The use of an arterial prosthesis with a tapered lumen has several important advantages; for example, improved stability of flow, increased wall shear and better matching of its size with that of the host vessel. Tapering may, however, lead to increased energy losses, particularly if the angle of taper is large and the flow is high. This study is concerned with the determination of pressure drop for steady and laminar converging flow through rigid wall models of tapered arterial grafts. The angles of taper examined ranged from 0.5 degrees to 1.0 degrees. Aqueous solutions of polyacrylamide, with non-Newtonian viscous properties similar to those of blood, were used. The pressure drops across the tapered tubes were measured and the data were related to the pressure loss in cylindrical tubes of equivalent dimensions. Expressions for the ratio of the pressure drop in a tapered tube to that in a cylindrical tube for steady flow of a power law fluid were derived; there was good agreement between the predicted and the measured pressure drop ratios over a wide range of flows. The results of this study may be applied to the design of tapered arterial grafts. The pressure losses to be expected in tapered bypass grafts having various dimensions can easily be computed.