The problem of the numerical solution by the method of characteristics of the two- and three-dimensional flow of an inviscid nonequilibrium gas is formulated. For unsteady flows the method is limited, for practical reasons, to the solution of hypersonic or other flows which have short transient times. The specific details of the finite difference network and the associated finite difference equations are presented. It was found that numerical instability was possible with certain previously proposed networks. An existing stability criterion was found to apply to the multidimensional method of characteristics and this criterion was used to synthesize a stable finite difference network. High speed digital computer programs are presented which perform the numerical calculation. Four example cases of the flow about a circular cylinder with its axis perpendicular to a Mach five free stream are presented.