A Vortex Theory for the Maneuvering Ship 



satisfies the condition 



r in n^ , 



curl V^,(M) - j 2U . . , 



^- ly xn n. . 



Consequently, V^jCM) depends in n^ upon a velocity potential. Let f^ojC?^) be 

 this potential. One has 



\;^r,(^) = -grad <D^2^M) in Q^ . 



In n., Vo2 does not depend on a velocity potential; but 



V02CM) - ^ iyAOM 



depends on a velocity potential 0: 



-grad 0CM) + \q^(U) = ^ iyAOM in fi. . 



Let us now consider the velocity potential <i>^2(^^ defined in n^ and in n^ by the 

 distribution of doublets y'(m) on s so that 



^o'.CM) - -i;jj >'^"^)d^i[^S(m), (13) 



with 



1 '/ N 



y 7 (m) 



1 rr SI 



"7— 7'Cm') -] dS(m') = -0(m^) + constant 



One has 



Oq2(M) = c/)(M) + constant in 0.-^ . 



Hence, one has 



-grad (I)o2(M) =-grad0(M) = - Vg 2(M) + - iy AOM in D. . 



Therefore 



-grad <I>o'2(mi) + Vo2(nii) = - iy AOm- on S- . 



When M passes through the boundary layer, from m. to m^, the normal com- 

 ponent of grad Op' 2 is continuous. The normal component of y^^ is continuous, 

 too. Consequently 



827 



