A stability analysis is presented of the electrothermal modes of perturbation of a spatially homogeneous, two-temperature and fully ionized plasma with wave-vector K perpendicular to the magmetic field. It is found that the wavelength of the fastest growing unstable, non-convective mode is of the order of (mi/ me)½ ae, with maximum growth rate of the order of (me/ mi) vei, where ae and vei are the electron Larmor radius and electorn-ion collision frequency respectively. Depending on the values of ωτ and β, this analysis also shows the presence of a convective mode which persists in the limit of long wavelengths. This theory has been consistently examined with homogeneous equilibrium plasma models constructed under the assumption of either a scalar or a tensor electric conductivity. A comparison of these results with the previous steady-state profiles obtained in parts 1 and 2 of this series indicates that current filamentation can take place as a result of electrothermal instabilities in the plasma.