Vertical-Axis Propeller Efficiency 53 
of y and (x,y) is periodic for |y| >R the contributions of the integrals cancel each other for 
ly| > R. This is no longer true for |y| < R. We obtain for the part E,, of E which belongs to 
the vertical boundaries 
+R 
E,=- 5p i {$(0,y) - $(2muR, yy} FO) dy. ay 
Using (3.7) and (3.11) we find after some reduction 
. y- 
et FP 0) U8) jain eae! sin py 
Eo = - £— a Ee dt 
Vv 87 (y'=7,,) “ ie ap ’ (3.15) 
m=0 0 0 cos ar ae cos uR 
where y = R cos y and €,, and 7, are from (3.4). 
Hence we find for the total kinetic energy 
27 © ,277 : 
F=E,+E,= | I ay) T(&) Liy, 8) dp dé (3.16) 
0 0 
where 
Cis a) eee 
ol M-1) sin W sinh a a - (“+ cosw) sin (to usa! 
ee ee eae 
(¥,6) Sr = bei (Yo~- 7m) ii (x,-&n) 
n= eo Ss —_—__— 
pR HR 
| 
ae A me a 
RSE ae Te a aS a a) ee 
C4) be Saas 
cosh ——_~ - cos — 
LR 
in which y = R cos w and x,, y, and €, 7, are from (2.1) and (3.4). 
It can be seen that for p = $ this function possesses a singularity of the form 
: aed 1 
eae L(y,8) ~ 4 a (3.18) 
Besides this singularity there are other ones which are the points of intersection of the 
cycloids with themselves (yz < 1), with each other (M > 1), or with the vertical lines 
x= 2knpR. The first term yields besides the singularity mentioned in (3.18) also singulari- 
ties for 
Veo Nes See Gos PATER, I Sy, 
(3.19) 
®< 27, yt, k=0, 1, 
