We consider the Euler equations of an incompressible homogeneous fluid in a thin two-dimensional layer , , with slip boundary conditions at z = 0, and periodic boundary conditions in x. After rescaling the vertical variable and letting go to zero, we get the following hydrostatic limit of the Euler equations    supplemented by slip boundary conditions at z = 0 and z = 1 and periodic boundary conditions in x. We show that the corresponding initial-value problem is locally, but generally not globally, solvable in the class of smooth solutions with strictly convex horizontal velocity profiles, with constant slopes at z = 0 and z = 1.