From Archimedes' principle, or equivalently, satisfying Equation 

 [26] to zero order in t,, 



gm = pg S(z)dz [28] 



and thus 



JO 

 gKz dS ^^ ^ Q^j^4 ^Qg p^j j29] 



■H ^^ 



while, from Equations [28] and [25], 



2m| = - p r [i|/(z - Zq) - 2u)^Ae^^ cos cot] S(z)dz + 0(R^ log R) [30] 



Let us denote: 



I = mk^ 



X = 



pHS(O) 

 .0 



Q 



Pri = — ( (z - z^)"" S(z)dz (n=l,2) 



.0 

 (K) = -H- e^z (z _ z^r S(z)dz {n=0,l) 



and note that ^, t, , and i^i must be sinusoidal with frequency co. The 

 equations of motion then become 



2^ + Pj 4j = - 2AQ cos wt [31] 



(1 - XKH)^ = A(l - XKHQq) sin Got [32] 



{^2 + ky -] 4j + Pi ^ = - 2AQj cos wt [33] 



12 



