The potential energy of a deformed lattice can be written in the form $V=V_0+V_1+V_2$ where $V_0$ is a constant (the energy of the undeformed lattice), $V_1$ the part linear in the displacements of the lattice points from their normal positions, $V_2$ the part quadratic in the displacements. The terms of higher order are neglected. In view of the requirement that the normal position of each lattice point be a position of equilibrium the linear part vanishes ($V_1=0\mathrm{}$) so that the energy is simply equal to $V_2$ (apart from the constant $V_0$). As the energy must be invariant with respect to rotations of the system, W. Voigt postulated the invariance of $V_2$ and derived from this assumption the so-called Cauchy relations between the elastic coefficients. A closer analysis shows that this conclusion is open to objection. The term $V_2$ represents the energy only because of the subsidiary condition $V_1=0\mathrm{}$ which, upon investigation, turns out to be not invariant with respect to rotations. Hence, $V_2$ is not invariant either: a fact which removes the theoretical basis of the Cauchy relations.

• Received 13 August 1946

DOI:https://doi.org/10.1103/PhysRev.70.915