Brard 



Po-/ (^) 



Figure 6 



(c) is equivalent to a distribution of normal doublets on the part Z(P, t ' - r') of 

 the wake surface ^(f) between the arc P'KP of flgi, and the arc PP^, ?;,?'. 



Obviously the two distributions on ^^(P) are equal and opposite. Hence the 

 vortex dpd^,r|(P,T') lying at the time t' on the arc P'm^,P and on the contour 

 PP^,P^,P' is equivalent to the sum of the distributions of normal doublets on 

 S'(P,t') and on 2(P, t' - t'). 



An identical reasoning applied to vortex dpd^, r'j(P,T') lying, at the time 

 t', on the arc Pm^,P' and on the arc P'P^,P^/P shows finally that, we have to 

 deal with two distributions of doublets, say on S'(P,t') and s"(P,r') and on 

 2(P, t'-r'): 



dpd^.WjCP, t') = dpd^,yi(M,t') + dpd^, y;(P,t') , 



(3) 



with 



dpd^,y;(M,t') = 43dpd 



U,.. ,. ' ^m' m'M 



'S'(P,r') 



/I 



r';(P,T') 5^^ — dS"(m") 

 s " ( p , r ' ) "^ m "M 



y (3') 



dpd^,y,(M,t') . - ^dpd^,r,(P,r') ^^ 



d J_ 



Z(P, t '- t') '^ ^ 



Now, let us consider at time t ' the potential 



d2(/x) 



vi(M,t') = -^/J -i(/x) d^^if^^/^) 



(4) 



S(t') 



834 



