Brard 
VIII - THE APPLICATION FIELD OF VORTEX THEORY IN SHIP 
HYDRODYNA MICS 
GENERAL - 
It has been seen in the preceding Sections that the vortex 
theory applies to any body moving in water whatever its motion may 
be. The methods to be used in practice may considerably vary with 
the shape of the hull, the motion of the body, the boundaries of the 
fluid domain. To perform the calculations, it may be advantageous to 
substitute normal doublet distributions for vortex distributions because 
a regular scalar Fredholm equation then replaces a vectorial singular 
Fredholm equation. This is why the vortex distribution kinematically 
equivalent to the body has been divided in Sections V and VI into 
two parts, one of them being equivalent to a normal dipole distribu- 
tion. When the motion of the body consists of a pure translation, the 
vortex theory leads to computations which are not more complicated 
than those involved when source distributions are used ; they are even 
simpler when the distribution of the pressure over the hull is needed. 
This can be of interest when there exists an incident unsteady flow. 
The theory extends to the case when there exists a free surface, at 
least when the condition on the free surface is linearized. Neverthe- 
less, some difficulties are to be expected when the hull pearces the 
free surface. It is necessary to close the vortex filaments by their 
mirror images with respect to the plane of the free surface at rest. 
This can lead to difficulties analogous to those encountered in the case 
of the Zero-Froude number approximation when the hull is replaced 
by a normal dipole distribution [6,7 ] 
One of the main features of the theory developed in the pre- 
sent paper is that it includes the case of bodies which are neither thin, 
flat nor slender. However, to the knowledge of the writer, the vortex 
theory is still used only in cases of thin lifting surfaces. There is 
thus a need for more general methods and one of the purposes of this 
paper is to give means to extend the field of applications. 
In this Section the present field of application is briefly out- 
lined. Yet the problem of maneuverability and control of marine vehi- 
cles is examined in a more detailed manner, for progresses in that 
domain seem to be strongly needed. 
D'ALEMBERT'S PARADOX 
There exist many proofs of this theorem. The following one 
may be of interest,for it clearly explains the physical meaning of the 
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