  The Vliegenthart-Lekkerkerker relation for the second virial coefficient
value at the critical temperature found in [G. A. Vliegenthart and H. N. W.
Lekkerkerker, J. Chem. Phys. \textbf{112} 5364 (2000)] is discussed in
connection with the scale invariant mean-field approach proposed in [V. L.
Kulinskii and L. A. Bulavin, J. Chem. Phys. \textbf{133} 134101 (2010)]. We
study the case of the Mie-class potentials which is widely used in simulations
of the phase equilibrium of the fluids. It is shown that due to the homogeneity
property of the $Mie$-class potentials it is possible to connect the loci of
the fluids with these model potentials in different dimensions.
