Vertical-Axis Propeller Efficiency 59 
where # follows from (2.7). However, the first term on the right-hand side of (5.14) is Q(I). 
Hence we find, making use of (5.9), 
att a+ 
(9) = i l'(9,'s) ds = | Peo t( OS) ds 
+ OCT) = B(o) + OCT). (5.15) 
From (5.11) we obtain 
' me I'(9) AN te : ' = 
) aca) ey = iii Ser ct 
CS Wee 
on V2(9) In R + 6(T),. (5.16) 
Because 0= Q(['4"') the second term on the left-hand side can be disregarded. Besides 
this by the choice a = —(1/4)¢ the coefficient of this term as well as the coefficient of 
6,(9) vanishes. Then equation (5.16) changes into 
i 2 [ . [2 ( s 3) | Se + Q(T). (5.17) 
Hence, for a more easy analysis of the propeller, it can be recommended to place the 
turning point T at one quarter of the chord length from the trailing edge. 
From this we see that, within the accuracy of the theory, we do not have to take into 
account the induced velocities (5.1), which give rise to a change of the angle of attack by 
an amount of Q(I). 
6. VERTICAL-AXIS PROPELLERS WITH AN INFINITE NUMBER 
OF BLADES OR QUICKLY ROTATING PROPELLERS 
When the number of blades or the rotational velocity of the propeller increases, the wake 
of the propeller becomes crowded with free vortex layers. Hence it seems natural to replace 
this complicated system by a continuous distribution of free vorticity. The bound vorticity 
can be replaced by a continuous distribution over the circumference of the circle. We as- 
sume for its strength per unit length the function I".(), where the index c indicates a con- 
tinuous distribution. 
For the free vorticity per unit length in the x direction, which is left in the strips be- 
tween y and +R (see Fig. 7) we find 
