m(s-,_ + a22_k-|_h) v + [s-|_ (c-|_ + BS) - a2ik-|_CS] v 



+ k^ (1 - a^^BS^) V - a^2^^G = 



(s^hm + a^^k^Ig) v + [s^LBS + a^^k^ (c^ + E^S)] v 



- a k LBS V + s G = 

 21 1 1 



and if V and G are proportional to e : 



BS 



{- oj^m (s^ + a^^k^h) + k^ - a^^k^ 



+ iu [s (c + BS) - a k CS] i V - a k G = 

 11 21 1 ' 12 1 



|- 0)^ (s^hm + a^^k^Ig) - a^^k^LBS^ 



+ io) [s^LBS + a^^k^ (c^ + E^S) ] [ v + s^G = 



Equating to zero the determinant of the coefficients of v and G and 

 dividing the imaginary part by iw yields: 



2 2 



s [-03 m (s + a^-ik h) + ^ - a2-|^k^BS ] 



2 2 



- a k^ [w (s,hm + a2-,k^Ig) + a2-|^k^LBS ] = 



^1 ^^1 '^1 * ^^^ " agi^l"^^^ 



+a k [sLBS+a k (c +ES)]=0 

 12 1 1 21 1 2 L 



Now multiply these two equations by k , substitute into them 



a a k k = s s from D = 0, and then divide the equations thus obtained 

 12 21 1 2 12 



by S-, on the assumption that s^ ^ 0. Equations [53c, d] with D set equal 

 to zero are the result. 



56 



