56 ie ¥ J. A. Sparenberg 
Fig. 6. The profile in the neighbourhood of the cycloid 
~ 
of the real profile are represented by points on the cycloid in such a way that s* = s. The 
leading and trailing edge of the profile are denoted on the cycloid by p and 9,. 
First we consider the velocities induced by the bound vorticity of the other blades and 
by the free vorticity along the cycloids C,,, m=1,...,M-—1. It is clear that for these ve- 
locities and hence for their resultant component v, ,(9,,s) normal to the profile, we have 
the relation 
v,1(9,s) = QI), a<s<atf, (5.1) 
At the end of this section it will turn out that this information about these velocities is suf- 
ficient with respect to the accuracy with which we have approximated the basic formulas 
(2.6), (2.8), and (2.9). 
Next we consider the velocity 3,(9,z) with components v,, and v,,, induced by the 
free vorticity along C,, 
x,z y,z 1 J es C(@)] 
< 
J 
hay 
< 
il 
CCH.) : 
T'(@) dl 
Dinas i. [z- €(8)] R(uw + cos 0 - i sin #) © oe 
The second integral formulation has the form of a Cauchy integral. We have to consider 
the behaviour of this integral when z is in the neighbourhood of €(9,). This is a well-known 
problem [4, ch. 4]; we find 
tit dt vd Lennar oltre nb ook (Oeil I Zieral- Bedi 
. i) pieleansio Are R(u + cos 9, - i sin 9,) the) ne) 
