Brard 
T being determined in that manner, the velocity V' gene- 
rated by the vortex distribution aa on )> and te Wp inside Dj; 
is equal to VE within Dj Therefore, as the jump of v' eaegn 
‘Ss is purely tangential, one has n V' = n Ve on os and V' 
evidently coincides with the velocity V_ of the irrotational fluid mo- 
tion inside D, 
The above results can be extended to the case when there 
exists inside D, some incident flow of velocity Vig . It suffices to 
replace VE oy TS - Ve into the right side of (3. ‘9). The resulting 
velocity is 
i= inside D. ; 
V inside D 
e 
If De were bounded by solid walls and (or) a free surface, 
one would have to add singularities distributed beyond the boundaries. 
These singularities would be linear functionals of T and therefore 
T would be found on the right side 6f (3.9) too. 
Furthermore, it is seen that the incident flow V_ could be 
due totally or partly to free vortices shed by the hull itself. In such 
cases, the right side of (3.9) would depend upon the history of the 
motion of the body. 
Various remarks 
(i) It is to be noted that the vorticity inside Dj; could be chosen 
1 2a 
1° 49 p; MM! oe 
The possibility condition (3. 10) would still be fulfilled. And, for the 
same reasons as above, T would still be determined uniquely by 
(3.9) -and.(3..9"),, But, inside D; , the resulting velocity could no 
longer be identical with VE ; 
arbitrarily. Te should be replaced by a 
In all the cases, the resulting velocity Vv on), = 1. ey 
between )); and >, - is given by 
V (M) => | Faw) " ¥ (a) (3. 12) 
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