We examine the convective stability of hydrodynamic discs with full stratification in the local approximation and in the presence of thermal diffusion (or relaxation). Various branches of the relevant axisymmetric dispersion relation derived by Urpin are discussed. We find that when the vertical Richardson number is larger than or equal to the radial one (i.e. |Riz| e |Rix|) and wavenumbers are comparable (i.e. |kx| 0 and Riz > 0. The origin of this resides in a hybrid radial?vertical Richardson number. We propose an equilibrium profile with temperature depending on the radial and vertical coordinates and with Riz > 0 for which this destabilization mechanism occurs. We notice as well that the dispersion relation of the ?convective overstability? is the branch of the one here discussed in the limit |kz| k |kx| (i.e. two-dimensional disc).