The cascade mechanism of electromagnetic shower evolution in a strong magnetic field is considered. It is based on the kinetic equations for distribution functions of such particles as electrons, positrons and photons. The differential-difference equations for Mellin transforms of the distribution functions are derived and solved in both adiabatic and modified adiabatic approximations. The authors calculate the differential and integral spectra of particles and other characteristics of the shower, i.e. the 'track' length, the position of the centre of gravity, the longitudinal spread, and the positions of the maxima of the distribution functions.