It is studied how to analyze the magnetostatic problems by employing the boundary integral equations. There have been reported a lot of papers about the magnetostatic analysis by these equations. When the permeability is low, most of the magnetostatic problems are solved adequately. However, as the permeability becomes higher, the computed results become poorer especially in the case of the problems without air-gap (multiple-connected problem) or the problem with small air-gap. In order to overcome these difficulties, we derive two new kinds of the boundary integral equations and check their adequacy and effectiveness by solving the multiple-connected problem. Further we investigate their capability to solve the problems concerning the magnetic core with small air-gap.