Minimum Wave Resistance for Dipole Distributions 105 
&,(x) = uncrseit p7 x) (B3) 
J1/4 - x? 
where h(x) is a regular function in -1/2<x< 1/2. The singularity in g(x) at x = 1/2 in- 
troduces a loss of accuracy into the numerical solution of (B2); we therefore convert (B2) 
into an equation for the regular function, h, by means of the following change of variables: 
x= 172 sin £, x = 21/2) sin 6. (B4) 
Equation (B2) then becomes 
a/ 2 
H(B') rae |sin 8 - sin f' ) 
i) 
fale [F(sin B+ sin ey] dB' = -1 (B5) 
where H(8) = h(1/2 sin 8). Thus the unknown in (B5) is a regular function. 
To perform the integration in Eq. (B5) numerically, we first divide the interval of in- 
tegration, 0 < B' < 7/2, into an even number, n, of equal subintervals of length 7/2n. We 
denote the values assumed by f' at the endpoints of these subintervals by Bj =1, 2,..., 
n+1). Then 8, =0and B,,,= 7/2. We further denote by H; the values of the unknown 
function H(8') at B,, i.e., H; = H(8;). Finally, we allow the variable 8 to assume the 
values B(i=1,2,...,n+ 1). We thereby obtain n + 1 linear algebraic equations for the 
n+ 1 unknowns H,, each equation corresponding to a different value of the index, i. These 
equations are constructed as follows: 
The part of the integration in (B5) that involves Y,[F/2(sin B + sin B’)] can be per- 
formed at once by Simpson’s rule: 
1 / 
| 
0 
2 
H(B') ¥, [Fcsin 6, + sin B")| dg' = 7 {i [Ecsin B,+ sin 6,)| 
+ 4H,Y, Hes B; + sin p.)| + 2H,Y, | (sin B, + sin p3)| 
Fa.» gH ¥,|£(sin 8, + sin Pau] (1 = 2,3, .<. mt 1)s Fe) 
(We have excluded the case i = | in (B6) because it is discussed separately below.) 
On the other hand, the logarithmic singularity in Y, prevents the remaining integral in 
(B5) from being represented by a formula like that of (B6); when i = j, the expression 
Y,(F/2|sin 8; — sin B,|) is meaningless. We therefore break up the integral and write 
(when i = 1 or n + 1) 
