The early treatment by Townsend of the influence of a crossed magnetic field on the movement of an electron swarm in a uniform electric field is extended to high electric fields when ionization occurs and secondary processes can no longer be neglected. It is shown that the discharge moves bodily in the ev×B direction with a velocity, when secondary electron emission by positive ion bombardment of the cathode takes place, given byv = v_T⁺ (tan θ⁻ + tan θ⁺)(τ₊/τ_s + τ₊) in which ν_T⁺ is the transverse ion drift velocity, θ⁻ and θ⁺ are the electron and ion deflection angles and τ₊ and τ_s are the ion transit and site times. The site time is the delay which occurs between the impact of a positive ion on to a surface and the subsequent emission of an electron; the transit time is the time it takes for the ion to cross the gap. When secondary electron emission by photon bombardment takes place, the propagation velocity of the gaseous discharge is given by v = v_perp(v*/v_T⁻ + v*) in which ν* is the photon velocity and ν_perp^- and ν_T^- are the electron perpendicular and transverse drift velocity respectively.Because of the importance of θ^-, the dependence of the perpendicular and transverse electron drift velocity on collision cross section, mean energy, E/p and B/p is examined in detail over a wide range and quantitative results obtained for hydrogen. Whereas ν_perp⁻ reaches a maximum with increase in B/p when v₀ approximately equals eB/mp, tan θ⁻ continues to increase linearly with B/p from low to high magnetic fields. Good agreement is obtained with some but not all of the experimental data reported in the literature. The need for similar data on positive ions is pointed out.