Although the quadratic terms in Equation (141) seem to be troublesome in 



30 

 evaluating the above integral, a theorem which has been proven by Tuck 



becomes a powerful aid. The theorem is proved in Appendix B. Consider a 



velocity vector V of an irrotational motion of an inviscid fluid outside 



the hull, which satisfies the boundary condition on the hull surface such 



as 



V • n = (143) 



where n is a unit vector along the outward normal to the hull surface. 

 Next we define the vector m by the relation 



9V 

 m = - 3- (144) 



Further we define vectors n and m by 



n = r x n 



\ (145) 



5 " - to ( ^ Xn) J 



where r is the position vector (x,y,z). Then, the following relations are 

 valid. 



If 



[mcjH-n(V'Vcj>)] dS = - 



n d> w d s 



S L o 



(146) 



[m (J)+n (V- 



VV.) | dS - ! n (j) w d s 

 L 



where w is the z-component of V and L is the still waterline of the hull. 

 The line integral on the right-hand side is omitted when the body is 



55 



