This paper is a theoretical and experimental investigation of the onset of the buoyancydriven longitudinal roll cells that occur when a liquid layer flows over a heated horizontal plate. Linear stability theory is applied under the assumption that the spatially developing temperature profile can be treated locally, that is at each axial position, as being ‘frozen’. Using the film thickness as the length scaling factor, the critical Rayleigh numbers associated with the onset of longitudinal rolls are found to be considerably lower than measured values. A modified local stability analysis using the thermal boundary-layer thickness as the scaling factor is shown to agree with experiments. Predicted wavenumbers and the position of the onset of cellular convection are in agreement with wavenumbers measured by flow-visualization techniques. The position of the onset of cellular convection is also obtained from heat-transfer measurements at the heated surface. In the asymptotic limit of a linear undisturbed temperature profile the classical solutions of the Rayleigh-Bénard problem for the critical Rayleigh number and wavenumber are recovered, the only effect of the flow being the structure of the secondary flow that occurs when the system is unstable. Amplification theory is also compared with experimental data for the position at which the thermal effects of the convection are detectable and for the wavenumbers measured by flow visualization. The thermal amplification ratio $\overline{Nu}$ and a velocity-disturbance amplification ratio $\overline{w}$ are used to interpret the onset of discernible cellular convection. The data are not consistent with any single amplification ratio over the range of Rayleigh numbers studied (103 < Ra < 3 × 105), and the theoretical and experimental results suggest that a band of wavenumbers rather than a single wavenumber is encountered when cellular convection occurs.