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Experimental measurements on the transverse velocity distribution in cylindrically symmetric electron beams emergent from magnetically shielded cathodes are reported. For strong magnetic fields the transverse velocity distribution at the beam center has been found to be composed of discrete classes. These results are explained on the basis of a beam model with nonlaminar flow in which thermal velocities and space charge are neglected. The mechanism depends on the periodic passage of electrons through or very near the beam axis and the fact that the wavelength of this periodicity is a decreasing function of the distance between the point of entry into the magnetic field and the axis. When the space charge and magnetic forces are comparable, no discrete transverse velocity distribution was observed. It is shown, however, that the beam should still be in the same type of nonlaminar flow as above.