  We have examined the atomic theory behind recent constraints on the violation
of the Pauli Exclusion Principle derived from experiments that look for x rays
emitted from conductors while a large current is present. We also re-examine
the assumptions underlying such experiments. We use the results of these
studies to assess pilot measurements to develop an improved test of the
Principle. We present an improved limit of $(1/2)\beta^2 < 2.6\times10^{-39}$
on the Pauli Exclusion Principle. This limit is the best to date for
interactions between a system of fermions and a fermion that has not previously
interacted with that given system. That is, for systems that do not obviously
violate the Messiah-Greenberg symmetrization-postulate selection rule.
