Minimum Wave Resistance for Dipole Distributions 107 
By these procedures, we can perform numerically the integration on the left side of (B7) 
for i= 2,3,..., 2. Adding the result to the right side of (B6) and (in accordance with the 
integral equation (B5)) equating the entire sum to —1, we obtain n — 1 linear algebraic equa- 
tions for the n+ 1 unknowns H;(j=1,2,..., 2+ 1), each equation corresponding to one of 
the i values, i = 2,3,..., 7. 
The remaining two equations, associated with i = 1 and i =n + 1, are arrived at sepa- 
rately. When i = 1, 
y,(E |sin B= Sin 6'|) = AG |sin By + Sin 6'|) 
since 8, = 0; thus (B5) becomes 
mn / 2 
2H(B') (4 sin 2’) dp’ = -1 (i = 1). (B10) 
This equation can be rewritten 
Bo - Bn+1 : 
2H(") ¥, ( sin 6’) dB’ + | H(A") Y, (sin 6" dp’ = -1. (B11) 
By B 
2 
The first integral on the left is approximated by setting H(8‘) = H, and expanding Y, 
about 8'=0. The result is 
5 Rid é ea 2H 1y'F 72F2 
i 7% »¥6(5 fe s') Te Pian oe ( on ) ( i rtd 
2H, er Oy F2 
sien haw iong Bot BS B12 
n [ arr (4 =) roe 
The second integral in (B11), which is free of singularities, extends over an odd num- 
ber of subintervals and is therefore evaluated by the combination of the trapezoidal and 
Simpson’s rules already described. Upon adding the result to the right side of (B12) and 
substituting into (B11), we obtain the algebraic equation corresponding toi = 1. Finally, 
when i = n+ 1, (B5) becomes 
Bn+1 F 
j we'yy, (4 lsin Bue isin a"|) dB' 
1 
Ba+t 
+ \ H(B') ¥, E (sin 6,4, + sin 6’) | dp'=-1 (i= n+1). (B13) 
1 
