20] RELATING TO THE MOON'S ORBIT. 135 



Now it is remarkable that every one of the steps of this process 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 h and C be 

 r 



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 shewn in the following manner. 



fJv* 



At the apsides -j- = 0, and therefore the equation (C) gives the follow- 

 ing equation for finding the apsidal distances, 



m' 



Now if a be the mean distance, and e the eccentricity, the apsidal distances 

 are a(l+e) and a(le). 



Substituting these values for r in the above equation, and developing 

 the small term to quantities of the fourth order, we obtain 



Ca 2 (l + 2e + e 2 ) j^ a* (I + 4e + 6e 2 ) = 0, 

 and 



whence it follows that 



A 2 - 2fM + Co? (1 + e 2 ) - a a 4 (1 + 6e 2 ) = 

 and 



m! 



