The change from subcritical to supercritical linear flow in prismatic channels with constant slope is gradual and spontaneous only in the presence of a critical slope. Yet, generally, such changes occur when channel's slope varies. When this happens, the gradual change from subcritical to supercritical linear flow is spontaneous while the opposite change, from supercritical to subcritical linear flow, must necessarily be determined and regulated with the help of properly located devices along the channel. However, the gradual change of the linear flow becomes complete in the control section, where flow depth equals critical depth, curvatures of streamlines are not negligible, pressure's hydrostatic distribution breaks down as well as flow linearity. The analysis of non-linear motion can be conducted by replacing the first order differential equation governing gradually varied flows with a third order differential equation which is obtained by applying Boussinesq's hypothesis referring to the variation of the curvature of any streamline passing through the same transversal section of the flow. Limit values, a priori unknown, must be assigned to the equation. This study proposes a method for defining such limit values and presents some results (not only of purely conceptual interest) obtained with such equation.
