84 Samuel Karp, Jack Kotik, and Jerome Lurye 
(16) or (29) represent valid formulas for the drag of the extended class of shapes 
o(x,y; V/2L).* 
To minimize this drag among all shapes o(x,y; V/2L), we must determine a function 
g(x) that minimizes C,, in Eq. (29), subject to the side condition Eq. (26). This is a 
straighforward problem in the calculus of variations and leads to the following integral equa- 
tion for the minimizing function g(x): 
1/2 
| é(x') Y,(F|x-x"|]) dx’ = A (30) 
-1/2 
where A is a (constant) Lagrange multiplier whose value is determined by the use of Eq. (26). 
It is easily shown that Eq. (30) is the necessary and sufficient condition for g(x) to minimize 
Eq. (29). 
Once Eq. (30) has been solved, subject to Eq. (26), the resultant g(x) is substituted 
into Eq. (25) and the potential (x,y,z) obtained. The minimizing shape, o(x,y; V/2L), is 
then given by the set of closed streamlines associated with the potential cLx + (V/2L) 
u(x,y,z). As already noted, such a shape will not be a true strut; it is a strut only to the 
first order in V/2L. 
It has been proved [10] that any solution to the integral Eq. (30) must become singular 
at the endpoints x = +1/2, the singularity being of the form 
Thus the solution cannot have a first derivative which is integrable over the closed interval 
-1/2 < =< 1/2, and therefore cannot belong to the set of functions C(x). It is for this rea- 
son that we introduced the extended class of functions g(x). 
Since the minimizing g(x), i.e., the solution to Eq. (30), becomes infinite at x = +1/2, 
it may seem that such a function can have little to do with the actual minimizing shape, 
o(x,y;V/2L). However, Eq. (28) indicates that (V/2L) g(x) = 0, is the first-order (in V/2L) 
approximation to a. The explanation is that the approximation of o by o, is not uniform in 
x, and gets worse and worse toward the endpoints x = +1/2. Thus, even for very small 
V/2L, the shape will be well approximated by o, only if x is not too near the ends. The 
shape is determined by a set of streamlines, as stated above, and its complete determination 
is discussed in Sections 3 and 5. 
We have interpreted g(x) as a measure of the density, —(c/2m)(V/2L) g(x), of the dipole 
distribution that gives rise to the shape o(x,y; V/2L). In this view, the minimization process 
consists in the following: among all distributions of x-directed dipoles in the half-strip al- 
ready defined, whose density is a function of x only and satisfies Eq. (26), we choose that 
*Note that if g(x) = const. the body in an infinite fluid would be a Rankine oval, and delta function 
singularities would have been necessary in Eq. (7). 
