The stability of electron orbits in a synchrotron with a frequency modulated r-f has been investigated. The method consists in expanding the equations of motion and in solving them to zeroth and to first order of approximation. Attention is centered on the relatively low voltages—up to 200 Mev—and consequently the effects of radiation damping are of minor importance and may be neglected. With the correct frequency modulation, the electrons, in zeroth order, move in circular orbits with a constant radius. In first order their $r$ and $\theta$ coordinates oscillate with the two frequencies ${\omega }_1$ and ${\omega }_2$ while the axial coordinate z oscillates with the frequency ${\omega }_3$. It is shown that as the energy of the electrons increases, the amplitudes of the ${\omega }_1$ and ${\omega }_3$ motions decrease approximately as $\frac{1}{E^{\frac{1}{2}}}$ while the ${\omega }_2$ amplitude decreases as $\frac{1}{E^{\frac{1}{4}}}$. Thus the electron orbits are stable. A numerical example is calculated in detail for a machine where the injection energy is ½ Mev and the final energy 150 Mev. The accuracy of the solution is discussed and it is concluded that it will be correct to better than one percent provided the frequencies ${\omega }_1$, ${\omega }_2$, and ${\omega }_3$ are not commensurable. In the latter case, under certain conditions, secular changes in the orbits could occur which would destroy their stability.

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