The method developed by the authors referred to in the Preface has been improved in a number of minor respects, to make possible calculations at Taylor numbers up to about 106, for invariant manifolds up to dimension 14, and to suppress axial flow. The method was programmed for the Cray-1 computer and applied to two series of calculations, one for a narrow gap between the cylinders and one for a wide gap. The calculations exhibited the first few bifurcations and the corresponding modes of motion and determined the stability of those modes. No aperiodic motions were found. There is evidence for an additional subcritical bifurcation from the wavy vortex mode at a Taylor number ï¿½ï¿½ 1,148 x 106 in the wide gap problem. Orbit calculations at 1.15 x 106 failed to find further attractors and indicated an explosive transition to regions of the configuration space beyond the applicability of the present method.