Highly relativistic electrons (3--10 MeV) at times are observed to populate the earth's magnetosphere near the geostationary orbit (r~6.6 RE). Electron fluxes and energy spectra are shown which were measured by two high-energy electron sensor systems at 6.6 RE from 1979 to the present. Large, persistent increase in this electron population were found to be relatively infrequent and sporadic in 1979--1981 around solar maximum. This situation contrasts markedly with the frequent occurrence of very large flux increases from late 1981 to the present. Furthermore, in this latter period, which constitutes the approach to solar minimum, it is observed that the highly relativistic electrons occur with a relatively regular 27-day periodicity and are well associated with the reestablished solar wind stream structures. On the basis of an examination of long-term flux variations as a function of solar rotation we find strongly periodic behavior beginning with Bartels rotation ~2035. Through a superposed epoch analysis technique we study empirical particle lifetimes as a function of particle energy. An energetic electron enhancement typically rises on a 2- to 3-day time scale and decays on a 3- to 4-day time scale at essentially all energies above ~3 MeV. The present analysis suggests either that a regular pulsing by high-speed solar wind streams makes the magnetosphere an efficient relativistic electron accelerator or else that the Jovian magnetosphere is a recurrent source of a significant electron population in the outer terrestrial magnetosphere.