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This paper deals with the calculation of the force and the stiffness between two ring permanent magnets whose polarization is axial. Such a configuration corresponds to a passive magnetic bearing. All the calculations are determined by using the Coulombian model. The paper also discusses the optimal ring dimensions that provide a large force or stiffness between the rings. Such properties are commonly sought in passive magnetic bearings. We propose a three-dimensional method for optimizing these parameters. An important result is established in this paper: the exact relative position of the rings for which the force is the strongest depends on the air gap dimension. As the expressions in this paper give this exact relative position, manufacturers can easily optimize their passive magnetic bearings. This result is new because the curvature effect is taken into account. Such semianalytical expressions are more precise than the numerical evaluation of the magnetic forces obtained with the finite-element method. In addition, semianalytical expressions have a low computational cost whereas the finite-element method is computation-intensive. Such calculations make it easy to optimize quadripolar lenses and other devices utilizing permanent magnets.