Transition to Taylor vortex flow between two conical cylinders, with the inner one rotating and the outer one stationary, is studied by the numerical method in this article. It is found that the basic flow becomes unstable with increasing of Reynolds number (Re) above a certain critical value Rec=117 and with the further increase of Re to about Re=300, the first stable vortex is formed near the top of the flow system. The annulus is filled with six pairs of vortices at about Re=800. These are confirmed by experimental observations. Moreover, the local extreme values of pressure and velocity are reached at the adjacent lines between neighboring vortices or at the medium lines of large vortices. The local minimum values of velocity and the local maximum values of pressure are reached at the same point, whereas the local maximum values of velocity of the flow are assumed at the point of inflection for pressure.