62 J. A. Sparenberg 
This result can be checked easily by calculating the external force necessary for the in- 
crease of momentum which follows from (6.11). 
Isay [2] uses a formula, Eq. (1) in his work, which is not in agreement with our formula 
(6.1). He gives also another formula, Eq. (12), which he thinks to be faulty, but which, in 
the opinion of the author, is the correct one. That something is wrong with Eq. (1) of Isay 
follows from the fact that in it the velocity of the incoming fluid does not occur. It. is clear 
that when this velocity increases, the density of the free vorticity in the wake decreases, 
when the bound vorticity distribution remains the same. 
7. THE ANGLE OF INCIDENCE IN THE CASE OF 
CONTINUOUS DISTRIBUTIONS 
We first consider the induced velocity by the two half infinite equidistant rows of vor- 
tices which form the wake. ee elementary calculations we find that the component of the 
velocity v, 1, in the direction 7, normal to the cycloid, at some point 9 of the circle ESS: 
7) amounts to 
Mea pik AO ea UM NE ER le" sin 9 
8oRU 1 + uw? + 2u cos op 
1 1+ cos 9 
Enea ae 
— (4 + cos 9) In (js 2), 
Pe: Oe OM ONES 
Boas bl et ae ae (7.1) 
The component of the velocity normal to the cycloid, induced by the bound vorticity (6.9) on 
the circle, becomes 
v = cos ta 1 + cos 9 
n,2 ~ ~ ga ey 1 - cos 9 (7.2) 
V1 + up? + 2 cos o 
Finally we write down the normal component of induced velocities by a constant density Fs 
of bound vorticity on the circle, 
Py e 
<b sin 9 
Eo (7.3) 
2/1+ uw? + 2 cos o 
The velocities (7.2) and (7.3) are the mean values of the velocity components normal to the 
cycloid in points just outside and just inside the circle. Hence we find for the total induced 
normal velocity 
Vina Vian estan (a at MB ene (7.4) 
When the propeller consists of M blades, each blade has to possess a bound vorticity 
