The authors previously proposed the boundary integral equation method (BIEM), whose integral equations were formulated from the viewpoint of the two-dimensional theory of elasticity, and applied this method to stress and displacement analyses of single connected domain problems such as thick-rimmed external and internal spur gears. For the purpose of the further development of BIEM, we present ways of improving the analytical accuracy and the procedure for solving multiply connected domain problems. In this report, a thin-rimmed spur gear with circular holes is analyzed as one of the multiply connected region problems and the calculated results are compared with those obtained by the boundary element method (BEM). It has been said to be difficult to solve such complicated problems using traditional methods based on the two-dimensional theory of elasticity. However, stress and displacement distributions of thin-rimmed spur gears can easily be analyzed by BIEM under the condition that its model has a semi-infinite gear tooth domain with circular holes. As a result, it is shown that this new method, is applicable not only to single connected domain problems but also multiply connected domain problems.