82 Samuel Karp, Jack Kotik, and Jerome Lurye 
a _ : k V 
ZS zz a) ’ (23) 
its expansion in powers of V/2L is assumed to be (see, for instance, Ref. 14) 
Byped Ziy Wags 
s(x.¥: a mya C(x) + smaller terms. (24) 
Equation (24) reflects the fact that the so-called strut is a strut only to the first order 
in V/2L. The actual shape as determined by the closed streamlines of Eq. (22) has a depth 
dependence; however, we expect that the shape approaches a strut as a limiting form for 
great depths.* 
At this point it is important to observe that the leading term in the approximation to C,, 
for the shape s(x,y;V/2L), as V/2L + 0, is given by Eqs. (7) and (16). 
The basic idea, on which all our results depend, is to consider a class of potentials 
more extensive than that defined by Eq. (21) but including the latter as a subclass. Specif- 
ically, let 
(0 1/2 
V c V OG(x,y,z;x',y' ,0 
aL M(X.¥+2) = ar ar | dy' | &(x') —— dx' (25) 
0 -1/2 
where g(x) is integrable and satisfies Eq. (11), i.e., 
1/2 
ACD) he = ie (26) 
-1/2 
Otherwise g(x) is to be arbitrary. Equation (25) resembles Eq. (21) integrated by parts. 
Evidently, p satisfies Eqs. (20) and (20a) with d¢/dx replaced by dg/dx. The derivative 
dg/dx need not be integrable, and g(+1/2) need not vanish; hence the potentials p form a 
larger class than the potentials ¢, since the functions ¢(x) appearing in Eq. (21) were as- 
sumed to have integrable first derivatives and to vanish at x = 1/2. (Obviously, the class 
of function €(x) is included in the class g(x).) Our expectation is that among the ship forms 
defined by this larger class of potentials there will be found one with a minimum wave re- 
sistance. That this is so will be established shortly. 
As in the case of the forms s(x,y; V/2L) associated with qd, the extended class of forms, 
a(x,y; V/2L) say, associated with p, are made up of the closed streamlines! of the flow 
*For instance a depth equal to a wavelength of a surface wave having velocity c. 
tAlthough we have not proved it rigorously, we believe that there will always be a closed body (i.e., 
a surface of closed streamlines) formed in the flow defined by Eqs. (25) and (27) whenever the total 
moment of the dipole distribution on the x axis is negative. Since, as discussed above, the density 
of the distribution is —(c/2m7)(V/2L) g(x), the assertion is that there will be a closed body if 
1/2 
{ g(x) dx > 0. 
-1/2 
We now observe that all the distributions considered satisfy this latter condition because they all 
satisfy Eq. (26), the right side of which is always positive. Thus, all the flows defined by Eqs. (25) 
and (27), subject to Eq. (26), lead to closed bodies. 
