It is shown that, formally, the equations governing the flow of an upper convected Maxwell fluid can be reduced to the (compressible) Euler equation in the high Weissenberg number limit. A difference arises, however, in the ＇equation of state＇: instead of a relation between pressure and density, we obtain a relation between the ＇density＇ and the divergence of the ＇velocity field＇.