Equation (4.6) can then be expressed in the form 



C(j)(|) = ij;(t) - Lit-A) (4.7) 



where C is given by Equation (4.5), ijj(f) is the potential defined as 



^(t) = I GV^({)dv - r G[3^-(f-iF9^+i£)^](f)dxdy 

 d a 



+ G8(J)/3nda + F^ Gn t 8(})/8nd£ (4.8) 



h c 



and L(C;cj)) is the linear transform of (j) defined as 



L(t;(j)) = j (f)8G/anda - 2i(f+ie)F f G4)t d£ 

 h c 



+ F^ I [(l)3G/8x-G(t 9(t)/8£-n t 9(})/9d)]t d2, (4.9) 



J X z y y 



Use of Equations (3.3a and b) and (3.4a and b) in expression (4.5) for C then shows 



_ -*- 



that we have C = 1 if the field point ^ is strictly outside the hull surface h, in d 



_ -)■ 

 or on a, whereas we have C = if ^ is strictly inside the ship surface h. It can 



also be seen from Equations (3.3) and (3.4) that we have C = 1/2 if the point C is 



exactly on the hull surface h or on its intersection c with the plane z = 0, at 



least for points ^ where the hull h + c is smooth; more generally, the value of 



47tC (or 2ttC) at a point ^ of h (or c) is equal to the angle at which d (or a) is 



->■ 

 viewed from the point C- We thus have 



16 



