50 LECTURES ON THE LUNAR THEORY. [LECT. XI. 



Multiply the second by _- -,-- , and add to the first ; this will 

 eliminate 6 21 , and gives 



- 4 ( n - ri) (n -p) ( 1 



Lastly the equations obtained by equating the coefficients of 

 c cos (2t/ + ut ra) and <" sin (2i/ + nt vr) 



to zero are 



a 

 + |(8n - 2n' -p)- + ~\ a,, - 2n (3n - 2n' - p) b,, = 0, 



and 



4 ( - n') (n -p} ( 1 + &) a a + 2(. - ') ( - p) b, + 3w' 2 ( 1 + & ) 



- (3n - 2n' -_p) 2 /; + 2, (3 - 2' -p) a,, = 0. 



Multiply the second by ---- and add to the first ; this will 



3ii - 2n' p 



eliminate b^, and gives 



I" _ 2 (n - n') (n -/>) + 5 ^ - - - (n - n') (n -p} ( 1 + 6.) 1 a, 



L 2 a j 3n 2n p 



+ [ - 4 ( - ') ( - ,>) ( 1 + (..) - to -*?-_-( - ') ( -P)] >>. 



These six equations are to be solved by successive approximation ; 

 taking the first rough values of p/n and b a , we find from the last two 

 pairs values for a ai , b. a , a w , &.,,; these are substituted in the first pair and 

 yield more approximate values of p/n and 6 , and so on. 



It will be noticed that this complexity is made necessary by the fact 

 that OH, &, are found by means of a small divisor (n 2n'+pf 4 a + , 



or 



