Vortex Theory for Bodtes Moving in Water 
Hence D' does not include the arc ds, . We consider the relative 
motion of the fluid with respect to the system of axes just defined. The 
velocity may be written in the form 
Wee Va - BV") OVE , 
ae Viol 6V') 
where Vo defines some incident flow, V' is the velocity induced by 
Yana 5 V' that induced by the part ds, of LF Let dE' be the 
set of fluid points inside D', and dE, the set of fluid points belonging 
to ds, . One easily sees that the momentum I (dE|) is null, and, 
therefore, that, I (dE') being the momentum of dE' , 
d+ 1 spe niGll 1 te a. deere is 
af 1 GE) =) Tae rae)= ff p V, @V,) as’; 
(m is in the outward direction , and S' is the boundary of 
(D'+ds). 
One readily obtains : 
Iimeud 1 = 
ee? (de) = —— Pr ee led : 
R50 dt Co ae Arata bon Tsk (7. 44) 
where [ is the intensity Pe ree are aay j ie being the unit 
vector tangent to ds, and Vj the finite incident velocity on the arc 
ds 
1 
On the other hand, by the momentum theorem, one has 
oer (dE) = dE f] -pu dS", 
dt ia an 
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