cos(n,x') = - cos + O(R^) 



cos(n,z') = -^ + 0(r2) 

 dz 



and the forces along the (x, z) axis are related to the forces along the 

 (x' , z' ) axis by 



^x " ^x' ^°^ "^ ^ ^z' ^'" ^ = F^, + 4) F^, + 0(4;2) 

 F^. = F^, cos 4j - F^, sin ^ = F ^, - i\j F^, + 0(^^) 



Thus the equations of motion may be written in the form 



rZn ,^*-t. + x> . 

 mi = (- cos 9 + 4j ^j pRdz' de' 



.Ztt ^^*-^,+ x> .^ . 



m(t, + g) = I j^+ ^j; cos e] pRdz' de' 



.Ztt -;*-C + x'4; 

 14/ = [{z' - Zp) cos(n,x') - x' cos(n,z')] pRdz' de' 



-'0 J-H 



.Ztt rtj^-l + x'^ 

 = - \ (z'- Zq)cos e pRdz' d0' + O(R^) 



where ^ is the free surface elevation at the body. Substituting Equation 

 [21] for the pressure and neglecting second-order terms in the oscilla- 

 tory displacements |, ^, 4'. and A, we obtain 



m^ = - Pg I I [- cos 9' + 4^ ^] (z' + ; - 4jR cos e')Rdz'de' 



j cos 0' [| R cos 9' + L|JR(z - Zq) cos 9' 

 •'-H 



+ gAe^^' sin wt - Zw^Ae^^ R cos 9' cos wt] Rdz' d9' 



10 



