Brard 
—_> 
where -dF* is the force exerted on dE' by dE, . From Euler's 
equation (4.1), we have inside D': 
¥ = a5 (curl V_) A(V, + 6V") -V : (V+ v1)? 
Bo wae at o i a ate : 
This gives, when R——-0O, Pq =tS4p V; ’ 5V' + constant on S, and 
é a =>, 1 =F a 
iim — T(E") = —-dr -— 2 AV Xde (7.45) 
dt n: SZ i 1 
R>0 
dF. Set oa whi eli (7.46) 
dF ip is the force exerted on the arc ds, of the bound part 
Ly of & by the adjacent sets of fluid points. The force vanishes 
with Vj, that is when the arc ds, moves with the fluid. 
Formula (7.46) is of practical interest when the vortex dis- 
tribution equivalent to the hull is replaced by a unique concentrated 
vortex and a suitable distribution of sources or normal dipoles on the 
hull surface. 
This formula does not imply that the fluid motion around the 
bound arc of the vortex filament is steady. If [ varies with time, one 
has to consider that another vortex filament &#’, of intensity dI , 
appears in the time interval (t, t+ dt). & is distinct from L although 
their supports have a common part ; the free part of &" does not coin- 
cide with the free part of ae 
Let us consider now a flat vortex tube inside the bound vortex 
sheet over a hull ye . This tube has athickness ¢€ , a width do 
Let ds, be the element of arc of the tube. Applying (7.46), we have 
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