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1. Introduction. It has been known for some time that the stress distribution in a plane bounded region with an irregularly shaped boundary may be accurately approximated by finite difference methods. Indeed, some twenty years ago, Southwell and his co-workers [1,2] utilized finite difference approximations, and applied Southwell's "relaxation" method, a noncyclic or free-steering iterative procedure, to the approximate solution of interesting engineering problems. Though this "relaxation" method has been known to work well when human insight guides the course of calculations on a desk calculator, such noncyclic iterative techniques become extremely laborious on desk calculators when results of high precision are required. More damaging is the fact that such noncyclic iterative techniques can rarely be adapted efficiently to high-speed digital computers.