Linear (gradient) filtration law Dupuis transformed for non-isothermal barotropic flow of a perfect gas. Integral characteristics of the filter (the coefficients of diffusion and filtration) are related to the intensity of heat transfer in the course of filtration. Isothermal filtration is obtained as a limiting case of isothermal filtration at a value of the polytropic index  n→1  ±0. We prove the non-linearity of the pressure distribution and passive impurities in the course of isothermal filtration.