Opttmum Shapes of Bodtes in Free Surface Flows 
or similar ones. The end conditions are usually required on physical 
grounds in order that the fluid pressure is well behaved at the end 
points £ = + 1, at which the free boundary meets the wetted body 
surface. 
The solution to the Dirichlet problem (3), (5), (6), ive. 
and the solution to the Riemann-Hilbert problem (4), (5), (6), given 
by 
ly a7 : 
1 2 t)dt 
OU te aee - 1) | (8a) 
i at (ee) 
with 
1 
l ne aon (8b) 
t (l-¢,) 
are A ae baa to each other, as can be readily shown. Here, the func- 
tion (cee - 1 2 is one-valued in the ¢ =plane cut from f£ = =1 to 
5 = 1, On the body surface, we deduce from (7), by applying the 
Plemelj formulas, that 
1 
i i f Tet su] (lel<1) (9) 
rats 
