We have considered the infinite element method for a class of elliptic systems with constant coefficients in [1]. This class can be characterized as: they have the invariance under similarity transformations of independent variables. For example, the Laplace equation and the system of plane elastic equations have this property. We have suggested a technique to solve these problems by applying this property and a self similar discretization, and proved the convergence. Not only the average convergence of the solutions has been discussed, but also term-by-term convergence for the expansions of the solutions. The second convergence manifests the advantage of the infinite element method, that is, the local singularity of the solutions can be calculated with high precision.