  Inversions to determine the squared isothermal sound speed and density within
the Sun often use the helium abundance Y as the second parameter. This requires
the explicit use of the equation of state (EOS), thus potentially leading to
systematic errors in the results if the equations of state of the reference
model and the Sun are not the same. We demonstrate how this potential error can
be suppressed. We also show that it is possible to invert for the intrinsic
difference in the adiabatic exponent Gamma_1 between two equations of state.
When applied to solar data such inversion rules out the EFF equation of state
completely, while with existing data it is difficult to distinguish between
other equations of state.
