In the study of transonic flow, one of the most illuminating theorems to prove would be:Given an airfoil projle and a continuous two-dimensional irrotational transonic compressible inuiscidflow past it with some given speed at i n j n i g , there does not exist a corresponding Jlow with a slightly dtjfeerent speed at infinity.Although this theorem was first formulated in 1954, on the basis of conjectures of Frank1 and Guderley, see [l], it has not yet been established. Strong evidence that the theorem is true and proof that smooth transonic flows do not exist generally are given by non-existence theorems in which the profile is varied in the supersonic region and the speed at infinity kept fixed.In [2], Part I, there is such a %on-existence" theorem for continuous transonic flows which are considered as disturbances about a given smooth flow. Except for considering only a linear perturbation this theorem is quite general and complete but the proof is tediously long. In [2], Part 11, allowance is made for the non-linearity at the expense of still further complication. It seems worthwhile to present here a "non-existence" theorem which covers the physically interesting situation and which is fairly simpIe to prove. The proof will be made even more elementary by the addition of a few assumptions on the pressure-density relation.In Section 1 we describe the unperturbed flow and the assumptions, in Section 2 the perturbation flow, in Section 3 the non-existence theorem, in Section 4 the underlying uniqueness theorem. We begin by discussing the flow which is to be varied by varying the airfoil profile.
