The title problem is investigated with the view of clarifying the nature of the separation in an adverse pressure gradient when the suction υ0 is very strong. A linear model is investigated, but it is argued that this must contain many of the essential features of the exact (non-linear) problem. With a free-stream U = 1 − x the similarity solution u = (1 − x)f(y) is appropriate when (υ0x − y) ≫ 1, x = O(1), 1−x = O(1). There is an O(1) layer centred on y = υ0x in which more complex x-dependence first appears and then, when (y − υ0x) ≫ 1, the solution assumes its final asymptotic behaviour. The y = υ0x layer is non-uniform in x and exhibits a marked change in behaviour when (1 − x;) = O(l/υ0). The solution in this narrow rectangle is singular like (1 − x)−2 at the trailing edge. It is these singular terms that induce separation and reversed flow in an exponentially small neighbourhood of the trailing-edge.