In this paper we give an improved treatment of the multiplication and absorption of cosmic-ray electrons and gamma-rays in the atmosphere. After a brief recapitulation, in I, of the principal processes involved in the cascade theory of showers and the resulting diffusion equations, we give, in II and III, a solution of these diffusion equations, making no further approximations. In IV, an expression for the probable number of charged particles is given; the values of the functions necessary for numerical calculations are given in a tabular form. In V, it is shown how the Bhabha-Heitler cut-off energy method for estimating the effect of ionization losses may be consistently used. In VI, a comparison is made with the Carlson-Oppenheimer calculations. The differences near the top of the atmosphere are small and in a direction to improve the agreement with observations. At sea level the new calculations give an intensity roughly $\frac{3}{5}$ of the Carlson-Oppenheimer results.

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