  We consider the flow of an electrically conducting fluid between
differentially rotating cylinders, in the presence of an externally imposed
toroidal field B_0 (r_i/r) e_phi. It is known that the classical, axisymmetric
magnetorotational instability does not exist for such a purely toroidal imposed
field. We show here that a non-axisymmetric magnetorotational instability does
exist, having properties very similar to the axisymmetric magnetorotational
instability in the presence of an axial field.
