Two wide classes of velocity distribution functions suitable for confinement in "open-ended" containment fields may lead to electrostatic (linear) instability of an infinite homogeneous plasma in a uniform magnetic field. They are the anisotropie distributions with a preponderance of the particle energy being directed perpendicular to the magnetic lines (distributions which may also be set up by such situations as resonant heating, charge exchange losses and energetic particle injection) and the double streams resulting from the axial oscillations of the trapped particles between two magnetic mirrors. We are concerned here with a plasma constituted by electrons which spiral around the magnetic lines of force with a large (but not relativistic) perpendicular velocity and by ions which move almost axially. We consider in particular the effects associated with the finite radius of the streams. The discussion of the stability of such a system is of some relevance to the injection and accumulation experiments in progress at Saclay in the Pléiade and Circé machines. The validity of the electrostatic approximation made throughout the paper is investigated in an appendix.