Brard 
intensity of a flat tube or of a tube whose all transverse dimensions 
are null can be finite. This is the case of the vortex tubes on a vortex 
sheet and of the isolated concentrated vortex filaments. 
Equation (1.1) can be solved with respect to V. Poincaré's 
formula gives V_ when the vorticity is known. A particular case is 
the Biot and Savart formula which expresses the velocity ''induced" 
by an isolated vortex filament. A consequence of Poincaré's formula 
is that the perturbation flow due to a body moving in an inviscid fluid 
can be regarded as generated by a vortex sheet distributed over the 
hull of the body and fulfilling the condition that the fluid adheres to 
the hull. There is a kinematical equivalence between the body and 
the vortex sheet. 
(ii) The second part deals with the evolution of the vorticity with 
time under the assumptions that the fluid is inviscid and that the 
exterior force per unit mass is the gradient of a certain potential. 
The basic theorems are due to Cauchy and Helmholtz. The intensity 
of every vortex filament is independent of time and the vortex fila- 
ments move with the fluid. This means that every vortex filament is 
composed of an invariable set of fluid points. Lagrange's theorem 
follows according to which the fluid motion is irrotational if its 
starts from rest under the effect of forces continuous with respect 
to time (shock-free motions). This theorem seems to be contradict- 
ed by the possible existence of vorticity in the motion of an inviscid 
fluid, but the difficulty can be overcome by considering such a motion 
as the limit of the motion of a real fluid when the viscosity goes to 
zero. Although the second part of the theory is based on the Euler 
equation, it only deals with fluid kinematics. 
(iii) The third part of the theory concerns the dynamical interaction 
between flow and vorticity. If the set of fluid points belonging to an 
arc of vortex filament does not move with the fluid, this interaction 
cannot be null. The concept of force exerted by the flow on every 
bound arc of a vortex filament is now classical. Conversely, the set 
of fluid points belonging to this arc exerts a force equal and opposite 
on the adjacent sets of fluid points which proceed with the general 
flow. 
As it has been shown by Maurice Roy [1] , the system of 
forces exerted by a steady flow on a body in a uniform motion can be 
obtained in this way. This led to an important generalization of the 
Kutta-Joukowski theorem. Later von Karman and Sears have success- 
fully solved the problem for wing profiles in a quasi-rectilinear non 
uniform motion [2] . The pressure distribution on such profiles 
has been calculated by the present writer [3] . There exist now 
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