Optimum Shapes of Bodtes tn Free Surface Flows 
and oc. are independent and arbitrary, the last integral in (17)! 
and the factor in the parenthesis of the first integrand must all vanish, 
hence 
(22) 
The nonlinear integral equation (22) combines with (9) to give a pair 
of singular integral equations for the extremal solutions, This is one 
necessary condition for I ia to be extremal ; it is analogous to the 
Euler differential equation in the classical theory. Presumably, cal- 
culation of the extremal solution ['(¢&) from (22) and (9) can be 
carried out with h,, ... Ayqy regarded as parameters, which are 
determined in turn by applying the M constraint equations (12). 
While we recognize the lack of a general technique for solving the 
system of nonlinear integral equations (9) and (22), we also notice 
the difficulty of satisfying the end conditions (6), as has been expe- 
rienced in many different problems investigated recently. The last 
difficulty may be attributed to the known behavior of a Cauchy integral 
near its end points which severely limits the type of analytic proper- 
ties that can be possessed by an admissible function I ( & ) and its 
conjugate function 6 (é&). 
Supposing that these equations can be solved for [I (£; Cy > 
Cor +e C,), we proceed to ascertain the condition under which this 
extremal solution actually provides a minimum of I [r] . From the 
second variation 6°1 we find it is necessary to have 
1 
[' a°F } ac;) dE>0" 5 (23) 
=i 
1119 
