Brard 
REPLY TO DISCUSSION 
Roger Brard 
Basstn d'Essats des Carénes 
Parts, France 
In reply to Dr. Treshchevsky's comments, I would like to 
draw his attention to the fact that the equation giving the vortex dis- 
tribution on the hull is not singular when the motion of the body con- 
sists of a pure translation. If no free vortices are shed, the vortex 
distribution on the hull is equivalent to a normal dipole distribution, 
The density of the latter distribution is yielded by a regular Fredholm 
equation expressing that the interior determination of the velocity po- 
tential coincides with cx + constant, if the body moves in the x-direct- 
ion with the speed c. Hence one deals with the Fredholm equation for 
the Dirichlet interior problem. 
If the angular velocity of the body is not null, then the total 
vortex distribution on the hull is the sum of two distributions. The 
equation giving one of them is regular, The other one is singular. But 
it is possible to show that the irregular vectorial equation is equi- 
valent to a scalar Fredholm equation for an interior Neumann problem, 
The solution of this equation is much simpler. This point has been 
omitted in the preprint of my paper, but it will be included in its final 
version, 
When free vortices are shed, a point of importance is to de- 
termine the position of the shedding line. It may occur that several 
shedding lines exist simultaneously. The problem would be under- 
determined if no account were taken for the condition concerning the 
continuity of the pressure on the hull through each of the shedding 
lines. 
It is also to be pointed out that, if the shedding line is unique, 
then the vortex distribution on the hull given by the regular Fredholm 
equation and the free vortex distribution can be combined in sucha 
way that they can be replaced by normal dipole distributions. This 
probably leads to important simplifications. 
When one deals with a wing of finite thickness, the method 
using the acceleration potential does not apply in a simple manner, 
except, perhaps, if one can combine source and vortex distribution. 
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