16 LECTURES ON THE LUNAR THEORY. [LECT. 



Assume as a first approximation 



,. sin 2$, say ; 



- = - fl +a,cos 2\jj], 

 r a L 



and we shall suppose a.,, 6 2 so small that in the first instance we may 

 neglect their squares and products. 



Substitute in the equations ; then 

 4 (n - n'Y a 2 cos 2$ - {n 2 + 4n(n- n') b, cos 2$} + ^ {1 + 3a 2 cos 2t/} 



4 (n n')" to sin 2\fi + 4 (n n') na. 2 sin 2$ - n' 2 sin 2t/. 



A 



Hence, equating the coefficients of similar terms, we have 



= n * + l n >* 



a" 2 



which gives the relation between n the Moon's mean motion, and -, the 



QJ 



mean of the reciprocal of the distance ; also 



n-n')b a _= 3 -ri* ............... (l), 



-w')a 4 = -\n H ............... (2). 



2i 



From (2) 4n (n - n') b, - 4n*a, = nn ' 



Add to (l), and substitute for /it/a 3 ; 



t> i A* 3 ,,"] 3 . 2n n f 



t (n -nj-n> + -n"\a, = -n"-^, 



3 2n-n' 



