Vortex Theory for Bodtes Moving tn Water 
a vortex filament 292 ~ interior to Dj necessarily belong to the 
same line £ '. 
— ae 
The vortex family (ee a oa oe a ai ) 
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The-velocity V'' induced by this vortex distribution is given 
m (rT). my, | T.(M') 
V''(M) = oof ff adu(M') aed, aarti 2,009] , 
f (5,13) 
V" is irrotational everywhere except on oz and on ae Re On, =e ; 
one has V"(M,) = (V - Vo) Me and, on Qj, V"(M,) = (V,-V'-Vo) us, 
Accordingly : 
V'"'(M_)-V"(M.) = V2 (M,) 4 WiUM. (5. 12) 
Let's, -0 _denote two unit vectors on oy , t (M) being in the di- 
rection of (T - T')y , so that the three directions (Se Sar} 
make a right-handed system. It follows that @ (M) is in the direc- 
tion of Vr(Me) + V'(M;). The curves @ tangent everywhere at 6 
and the vortex filaments LY on be define a system of orthogonal co- 
ordinates (o, s) increasing in the directions of @ and 7 ,» respec- 
tively. 
We may consider V" as gene'rated by a normal dipole distri- 
bution on >3 + yoy . Let # and yw ¢ denote the density of the distri- 
butions on) and a respectively. We have: 
© (M) ff bw (M"') a ae aos ia) (5. 13) 
2, 
lg1s 
