136 ON PROFESSOR CHALLIS'S NEW THEOREMS [20 



These equations give the relations between the arbitrary constants h 

 and G, and the new constants a and e by which the former may be re- 

 placed. 



From the second of them, we find 



:s 



m' a 3 



or, putting for a in the small term its first approximate value p, 



p. mV 

 ~<7 a' 3 C 4 ' 



which agrees with Professor Challis's expression in p. 281. 



Now apply a similar process to the equation 



dr\* h s 2/i >'<* 1 ,fi_ n 

 dt) + ^""2a*"* U) 



which differs from the equation (C) in having a put for r in the small 

 term. In this case, we find 



' 



m 



h" - 2^a + Co? ( 1 + e 2 ) - " > 3 a 4 ( 1 + e 2 ) = 0, 



and 



m' 



from the latter of which equations it follows that 



u, m' a 3 

 a = 77 H / 3 77 > 



a m' u 3 

 or a = H >3 ^ , 



to the same degree of approximation as before. 



Hence we see that the values of a, in the two cases supposed, differ 

 by a quantity of the second order. Consequently the difficulty into which 

 Professor Challis is led by the conclusion that these values are the same, 

 disappears, and the solution of the difficulty with it. 



