A dynamic theory of the cubic-to-tetragonal transformation in spinels containing octahedrally coordinated d9 and d4 Jahn-Teller ions is presented. Following Kanamori, the model assumes that the dominant coupling between the doubly degenerate electronic orbital and the lattice is via the elastic strain. The temperature dependence of the splitting of the degenerate electronic level and the changes in the elastic constants due to the interaction is determined. For linear Jahn-Teller coupling, the phase transition is second order. The soft mode is the acoustic shear mode propagating in the  direction. The frequency of this mode goes to zero at the transition temperature and remains zero in the distorted structure. Including third-order Jahn-Teller coupling, the phase transition becomes first order. The shear sound velocity changes discontinuously at the transition temperature and then increases with decreasing temperature. © 1971 The American Physical Society.