In a recent paper i the writer, discussing the significance of J. Clay's 2 observations of the cosmic radiation down to depths of 27o meters below water, amplified a view suggested in former communications to the effect that the primary cosmic radiation of corpuscular kind produces no ions until by the production of showers (singles, doubles, and many-ray groups) it has fallen in energy below a certain critical value. On this view the primary corpuscular radiation is to be regarded as passing through our atmosphere with a density (measured by numbers of rays per square centimeter, per second) which is independent of altitude. The apparent increase of intensity with altitude becomes really an increase in the number of secondaries (showers) resulting from increase of the energy (not number) of the individual primary rays with altitude. The purpose of this note is to amplify further this view. To avoid complexity we shall retain only the fundamental elements and shall confine ourselves to the approximation that the range of the secondaries is small compared with the effective thickness of the atmosphere, so that the ionization at any altitude is determined by the number of secondaries produced per centimeter per second at that altitude. We shall consider that the showers are all of the same kind.
We shall first consider the simplest case, where the number of showers N, produced per c.c, by the primaries falling