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One of the factors influencing the properties of a pulsed beam is the growth of the charge bunches due to space charge forces. The growth along the direction of motion of the bunch degrades the time-structure of the pulsed beam and the growth transverse to the direction of motion de-focuses the beam. Although the space charge effect is only important at high current densities, it must be considered a limiting factor in any pulsed beam system. A general method of obtaining the field in the bunch to second order in v/c was developed. From this the motion of charges in the bunch was obtained by numerical integration for the case of a Gaussian distribution of space charge under the assumption that the charge distribution remains Gaussian. Solutions are presented in a dimensionless, graphical form which can easily be scaled for a particular problem.