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The relativistic differential equations of motion of an electron in the field of a linear accelerator are considered. Treating only electrons near the axis, the field components are expanded in powers of distance from the axis r, and terms in r and r˙ of order two higher than the lowest order terms are neglected. The first integral of two of the equations of motion are then obtained, and together with the equation of motion for the r coordinate, give a second‐order nonlinear differential equation for r as a function of the phase of the electron with respect to the peak of the accelerating electric field. Utilizing earlier results giving the dependence of phase on distance along the axis of the accelerator, a choice of a particular dc magnetic focusing field allows this differential equation to be integrated twice, from which r as a function of distance along the accelerator is plotted. The need for a focusing field is indicated, and it is shown that the particular field chosen has the advantage that with it, the electron will oscillate between known fixed limits of r. The necessity of shielding the electron emitter from the focusing field is shown, and an indication of the magnitude of the field and length of the accelerator for which it should be used, is given for the Purdue accelerator.