Eggers 



where v is the flow vector in any system of reference either at rest or in uni- 

 form translatory motion and n is the unit normal vector directed outward. If 

 we now select 



and define fi^ as the X component of 



ff = /o [(V • n)v - (v . v)/2n] dS , 



■Jn 



(17) 



where integration has to be performed over s^ , the hull surface up to z= 0, 

 then we have from Eq. (15), returning to nondimensional quantities, with e^ as 

 the unit vector in the X direction. 



R. 



pc2(L/2)2yg 



1 



V/JJ02 ^^^'n - 7o 



'/'xS^Y ^'^^^ 



.s 



^x'^7 ^^^'n + 7, 



IJ-JJ 



V+0z' 



dr]di 



(18) 



Reference to conditions C and D shows that the surface S^, s^ , and Sj^ may be 

 left out. The integral over s° may be transformed to line integrals along the 

 boundaries and an integral containing the function &(X,Y) in a similar way as 

 was done for the potential given by Eq. (2). The contribution from S^, including 

 the line integral from 5^° , tends to zero as x^ ^oo due to F. Thus, we are left 

 with 



-i 



K^^V) ^xi^'V^O) d^dr, 



-I 



V'x' [^.eF(^,0),0) dr, 



2 + ^0 2 ^^ 



-'-H 



dr] , 



(19) 



where the line integral over the load waterline Lp is again in counterclockwise 

 direction when viewed from above. 



There is no reason to assume that the contribution from s °, though 

 bounded in magnitude, should tend to a definite limit as x^ ^-a, nor can we pos- 

 tulate this for the contribution of the region X= x^. It is only by some property 

 of the Green's function G involved that we shall be able to evaluate the contribu- 

 tion of S(X,Y) to the resistance by an integration on Sf° in the vicinity of the 

 ship only. We shall now look for a relation between the quantities defined as R^ 

 and fig. To achieve this, we add an expression to the integrand of Eq. (17) 

 which has no component in the direction of e^.. We set 



658 



