Based upon a variational principle derived in a preceding paper, expressions for the magneto-elastic buckling values for ferromagnetic or superconducting systems are given. These relations are evaluated for systems of slender beams. Explicit buckling values are calculated for a single ferromagnetic or superconducting beam of arbitrary cross-section, and for systems of two parallel ferromagnetic or superconducting rods. In the analysis needed for the calculation of the intermediate (i.e., rigid-body) and the perturbed magnetic fields, an intensive use of methods inherent in the theory of complex functions is made. In conclusion our results for a set of two superconducting rods are compared with the results of a mathematically less complicated, but also less rigorous, theory.