I (t)V^Gdv - r (|)[8^-(f+iF3^+ie)^]Gdxdy 



(t)dxdy 



+ I (G8())/8n-(})3G/8n)da 

 h 



+ F j [2i(f+i£)G(|)+F(G9())/9x-(})3G/3x)]dy + I^ (4.3) 



where the term I^ is given by the integrals 



I = 



r ((f)3G/3n-G8(j)/8n)da + F [2i(f+ie)G(t)+F(G3(j)/3x-(|)3G/3x) ]dy 



We have G = 0(l/r) and (j) = 0(l/r) as r E (^^ 2_^^2^1/2 ^ ^ ^^ ^^^^ ^^^ ^^^^ ^ 



' oo 



vanishes as the large surrounding surface h^ is made ever larger. The term I can 



then be ignored if the finite domain d'' and the finite region o^ of the mean sea 



plane are replaced by the unbounded mean flow domain d and the unbounded mean sea 



surfspp n outside the mean hull surface h and its intersection curve c with the plane 



z = 0, respectively. 



By expressing the potential cj) in the integrands of the two integrals on the left 



side of Equation (4.3) in the form <p = <^^ + ('p-'t'^) , where (j) E 4)(x) as was defined 



->■ _ ->■ 



previously, and (J)^ represents the potential at the field point ^, i.e., (f)^ = (})(C), 



we may obtain 



I* (j)V^Gdv - r (})[3^-(f+iF3^+ie)^]Gdxdy = C(j)^ + C' (4.4) 



14 



