The plasma in the dayside magnetosheaht exhibits a persistent pressure anisotropy in the sense p⊥>p∥. A likely source for this anisotropy is the effect of field compression and plasma depletion along field lines as magnetosheath plasma flows toward the magnetopause. The model of Zwan and Wolf describing this effect for the case of isotropic pressure is combined with the double-adiabatic fluid equations to predict the behavior of the anisotropic pressure. For a fluid element following a streamline inward from the bow shock, we find theoretical pressure anisotropies of magnitude large enough to trigger the mirror instability over most of the dayside magnetosheath. These findings are supported by the observations of Kaufmann et al. of large-amplitude hydromagnetic waves in the inner magnetosheath believed to be generated by the mirror instability.