The problem of the instability of counterstreaming beams of charged particles is extended to cylindrical and spherical geometries. For well‐focused configurations it can be solved by complex contour integral representations. The effects of the convergence of the flow and the density gradient along the trajectories of the particles are considered. The linear spectrum for the cylindrical case is obtained, together with the proof that the solution has finite energy and satisfies two physical matching conditions through the origin. The properties of the special functions which solve this problem are presented. Although the density of the ideally focused model diverges as 1/rat the origin, the growth rate of the instability, for a system of radiusR, is given by &ohgr;2pR/V02&xgr;n, whereV0is the beam velocity, &xgr;nare the zeros of the Bessel function of zeroth order, and the plasma frequency &ohgr;pis evaluated at one‐half the average density of particles.