  I give an explicitly verifiable necessary and sufficient condition for the
uniqueness of the eigenform on finitely ramified fractals, once an eigenform is
known. This improves the results of my previous paper [14], where I gave some
necessary and some sufficient conditions, and with a relatively mild additional
requirement on the known eigenform. The result of this paper can be interpreted
as a uniqueness result for self-similar energies on the fractal.
