The spatial structure of fast magnetosonic eigenoscillations of an axisymmetric magnetosphere is investigated theoretically. It is shown that transparent regions of the eigenmodes where the bulk of the fast magnetosonic oscillation energy is concentrated lie near the equatorial plane. The transparent regions have a narrow channel from the magnetosphere to the solar wind near the equatorial plane. This enables the magnetosonic waves to penetrate from the solar wind deep into the magnetosphere, as well as some of the energy of intramagnetospheric magnetosonic oscillations to escape to the solar wind. Some of the eigenmodes have an isolated portion of the transparent region above the plasmapause that forms a well-known magnetospheric cavity for fast magnetosonic waves. ¿ 2000 American Geophysical Union