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In the ‘polytron’ a series of cusp magnetic fields is superimposed on a toroidal discharge. Stability is improved by favourable field curvature, and containment will be good if the plasma acquires a substantial velocity relative to the cusps. This can be achieved by a Hall acceleration mechanism. In this paper a hydromagnetic analysis is presented in which realistic time variations of electric and magnetic fields are allowed, and plasma self-fields are included. An upper bound is set to the acceleration obtainable from a given cusp system and expressed as a critical electric field Ec. The existence of Ec is confirmed experimentally, and it is possible to reproduce experimental waveforms approximately by inserting appropriate parameters into the equations. The effect of a toroidal magnetic field is also considered and a criterion derived for its maximum permissible value. In designing a fusion device this criterion could be an important consideration.