where C and C' are defined as 



C = j V^Gdv - r [8^-(f+iF9^+ie)^] 



Gdxdy 



C = f (())-()) j^)V^Gdv - r ((f)-(l)^)[9^-(f+iF9^+ie)^]Gdxdy (4.5) 



It may be seen from Equations (3.3) and (3.4) that we have C = if ^ - ^^ = ({"(x) 

 - ^(K) ->■ as X ->■ 5, that is if the potential is continuous everywhere in the solu- 

 tion domain d and on its boundary a + h + c, as is assumed here. Use of Equation 

 (4.4), with C' = 0, in Equation (4.3) then yields 



C(})^ = f GV^(l)dv - r G[9^-(f-iF9^+i£)^] 



({)dxdy 



+ j (G9(j)/9n-ct)9G/9n)da 

 h 



+ F [2i(f+ie)Gcl)+F(G9(j)/9x-(t)9G/9x)]dy (4.6) 



Let t(t ,t ,0) represent the unit vector tangent to the curve c oriented in the 



X y 



clockwise direction, as is shown in Figure 1. On the mean waterline c, we have 



dv = t d£, where dl is the differential element of arc length of c. Furthermore, we 



y 



have 9<})/9x = V({)*i, where i (1,0,0) is the unit positive vector along the x axis. This 



then yields 94)/9x = [n9(t)/9n+t9(l)/9£+(nxt)9<t)/9d] -i = n^9(l)/9n + t^d<^/d!i - n^t^9(})/9d, 



where (n .n .n ) are the components of the unit outward normal vector n to the hull 



x' y' z ^ 



surface h, 9())/9«, is the derivative of (J) in the direction of the tangent vector t to 



->- ->- 

 c, and 9(l)/9d is the derivative of (}) in the direction of the unit vector n x t, which 



is tangent to h and pointing downwards. 



15 



