The emissivity of a body is generally a function of temperature and this makes calculation of the heat exchange difficult. Show that if the emissivity varies linearly with temperature, that we can recover the form of the two-body exchange law,

\[Q_{12}=\left(\varepsilon_{1}\right)_{r e f} \sigma A_{1}\left(T_{1}^{4}-T_{2}^{4}\right)\]

provided the emissivity is evaluated at a reference temperature defined by:

\[T_{r e f}=\frac{T_{1}^{5}-T_{2}^{5}}{T_{1}^{4}-T_{2}^{4}}\]