This paper is concerned with a new method for electron acceleration. The principles of the method, which was successfully accomplished for the first time at the University of Illinois (1, 2, 3), will be described briefly, since the type of accelerator used, the betatron, should find worthwhile applications in deep therapy. Some experimental results form the subject of a second paper (p. 120). In the betatron energy is transferred to electrons by the accelerating effect of a time-varying magnetic field. Since a betatron is a powerful magnet, between the poles of which the electrons circulate in essentially one plane, the apparatus looks somewhat like a small cyclotron. It operates, however, with alternating current instead of direct current, and the process of acceleration is entirely different from that in the cyclotron. The betatron can accelerate particles whose velocity is very close to the velocity of light, such as electrons with energy in excess of half a million volts. Particles accelerated in a cyclotorn, on the other hand, must have a velocity much less than that of light, and therefore only heavy positive ions can be accelerated in it to appreciable energies. Electrons from an electron gun or injector are shot into a circular path within a doughnut-shaped vacuum tube, while the magnetic field intensity is small. As these electrons circulate between the poles of the magnet, the magnetic field is increasing, and the time-rate of change of flux linking the orbit produces an energy gain per revolution equal to that produced by the voltage which would be read on a voltmeter connected to a one-turn coil placed at the orbit and recording instantaneous voltage. Because of the great number of revolutions described by the electrons while the flux linkage is increasing, the energy in electron volts which is reached is roughly the same as the voltage generated in a secondary coil of the same number of turns placed around the magnetic core of the betatron and acting like a transformer secondary. Thus a betatron is similar to a transformer, but has the advantage that it is unnecessary to produce full voltage on a secondary coil and then apply that voltage to a high-vacuum x-ray tube. The electromotive force is instead continually applied directly to the electron stream. Figure 1 shows the vacuum “doughnut” in which the electrons circulate many times, having traveled as far as 200 miles when they finally strike the injector, where they produce x-rays and scatter out of. the doughnut into the room. The orbit-expanding coils are not energized until after the electrons have been accelerated; they disturb the flux distribution near the electron path, causing the electrons to spiral outward until they hit the first obstacle, the injector, which acts as the target. The injection time is indicated at A on the H curve in Fig. 1 (H = the magnetic field), and the orbit is expanded to the target at the time indicated by C, when the energy is at a maximum. These processes are repeated in each cycle with a period of three-fourths of a cycle when no electrons are in the doughnut.