Brard 
it follows that 
Y 
<| 
curl ( + SAV) = 0. 
at 
Taking into account the continuity equation, one obtains the 
basic Helmholtz equation 
ae ey sae 
—(——) = ses Re NE (1) (4. 3.) 
This equation entails the following consequences : 
(i) If F and the boundary conditions are continuous with res- 
pect to time, and if the fluid starts from rest, then 
@=0 everywhere at every t. 
(ii) Every vortex filament is made of an invariable set of fluid 
points. 
(iii) The circulation of the velocity in any closed fluid circuit is 
time-invariant. 
According to consequence (i), the absolute fluid motion should 
be irrotational everywhere. This explains why the concept of velocity 
potential is of importance in the motions of inviscid fluids. However, 
this consequence does not hold if there exist regions where F is not 
the gradient of a potential . This is obviously the case when the 
fluid is submitted to the condition of adherence to solid walls. 
For this reason, the vortices existing in the motion of an 
inviscid fluid about a set of solid bodies have to be considered as ori- 
ginating on the surfaces of these bodies, Thus every vortex filament 
is made of two sets of fluid points : one of these two sets is at rest 
with respect to the surface of a solid body, the second one is free and 
moves with the fluid. 
(1) This equation holds if the fluid is not incompressible, provided 
that 1 
h (p) 
1204 
