The authors discuss the stability of the Nystrom method for the Sherman-Lauricella equation. They first study the Sherman-Lauricella equation on contours with corner points. They consider a series of numerical examples that show an excellent convergence of the sequence of approximate solutions for smooth and piecewise smooth contours. In addition, the authors monitor the transformation of the Sherman-Lauricella equation solutions when a smooth contour is continuously transformed into contours with angular points of various magnitude.