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The electron on a trajectory around a nucleus radiates energy since it is accelerated and, following the classical theories of physics, should fall on the nucleus. But since it does not fall, it exists a cause, not taken into account by these theories, that prevents it from this fall. Indeed, in our precedent publications we showed that the fields and the particles are changes in a specific elastic medium called "ether". In particular, that a particle is a globule called “single-particle wave” that vibrates in the ether and is denoted  . This  that moves at the velocity V of the particle, contains all the parameters relative to it, and modulates the amplitude of the superposition of a wave called “phase-wave”, denoted , (and not ) of phase velocity V_p . In the present paper, one considers that the electron moves on a circle of circumference C, under the attraction of a nucleus.  is then present simultaneously several times, e.g., N times, at each point of this circle, that is, in , which is influenced by these N superpositions, due to the fact that  interferes with itself at each round. It appears then the two possible cases: the resonant and the nonresonant. In the resonant case, C contains an integer number of wave lengths  of  that therefore superposes itself at each round in a constructive addition, and creates a relatively large  , i.e., of a relatively large energy. The electron that radiates then only a limited percentage of its large energy, does not fall on the nucleus. In the nonresonant case, C contains a non-integer number of , then  superposes itself at each round in a destructive addition, and creates a much smaller , i.e., of much smaller energy. The electron that radiates then almost the same energy as in the resonant case, loses in fact a much larger percentage of its energy than in the resonant case, cannot remain in this state.

The Existence of Stable Electron Trajectories in the Atom Proved by the Ether Theory