Upcoming missions such as Messenger or BepiColombo will revolutionize our knowledge about planet Mercury and its magnetosphere. To contribute to a comparison between the terrestrial and the Hermean magnetosphere the system of partial differential equations describing coupled directional and isotropic eigenmodes of a magnetosphere filled with a multi-component plasma is derived and investigated to study eigenoscillations of Mercury's magnetosphere. As anticipated eigenfrequencies are smaller, but comparable to the ion gyrofrequencies of the plasma assumed, an MHD treatment as used by Dungey <1954> is not appropriate. Instead, a cold plasma approximation is used keeping the off-diagonal components of the dielectric tensor thus allowing higher frequencies to be tackled. Unlike in the terrestrial magnetosphere, the directional and the isotropic modes in the Hermean magnetosphere remain coupled even for axisymmetric perturbations with azimuthal waves numbers m = 0.