The poloidal mode is a coupled Alfv¿n and fast magnetosonic mode which has Alfv¿n-like dispersion in the limit of large azimuthal wave number. In this brief report we discuss the assumption of incompressibility. From the point of view of ideal MHD, incompressibility is, in general, a bad assumption for the poloidal mode when the magnetic field is curved. Assuming an adiabatic equation of state for the pressure p and that it is acceptable to drop the contribution of v∥ (a low beta assumption), an approximate condition for the incompressibility assumption to be valid in the magnetosphere is that the equilibrium pressure scale length Lp≪LRE(1+&ggr;&bgr;/2)/(6&ggr;), where LRE is the geocentric distance from the Earth, &ggr; is the ratio of specific heats, and the plasma beta &bgr;≡2p0/B02. However, the results of kinetic theory (only discussed here) indicate that the most applicable equation of state for an odd magnetospheric poloidal mode is an incompressible equation for p. This is not because the divergence of the bulk velocity equals zero. The high-temperature hot particles which supply the plasma pressure have motion which is dominated by their gradient and curvature drifts and bounce motion; this motion is different from that of the bulk plasma which moves only because of the E¿B drift. ¿ 1998 American Geophysical Union