Vertical-Axis Propeller Efficiency 61 
R aT 
a | [r.(@) r r,(2n- 9) [ 4409) = é,(27- 9) sin 9 do 
0 
+ Apw R? \ 6&,(9) singdg=0 (6.6) 
0 
where A is a still unknown constant. 
This equation can be written in the form 
1 17 27 
Fe za | [r. (o) 3 r(27- @)| 6,(9) sin 9 do 
fe 0 7 
27 
+ A\wR 6,(9) sin gp dg=0. (6.7) 
o 
Because g,(9) is arbitrary we have 
0<qQo<7 
. 6.8 
pe (6.8) 
s [F. (@) = r,(27- 9)| == hau 
From this it follows that the only condition which we have to satisfy is: the difference of 
the bound vorticity at the front and the back of the circle for the same values of y is a con- 
stant. Hence we may for instance assume I",(9) = -I",(27—@) = const. From (6.4) it follows 
0 < <7 
y (6.9) 
K, 
EE CO) eae Ha > : 
This agrees with the result of section 4 where it was found that the optimum bound vorticity 
was an odd function of 9 plus some arbitrary constant. In this case of continuous vorticities 
it is allowed to add an arbitrary even function of 9 to the distribution in (6.9). 
From the above it follows that the vorticity in the wake is concentrated wholly on the 
lines y= +R. The strength y per unit length in the x direction becomes 
ws iK 
ae aa y= FR. 
i 20 RU _ 
This means that the induced velocity u in the wake is, parallel to the x axis, independent of 
y and amounts to 
iain aS 
2p RU : (6.11) 
646551 O—62——_6 
