X.] THE ANNUAL EQUATION. 43 



Hence we get the equations following : 

 Equate to zero the coefficients of cos a, sin a : 



'* + ^} a, - 2n' ( 1+ n') &. + 2 (2 - n') a.& 6 + ^ . | a 2 a 6 

 ct I ct 



- 2 (2 - ') &A + ^ w' 2 ^ + 2 (2 + n') a,a 7 + % . | 2 a 7 



*i Ct Zi 



7 I /yi' 2 /i 'w' 2 /I *-l/\ 



n'\ 3n'-bJb 6 + 2 ( 1 + n') n'a 5 + 2(2 n') a,J} 6 



- 2 (2 - n') bM, - - n'\ -2(2+ n') aj) 7 

 3 



Equate the coefficients of cos(2i// a), sin(2i| a): 



(2 - n')" + 3 ^ a, - 2 ( 1 + ?/) (2 - w') 6 6 + 2i'.,o, + 3 ~, . - a.,a 

 ' a j 2 



3 21 



- (2 - w') 2 6 6 - 3w /2 &A + 2(1 + n') (2 - n') a, + 2'a.,6 5 - 2w'6 2 a 5 



3 W7 21 



Equate the coefficients of cos(2/ + a), sin (2// + a) : 



(2 + n') 2 a 7 + 3 ^ 3 a 7 - 2 (1 + n') (2 + n') 6 7 - 2w'o i a k + 3 



- (2 + w') s 6 7 - 3n' 2 &A + 2 ( 1 + n'} (2 + n') a 7 + 2n' 



/ 



In equations of this class, as a general rule we would determine 

 , 6 5 approximately from the first pair, substitute them in the second 

 pair and determine a 6 , 6 6 approximately, and similarly a,, 6 7 from the 

 third pair, and repeat this approximation as often as might be necessary. 



62 



