Measurement of the total normal thrust generated in steady torsional flow between parallel plates enables one to determine the normal stress quantity [P 11 −2P 22 +P 33 ] as a function of shear rate γ̇ for polymer melts. This measurement has been carried out for two linear polyethylene melts of different molecular weight using a modified Maxwell orthogonal rheometer. It is found that the normal stress curves have similar shape but are shifted from one another both horizontally along the γ̇ axis and vertically. A strong correlation exists between [P 11 −2P 22 +P 33 ](γ̇) and 2G′(ω), where G′(ω) is the dynamic storage modulus in shear, and the two quantities have identical molecular weight dependencies. The manner of treatment of normal stress data for polymer melts of differing molecular weights is explored, with particular regard to the application of the method of reduced variables. In addition, the predictive ability of the Springs four‐constant equation as regards normal stress behavior of polyethylene melts is discussed.