The second-order radial differential equations for the relativistic Dirac hydrogen atom are derived from the Dirac equation treated as a system of partial differential equations. The quantum operators which arise in the development are defined and interpreted as they appear. The splitting in the energy levels is computed by applying the theory of singularities for second-order differential equations to the Klein-Gordon and Dirac relativistic equations. In the Dirac radial equation additional terms appear containing a constant, which is shown to be (...) the “radius of the electron.” It is concluded that the minute perturbation of the radial eigenfunction in the vicinity of the proton brought about by the extension of the elementary particles, which appears naturally out of the Dirac equations, results in the prediction of the observed splitting of the hydrogen atom energy levels by the Dirac theory. The extension of the particles arises even though the Dirac hydrogen atom is originally formulated for point charges. (shrink)