Vertical-Axis Propeller Efficiency 55 
From this result and from (4.2) we find that we have to take 
ry) =0. (4.6) 
Hence I(w) has to be an odd function with respect to w. We suppose 
N 
Mw) = Teg) 2 ah a, sin mp. (4.7) 
n=1 
Condition (2.13) yields 
2K 
a : P 
Laas RM (4.8) 
Using (4.7) the energy E becomes 
N 27 27 
5 Si De qa, 4, | | sin pw cos qét L(y,#8) dy dé. (4.9) 
p,q=l1 0 0 
Differentiation of this expression with respect to a, (n = 2, ..., N) yields 
N 
Poa 
0a, q 
277 
q=1 0 
27 
J (q sin ny cos qé 
0 
+ n sin qy cos n@) L(y,#) dy dé = 0, Gnesi Dye eda do N) a w(4 0) 
These are N — 1 linear equations for the N — 1 unknowns a,, ..., ay. 
Without altering the kinetic energy in the wake we may add a constant value to the cir- 
culation I'(w). 
From the fact that the optimum bound vorticity (yw) has to be an odd function of W plus 
an arbitrary constant we find the following property, mentioned already in the introduction, 
(y) + T(-W) = const. (4.11) 
5. THE DETERMINATION OF THE ANGLE OF INCIDENCE 
Our aim is to derive formulas which determine the angles of incidence of the profiles 
as functions of their position. 
Because all blades are equivalent we consider the profile whose turning point T moves 
along the cycloid C,. On this profile we introduce a parameter s* (Fig. 6) which measures 
the distance of a point of the profile to the turning point 7; s* is positive in the direction 
of V(~). We can calculate the normal velocities on the profile caused by the translation 
V() and a rotation around 7, which have to be compensated by a suitable vortex distribu- 
tion on the profile. Next we assume a kind of linearisation procedure which consists of 
compensating the normal velocities not on the real profile, but on the cycloid. The points 
