M2 = rC-a^Mi + a^Mg), 



where r = (a^p^ - a^p^)'-^. Using (30a) to (30d), the factor r is 

 (P1/P2) to order e and 



h^ = [0 + (H3^/D)0g](PiP2) "^ 



h2 = [(eH2/D - 1)^ + (H2/D)^g](piP2) '^ 



(33) 

 Ml = (Hj^/D)Mg + M^, 



M2 = (H2/D)Mg - M^. 



Using (31) and (32) the coupling terms defined by (13) can be 

 approximated by, 



Bg = rgrge^^V(Hj^/D), 



B^ = -gri0gV(H3^/D), 



(34) 

 Cg = r€M^-V(Hi/D), 



q = -Mg-V(H]^/D). 



These forms show clearly that the external mode is influenced by 

 the internal mode and vice versa when H^/D is variable. 



It can be shown by substituting (34) in (23) that the energy 

 transfer between modes, T, is zero as required. 



Finally, the forcing terms G can be approximated by using (5), 

 (8), and (31) as: 



15 



