Vortex Theory for Bodtes Moving tn Water 
® is therefore generated by a distribution of normal dipoles over ). ; 
with the constant density [.. Of course, ® is not single-valued. 
If C is a closed circuit intersecting a, at P and surrounding 
one time, then the circulation of V in that circuit is 
[Fe A Adrabeae wiTe ths Feet the pete, pies 
G 
Application to vortex sheets 
Let us consider the partys ye of a vortex sheet. The contri- 
— 
bution of ‘aa to the velocity V is given by 
av =— curl ee |] Sa Neer dd (M (2:2) 
III. VORTEX DISTRIBUTIONS KINEMATICALLY EQUIVALENT TO 
A HULL SURFACE 
Let be denote the hull surface of a solid body completely 
submerged in an unbounded, incompressible fluid at rest at infinity. 
Let Vy denote the velocity of the body and Q E its angular velocity. 
One has 
ay = we = 
On curl E 7K): E 
Let D- - resp, D, - be the intervor.- reap. the exterior — of ay , and 
Se ; o. the Nie sides of yu: For convenience, i ‘S D; and 
de @ De.) The-unit- vector normal to > is in the Saeed By eeemar 
et V ener the velocity of the fluid inside D,. One has 
nV_ = a V -om yi 2 Gu 
Poincaré's formula applied to VE inside D, gives : 
EES 
