Prof. Challis on the Theory of the Moon's Motion. 379 



and if cn= — T . the apsidal motion in one revolution of the moon 



is27r(l-c). 



Since by the second approximation 



, _ nP _ Sm? 



rt _ 1 + e 2 £ _ 2_ cf_ _ ~2~ 



h*~ ri 1 ' n~ n* ' C ~ n? ' 

 the foregoing values of h 1 ' 2 , fj,', and C become 



;/2 11 ( *i 3m 2 45 „ 3mV\ 



h = h V--T-T B *-—) 



lt'=/*Cl-* 2m 2 - J^ n? + mA 



C' = C (l-3m 2 - ^mH |^" 4 ). 



In these values the terms of ike fourth order are to be rejected, 

 because the present approximation extends only to quantities of 

 the third order. Hence it will be found that 



2 . Ch\ m* 



This result proves that the expression for e 2 contains no term 

 involving m 3 . 



Again, in obtaining the value of c, terms of the third order 

 are to be omitted, because, for reasons already adduced, this 

 third approximation can give the value of c only to terms of the 

 second order. Now to that order we have, as in the second 



approximation, nd i — hll+— J. Hence 



9m 2 



fji'n fj.h 



(l +£)(!-**)' 



and as « 2 = £, (1 + 2m 2 ) and ^ = 1 + L - 2L nearly, it will 



be seen that 



, 3m 2 



which is the known approximation, obtained by a strictly deduc- 

 tive process. 



