Let us show that Eq. (24) can be considered as some consistency condition between the scaling of for the LG Hamiltonian (1) and that for (20). Note that Eq. (25) represents in particular the scaling of the molecular size n$_{b}$ and corresponding scaling of energy scale given by T$_{B}$ or equivalently T$_{Z}$ . To connect the LG model with the LJ fluid where the interaction has the form (20) we take into account that under the scale transformation the loci of the corresponding critical points of these systems should be scaled consistently. Indeed, according to the structure of the LG Hamiltonian (1) the analog of Eq. (26) for the critical point of the LG is as following: Then the consistency condition between the scaling of the critical point coordinates of the LJ fluid (19) and the Lattice Gas (27) is: In accordance with (8) we get: In the following Section we use (29) to calculate the locus of the critical point of the fluid where the interaction potential is of the LJ type (18). To test the predictions (29) we compare them with the results of the computer simulations for the potential (18) for 2 D , 4 D and 5 D dimension geometries available in [19, 20]. In accordance with the results of Section we determine the packing parameter η$_{b}$ = n$_{b}$ σ d so that: where the particle volume v$_{d}$ ( σ ) and U$_{0}$ are put to units. The result are placed in Table I. x 2 c t$_{c}$ = const ⇔ d t$_{c}$ d x$_{c}$ = − 2 t$_{c}$ x$_{c}$ . (27) T$_{c}$/T$_{Z}$ n$_{c}$/n$_{b}$ ( 1 + ε d ) = 2 ⇒ 1 z = 1 + ε d (28) T$_{c}$ T$_{Z}$ = 1 2 + ε d , n$_{c}$ n$_{b}$ = 1 + ε d 2 ( 2 + ε d ) . (29) The case of the attractive potential of the van der Waals forces in d -dimensions corresponds to ϵ = d , z = 1 / 2 and we get the estimates: T$_{c}$ = T$_{Z}$/ 3 , n$_{c}$ = n$_{b}$/ 3 . (30) T$_{Z}$ = T ( vdW ) B = a/b = 4 d 6 − d , a = 2 d − 1 4 d 6 − d , b = 2 d − $^{1}$, (31) THE LOCUS OF THE CRITICAL POINT FOR THE LENARD-JONES FLUID 8