THE VARIATION. 



/ 7 j. \ n 



in' -J2,) cos 2(6- 6'} = TO-' [(2 - TO) b, + cos (2 - 2m) 6 + (2 - 3m) b, cos (4 - 4m) 0], 



\ <**7 



= [1 + h.; - 2h., cos (2 - 2m) d + (h? - 2h 4 ) cos (4 - 4m) 0]. 



-fj. Iv 



Substitute these in the equation above, and equate the coefficients of 

 corresponding terms, 



13 33 UM 



1 + - m 2 + - TOT (2 m) o., + - m 2 a 2 + - m 2 (2 2m) = ~ (I + h?) 



(2 2m)- a,+ ( 1 + - m'J , + - ?n 2 + (2 2m) m*h, = ~ ( 2h 3 ) 



(1 \ 3 33 



1 + - wr 1 a t + '- m" (2 3m) b., + - m-a,, - m" (2 2m) a, 

 2/2 44 



If we neglect at first terms of the fourth order, we find from the 

 first of these equations 



fM 1 



h 2 



From the earlier set of equations we have 



(2-2m)b 1 = -h,-2a 2 ; 

 substitute this in the second equation above. We get 



- (2 - 2m)- + 1 + - TO- - 2m- a., - m%, + - m" = ( 1 + - TO 2 ) ( - 2/i.,), 

 L J 2 \ 2 ] 



or 



11. ...\_ 8_ 3.3 m 2 3.._,2-TO 



so that 



and 



3 2-m I 



a., = ^ 



2"' 1-m Q 11 ,' 

 3 8m + TO" 



Ll 



l-m 2-2m 



1 3m 2 



ft, 



8 (l-m) 2 ' 



42 



