When a nonhomogeneous solid is melting from below, convection may be induced in a thermally - unstable melt layer. In this study, the onset of buoyancy-driven convection during time-dependent melting is investigated by using similarly transformed disturbance equations. The critical Darcy - Rayleigh numbers based on the melt-layer thickness, Ra-H,Ra- c, are found numerically for various conditions. For small superheats, the present predictions show that Ra-H,Ra- c is located between 27.1 and 4 pi(2) and it approaches the well-known results of the original Horton - Rogers - Lapwood problem. However, for high superheats, it is dependent on the phase change rate. and the relation of Ra-H,Ra-c lambda = 25.89 is shown.