Recently we have proposed (J. Chem. Phys. 128 (2008) 134508) a new rescaling of fluid density rho by its critical value p(c)(2/3) to apply the corresponding states law for the attractive Yukawa fluids study. Analysis of precise simulation results allows us to generalize this concept to the case of simple fluids with different interparticle interactions, like Mie(n, m) and Sutherland pair potentials. It is shown, that there is a linear relationship between the critical pressure and critical temperature, as well as the critical density and inverse critical temperature for these frequently used pair potentials. As a consequence, the critical compressibility factor of these model fluids is close to its universal value measured experimentally for different real substances. (c) 2008 Elsevier B.V. All rights reserved.