Conditions under which the compressibility can be neglected for the magnetospheric ballooning instability, which arises in the shear Alfv¿n branch, are clarified in the context of ideal magnetohydrodynamic plasmas and stochastic plasmas by using the normal mode analysis and the energy principle. An expansion in the small parameter, which is equal to the ratio of the ∇⊥B0 scale length to the field line curvature radius, shows that the incompressible assumption is valid for the ballooning instability with long-thin perturbations in the low-frequency regime and in the long-thin magnetospheric equilibrium, in which the field line curvature radius is much larger than the ∇⊥B0 scale length. When the long-thin assumption for the equilibrium is not satisfied near the equator, the calculation of the energy functional for a trial function shows that the strongly localized ballooning mode is essentially incompressible if the plasma &bgr; at the equator is much larger than 6/&Ggr;, where &Ggr; is the ratio of specific heats. For the stochastic plasmas near the equator the strongly localized ballooning mode is essentially incompressible irrespective of the &bgr; value. These results justify the incompressible assumption made in a previous ballooning stability analysis for the long-thin magnetospheric equilibrium. It is suggested that before the substorm onset, the near-Earth plasma sheet becomes more taillike, and the long-thin assumption for the equilibrium becomes more likely to be satisfied on average, and thus the near-Earth plasma sheet becomes more favorable to the onset of the ballooning instability without the strong stabilizing influence of the compressibility. ¿ 2000 American Geophysical Union