1104 
boundary layer (parallel to the surface), wl —= vorticity. Further 
aU is written for the stream function of v,U, and BU for the 
potential. 
The stream line «=O is the line of symmetry of the motion; 
in front of the body it splits into two branches. These two unite 
again at the back of the body. At the branch points the values of 
8 are 8, and 8, For a circular cylinder we have f.i. (when the 
radius of the section is a): 
«=y(1-5) : p=e(1+5). 
r r 
The line a=O consists of the x-axis (y = 0) and the circle (r = a); 
at the points of intersection a — +a, so that: 
B= 2a; 8 at AE 
II. The differential equation for the vortex motion in the boun- 
dary layer is: 
dw vy dw 
08 Uda 
which is a shortened form of equation (21). When the latter was 
derived from equation (20*) it has been divided by: 
sa alo Guan 
(a) + (Qa 
which quantity is different from zero everywhere but at the points 
B, and 9,, where the stream line «a =O splits. When at these points 
(21) is satisfied, then necessarily the original equation (20*) is also 
satistied. In the neighbourhood of these two points — at least in 
that of 8, — the boundary layer may no longer be treated as infi- 
nitely thin, so that there the simplification of (21) is not allowed; 
in the determination of v too we find here a difficulty. With increa- 
sing A however the allowable limit of |—,| decreases. 
III. As a solution of (22) we may take for B, <@ <8, and for 
a 20 (that is for the left side of the surface of the body): 
(22) 
A (8) — a’ 
Inta bgg tE le, I 
Vai. Ae ie 
w=— fd 
1 
where k= v/U. Because of the symmetry the same expression may 
hold for the right side of the surface, with the opposite sign. 
A (5) determines the quantity of vorticity that leaves the surface 
in the neighbourhood of the point B=6 in unit of time; it diffuses 
in the direction perpendicular to the surface and at the same 
time it is washed backward in the direction parallel to the sur- 
