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Most previous treatments of long high‐density electron beams have assumed laminar flow. Recent experiments show that in many cases this assumption is unjustified. The present theory is based on a non‐laminar optical model which treats thermal velocities as an integral part of the motion. A Maxwellian distribution of initial transverse velocities is assumed at the cathode, and a first‐order focusing theory is applied in order to calculate trajectories at any point in the beam. It is shown that whenever a long beam is confined by a focusing field, images of the cathode are formed repeatedly along the axis. When applied to uniform magnetic focusing fields, the theory predicts the periodic formation along the axis of cathode images and crossovers, and a relative rotation of successive images. Such effects have been reported. Values of required focusing fields are derived for given space‐charge and thermal‐velocity conditions, and relations are established between beam spread and transverse velocities. The resulting picture differs radically from the laminar model of beam flow.