  The moist-air entropy is defined in Marquet (QJRMS 2011, arXiv:1401.1097) by
$\boxed{s = s_{ref} + c_{pd} \: \ln(\theta_{s})}$ in terms of two constant
values ($s_{ref}$, $c_{pd}$) and a potential entropy temperature denoted by
$\theta_s$. It is shown in Marquet (2011) that a quantity denoted by
$({\theta}_{s})_1$ plays the role of a leading order approximation of
${\theta}_{s}$.
  The aim of this note is to demonstrate in a more rigorous way that
$({\theta}_{s})_1$ is indeed the leading order approximation of ${\theta}_{s}$,
and to derive a second order approximation which may be used in computations of
values, gradients or turbulent fluxes of moist-air entropy. Some impacts of
this second order approximation are described in this brief version of a note
to be submitted to the QJRMS.
