Degenerate scales of different shapes have been studied by many researchers. BEM/BIEM may lead to a degenerate scale for a two-dimensional Laplace problem. In this paper, we study the problem of the two tangent discs with different radii. We use the conformal mapping of complex variables and the unit logarithmic capacity to examine the degenerate scale. Although the degenerate scale of infinite plane containing two circular holes has been derived by using the degenerate kernel in terms of bipolar coordinates, this shape cannot be solved by using the degenerate kernel. Finally, the numerical results of BEM were also compared with those of analytical formula.