S2 Mr. J. C. Adams on Professor Challis's new Theorems 



The equation above deduced from Professor Challis's differs 

 from this by the omission of the last term, which gives rise to 

 the variation inequahty. In order to find the evection, which is 

 also an inequality of the second order, it would be necessary tQ 

 carry the approximation one step still further than we have he|?g 

 done. 



This shows how unfitted equation (C) is for giving any acq^^ 

 rate information 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 hori- 

 zontal parallax, given in the Nautical Almanac, is sufficient iii 

 show how far this is from the truth. - r .r ,v';~ -\\ 



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

 Professor Challis's Theorem II. cannot be inferred from his equa- 

 tion (C). The process by which he attempts so to infer it is of 

 the following nature, lijf 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 supposition that the 

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

 tries to show that, in order that this condition may be satisfied, 

 the arbitral^ constants h and C must have a certain relation to 

 each other, fi*om which it would immediately follow that the 

 eccentricity must have the value assigned to it in Theorem II. 



Now it is remarkable that every one of the steps of this pro- 

 cess is unwarranted. The difficulty to which Professor Challis 

 is led is purely imaginary; the supposition that the equation (C) 

 contains the disturbing force as a factor is wholly unsupported 

 by any proof ; and even if that supposition were well founded, 

 it would not follow that the constants h and C must have the 

 relation assigned to them by Professor Challis. 



The supposed difficulty is founded on the inference at the 

 bottom of p. 280 of Professor Challis's paper, " Hence we must 

 conclude that the mean distance and mean periodic time in this 

 approximation to the moon's orbit are the same as those in an 



elliptic orbit described by the action of the central force ^." But 



this is not a correct conclusion : if A and C be supposed to have 



the same values in equation (C) and in that obtained from it by 



putting a for r in the small term, the values of the mean distances 



in the two cases would not be the same, but would differ by a 



quantity of the second order. 



This may be readily shown in the following manner, ii ii&i^i 



dr 

 At the apsides ^=0, and therefore the equation (C) gives the 



