relating to the Moon's Orbit. ■ '^ ' --■' 31 



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 equa- 



(dr\^ 

 -^ I in terms of r, but 



that it does not profess to include terms of the third order. 



dr 



Now -J- is itself a quantity of the first order, and consequently 



at /'dr\^ 



an error of the third order in ( -^ j leads to one of the second 



order in -r, 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 result- 



dr 

 ing equation becomes divisible by -j-; dividing out, we obtain 



d^r ^^ .H' ^'^ __n 



dF'^'i^'^^''2^ 



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-\-6 for 6, and a for r in the term of the second order, we find 



c,d6 , 3 m' fl^ ,^ ^-j ,^ 



r^-Tr = h+--i^—cos(2nt-i-e'-'2n't + €'), 

 dt 4«'^ n ^ ' 



The value of the constant A, expressed in terms of the system of 



constants before used, is 



Hence 



r^(^y==;i2^ |^a4 cos (25^7+7-2^7+?), 



(ddy h^ 3 m' 



^\'5^/'^^'^2^^^^^(^^^"+"^""^"'^ + ^')' 

 putting, as before, a for r in the small term. Substituting this 



value of r( -^ j in equation (2), we find 



<^r h^ , fJL wir „ m' ,~ ^-7 — -^^ ^ 



