The solutions of the incompressible quasi-cylindrical momentum-integral equations describing the flow in the viscous core of a wing-tip vortex are obtained in a closed form and are shown to have two distinct branches. The discontinuities of these solutions have infinite axial gradients and therefore, following Hall, are assumed to signal the inception of the vortex breakdown. Benjamin's finite transition, with its excess flow force dissipated, is shown to give results equivalent to a sudden cross-over, upstream of the discontinuity, from one branch solution to another. The critical point of such a cross-over is downstream from the cross-over, at the discontinuity. Sarpkaya's experimental data, and the nature of the solutions ahead of the discontinuity, suggest that the physical manifestation of the discontinuity is the spiral breakdown, whereas the cross-over seems to be related to the rapidly expanding and subsequently contracting axisymmetric bubble. This therefore implies that the beginning of the spiral breakdown is the all important disturbance which triggers off not only the downstream asymmetric departure of the flow from its quasi-cylindrical form but also the formation of the upstream axisymmetric cross-over bubble. Solutions for the turbulent flow downstream from the spiral breakdown indicate that the wing-tip vortex breakdown can result in an appreciable reduction of the maximum circumferential velocity and should thus lessen the danger that trailing vortices present to following aircraft.