An axial annular‐flow instrument was used to determine the secondary normal stress difference,N2=σrr−σφφ,in case the velocity field is expressed asvr=0,vφ=0,andvz=vz(r).The pressure difference, at a fixed axial position, between inner and outer walls of the annulus was measured at various shear rates or axial pressure gradients. The pressure at the outer wall is found to be higher than that at the inner wall throughout the experiment. This meansN2is tensile. The data of pressure difference were analyzed to obtainN2as a function of shear stress.N2increases with the square of shear stress at a low shear stress but at a rate slightly less than that when shear stress is above 0.08 MPa or so. Also,N2is compared with the primary normal stress difference,N1,which had been determined from the measurement of exit pressure with a capillary‐type instrument. The magnitude ofN2relative toN1is found to be 20–40% for the polymer melts we used.