Abstract: Boundary element method (BEM) is usually done by only one boundary integral equation (BIE).The ordinary BIE for potential in the plane is of the second kind for potential U, but is of the first kind for potential derivative ∂_Ω.In order to result in diagonal dominance of the control system of equations of BEM for mixed boundary value problem, we present an improvements that using the ordinary control system on the NEUMANN boundary segment,but the control system of the conjugate harmonic function on the DIRICHLET boundary segment.Some examples for computing the capacitance density of two-wire transmission lines are given.which indicate the method of conjugate function has higher accuracy than ordinary BEM.