In this paper, the general hydrodynamic theory of visco-inelastic, incompressible, non-Newtonian fluids, developed in a previous paper (l), is applied to the problem of the flow of such a fluid through a tube of circular cross-section. It is found that the absolute values of the normal stress components are no longer uniform over a cross-section of the tube normal to its axis, as in the case of Newtonian fluids obeying the laws of classical hydrodynamics.However, the pressure difference, between points at equal radii on two planes normal to the axis, is independent of the position of these points on the planes.