48 J. A. Sparenberg 
l'(9,s) + 0, azs<¢a+t+4. (2.3) 
In the following we will use the notation 
‘ a9, 2 
SF LE: {SL =i’ (ons). (2.4) 
When the elementary bound vortex I'(~,s) ds changes its strength when 9 increases, it 
leaves behind at the place 9,s free vorticity of the density 
" F(9,s) ds 
2 1 (2.5) 
RV1 + u* + 2n cos @ 
The total density y(P,s) of the free vorticity which is left behind at the point 9,s by 
the bound vorticity of the profile which has passed this point is then given by 
att = 
¥(9,S) = - | : l'(@,0) do 
EE OD Gee o (2.6) 
After the partial differentiation in the integrand of (2.6) we have to consider } as a 
function of 9, s, and o by 
c= s+R[{ Jit p+ Qu cos x ox. (2.7) 
6 
A necessary condition on the bound vorticity follows from the mean value K of the x 
component of the forces which act on the bound vortices. This mean value has to possess 
some prescribed value which depends on the velocity of the ship. Calculating the angle X 
(Fig. 3) and using the fact that the profile has the velocity V(p) we find for the M blades by 
the law of Kutta-Joukowski = 
27 att 
/ 2 
K = —a | | l'(9,'s) sin ge Vit w+ 2 cos Oo 4, do 6 (2.8) 
0 2 V1 + uw? + 2u cos 
where 8 = 0(9,s) by (2.7) with o = 0. 
Another formula which will be used later on gives the difference of the value of the 
velocity potential ¢(x,y) across the cycloid (points D,, D_, Fig. 3) when the profile has 
passed. This difference at the place w is 
ath 
ew) - 0) = [  T(8,0) do. (2.9) 
where @ = &(/,c), defined by (2.7) when we puts = 0 and ~ = w in that equation. In order to 
use a potential for the whole fluid it is necessary that the lines covered with free vorticity 
