LD R. Timman and G. Vossers 
where 
depends on the Froude number. We introduce variables ® and ®, by cos #= €, cos #, = & 
and have to minimize the integral 
7 77 
j= | ao { de, f(#) f(8,) sin ® sin @, ¥,(y,|cos 8- cos 0,1) = J(f,f) 
0 fr) 
with the condition 
f(97) do=C. 
0 
REDUCTION TO AN INTEGRAL EQUATION OF THE SECOND KIND 
Suppose that f(®) is the minimal function and consider a neighbouring function 
f(®) + eh(*) 
where € is a small parameter. Then 
JCf+ ch) = J(f,f) + 2eJ(f,h) + ©7J(h,h) 
and 
V(ft+eh) = Wf,f) + 2eV(f,h) + €?V(h,h). 
Since J(f,f) must be a minimum for f, with constant V, we consider 
JC f+ eh) - VC f+ ch) 
where \ is a Lagrangian multiplier. The expression 
J(£+ eh) - W( f+ ch) = JC f, f) - WCF, F) + 2e(JCf,A) - AV(E,A)) 
+ e7[J(h,h) - \V(h,h)] 
can only be an extremum if 
JCf,h) - VC f,h) 
