An improvement is introduced to solve the plane problems of linear elasticity by Boundary Element Method for orthotropic materials, It is considered that the problem to be solved is the first fundamental problem and the region concerning the problem is multiply connected. The kernels of the standard formulation are the functions of two complex variables depending on the coordinates and the material constants of the orthotropic medium. Boundary of the region is idealized as a collection of line segments. The end points of these segments are nodal points. N is the number of the nodal points. 2N integral equations can be written by assuming there is an singular loading at every nodal point on each direction. In these integral equations, the integrals over the boundary are reduced to the summation of the integrals over the line segments. In addition to these, an artificial boundary is defined to eliminate the singularities. A new algorithm is introduced to calculate multivalued complex functions. The chosen sample problem is a plate having a circular hole stretched by the forces parallel to one of the principal directions of the material. Results are compatible with the solution given by Lekhnitskii for an infinite plane. Three different orthotropic materials are considered.