  A number of recent surveys for gravitational lenses have found examples of
double Einstein rings. Here, we investigate analytically the occurrence of
multiple Einstein rings. We prove, under very general assumptions, that at most
one Einstein ring can arise from a mass distribution in a single plane lensing
a single background source. Two or more Einstein rings can therefore only occur
in multi-plane lensing. Surprisingly, we show that it is possible for a single
source to produce more than one Einstein ring. If two point masses (or two
isothermal spheres) in different planes are aligned with observer and source on
the optical axis, we show that there are up to three Einstein rings. We also
discuss the image morphologies for these two models if axisymmetry is broken,
and give the first instances of magnification invariants in the case of two
lens planes.
