78 Samuel Karp, Jack Kotik, and Jerome Lurge 
If we introduce the dimensionless variables x and x’ into (5), we get 
1/2 
1/2 
d d 
C= 4 | | — aise, ) K [F(x- x ')] dx' dx (7) 
1/2 1/2 
where C(x) is defined by 
tx) = C,(Lx) (s) 
and 
uh a | 
fark FE he pn" (9) 
The quantity f is called the Froude number. 
The problem is to find the function ¢€(x) that minimizes Eq. (7) subject to a suitable 
constraint. In the case of a strut, a natural constraint is the requirement that a strut of 
given length L have a given volume V per unit draft. The side condition is thus 
L/2 
20(x) dx = V. (10) 
-L/2 
In terms of €(x), this becomes 
1/2 
G(x) Mdxa= 1). (11) 
-1/2 
The problem therefore is to minimize Eq. (7) among all functions ¢(x) that satisfy the 
following conditions: 
1. d¢/dx must be integrable in —1/2 < x < 1/2 (otherwise Eq. (7) would be meaning- 
less). 
2. €(x) must satisfy Eq. (11). 
3. ¢(+1/2) =0. 
It is at this point that the central difficulty of the problem appears, in that the minimiz- 
ing ¢(x) will now be shown to obey an integral equation that has no solution. Using stand- 
ard procedures in the calculus of variations, we have at once that if ¢(x) satisfies the three 
above conditions, it must also satisfy the equation 
1/2 
| BED Erle. | ees (12) 
1/2 
dx' 
