Summing up the above equations, we obtain 



= I (w n.+u n n +v n„) tbdJ 

 JJ z 3 z 1 z 2 



-IS 



|^cf,dS 

 dn 



Then the relation of Equation (320) follows when 1=1, 2, 3. It can be 



shown that a similar derivation is applied to the cases of i = 4, 5, 6, 



if we introduce the definition of n. , n c , n, in Equation (314). 



4 _> b 



Reverse Flow Theorem and Haskind's Relation 



Applying the above theorem to Equation (318) and making use of the 

 relations 



8$. 



Sty. 



= n ,-> 



(321) 



dn "!' 9n 

 one can express the forces and moments in the form like 



F.. = e 1Wt P C (ico$.+iiO |- (ia>*.-i|>.) dS 



ij i JJ K i y i' 9n j V 



b 



where the simple harmonic displacement is expressed by e £.. The line 

 integral term vanishes because we have assumed that w = on the plane 

 z = 0. Now we assume that $ . and lit. satisfy the linearized free surface 

 condition such as 



2 3 *i 2 d \ 



- wO. + 2icoU -^ — + U — x- 

 l 3x ~ 2 



dx 



2 ^i 2 ^J 



- CD ill. + 2icoU -r-^- + U — -^ 



i dx ^2 



9x 



+ s ar-=o 



+ siz-=° 



(322) 



and the radiation condition. Next we define velocity potentials $ 



* 1 



and ty which satisfy the same boundary condition on the hull surface as 



114 



