A Vortex Theory for the Maneuvering Ship 



AX;(t') = -X[^'\t') + A'x;(t') (29) 



with 



and 



X;^'\t') = "/^Ldt^ (u) , If >''i^'"'0+) n(m)i^dS(m), 



(30) 



A'x;(t') = -p^ 



(uLlf ^ S7;('n.t')n(.)i,dS(.) 

 " s 



+ [ ^ (u) , '^'^' JJ bT^ ^^l'^'"' ^'"^'^ n(m) i^(m) dS(m) 



This expression may be written: 



A'Xi(t') = |pAU2a,|(g)^^>A(0) +1 (Ut ^^^''-"'^^"'}' (2^) 



with 



"^(^'^ " ALUi: JJ ^ Sr:(m,t') n(m)i^dS(m). (32) 



^ s 



The first formula (25') shows that, at t ' = + , the deficiency is, in case 

 "a J," less than 1. 



We have, moreover, components z^ (t'), x[ (t') and -A'zJ(t ') , 



A'x;(t'). 



Although 7*(m,o+) - 0(1), the component z[ ^ (0+) is small, because firstly, 

 y*(m,o+) has significant values only when m is close to Q^^, and secondly, in this 

 case, n(ni) is nearly normal to i^. The components Xj<^^Vt') and-A'Xj(t') are 

 small too, for the first of the previous reasons, and also because the projected 

 area of the body on the (y, z)-plane is small with respect to A. 



Now, let us consider the moment of the momentum. It is 



Wi(t') = w;(t') + w';(t') , 



with 



855 



