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The problem of focusing long electron beams by means of magnetic or electric fields which vary periodically along the beam is considered. Four specific cases are discussed: periodic fields of the axially symmetric and quadrupole type with either electric or magnetic fields employed. The equations of motion are written and solutions obtained which show beam trajectories corresponding to essentially parallel flow. Under certain specified conditions only small ripples are present. Actual trajectories as plotted by the analog computer are shown and charts and equations are presented to aid in the design of practical periodic focusing systems.