New York, NY: American Institute of Physics [AIP] January 1956. First Separate Edition. -43 pages. 10 1/2 x 7 3/4 inches. Publisher’s original tan printed wrappers. Sun fading to the rear panel. Near Fine. Wraps. 
The earliest known form of this paper is an internal Bell Labs reproduced typescript. The Journal of Applied Physics, Vol 27, No. 1, pp. 42-43, January 1956 first published this paper, for which an offprint is known (as here). Also later printed in the Bell Telephone Systems Monograph Series (#2547: March 1956).
Abstract: "It is proved that any network of linearly wound potentiometers and fixed resistors have a curve of resistance versus shaft angle which is concave downward." and later "As part of a computer, a rheostat having a resistance that was a concave upward function of the shaft angle was needed. After many attempts to approximate it with networks of linearly wound potentiometers and fixed resistors, it became apparent that either it was impossible or that we were singularly inept network designers. Rather than accept the latter alternative, we have proved the following theorem..."
In short, Shannon and Hagelbarger proved it was impossible to build a needed electrical circuit with ANY network of linearly wound potentiometers and fixed resistors. This mathematical result is an excellent example of where a practical engineering solution was needed and could have been experimented upon endlessly had the mathematicians not stepped in and proved it impossible - a hallmark of much of Shannon’s work.
Paul J. Nahin surmises that this paper, whose results are sometimes called the Shannon-Hagelbarger theorem, were probably related to analog computers and not a digital machine. Shannon worked for a time on the Differential Analyzer, "the most advanced electromechanical computer in the world." He also notes that H. M. Melvin advanced an alternative proof for this paper in "Journal of Applied Physics,” June 1956, pp 658-659. Further research might reveal just what practical application Shannon and Hagelbarger were working on, but we haven't been able to discover it thus far.
It is interesting bibliographically that all three forms of this paper (reproduced typescript, Journal of Applied Physics offprint, and Bell Monograph) in Shannon's files were slightly different. The (presumed earliest) reproduced typescript has some apparent typos and incorrectly uses the phrase “concave downward function of the shaft angle" as the desired solution in the first sentence. The Bell Monograph paper corrects that to "concave upward function," as does the Journal of Applied Physics offprint. Lastly, the Journal of Applied Physics offprint includes a second sentence in the first paragraph of Corollary II that starts "This is also true if..." which is present in the reproduced typescript but not in the Bell System Monograph. The completist will want one of each.
PROVENANCE: The personal files of Claude E. Shannon (unmarked). There were multiple examples of this item in Shannon's files.
Sloane and Wyner, "Claude Elwood Shannon Collected Papers," #98
Nahin, Paul J., "The Logician and the Engineer: How George Boole and Claude Shannon Created the Information Age," p 5.