Comparison of Calculations on Cascade Theory

The diffusion equations of the cascade theory of electron showers are solved using improved mathematical methods. The results obtained agree essentially with those obtained earlier by Carlson and Oppenheimer, by Snyder, and by Serber, but do not agree with those obtained by Bhabha and Chakrabarty. Solutions are given corresponding to a single incident electron, a single incident $\gamma$-ray, and a $\frac{1}{E}$ spectrum of $\gamma$-rays. The total number of particles expected under a given thickness of material are given in tables for various incident energies and each of the above initial conditions. Other tables and formulas are included which enable one to calculate the spectrum of the particles and $\gamma$-rays for various initial conditions.