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The calculation of the multiplication of the soft component of the cosmic rays in the upper atmosphere is refined by use of the exact high energy radiative formulae of Bethe and Heitler, and the resulting diffusion equations are solved by the method of Snyder. Comparison with the vertical counter work of Pfotzer, and of Carmichael and Dymond, and with the ionization chamber data of Bowen, Millikan, and Neher, shows that the calculated position of the maximum of the multiplication curve is now in good agreement with that observed. The latter comparison also indicates that the penetrating component is at least largely of secondary origin.
Robert Serber (1938) studied this question.