The third paper of this series provides a theoretical basis for analysis of precision measurements of the fine structure of hydrogen and deuterium. It supplements the Bechert-Meixner treatment of a hydrogen atom by allowing for the presence of a magnetic field, as well as radiative corrections. The theory of hyperfine structure is somewhat extended. Stark effects due to motional and other electric fields are calculated. Possible radiative and nonradiative corrections to the shape and location of resonance peaks are discussed. Effects due to the finite size of the deuteron are also considered. A theory of the sharp resonances 22S12(ms=12) to 22S12(ms=-12) is given which leads to an understanding of the peculiar shapes of resonance curves shown in Part II. In this connection, a violation of the "no-crossing" theorem of von Neumann and Wigner is exhibited for the case of decaying states.