This report proposes a new element, a polygonal line element, for two dimensional elastostatic analysis by the boundary element method and shows special technique for analytical integrations to overcome singularities of integrants. In addition, the stresses on boundary can be obtained by this method without using the numerical integrations. In order to evaluate the accuracy for the proposed element, the author analyzes the stress concentration factor for bending of a plate with a deep hyperbolic notch. It is made obvious that sufficiently accurate solutions can be obtained with better efficiency than that by using linearelements.