The theory of the fixed frequency cyclotron is developed and illustrated at each step with numerical results for an example. The conditions for ion transmission with respect to horizontal motion are derived, and it is shown that they are satisfied by all ions in most cyclotrons. A more correct and more complete focusing formula than that found in the literature is derived and used to integrate the vertical motion. It is characterized by a region in which the net force is defocusing causing the ions to diverge exponentially from the median plane.

The theory is applied to find the maximum energy attainable as a function of dee voltage and magnetic field fall‐off. The problem of orbital precession is treated quite generally, and equations for calculating the center of curvature as a function of time are derived. Some considerations in the magnetic field design are presented. A proof is given that ions experience strong radial grouping. The effect of ion source ``feelers'' is discussed.

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