Brard 
ship hulls could probably be found. A fact is that an arc of vortex fi- 
lament does not generate a velocity potential and in practice the re- 
presentation of the hull by a source distribution can be simpler than 
its representation by a vortex distribution. However the vortex theory, 
which allows the fluid to satisfy the physically meaningful condition of 
adherence to the huil, certainly offers mathematical models closer to 
the reality than those drawn from the other types of hull representa- 
tion. Furthermore, it leads more rapidly to the determination of the 
distribution of the pressure and of the velocity on the hull surface. 
The present paper is therefore an attempt to explain the 
fundamentals of the vortex theory and of its application to the kine- 
matics and dynamics of bodies moving in water. 
After a brief survey on the various aspects of the vortex 
theory in inviscid fluids (Section I), one will find in Section II the 
Poincaré formula which permits the calculation of the velocity ina 
fluid domain when the vorticity inside the domain and the velocity on 
its boundary are known, and in Section III the application of Poincaré's 
formula to the determination of vortex distributions kinematically 
equivalent to any given ship hull. The class of these distributions is 
infinite. Each consists of a volume distribution inside the hull and of 
a surface distribution over the hull. The volume distribution can be 
chosen arbitrarily. The surface distribution associated with it is de- 
termined by means of a singular vectorial Fredholm equation of the 
second kind. 
Section IV gathers material to be used later to solve the 
dynamical problem. 
Section V is devoted to the study of the structure of the vor- 
tex distribution which permits the fluid to adhere to the hull surface. 
The surface distribution is the sum of infinitely flat vortex tubes call- 
ed here ''vortex ribbons". The vorticity inside the hull is twice the 
angular velocity of the body. Thus, if the angular velocity is not null, 
the intensity of each vortex ribbon is not a constant along its length. 
Furthermore, if free vortices are shed by the hull, some of the vor- 
tex ribbons do not close on the hull. To overcome the difficulties aris- 
ing from these circumstances, the vortex distribution generated by the 
body is divided into two distinct families almost independent of each 
other. One consists of the volume distribution and of the surface dis- 
tribution associated with it so that the velocity induced by this first 
family outside the hull be null. The second family consists of a vortex 
sheet entirely located over the hull when no free vortex-sheet is shed 
by the hull. In the opposite case, it includes the free vortex-sheets. 
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