+ dx {i(jJ g;L (x)+Ug^(x)} f (l+Vz) |^r dc 

 J ^C(x) 



(151) 



where C(x) is the contour of each section along which the integral with 

 respect to c is taken. On applying the above mentioned theorem to the 

 second term, we obtain 



,(2D) 3z , 

 F„ = p|dx| i w <Jr y -»— r dc 



,< 2D > dc 



J J C(x) 



J ^C(x) 



+ p j {ico gl (x)+Ug^(x)} {-B(x)+VS(x)} dx (152) 



The last term is derived by the application of Gauss' theorem in which 

 B(x) and S(x) are breadth at the waterline and sectional area of each 

 transverse section, respectively. If we write the z-components of X and i£ 

 by x an( i *P » respectively, the two-dimensional part of the radiation 

 potential is expressed as 



(J) (2D) = {iw(z -xij;)-lty} x + U(z -aap) ip (153) 



g z g z 



Then, the vertical force is written as 



3X 



r 2 f dx z 



= - p I dx {(*) (z -xty)+iwlfy} x TT dc 



j C(x) 



r r 3x z 



+ i p a) U dx (z -xifO t\) tj— r dc 



J J c ,_ 



U dx (z -xlJO 



U 2 jdx (z g -x*) f ^ 



J C(x) 



- i p to U | dx (z^-xip) | x "a - r dc 



X z 

 C(x) 



- p U | dx (z„-xi|0 I ^ _-£. dc (154) 



(cont. ) 



57 



