We consider a limit of the upper convected Maxwell model where both the Weissenberg and Reynolds numbers are large. The limiting equations have a status analogous to that of the Euler equations forthe high Reynolds number limit. These equations admit parallel shear flows with an arbitrary profile of velocity and normal stress. We consider the stability of these flows. An extension of Howard's semicircle theorem can be used to show that the flow is stabilized if elastic effects are sufficiently strong. We also show how to analyze the long wave limit in a fashion similar to the inviscid case. (C) 2008 Elsevier B.V. All rights reserved.