  MHD in protostellar discs is modified by the Hall current when the ambipolar
diffusion approximation breaks down. Here I examine the Balbus-Hawley
(magnetorotational) instability of a weak, vertical magnetic field within a
weakly-ionized disc. Vertical stratification is neglected, and a linear
analysis is undertaken for the case that the wave vector of the perturbation is
parallel to the magnetic field.
  The growth rate depends on whether the initial magnetic field is parallel or
antiparallel to the angular momentum of the disc. The parallel case is less
(more) unstable than the antiparallel case if the Hall current is dominated by
negative (positive) species. The less-unstable orientation is stable for chi
less than about 0.5, where chi is the ratio of a generalised neutral-ion
collision frequency to the Keplerian frequency. The other orientation has a
formal growth rate of order the Keplerian angular frequency even in the limit
that chi tends to zero! In this limit the wavelength of the fastest growing
mode tends to infinity, so the minimum level of ionization for instability is
determined by the requirement that a wavelength fit within a disc scale height.
In the ambipolar diffusion case, this requires chi > v_A/c_s; in the Hall case
this imposes a potentially much weaker limit, chi > v_A^2 / c_s^2.
