134 ON PROFESSOR CHALLIS'S NEW THEOREMS [20 



The value of the constant h, expressed in terms of the system of constants, 

 before used, is 



Hence 



and 



d0* h* 3m' 



. O -T- -,. . 



= -r + g a /i cos (2n + e - 2 -/i' + e'), 



putting, as before, a for ? in the small term. Substituting this value of 



/dd\* . . /riX j 



rl-T-l in equation (2), we find 

 at 



The equation above deduced from Professor Challis's differs from this by 

 the omission of the last term, which gives rise to the variation inequality. 

 In order to find the evection, which is also an inequality of the second 

 order, it would be necessary to carry the approximation one step still further 

 than we have here done. 



This shews how unfitted equation (C) is for giving any accurate infor- 

 mation respecting the Moon's orbit. 



As a matter of fact, it may be observed that this equation would make 

 the Moon's apsidal distances to be constant. A simple inspection of the 

 calculated values of the Moon's horizontal parallax, given in the Nautical 

 Almanac, is sufficient to shew how far this is from the truth. 



I now proceed to make good my second assertion, viz. that Professor 

 Challis's Theorem II. cannot be inferred from his equation (C). The process 

 by which he attempts so to infer it is of the following nature. He first 

 finds that a method, apparently legitimate, of treating the equation (C) leads 

 to a difficulty. To get rid of this difficulty, he makes the strange suppo- 

 sition that the equation (C) contains the disturbing force as a factor, and 

 then tries to shew that, in order that this condition may be satisfied, the 

 arbitrary constants h and C must have a certain relation to each other, from 

 which it would immediately follow that the eccentricity must have the value 

 assigned to it in Theorem II. 



