Eggers 



For the exact potential i//, no such condition holds. But we define a function 



S(X,Y)by 



7o02 + ^xx = S(X,Y) . 



C. On S( and s^ we have 0y= 0? ^y^^ " ^' ^"^^ ^y = ^• 



D. On Sj^ we have ^z = 0, sA^^^ = 0, and G^ = 0. 



E. On S^ we have 4j^- ±eFy^/^/e^¥^[T7WJTT, where ± stands for rj positive 

 or negative and the index n stands for derivation in normal direction out of the 

 fluid's domain D. On S° * , the projection of s^° on the plane Y= 0, we have 

 'l^Y^^- ±eFx for the first-order potential. 



F. For fixed P' we have 



G = 0( 1 ) as X - -00 ; G = 0(X" ^ ) as X - +C0 , 



Gx = 0( 1 ) as X -> -co ; G^ = 0(X- ^ ) as X -> +co . 



Application of Green's theorem shows that // '' and Z- can be defined sub- 

 ject to the same modes of asymptotic decay, provided the quantity H^.Y) will 

 then turn out to be well behaved. The symmetry of functions 'P and 0^ '^ with 

 regard to the lateral coordinate Y will be taken for granted by the symmetry of 

 the ship sections and the tank profile. 



We should bear in mind that (3) for the function P existence as a harmonic 

 function is — if at all — guaranteed only in domain D but not necessarily in D°. 

 However, low-order approximations have been derived (3) which are found to 

 exist in the whole interior of D° . For the moment we seek terms up to second 

 degree only; we shall in the following formulate the problem for the domain D° 

 with boundaries known a priori in favor of a less intricate analysis, and derive 

 approximate solutions to this auxiliary problem by perturbation techniques. 

 Then, for point P within D° , we may apply Green's theorem to functions p and G 

 to find a representation of 4j(P) as 



0(P) 



= 4;; J [^n 



,(P') G(P,P') - 0(P') G„(P,P') 



dS' , 



(1) 



where the closed boundary S' is composed of s° , 5^° , s^ , s^, S^ s^, and S^ 

 and the subscript n stands for normal derivative outward in P' space. 



From conditions C and D we may conclude that the contributions of S^, S^., 

 and S^ can be omitted on the right-hand side. 



The integral over s/*, where the normal derivative is in z direction, may 

 be transformed by integration regarding ^ and use of B: 



652 



