In this paper we present a method for the evaluation of the finite part or Cauchy principal value of singular surface integrals occurring in boundary element methods. In section 1 we briefly discuss Somigliana's identity. This identity is the starting point for the calculation of elastic deformation and stress in elastostatics by use of boundary element methods. For details we refer e.g. to the book of C.A. Brebbia and J. Dominguez (1989).In section 2 we describe the boundary element approximation of the geometry of the elastic solid under consideration. If the source point of the fundamental solution of the Navier-Cauchy equilibrium equation is a vertex of a triangular element of the boundary element mesh, then the integration over that triangle must be performed analytically. For the evaluation of the finite part or Cauchy principal value of singular integrals we use a two-dimensional version of the theorem of Gauss. This idea for the evaluation of the finite part of singular integrals in boundary element algorithms may be useful in other applications also. In section 3 we present the formulas for these integrals for implementation in boundary element algorithms in elastostatics based on Somigliana's identity.