Application of a theory developed elsewhere enables us to generate good numerical normal mode eigensolutions for the translational oscillations of the solid inner core of a rotating, slightly elliptical earth model. Examination of the particle motions associated with these normal modes shows clearly the profound influence of the earth's steady rotation, particularly on motions in the fluid outer core. The variation of eigenfrequencies with core structure shows clearly how stable (or unstable) stratification in the outer core acts to increase (or decrease) the associated eigenfrequencies. A study of the energy partition during one cycle of motion shows that compressibility (as opposed to stratification) in the fluid outer core is of little significance but that inner core elasticity has a very significant effect. Normal mode excitation calculations show that the gravimeter signal available at the earth's surface as a result of the excitation of such a motion by some seismic event is intimately tied to the stability or instability of the stratification of the fluid outer core. We currently have little knowledge about this very important aspect of the earth's deep interior. We cannot currently preclude the possibility of observing this normal mode. We discuss how to estimate the damping of these modes by viscosity or anelasticity.