation between the radius-vector and 

 the time than that expressed by the equation (C). Mr. Adams's 

 objection to the correctness of the theorem will be met by enun- 

 ciating it with more precision as follows : — " All small quantities 

 of the second order being taken into account, the relation between 

 the radius-vector and the time in the moon's orbit, as expressed 

 by a differential equation of the first order, is the same as that 

 in an orbit described by a body acted upon by a force tending 

 to a fixed centre." 



Mr. Adams then goes on to consider the reasoning by which 

 Theorems I. and II. are arrived at, and makes this remark : 

 " All this reasoning is based on the equation (C), the truth of 

 which, he (Prof. Challis) says, cannot be contested." The rea- 

 soning to establish Theorem II. is certainly based on that equa- 

 tion, but Theorem I. is merely a verbal statement of what the 

 equation indicates. Again, I said (Phil. Mag. for June, p. 430) 

 that the proof of the equation cannot be contested; and if I said 



