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Minimizing the bearing reactions in statically balanced linkages reduces the loads experienced by the frame; and making the bearing reactions constant in all configurations makes the linkages run smoothly. Current techniques for static balancing of a linkage using only springs and no auxiliary bodies have free parameters to be chosen by the designer. We describe methods that utilize these free parameters to impose additional conditions that make the magnitude of all bearing reactions constant in all configurations. We also show additional conditions that also minimize the constant bearing reactions. The conditions required for constant bearing reactions for a statically balanced planar 1R crank and a statically balanced planar 2R linkage are derived in this paper. These results are then generalized for any planar linkage with revolute joints. The main contribution of this paper is in overcoming inherent disadvantages of static balancing using only springs. One disadvantage is concerned with large bearing reactions that vary with the configuration of the linkage. The second disadvantage is large pre-load in the balancing springs. The former is addressed in this paper and the latter was addressed in a prior work. In this paper, we show that simultaneous minimization of constant bearing reactions and pre-loads is not always possible. Hence, we discuss a tradeoff between the two as a practical way to regulate bearing reactions and spring loads. Practical examples of linkages are used to illustrate the application of the proposed methods.