20] RELATING TO THE MOON'S ORBIT. 133 



the truth of which, he says, cannot be contested. In speaking of the truth 

 of this equation, Professor Challis cannot mean that it is anything more 

 than an approximation to the truth, since in forming it he avowedly neglects 

 all quantities of orders superior to the second. 



Now what I assert is, first, that the degree of approximation attained 

 by the equation (C) is not sufficient to justify Professor Challis in inferring 

 Theorem I. from it ; and secondly, that Theorem II. does not follow from 

 that equation at all. 



To prove the first of these assertions, I remark that the equation (C) 



(drY 



gives an approximate value of I y- ] in terms of r, but that it does not pro- 

 fess to include terms of the third order. Now -,- is itself a quantity of 



/dr\" 



the first order, and consequently an error of the third order in ( j- ) leads 



\at I 



dr 

 to one of the second order in , -, and therefore to one of the same order 



in the value of r expressed in terms of t. Hence Professor Challis is not 

 entitled to infer that the relation between the radius-vector and the time 

 in the Moon's orbit is the same, to quantities of the second order, as that 

 which would be given by the equation (C). 



We may test the degree of accuracy to be attained by the use of 

 this equation in the following manner. 



By differentiation, the constant C disappears, and the resulting equation 



ft')'* 



becomes divisible by -^-; dividing out, we obtain 



d"r h? ju, m'r 



| T- *~~ \J t 



ft f " / Y^ f ^ *} Cl 



This is a strict deduction from Professor Challis's equation ; we will 

 now obtain directly from the equations of motion given above, an expression 

 to be compared with it. 



Integrating equation (1), and putting, with Professor Challis, nt + e for 6, 

 and a for r in the term of the second order, we find 



r> -h ^^L^L 

 dt 4 a" n 



