A new method of instrumentation for normal-stress determinations was developed, making possible accurate unsteady state as well as steady state measurements, eliminating errors arising from fluid-filled pressure-tap holes, and permitting the determination of the complete stress state in a single cone-and-plate shearing-geometry experiment. Sensitive, nonflow semiconductor pressure transducers mounted at several radial positions with their pressuresensing diaphragms flush with the plate surface provide data for the normal-stress distribution. The normal-stress distribution, together with the total normal force from the single-geometry experiment, enables determination of the primary and secondary normal-stress differences by two independent methods of analysis while the transmitted torque enables determination of the viscosity, each as a function of shear rate. Only the normal-stress distribution is required if an independent check on the normal-stress determination is not desired. Similar advantages arise in the application of the instrumentation to a parallel-plate shearing geometry, The new instrumentation was used in the determination of the complete rheological stress state of three aqueous and two “Tetralin” solutions of polymers in a cone-and-plate shearing geometry shear rates of 0.02 to 450 s−1 on a Model R-17 Weissenberg Rheogoniometer. The normalstress differences computed by means of two methods of analysis are in surprisingly good agreement. The ratio of the secondary to the primary normal-stress difference was negative. The absolute values of this ratio decreased with increases in the shear rate, the maximum observed value being 0.4.