The stability of a uniformly-rotating, electrically-conducting, stratified fluid is discussed under conditions of slight tidal straining. The ensuing steady flow consisting of elliptical streamlines and magnetic field lines is linearly unstable through the resonant coupling of two free waves of the system. New families of linear waves consisting of modified Poincare, “slow” hydromagnetic and, in the case of axial stratification, internal evanescent waves are isolated upon a rotating fluid bearing an axial current. All possible resonant couplings between these waves are examined. Stable radial stratification is found to be a destabilizing influence on some elliptical couplings and always so for slow hydromagnetic waves. Internal evanescent waves can be stimulated elliptically in the case of axial stratification and typically possess boundary-layer structure. In all cases studied, including in the presence of a central core, the preferred mode of disturbance is a tipping over, or spinover, of the basic vorticity and current into the plane of elliptical distortion. The elliptical instability preferentially excites westward-propagating waves but growth rates for the slow hydromagnetic waves appropriate to the geomagnetic secular variation are found to be too small to be significant.