20] RELATING TO THE MOON'S ORBIT. 131 



know that our first approximation was a good one, and that the true and 

 the only true solution of the differential equations has been obtained. 



On the other hand, no solution can be a true one, which does not 

 contain the proper number of arbitrary constants ; and any person who 

 asserts that one of the constants usually considered arbitrary is not so, is 

 bound to show by what other really arbitrary constant the former is replaced. 



I will now proceed to consider Professor Challis's two theorems, which 

 are thus enunciated by him. 



Theorem I. 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 is the same as that in an orbit described by a body acted upon by 

 a force tending to a fixed centre. 



Theorem II. The eccentricity of the Moon's orbit is a function of the 

 ratio of her periodic time to the Earth's periodic time, and the first ap- 

 proximation to its value is that ratio divided by the square root of 2. 



I will endeavour, in the first place, to show that these theorems cannot 

 possibly be true ; and secondly, to point out the fallacies in the argument 

 by which Professor Challis attempts to establish them. 



The problem will be simplified by supposing the Moon to move in the 

 plane of the ecliptic, and the Earth's orbit to be a circle. On these sup- 

 positions, Professor Challis's fundamental equations become 



d*x ax m'x 3mV , . A 



df r 3 2a' 3 2a' 3 



Multiply these equations by y and x respectively, and subtract the results ; 

 and again multiply by x and y, and add the results together ; thus we 

 obtain, after expressing x and y by means of polar coordinates, 



d 



d*r [ddV IL m'r 3m'r 



= -r> + M> + ~2a>*- 



172 



