Minimum Wave Resistance for Dipole Distributions 101 
The dipole moments, m,, are related to the source strengths, 7,, by 
a (s = 2p- 1) 
Gari |G s = = 
h % e 
(A5) 
-—=4q, (s = 2p). 
Introducing Eqs. (A4) and (A5) into (A2) and (A3), we obtain 
n/2 me _2 
1 a ~ a “vy ,cos u 
I= z Mm, {eos (vx, cosh u) - cos [v< x, + h) cosh ul} e (A6) 
p=1 
we a o -vy cosh*u 
Jn 1 m,{sin (vx, coshu) - sin ck, + h) cosh u]} an * , (A7) 
h p=1 
For small h, 
cosh u) 
cos|v(X, + h) cosh u] = cos (VX, 
- hv cosh u sin (vx, cosh u)_ (A8) 
sin[v(%, + h) cosh u| = Sin (vx, cosh u) 
+ hv cosh u cos (vx, cosh u). (A9) 
Therefore in the limit, as h + 0, 
n/2 K, -v¥_cosh?u (A10) 
I, = v cosh u m, sin (vx, cosh u) e th 
p=1 
me A -vy cosh-a 
J, =7v cosh u a m, Cos (vx, cosh u) e E ‘ (All) 
p=1 
These equations represent the forms assumed by /, and J, when the original sources 
(and sinks) are combined so as to form a collection of n/2 x-directed dipoles in the x-y 
plane, the dipole moments being m, (p=1,2,..., /2). The wave resistance of such a 
collection is obtained by substituting Eqs. (A10) and (A11) into (Al). Before substituting, 
we go to the limit of an infinite number of dipoles of infinitesimal strength, continuously 
distributed with a density m(x) in the half strip —L/2 <%<L/2, 0<y<~, Z=0. To this 
end, it is convenient first to imagine the n/2 dipoles arranged in a rectangular array of r 
rows each containing g dipoles. Then (A10) and (Al1) can be replaced by the double sums 
