Optimum Shapes of Bodtes tn Free Surface Flows 
from (26), giving 
(30a) 
where 
22) = ae ree et! , v(t) = -H, [a] er on) ae 
u (3 0b) 
After this Carleman equation for + is solved, a second Carleman 
equation results immediately upon elimination of B between the ex- 
pression for + and (9). The details of this analysis are given by 
Wu and Whitney (1971). These analytical solutions are of great interest, 
since in their construction there are definite, but generally very li- 
mited degrees of freedom for choosing the strength of the singularity, 
or the order of zero, of the solution [I[(é) and £6(£) at the end 
points §& = * 1. It is in this manner that the analytical behavior of 
the solution [(&) and £6(£) can be explicitly and thoroughly 
examined. This procedure will be demonstrated later by examples. 
V. THE RAYLEIGH-RITZ METHOD 
The central idea of this method, as in classical theory, con- 
sists in expansion of I(&) and £(£&) ina finite Fourier series 
B's ) = es Y 4, ©O8 a (31b) 
This expansion is noted to satisfy (9) automatically. The functional 
FIRS 
