A Vortex Theory for the Maneuvering Ship 



(21) 



Of course 



y,m = yoo^^oijj 



^oo being due to r^^ and VgiCw/U) to TpjCw/U), and, similarly: 

 When u/u , Lc^/U :|= , we have: 



u w Lq 



, , u , w , Lq . 



(22) 



with 



'Poi = '^oi + >^oi' (i = 0,l,2) 



(23) 



Now let us assume that y^g is known. Since I'oo must satisfy the condition 

 WqqCM.) = constant, when M. is in fi^, the density yoo(^^ on S is given by the 

 Fredholm's equation of the 2nd kind: 



i'yoo(^) - ^ \i yooi"^')^, 4-dS(m') = -XooC'"!) = -^oo('") • (24) 



1^ and m being on the same normal to S and infinitely close to one another. 

 The solution of Eq. (24) is: 



7oo(m) = -2yoo(m) + JJ A(m,mi) y^oi^^i) dSCmj) , 



(25) 



where A(m,mj) is the "solving nucleus" of the Fredholm's equation. 



We observe that VooC'") is discontinuous when M is crossing through 1; the 

 discontinuity is: 



^oo(M') - Yoo^M") = TqCm) , with M'M" = en (e>0). 



831 



