In a gas-like ether, the duality between the oscillating electric and magnetic fields, which are transverse to the direction of propagation of electromagnetic waves, becomes a triality with the longitudinal oscillations of motion of the ether, if electric field, magnetic field and motion are coexistent and mutually perpendicular. It must be shown, therefore, that if electromagnetic waves comprise also longitudinal condensational oscillations of a gas-like ether, analogous to sound waves in a material gas, then all three aspects of such waves must propagate together along identical wave-fronts. To this end, the full characteristic hyperconoids are derived for the equations governing the motion and the electric and magnetic field-strengths in a gas-like ether, in three space variables and time. It is shown that they are, in fact, identical. The equations governing the motion and the electric and magnetic field-strengths in such an ether, and their common characteristic hyperconoid, are all invariant under Galilean transformation.