Vertical-Axis Propeller Efficiency 49 
y(~,a) do not intersect each other. This can be obtained by a simple artifice; viz., in the 
neighbourhood of points of intersection of the cycloids we assume that the bound vorticity 
of the blades becomes zero. Then the bound vorticity of the profile is left behind as free 
vorticity; however, a little farther, free vorticity of opposite strength is created which com- 
pensates the effect of the first one. Then (2.9) can be used with arbitrary accuracy when the 
distance between the two free vorticities just mentioned becomes small enough. 
We now introduce two independent order quantities, viz., the length 4 of the chord of 
the profile and I the total strength of the bound circulation of the profile at a reasonable 
point. We consider the case 
2 
l1 - pl 
an , as &4) (2.10) 
hs te are 4 << R 
where p # 1 means that the cycloid has no cusp, and the second condition means that the 
chord of the profile is very small with respect to the smallest radius of curvature of the 
cycloid. 
The function I'(9,s) exhibits, as a function of s, large variations over the chord. In 
fact, it becomes infinite at the leading edge s = a + % and zero at the trailing edge s = a. 
As a function of 9 it is clear that the variations of I'(Q,s) are almost everywhere small when 
4 + 0 and 9 changes with an amount of the order 4. In formulas: 
(9, ste) + I(9,s) + eF'(9,'s), 
‘ (2.11) 
T(o+e,'s) = [(9,'s) + el(9,s), e= &4). 
From this we can develop, with respect to 4 and I’, Eqs. (2.6), (2.8), and (2.9), which 
are important for our theory. Only lowest order terms are taken into consideration. 
We find from (2.6) for the density y(y) of the free vorticity at the end of the profile, 
s = ain (2.6), within higher order terms 
at+Z e 
def DP. do 
Mey = ylwed) == J vied (AMA) i EO serdenb 
A RV1+pu?+2u cos 6 
att 
= ce Eas Sa | P(p,o) do 
Ya 
RV 1+ py? + 2u cos 
Hegiry oe te PG) Fs OP): 
212 
RV1+ pp? + 2y cos w oe 
The mean value of the component K in (2.8) becomes 
277 
RM 
K = a | (9) sin 9 dg + QI), (2.13) 
