  Percus showed that approximate theories for the structure of nonuniform hard
sphere fluids can be generated by linear truncations of functional expansions
of the nonuniform density rho (r) about that of an appropriately chosen uniform
system. We consider the most general such truncation, which we refer to as the
shifted linear response (SLR) equation, where the density response rho (r) to
an external field phi (r) is expanded to linear order at each r about a
different uniform system with a locally shifted chemical potential. Special
cases include the Percus-Yevick (PY) approximation for nonuniform fluids, with
no shift of the chemical potential, and the hydrostatic linear response (HLR)
equation, where the chemical potential is shifted by the local value of phi (r)
The HLR equation gives exact results for very slowly varying phi (r) and
reduces to the PY approximation for hard core phi (r), where generally accurate
results are found. We try to develop a systematic way of choosing an optimal
local shift in the SLR equation for general phi (r) by requiring that the
predicted rho (r) is insensitive to small variations about the appropriate
local shift, a property that the exact expansion to all orders would obey. The
resulting insensitivity criterion (IC) gives a theory that reduces to the HLR
equation for slowly varying phi (r), and is much more accurate than HLR both
for very narrow slits, where the IC agrees with exact results, and for fields
confined to ``tiny'' regions that can accomodate at most one particle, where
the IC gives very accurate (but not exact) results.
