52 J. A. Sparenberg 
1 
1+ up? + 2u cos 
n=(siny, 1+ cosy) * (3.10) 
The velocity (x,y) which belongs to the poten- 
tial } (x,y) (3.3) is 
M-1 27 
ee, ake 
rr Amu j ¥(#) 
m=0 0 
* (Oe as) (=>) 
$1 Tapp ese el) | ee 
pR pR 
sh s 
Fig. 5. The path of integration HR LR 
along the cycloid for m = 0 
- J/1 + u2 + 2 cos 8 dd (3.11) 
where €,, and 7,,, are defined in (3.4). By (3.10) and (3.11) we find for the normal velocity in 
a point x, y, of the cycloid with m = 0 (2.1) 
M-1 27 
od 1 
a ee 
m=0 0 
sin W sinh [2] - (u% + cos W) sin = 
‘ OZ sau) (%0- Sm) 
cos Pu Reds - cos a eat 
abies tt py? + 2u cos # dp. (3-12) 
v1 + py? + 2u cos W 
By (2.12), (3.9), and (3.12) we obtain 
M-1 27 27 
e.=- oD | f rote 
m=0 0 0 
Sin w sinh vee) - (% + cos W) sin Sony 
e -—|d@dy (3.13) 
(Vos ia) (X,7 Sn) 
ap ae ein os 
cosh c 
ns ‘aaqR brie 
where x,, y, and €,,, 7, are defined by (2.1) and (3.4). 
Next we consider the contribution of the vertical boundaries of the strip (Fig. 2) to the 
line integrals (3.8) of the kinetic energy E. Because 0 (x,y)/dx is periodic for all values 
