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The proposed use of intense beams of high-energy heavy ions for pellet fusion have renewed interest in the problem of space-charge limits in long periodic focusing systems. Recent analytical and numerical studies at Berkeley and NRL for beams with a K-V distribution indicate that instabilities may develop in such systems when the beam currents exceed certain thresholds. Since no relevant experimental data exists with which to compare the theory, we propose an inexpensive, small-scale experiment with a 5-20 kV electron beam to study this problem. The electron beam is injected into a periodic solenoid focusing system consisting of 20 solenoid lenses initially. The pertinent results of beam transport theory are reviewed and the design features of the electron beam experiment are presented.