In a previous publication the writer presented a rigorous solution of the Einstein-Maxwell equations which corresponds to a configuration of parallel magnetic lines of force in equilibrium under their mutual gravitational attraction. This "magnetic geon"-which it is appropriate to rename a "magnetic universe" because the magnetic-field energy falls off so slowly with distance-is unstable according to an elementary Newtonian analysis but is stable according to the analysis reported here. In this connection the general time-dependent equations for electromagnetic-gravitational fields in the case of cylindrical symmetry are discussed in detail and certain conclusions drawn. Then the solutions as functions of radius and time are found for the two gravitational potential fields and the electromagnetic field which appear when a magnetic universe is subjected to a radial perturbation. It is shown that the magnetic universe is stable (oscillatory) under such perturbations, i.e., all admissible frequencies are real. The converse is also true: All real frequencies are admissible. Every solution is a superposition of proper modes of which there are two types, a "g-type" and an "h-type" for each and every real frequency, and for each of the physically interesting fields, i.e., for the electromagnetic field and for the two gravitational fields ("Newtonian potential" and "C energy," respectively). The g-type modes may be identified roughly as "dominantly gravitational" since in them gravitational perturbations dominate electromagnetic both at small and large distances; the h-type may be identified as "locally electromagnetic" since in these modes, near the axis, the electric and magnetic perturbations dominate the gravitational perturbations. The radial dependence of each of the mode types for the various fields is expressed linearly, with radius-dependent coefficients, in terms of Bessel coefficients of order zero and one, respectively. The manner in which an initial perturbation is shaken off is analyzed, and causality is verified. The relevance of the analysis to the understanding of gravitational collapse is discussed. Pure magnetic (or electric) fields are the only presently known systems which resist gravitational collapse. It is suggested that extended magnetic fields may play a role in retarding and finally halting gravitational collapse of material systems.