  A number of recent determinations of $m_s$ using hadronic $\tau$ decay data
involve inclusive analyses based on the so-called $(k,0)$ spectral weights. We
show here that the OPE representations of the longitudinal contributions
appearing in these analyses, which are already known to be poorly converging,
have in addition an unphysical $k$ dependence which produces a significant
unphysical decrease in $m_s$ with increasing $k$. We also show how, using
additional sum rule constraints, the decay constants of the excited resonances
in the strange scalar and pseudoscalar channels may be determined, allowing one
to evaluate the longitudinal spectral contributions to the $(k,0)$ sum rules.
Taking into account the very-accurately known $\pi$ and $K$ pole contributions,
we find that longitudinal contributions can be determined with an accuracy at
the few % level, and hence reliably subtracted, leaving an analysis for $m_s$
involving the sum of longitudinal and transverse contributions, for which the
OPE representation is much better converged.
