Minimum Wave Resistance Problem Solution 73 
vanishes for any choice of h. This means that f must be such that 
77 77 
| de | dé, f(®) h(#,) sin ® sin @, ¥.[7o(cos ® - cos 9,)| 
0 0 
- rf f(%) h(®) d® = 0 
0 
for all functions h(#), or 
Tr T 
\ h(#,) d#,<sin @, | f(%) sin $ Y,|7.(cos $- cos $,)] dé-)f(#,) >= 0. 
0 0 
This can only be true, if f(#) satisfies the integral equation of the second kind: 
TT 
df(®,) = | 
f(®) sin ® sin ®, ¥,|7.(cos ® - cos *1)) dé. 
0 
Since the kernel is integrable, this is an ordinary Fredholm equation and the functions f(#) 
are the eigenfunctions of the corresponding equation, while » is the corresponding eigen- 
function. Apparently the solutions vanish for = 0 and $= 7, as was required. Moreover, 
the eigenvalue 2 is the value of the quotient J(f,f)/V(f,f) for the eigenfunction. For the 
ship hull only the first eigenfunction, which hag no zeros, will come into consideration. 
SOLUTION OF THE INTEGRAL EQUATION 
The integral equation 
7 
Af(#,) = ‘ f(®) sin 6 sin #, ¥,(y.leos $ - cos 9,1) dé 
0 
resembles an integral equation satisfied by the Mathieu functions. Therefore, at first a solu- 
tion in terms of a Fourier series in #, with coefficients depending on y, was looked for. 
This method, however, is only feasible for very small values of y,. For this reason, a di- 
rect numerical approach was used. 
