Vortex Theory for Bodtes Moving tn Water 
hull be at rest with respect to the body and by a complementary con- 
dition expressing that the pressure is continuous through the line of 
shedding of the free vortices. 
Only the second vortex family has a physical meaning. But 
both are necessary for determining the hydrodynamic pressure on 
the hull. This is not really surprising since both the fluid inside the 
hull and the bound vortex sheet over the hull must be in dynamical 
equilibrium. A consequence is that the classical expression for the 
force exerted by the flow on an arc of vortex filament which does not 
move with the fluid cannot be readily extended to the case when this 
arc belongs to the bound vortex sheet adherent to the hull. The 
hydrodynamic pressure on the hull is expressed in terms which only 
depend on the total vortex distribution. The dynamical problem is 
thus completely solved for any given hull in any given motion what- 
ever the incident flow may be. 
The theory developed in the present paper is quite general. 
Its application to practical ends does not seem to lead to insuperable 
difficulties provided that reasonable assumptions can be made con- 
cerning the position of the free vortex sheets with respect to the 
body and the possible variation of that position with time. In any case, 
it is shown in the last section that an older and less complete vortex 
theory is still useful in maneuvering. Thus it is hoped that the pre- 
sent one can guide the experimental and theoretical researches 
which are to-day urgently needed. 
I. A BRIEF SURVEY ON VORTEX THEORY 
The vortex theory can be divided into four parts. 
(i) The first part is, in fact, a chapter of Vectorial Analysis. 
The vector V is the velocity of the fluid points in a certain fluid 
motion at a fixed instant t and the vorticity » is defined as 
@= curl V = VAV. Cra) 
The starting point is the Stokes Theorem, according to which 
the flux of @ through an open surface is equal to the circulation of 
V in the closed circuit consisting of the edge of the surface. A con- 
sequence is that no vortex filament can begin or end in the fluid. A 
vortex filament is therefore a closed ring or its ends are located on 
the boundary of the fluid domain, or at infinity. A consequence is 
that the intensity of a vortex tube is a constant along the tube. The 
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