a^ = H2/D{l+j6 + €[(HiH2/d2)-J(Hi/D)] + Q{e'^)}i30c) 



/3i = -Hi/d{1+€[(H3^H2/d2)-J(H2/D)] + Oie"^)} (30d) 



Note that positive roots are chosen except for p^. Using a and /3 

 from (30) it can be verified that (25c) and (26c) are correct to the 

 order e. 



b) Normal Mode Equations 



To express Eqs.(ll) and (12) with all dependent variables in 

 terms of modes, (8) is rewritten for each mode as: 

 Me = ^e"l -^ ^^e^Z 



Mi = a^Mi + ^iM2 



*e = ^e^l^l ■" ''e''2^2 



(31) 



*i = "i^l^l * ^i^2^2 



Gi = GiFi + ^iF2. 

 It readily can be shown from (31) that 



Pl^i = r(^e0i - &j_4>^) 

 P2^2 = r(-ag*i + a^^g) 



Ml = r(0eMi - /JiMg) 

 14 



(32) 



