The Nature of Statistical Fluctuations with Applications to Cosmic Rays

I. The square of the mean deviation $D$ of the combined effect of several random processes releasing an average of $x, y, z\cdots$ particles per unit time and producing $a, b, c\cdots$ ion pairs, respectively, per particle, is $D^2=a^{2}x+b^{2}y+c^{2}z+\cdots$, regardless of whether the separate effects are added or subtracted by the experimental arrangement. For tube-counters, point-counters, scintillation screens and particle counting chambers, $a=b=c=1\mathrm{}$; for ionization chambers $a, b, c\cdots$ are unequal. II. From the standpoint of statistical fluctuations, the use of two identical instruments in a differential circuit is inferior to the use of a single instrument. III. The natural observational limit for the measurement of $x$ particles against a background of $y$ particles is $x=0.67\mathrm{}{\mathrm{}\left(y\right)\mathrm{}}^{\frac{1}{2}}$. IV. The statistical fluctuations in the ionization produced by cosmic rays in a spherical ionization chamber are treated rigorously and the fluctuations due to heterogeneity of range and to showers are derived. V. Application to existing data shows that the showers observed in cloud-chamber photographs of the cosmic radiation are also present in the ionization chamber in about the same frequency and multiplicity as indicated by the cloud-chamber results. The tube-counter investigations of the cosmic-ray flux are also in agreement with the deductions from the statistical fluctuations in the ionization chamber. An upper limit of 70±10 ion pairs per cm in air at 1 atmosphere is set for the total ionization along the path of an individual cosmic-ray secondary. The size and the relative frequency of occurrence of showers is appreciably greater at 14,700 feet elevation than at sea level. These showers are quite distinct from the ionization bursts or Stösse observed by Hoffmann, Steinke and others.