The time-dependent Vlasov-Boltzmann equation is analytically studied in mirror machines with perpendicular injection. The atomic and collisional processes are taken into account, with their explicit, rigorous forms. A uniform model is assumed by considering a square-well magnetic field configuration, and the mirrors are represented by related boundary conditions on the ion distribution function. It is shown that, the ion density increases from the initial value, drops by a certain amount, then increases again up to its final value, around which it performs damped oscillations. (C) 1996 American Institute of Physics.