A Vortex Theory for the Maneuvering Ship 



So Eq. (20) is quite analogous to the equation which yields the circulation 

 around an airfoil of infinite span in case "aj." 



Equation (20) is not convenient for numerical calculations. But, if the 

 nucleus H( t ' ) was really known, as in the case of the airfoil of infinite aspect 

 ratio, it would be possible to solve it after some transformations. Equation (20) 

 is equivalent to 



t ' e t ' t ' 



f A(t' -6)66^ F*(t') H(0- T')dT' = r A(t'- 6)66^^ A(T')dT', 



or to 



f F*(r') f A(t' - 0) n(e-T')d0 dr' = f A(T')dT'. 



The nucleus in the brackets is 



1 



t'-r') [ A[(l- \)(t' - r')] H[\(t' -T')]d\ = K(t'-T'). 



(22) 



If we choose A(t') in such a way that K(0) = l, what implies only 



A(t') = 0{F*(t')} 

 for t ' small, we obtain, deriving with respect to t ' : 



(23) 



I 



F*(t') + F*(t') K(t' - T')dT' = A(t'), 



(24) 



what is the wanted form of (20). 



Now consider the density 7*(m, t ') of the distribution of doublets on the hull. 

 Since 



when t' is small, and m close to d^^, one has 



7*(m,0 + ) = 0(1) . 



This result is compatible with the conditions 



(25) 



841 



