Hamilton's equations for the motion of an electron in a linear accelerator are integrated to find the final kinetic energy and phase of an electron injected with arbitrary initial kinetic energy and phase, after traveling down a fixed length of accelerator. The results are expressed in the form of a map of the initial energy-phase space onto a final energy-phase space. This map proves in practice to be very convenient for discussing the actual operation of the accelerator. The curves presented are calculated for numerical values appropriate for the M.I.T. accelerator.