54 J. A. Sparenberg 
and the second term yields singularities for 
Y= 7, €—,= 2kapk, O <b, 8 < Qn, * k =hOscthink aoe 
where as many integers k = 0, 1, ... have to be used as are compatible with the condition 
0<w, 8< 27. The order of these latter singularities is 
o{[(v. - p)* + (8, - 9)?] aa (3.21) 
where WS, , denotes the singular point in the w,# plane. 
4. THE VARIATIONAL PROBLEM 
We want to minimize the energy E left behind in the wake over a length of a period 
27pR. It seems most simple to start from (3.16) by using Ritz’s method. First, however, a 
general property of the function I‘(y) will be derived. 
We can split I'(W) into an even and an odd function with respect to = 7, 
ry) = Te) + TQ), Porte) = Pry) . 
(4.1) 
The condition (2.13) then becomes 
27 
fee —— | Po(p) sin y dp= K (4.2) 
0 
which does not yield any restriction on I',(w). Using (4.1) the kinetic energy (3.16) becomes 
27 277 
Ds | {[r.w P(e) + Py) Pe) # [row r (8) 
0 0 
+ Top) ro] L(y,8) dp dd. — (4.3) 
From (3.17) it can be derived that 
L(,8) = -L(2n-, 27-9). (4.4) 
By this (4.3) reduces to 
27 27 
E = | | rw) P(e) + Pop) r)| L(y,®) did. — (4.5) 
0 0 
Both terms between brackets in (4.5) yield a positive contribution to E, because each of 
them represents the kinetic energy which belongs to some bound vortex function I",(W) or 
ry). 
