Bethe's equation for the change of entropy in waves of finite amplitude is shown to be inapplicable for an initial state of temperature arbitrarily close to absolute zero, for a degenerate substance possessing a zero‐point pressure (or energy). Thermodynamic functions for such a substance at low temperature are formulated in general terms applicable to the Fermi‐Dirac gas, the Thomas‐Fermi atom, the Debye solid, and the Mie‐Gru¨neisen solid as special cases. The conditions under which the equation of state satisfies the Bethe‐Weyl conditions are given. Of the usual four basic properties of the shock transition under the Bethe‐Weyl conditions, two must be modified for the class of substances in question, for an initial state arbitrarily close to zero temperature. The argument follows from extension of Bethe's method, by Taylor expansion of the Hugoniot function about the initial state. The results are shown to be consistent with Weyl's procedure.