Permanent magnet bearings are useful for elements of super flywheels and super high vacuum pumps because of their simple construction and no losses. This paper describes a calculation method for forces between pairs of ring permanent magnets with small displacements from a center line of equilibrium. Test data on stiffness, thrust force and damping constant were used to verify the calculation. From this method, we developed a lower thrust force, or even repulsive type, permanent magnet bearing.