  Linearly polarized Balmer line emissions from supernova remnant shocks are
studied taking into account the energy loss of the shock owing to the
production of nonthermal particles. The polarization degree depends on the
downstream temperature and the velocity difference between upstream and
downstream regions. The former is derived once the line width of the broad
component of the H$\alpha$ emission is observed. Then, the observation of the
polarization degree tells us the latter. At the same time, the estimated value
of the velocity difference independently predicts adiabatic downstream
temperature that is derived from Rankine-Hugoniot relations for adiabatic
shocks. If the actually observed downstream temperature is lower than the
adiabatic temperature, there is a missing thermal energy which is consumed for
particle acceleration. It is shown that a larger energy loss rate leads to more
highly polarized H$\alpha$ emission. Furthermore, we find that polarized
intensity ratio of H$\beta$ to H$\alpha$ also depends on the energy loss rate
and that it is independent of uncertain quantities such as electron
temperature, the effect of Lyman line trapping and our line of sight.
