60 J. A. Sparenberg 
Fig. 7. The propeller with a continuous 
vortex distribution 
+ = (9) = (27= 9} , y=Rcos 9. (6.1) 
The velocity induced by two infinite strips with vorticity of strength given in (6.1) at 
y=R cos 9 is 
-= [r.co) - 1 (21-9)] . (6.2) 
By this the energy in an arbitrary rectangle of unit length in the x direction between y = +R 
and far behind the propeller becomes 
R 2 
= — [r. (9) - nie )| sin 9 dg. (6.3) 
2" J, 
The condition for the total component in the x direction of the force acting on the bound 
vorticity is 
27 
pw R? T.(9) sin 9 dp = K. (6.4) 
0 
We have now to minimize (6.3) under the condition (6.4). Introducing the perturbed bound 
vorticity 
D.(9) + ©, 8,() + &, 6,(9), (6.5) 
where g,(@) and g,(9) are arbitrary functions of Q, into (6.3) we have to differentiate E with 
respect to €,, while condition (6.4) must be taken into account. Applying the multiplication 
method of Lagrange we obtain 
