The hamiltonian for an ion which has a dynamic Jahn-Teller interaction with the many normal modes of a crystal can be separated in such a way that nearly all the coupling strength is concentrated in an interaction with a set of vibrations of one frequency only. All the rest of the hamiltonian, including the consequence of the effective vibration not being a normal mode, is expressed in a simple form, and treated as a second order perturbation on the energy levels of the first part. This procedure gives a good approximation to the lower states of the Jahn-Teller system, and can be used in studying phonon scattering. Some results are derived for the case of an E electronic doublet at a cubic crystal site.