A Vortex Theory for the Maneuvering Ship 



)U JJ [y;(m,0+) + hy\(m,+v.)] i^n(m) dS^m) I (J) 



+ Pl 



The difference between the two moments is 



A)ii;(t') - nioi (^) ^ - %[^"\t') + A')K;(t') 



(33) 



(34) 



with 



}K;^'\t') = -P ^ ^ (g) , Jj7'i(m,0+) [z(m) n(m) i ^ - x(m) n(m) i ^ ] dS(m) . ^35^ 



^ ' s 



In order to get ls'%[(t'), we will reason as above. 



The contributions of the first terms in ')Rj(t') and in !)lIoj(w/U) j- lead to 



puj r„/T^')xx(7]')xdT,j(J)^Jl-F*(t')] + I ^ (^)^^[l-F*(t'-r')]dT'| . 

 01 ° "^ 



The second term in ^K^ j(w/U) ^,, and the last term in ^[(t') give: 



PU Jj{(n)^^ [SrtCm.+co) - S7;(m,t')] 



+ J ^ (u) ,[S7*i(m,+oo) - Sy;(m, t '- r ' )] dr ' I i ^ n(m) dS(m) . 

 0+ '^ J 



Lastly, the terms in the 3rd and 4th lines in the expression of %[(t ') give 

 the contribution: 



X {z(m) n(m) i^ - x(m) n(m) i ^ } dS(m) . 



Integrating by parts, we obtain 



3t 



- Sy\(m, t'-r') 



dr' 



857 



