  The theory of waves and instabilities in a differentially rotating disc
containing a poloidal magnetic field is developed within the framework of ideal
magnetohydrodynamics. A continuous spectrum, for which the eigenfunctions are
localized on individual magnetic surfaces, is identified but is found not to
contain any instabilities associated with differential rotation. The normal
modes of a weakly magnetized thin disc are studied by extending the asymptotic
methods used previously to describe the equilibria. Waves propagate radially in
the disc according to a dispersion relation which is determined by solving an
eigenvalue problem at each radius. The dispersion relation for a hydrodynamic
disc is re-examined and the modes are classified according to their behaviour
in the limit of large wavenumber. The addition of a magnetic field introduces
new, potentially unstable, modes and also breaks up the dispersion diagram by
causing avoided crossings. The stability boundary to the magnetorotational
instability in the parameter space of polytropic equilibria is located by
solving directly for marginally stable equilibria. For a given vertical
magnetic field in the disc, bending of the field lines has a stabilizing effect
and it is shown that stable equilibria exist which are capable of launching a
predominantly centrifugally driven wind.
