A Vortex Theory for the Maneuvering Ship 



Let ^',17' be the absolute abscissa and ordinate of /j on ^t') . The density o-^dx) 

 on the area d^'drj' is the effect of the free vortices shed during the interval 

 (0,t') along the arcs PPj and P'PJ, with yp^-q' (or yp, - tj' when r/' <y(K)) and 



r' = t 



X'(m)-^' 



X'(m) being the absolute abscissa of P (or P'). Consequently 



and we may write 

 ^'ifM,t') --^\ dV r,(^^,t' - ^^ j^ ;iM^^^^)' (4a) 



•^y'(P;) ■'X'(7,')-Lt' ^ 



where x'c^?') is the absolute abscissa at t' of p (or p'), and XCt^') - Lt ' its 

 absolute abscissa at t' =0 (see Fig. 7). 



Figure 7 



An equivalent for this expression is: 



YjCM.t') = - 4^ f dyp [ r^(yp,r') ^ 



X {[x(M)-x(P) - L(t'-T')]2 + [y(M)-y(P)] 

 + [z(M') - z(P)]2} dr' . 



(4b) 



At time t ' , the potential yJCM, t ' ) due to the distribution of normal doublets 

 on s' and S" is: 



835 



