Brard 
= 
Within «D3'; .curl Vv! = curl V~. There exists therefore a velocity 
potential®, , such that 
This potential is then the solution of a Neumann interior problem. 
We can put 
-1 
1 = ! 
©, (M me) ff ial ) ade’, (6. 6) 
o(P', t) 
with 
it 
— o(M',t) 
5 deh) ae 
= ) 
ae BAS ie” 
Eq. (6. 6') is singular, but the cotton fa Vo dl,= Gis satisfied 
at every t and is determined up to an saat constant within D; 
Eq. (6.6') replaces Eq. (5.7), and it is much simpler. 
Let U, >» Us, Uy denote the components on the moving axes 
S of the velocity Vp (0) of their origin O and u, , us , Ug those 
of their angular velocity Q ,. According to (6.6) and (6.6'), we 
may write : 
6 
(iM, t). = 2 @ (Myra (ee (6. 7) 
In this formula, M is the point moving with the body which coincides 
witht Mi! tatiti= t,he DRheg @ lj 's only depend on the hull geometry 
and on the position of the system of axes S with respect to the body. 
Furthermore, it follows from the second equation (6. 1) 
that 
(ap) ! z= ! S, ! bi ' 
q (M t) ® (M t) © (M', t)) .(M', t), 
with M'cQo and M'M! = mie (0)... (6. 8) 
Consequently, the density uw of the normal dipole distribution on Pi 
which generates ao. the solution of the regular integral equation : 
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