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In I we discuss the status of the quantum theoretic formulae for pair production and radiation in the domain of cosmic-ray energies, and the relevance of these processes to an understanding of showers and bursts. In II we give a qualitative estimate of the course implied by the theory for a shower or burst built up by multiplication from a very energetic primary; we then set up the diffusion equations for the equilibrium of electrons and gamma-rays, and show how these can be simplified. In III we carry through the analytic solution of the diffusion equations, and find the distribution of electrons and gamma-rays as a function of their energy, the primary energy, and the thickness and atomic number of the matter traversed. We treat the effect of ionization losses on the shower, calculate the amount of radiation of low energy to be expected, and treat transition effects in passing from one substance to another. In IV we discuss the results of the calculations, and give a summary of the conclusions to which they lead, and the difficulties.