Vortex Theory for Bodies Moving tn Water 
Casec - 
The x'-axis is taken in the direction opposite to the mean 
velocity Vir of the origin of the system of axes moving with the body. 
Let uy be the absolute value of the x'-component of V},. The sup- 
port of the vortex sheet may be considered as generated by half 
straight lines starting in the positive x'-direction from a certain 
line on a . Let Py; and B; denote two points on the same gene- 
ratrix, By being infinitely close to G. We put & = |B} P}| . Let 
0 = - © be the time of the beginning of the motion of the body and 
t, the present time. 
If the fluid motion were steady as in case b, one would have 
M-(Py,t.) = u,(By,t,). Because it is unsteady, one has 
For reasons which will be elucidated later, we put 
(By 
’ > = ’ a 6M He ; 
pw AP? t 30+) = (Bet) eg 
' of ATS ! 0 ! emer 2 i 
(Pp fist t') uw (Bi t.) u(P rt st -t ), with (7.26) | 
One sees that, if u(Be, t) is a constant in the time interval (t',t ), 
then ys(P},t,; t,-t') is equal to that constant at t, , provided 
E<ug(to-t'). For > up(t,-t'), then pw ¢ at P} still depends at 
time t, on the variations of w,(Bj,t) for t< t'. In particular, the 
1241 
