50 J. A. Sparenberg 
while (2.9) changes into 
P(Y) - d.(Y) = TW) + OCP). (2.14) 
Hence with respect to the wake and to the mean value of K we can treat the profile with 
a sufficiently small chord as a concentrated vortex (9). This is no longer true when we 
calculate the angle of incidence of the profile; then we have to take into account the varia- 
tions of the induced velocities over the chord (section 5). 
3. THE KINETIC ENERGY LEFT BEHIND IN THE WAKE 
We want to derive the potential of the cycloidal free vortex layers with density y(w) 
(2.12). To this effect we start from the potential ¢*(x,y) of a row of equally spaced vortices 
of equal strength. The potential of such a configuration is known [3, p. 186]. Because in 
the remaining part of this paper we do not need the third space coordinate z, the symbol z 
will be used to denote the complex variable x + iy. We find 
$*(x,y) = Re xT ln sin a (z- (Gets) (3.1) 
where y is the strength of the vortices, h their mutual separation distance, z the point in 
which the potential is considered and €,, the location of some vortex (see Fig. 4). It can be 
easily seen that this potential is not periodic, we find 
-by, y>In£, 
¢ (xth,y) - ¢*(x,y) = (3.2) 
Vey ert 
bNo|= 
When we take h = 2myR and let the points ¢, =(€, + in,,) with m=1, ..., M, describe the 
parts of the M cycloids lying in the interval 0 < x < 27pR, the other points of the rows de- 
scribe the parts of the cycloids outside this interval. In this way we cover the whole wake 
of the propeller with free vortices. The potential }(x,y) of the wake flow can be written as 
y 
Ym 
() 
Fig. 4. Row of vortices with streamlines 
