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To study the characteristics of synchrotron radiation in magnetic fields ofaccelerators first the author was necessary to obtain a continuous solutions ofHill's equation. For this purpose the gradient or the components of magneticfield were developed in a series. The same procedure is followed now in thecase of storage rings. This approach proved to be interesting not only from thepoint of view of describing the motion of partiles in ordinarythree-dimensional space but also in the fact that we get new differentialequations. This brief review can be regarded as an introduction to the proposedmethod. The next step may be to add nonlinearities. This would be the bestapproximation to the determination of betatron oscillations in the existingaccelerators.

Linear dynamics of charged particles in the main lattices of storage rings