An approach to the kinetic theory of gas flows is developed which starts with Maxwell's original integral equations of transfer, rather than with the Maxwell-Boltzmann equation for the velocity distribution function itself. In this procedure the Maxwell-Boltzmann equation is satisfied in
a certain average sense, rather than at every point. The advantage of this method is that relatively simple distribution functions are utilized which contain a small number of unknown functions to be determined by
applying the conservation laws, plus several additional higher moments. For simplicity a "two-stream Maxwellian" is employed, which is a natural extension and generalization of Mott-Smith's function for a normal shock,
but differs from it in certain essential respects. As an illustration, the method is applied to linearized plane Couette flow and Rayleigh's problem. Reasonable results are obtained for macroscopic quantities such as mean
velocity and shear stress over the whole range of densities from free-molecule flow to the Navier-Stokes regime. This technique is now being applied to some typical non-linear rarefied gas flows
