Summary: This work studies the radial orbit instability for a family of anisotropic isochrone spheres by a numerical linear stability analysis, and then compares the results with N-body simulations. A new way of choosing basis functions for the modes is introduced that allows spatially infinite systems to be handled without truncation. Previous studies of the same models, by adiabatic deformation and by N-body integrations, found instability for \(<v^ 2_ r>/<v^ 2_{\theta}>\gtrsim 2\); the present computations indicate that very slowly growing modes persist for anisotropies as small as \(<v^ 2_ r>/<v^ 2_{\theta}>\simeq 1.4\). For the more unstable models, the shapes and growth rates of the modes are fairly well reproduced in N-body simulations. These results suggest that spherical models are unstable over a wider range of anisotropies than previously supposed.