Using a plane-geometry model developed previously for the study of flute instabilities having a general equilibrium state, and taking into account the finite dimension of the plasma along the magnetic field and the possible presence of a cold plasma between the plasma and the conducting plates, the author resumes the study of electrostatic instabilities which, when the plasma becomes infinite, give rise to instabilities propagated "perpendicular" to the magnetic field. He determines the dispersion relation for the flutes, for the Mikhailovski-Timofeev gradient instabilities and for the loss cone instabilities, which he analyses in greater detail since they entail particularly stringent criteria in infinite geometry. The stability limit is presented as a set of curves depending on two parameters. The values assumed by these two parameters can be divided into two quite distinct groups corresponding to longitudinal and transverse injection. The two groups of values lie in the plane to either side of a critical line that divides the stability analysis into two distinct regions. Starting from this dividing line it is possible to write in each region "approximate" criteria that are more simple but indicate general trends. Devices with longitudinal injection cannot be stabilized by cold plasma and finite length effects. Devices with transverse injection offer better stabilization possibilities; under present conditions, however, it is probably necessary to increase the plasma density by one or two orders of magnitude. The flute instabilities are stabilized at the same time. Since the effect of gravity appears to be slight, the results accordingly suggest that, for obtaining a stable plasma, "short" high-density devices with a simple mirror are preferable to "long" devices with a magnetic sink. It remains to be seen, however, whether the effect that has been described mathematically is physically acceptable.