Minimum Wave Resistance for Dipole Distributions 79 
where the constant & is determined by Eq. (11). Differentiation of Eq. (12) twice with re- 
spect to x yields 
Y,(F|x-x'|) dx’ = 0 (13) 
| dx' 
-1/2 
where Y(¢) is the Bessel function defined by 
@ 
2 cos At dv 
Y(t) =- 2i Se 
1 h? -1 
From Eq. (6) 
d?K(t) _ 
ape = Fol ts 
Dorr [7] has solved Eq. (13) with an arbitrary given right-hand side; i.e., he has solved 
1/2 
| h(x') ¥)(F|x-x'|) dx’ = p(x). (14) 
-1/2 
The change of variables x =—1/2 cos B, x'=—1/2 cos B’ converts Eq. (14) into 
| HCE.) Ys (F lcos B- cos p'|) dB’ = P(B) (15) 
0 
where H(B8') = (1/2 sin B') h(—1/2 cos B') and P(8) = p(—1/2 cos 8). 
Dorr then proves that the eigenfunctions of Eq. (15) are the even Mathieu functions of 
integral order, ce,(8, F2/4). Since over the integral (0,7) these are closed [8], and the 
kernel Y,(F/2|cos B — cos B'|) is quadratically integrable over the square 0< B' <7, 
0<B <z, it follows [9] that there is at most one solution to Eq. (15) in Ly. Moreover, 
there are no solutions to Eq. (15) not in Lo, as may be deduced from the work of MacCamy 
[10], who has shown that the only singularities of h(—1/2 cos 6’) in the interval 0< B'<7 
occur at 8’ = 0 or 7, these singularities being of the form 1/sin 8’. Thus 
H(B') = (2 san 6") n(-4 eae 6’) 
is bounded in the interval 0 < B' <7 and is therefore in L,. 
Summarizing, we have that Eqs. (15), (14), and (13) all have at most one solution. It 
follows that the unique solution of Eq. (13) is zero. Since any solution of Eq. (12) is a 
