A theoretical model is developed to describe blood flow in narrow capillaries, with inside diameters 3 microns to 6 microns. Each red blood cell is assumed to have axisymmetric geometry, and fixed surface area and volume. Cell velocities in the range 1 mm s-1 or higher are assumed, and the stress in the cell membrane is approximated by an isotropic tension. This tension is assumed to fall to zero at the concave trailing end of the cell, except in vessels whose diameter is near the minimum for passage of the cell. In the latter case, a separate analysis is used, in which the cell is effectively rigid and fully distended at each end. Lubrication theory is used to describe the plasma flow in the narrow gap between the cell and the vessel wall. Good agreement is obtained between predicted values of the tube hematocrit and apparent viscosity and published experimental values for these parameters.