46 J. A. Sparenberg 
constant. For instance, this constant may be taken equal to zero; then the blades pull as 
much as they push. It is also possible to give the constant such a value that the profiles 
are more active when they are in the front position. 
Also the case of a vertical-axis propeller with many blades or a high-rotational velocity 
is considered. Here the bound vorticity and the vortex layers in the wake are approximated 
by continuous vortex densities. Then we find the following condition on the bound vorticity: 
In order to obtain the highest efficiency it is necessary and sufficient that the difference of 
the bound vorticity density for two points lying on a straight line parallel to the direction of 
translation of the propeller is a constant (section 6). In this case it is possible to write 
down explicitly the angle of the incidence of a blade as a function of its position. 
It is intended to compare in a future paper numerical results of this theory with experi- 
mental results. 
2. STATEMENT OF THE PROBLEM 
We consider an unbounded fluid which is at rest relative to a Cartesian coordinate sys- 
tem x, y, z. In the direction of the positive x axis moves a circle with radius R (see Fig. 1). 
Its centre is on the x axis and has a velocity 
U. On the circle are M equally spaced identi- 
cal blades perpendicular to the x, y plane. 
The blades rotate with a constant angular 
velocity @ around C. Besides this they ex- 
ecute an oscillatory motion around the point 
T in order to preserve the desired angle of 
incidence. We simplify our considerations, 
as stated in the introduction, by assuming 
each blade to be two-sided and infinitely 
long. 
The orbits of the pivotal points T of the 
M blades are the cycloids C_, with 
x 
II 
R (uy 4 siny - rms \ 
us M 
Fig. 1. Scheme of a vertical-axis - 
i b 
propeller with M blades y, = Ros y, “= . (2.1) 
Several types of cycloids, as a function of p, are drawn in Fig. 2. 
As in lifting surface theory for airplane wings or ship screws we try to refer the profiles 
to a surface which moves through the fluid without disturbing it. For an airplane wing this 
surface is the projection of the wing on a certain adjacent flat plane with zero angle of at- 
tack. For a ship screw it is the projection of the blades on an adjacent helicoidal surface 
which does not disturb the incoming fluid when it rotates with the rotational velocity of the 
screw. In the case of a vertical-axis propeller a rigid reference surface does not exist be- 
cause the curvature of the cycloids is not constant. For this reason we start with a deform- 
able line segment A — B covered with bound vorticity (Fig. 3), which moves exactly along 
the cycloidal orbit of the turning point T. 
