Vortex Theory for Bodies Moving tin Water 
As shown in Subsection D, it may also include a volume vortex dis- 
tribution in the wake when the motion is unsteady. 
Section VI deals with the integral equations determining 
the two families of the total vortex distribution. The singular vectorial 
integral equation related to the first family can be replaced by a sin- 
gular scalar Fredholm equation for a Neumann interior problem. It 
can be solved once for all whatever the body motion may be. The in- 
tegral equation for the second family reduces to the scalar regular 
Fredholm equation of the second kind for a certain Dirichlet interior 
problem when the fluid is unbounded and at rest at infinity. In the 
most general case it becomes a Volterra equation expressing the so- 
lution in terms depending on the history of the motion. 
Section VII is devoted to the study of the system BA of 
hydrodynamic forces exerted on the body. As stated before the total 
vortex distribution determines inside the hull a fluid fictitious motion 
which coincides with the absolute motion of the body. For this kine- 
matical condition to be compatible with the dynamical equilibrium of 
the fluid, it is necessary to introduce a certain system of fictitious 
forces per unit mass inside the hull. 
The system od, of hydrodyhamic forces exerted on the body 
at t. can be written in the form 
where ae s, is the quasi-steady system of forces, that is the sys- 
tem to which qd would reduce if the motion of the body were 
uniform in a large interval (t', t,) eo is the system due to the so- 
called added masses ; it is independent of the free vortices shed by 
the body. There exists a difference between the structure of the free 
vortices at tp and at t=+00, the latter being evaluated under the 
assumption that the motion of the body is uniform to t > t,. This 
difference affects both the bound vortex distribution on the hull and 
the incident velocity on it. It entails the term - 1Sfe . The last term 
Pins ; : ; ¥ AAG 
‘a is an inertial effect due to the partial derivative =e at ty. 
of the bound vortex sheet. Fa reduces at t, + .0 to Za, 5 (t, -0) 
+ YF: (to + 0) if the body motion is uniform for - <t, - 0 and 
for t > to +0, but discontinuous between to - 0 and ty. 
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