- A linearized stability analysis is carried out for the breakup of small-diameter liquid filaments of dilute polymer solutions into droplets. Oldroyd's 8-constant model expressed in a corotational reference frame is used as the rheological equation of state. The crucial idea in this theory is the recognition that the liquid may be subject to an unrelaxed axial tension due to its prior history. If the tension is zero, the present analysis predicts that jets of shear-thinning liquids are less stable than comparable jets of Newtonian liquids; this is in agreement with previous analyses. However, when the axial tension is not zero, and provided the stress relaxation time constant is sufficiently large, the new theory predicts that the axial elastic tension can be a significant stabilizing influence. With reasonable values for the tension and stress relaxation time the theory explains the great stability observed for jets of some shear- thinning, dilute polymer solutions. The theory explains why drops produced from jets of such liquids are larger than drops from corresponding Newtonian liquids. The theory also appears capable of explaining the sudden appearance of irregularly spaced bulges on jets after long distances of t,ravel with little amplification of disturbances.