the TM-mode cyclotron maser instability for a hollow relativistic electron beam propagating parallel to a uniform axial magnetic field is investigated in this paper. By the method of local field expansion

the maser dispersion equation has been obtained within the framework of linear Vlasov-Maxwell equations

and the maser dispersion equation has been calculated using a digital computer. The cyclotron maser mechanism for TM-modes has been analysed. It is shown that the azimuthal bunching of electron is caused by the relativistic effect and the energy exchange between waves and bunched electrons results mainly from interaction between axial electric field and axial velocity of the electron beam. When wave group velocity Vg equals electron axial velocity V

the system remains stable. Axial bunching and non-relativistical instability are also discussed.