70 R. Timman and G. Vossers 
The volume of the ship is given by 
+L/2 T 
V= | dx | £(¢x,2)) dz. 
“L/2 0 
The problem then consists in the determination of the function f(x,z), which minimizes the 
resistance at a prescribed volume. 
SIMPLIFIED TREATMENT OF THE VARIATIONAL PROBLEM 
We put the condition that f(x,z)=0 for x= +L/2 and transform the resistance integral 
by integration by parts: 
© @ 
‘c2B2T2,2,4 a 
Coe te ae (ae ee a 
-1 -1 0 0 
NK T(CH,) = 
= er Seeds as (é>4,) 
1 py | 
In order to simplify the problem, a special class of ship hulls is considered which is so de- 
termined that both R and V appear as quadratic functionals of a function f(£). We assume 
/y _ £2 
f(E, 0) = rey a [OAS g |. 
f(¢) 
where f(&), the waterline function, is to be determined and g(y) is a fixed function of p 
which is zero for » greater than a certain value. If the factor /1—€2 did not exist, the as- 
sumption would mean that all cross sections of the ship were similar. If f(€) vanishes at 
the front and the stern of the ship, also the draft would vanish there. This is prevented by 
the present assumption, and moreover it will appear later that the addition of this factor ac- 
tually yields functions f(€) which vanish at these places. The volume of the ship is given 
by 
! +1 (4) +1 2 oe 
v=LT( ae. at (2 ‘[ohee | = LT) FO) Ge'( teqaylaae 
2 | j f(é) 2 iy 1- €? j 
The resistance is 
