Vortex Theory for Bodtes Moving tn Water 
powerful methods of computing the pressure distribution on a wing 
of finite aspect ratio in the same kind of motion (see, for instance 
[4] 
The case of bodies of high displacement/length ratio in an 
unsteady motion is sensibly more intricate than the case of the usual 
lifting surfaces and there was a need for a general theory. Poincaré's 
formula gives means for determining such a vortex distribution on 
the hull and inside the hull that the fluid adhere to the hull. This 
vortex distribution is kinematically equivalent to the body. But it is 
not the only vortex distribution with this property. Furthermore if 
the motion of the fluid about the body is unsteady, any vortex distri- 
bution kinematically equivalent to the body varies with time. Lastly 
the theory would be without practical interest if it were not capable 
to take into account the effect of the free vortices shed by the body 
and that of an arbitrarily given incident flow. This paper gives an 
answer to the problems arising from the afore-mentioned needs. 
(iv) The fourth part of the theory concerns vorticity in viscous 
fluid motions, but it does not fall within the scope of the paper. 
Il, POINCARE'S FORMULA 
VORTEX SHEET 
Let > be a part of a certain surface. The two sides Ds ’ 
Dre of » are distinguished from each other. The unit vector n 
normal to a is uniquely defined at every point Pot Se and in the 
direction from Mie towards Dos . We put PP, = np(0+) +¥ 
BP emis Blt): .oPsoand~P4jbeimg the ue eas a of np with 
deawandia Diawmespectivelys (Bigure t,and 2), 
Let Dee - year! denote ae surfaces described by the points 
Pi, Pj such that PP! = = $= Tp, PPE -— Tip , respectively. 
We pepe that a Toten w is Se isine 2 G1 aiepribapad between 
a and De . @(M)is normalto n,, at every point M of each 
rien P; P,. When e{o , €(M) has a finite limit 
T (P) tangent to. a . Let 6 p » Tp denote two unit vectors tangent 
to)’ at P, such that the directions (np. Op? Tp) make a right- 
handed system, with 7p in the direction of Tp. Aline ¥ tan- 
gent to 7 at each of its points is a vortex filament of the limiting 
distribution. Let oe L, be two lines Y close to each other. Let 
eC be a line orthogonal i the £ 's and containing P. It intersects 
12 at P, . We may put PP, = =do6_. For the limiting distribution, 
ak Pp ut o is the flux of the vortex through the area of the infinitely 
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