A progressive reduction of the observed field current below values expected from the empirical law for increasing values of the potential is attributed to space charge. The current density expected in the presence of space charge is calculated from the Fowler-Nordheim field emission theory using values of the cathode electric field obtained from a solution of Poisson's equation for plane electrodes with boundary conditions appropriate to field emission. The result is a generalization of Child's equation, and is asymptotic to it when the applied potential is large compared with the value required for appreciable field emission. Current densities predicted by the theory are compared with experimental values obtained for several values of the work function by use of the clean tungsten and the barium-on-tungsten emitting surfaces in high vacuum. The Fowler-Nordheim theory correctly predicts the observed average current density from clean tungsten for densities in the range from 2 amp/cm2 to 4×107 amp/cm2, thus extending a previous test of the theory which was limited to current densities less than 6×106 amp/cm2. Significant features of the electron emission pattern observed at high current densities are explained as effects of space charge.