  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.
