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Calculations are made of the electron trajectories in infinite strip beams constrained to move in a magnetic fleld parallel to the direction of flow. Space charge is taken into account, but nevertheless the analysis is only valid within certain well-defined limits. Different types of solution are obtained according to whether the beam is launched from a source outside or inside the magnetic fleld. When the gun is outside the field the type of flow also depends on whether the current is greater or less than a certain value. In all cases the beamwidth varies periodically along the length of the beam, and expressions for the periodicity and the ratio of maximum to minimum width are obtained. Both fundamental and practical limitations are then considered, and their effects on the electron flow discussed qualitatively. The importance of conditions outside the beam boundary is emphasized.