Vortex Theory for Bodtes Moving tn Water 
We can take for S, a surface entirely located within Dj . Therefore 
the flux of 2-2" through S, isnull. Let % be the unit vector 
normal to the first sacle so that the pees directions M, ; Me ; nj}, 
vy make a right-handed system and let v5 be the unit aegior norinal 
to the second rectangle such that the three directions — Mi Ms n, 
5 also make a right-handed system. The vectors , , ro ¢ are 
tangent to x, VO at M‘ and to Se at M*, , respectively, and 
both are in the aietind towards B°. hokee ; 
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rec ). = jares Mj Me) Mh Mi), (A) pe 
: LS eeD A, (LT TRL l 
TT! 
+ [area M'! M, M, M; V5 ( z ) Me * 
Z Z Z 2 2 
The intensities dQ of the vortex ribbon L, and dr, of 
the vortex ribbon L, are given by 
r - = = 
dT = hice M M' M! M. #(- ae ’ 
1 1 1 1 1 
dr, = lie M' M M. M! 74. tae ’ 
zZ 2 2 2 
respectively. 
Hence 
" = ie < - 
r (C.) qr, ae | vis 7) + dt ( ’ > T 5) 
Similarly, vs being the unit vector normal to the rectangle 
Mg, ME M}, Mr, and such that the three directions —_ Mr =, 
nf, v-¢ makea nieke- -handed system, we have : - 
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