  We present a simple theory of the thermodynamics of an incommensurate quantum
solid. The ground state of the solid is assumed to be an incommensurate
crystal, with quantum zero-point vacancies and interstitials and thus a
non-integer number of atoms per unit cell. We show that the low temperature
variation of the net vacancy concentration should be as $T^4$, and that the
first correction to the specific heat due to this varies as $T^7$; these are
quite consistent with experiments on solid $^4$He. We also make some
observations about the recent experimental reports of ``supersolidity'' in
solid $^4$He that motivate a renewed interest in quantum crystals.
