The Boundary Method is now well established as a valid numerical technique for the solution of field problems, equal to the Finite Element Method in generality and surpassing it in computational efficiency in some cases [1] In this paper is presented a Regular Boundary Element Method as applied to two dimensional stress analysis. It involves the formation of a system of regular integral equations obtained by moving the singularity outside the domain of the given problem [2]. It is shown that continuous elements may be used here after the manner of Finite Element Method [3].