This paper analyses the dispersion relation for a collisionless moving electron plasma, when the direction of motion is normal to the magnetic field and that of the wave propagation along the magnetic field. It is shown that, in strong magnetic fields, the one continuous allowed band of the left-handed wave (of the stationary plasma) splits into two, and the right-handed wave shows a second resonance besides the cyclotron resonance. In weak magnetic fields, the lefthanded wave develops a backward wave band, which shows resonance at its low frequency edge, and the right-handed wave also develops an extra band of propagation. The effect of the motion of the plasma, on waves of frequency much lower than the plasma frequency, is identical to a doppler shift, but, on those of frequency much higher than that, is negligible.