It is found that in the temperature-independent approximation the Dugdale-MacDonald formulation of the Grüneisen parameter differs from the Slater relation only because the former authors have implicitly considered the effect of the variation of Poisson's ratio with volume. The Slater assumption of the volume independence of Poisson's ratio is found to be untenable for any real crystal, and as a result, greater validity is attributed to the Dugdale-MacDonald approximation. The Dugdale-MacDonald approximation, however, is still found inadequate because of the simplicity of the model on which it must be based. Consequently, a new formulation for the temperature-independent Grüneisen parameter is presented for all monatomic cubic crystals in which the atomic forces are central. The new formulation includes the effect of the variation of Poisson's ratio with volume and requires only a knowledge of the lattice energy at zero temperature as a function of the lattice parameters.