Two previous communications dealt with the nature of the motion and the resistance to flow in capillary blood flow. In this final paper measurements are reported of the pressure required to force mammalian erythrocytes through pores having a diameter less than the cells. The cells, from fresh human or dog blood, were resuspended in acid citrate dextrose solutions. The final suspensions (about 1,000 times more dilute than whole blood) were immediately emptied into a millipore filter apparatus and the rate of filtration was measured. Filters having pore diameters of 5.0 and 3.0 microns were employed. The cellular concentration of samples of the original suspensions and of the filtrate was determined. It was observed that the rate of filtration decreased rapidly initially and then became constant. In the `steady state' the cellular concentration of samples of the filtrate was found to be approximately equal to that of the original suspension. A simple theory is presented which adequately describes the flow of a suspension through such filters. It is concluded that mammalian erythrocytes, particularly human and dog cells, will pass steadily, without hemolysis, through pores 5.0 or 3.0 microns in diameter under pressures of 4cm of water or less.