A new hydromagnetic theory is developed for describing compressional pulsations with large azimuthal wave number. It is assumed that there are two plasmas, one hot, in which pressure effects are important, and the other cold. The equations are derived in a general set of magnetic coordinates which allow realistic calculations including geometrical effects in the magnetosphere. The equations describe the three hydromagnetic modes which are coupled by the geometry. When the azimuthal wave number is large, the fast mode is strongly evanescent. This allows an expansion in 1/m in order to decouple the fast wave. The remaining equations describe the coupled transverse Alfv¿n and magnetosonic modes. Some of the puzzling features of the observations of polarization are discussed. ¿American Geophysical Union 1987