Perturbation theory of multi-plane lens effects in terms of mass ratios: Approximate expressions of lensed-image positions for two lens planes

Continuing work initiated in an earlier publication (Asada, MNRAS. 394 (2009) 818), we make a systematic attempt to determine, as a function of lens and source parameters, the positions of images by multi-plane gravitational lenses. By extending the previous single-plane work, we present a method of Taylor-series expansion to solve the multi-plane lens equation in terms of mass ratios except for the neighborhood of the caustics. The advantage of this method is that it allows a systematic iterative analysis and clarifies the dependence on lens and source parameters. In concordance with the multi-plane lensed-image counting theorem that the lower bound on the image number is $2^N$ for N planes with a single point mass on each plane, our iterative results show how $2^N$ images are realized. Numerical tests are done to investigate if the Taylor expansion method is robust. The method with a small mass ratio works well for changing a plane separation, whereas it breaks down in the inner domain near the caustics.