On Fluctuations and the General Distribution Problem in Electron Cascades

Recent developments of Bhabha, Friedman, and Jánossy are applied to the general probability distribution problem of electron cascades (cosmic-ray showers). In particular, it is shown that the master function giving the probability of any number of particles with any given energy distribution at a depth $t$ in a shower is determined, in principle, by the average energy spectrum. Certain theorems connecting the various functions are derived, and a discussion is given of the as yet unsolved problem of getting satisfactory analytic or numerical results from the formalism. Only the simplified model originated by Furry has been discussed, but the theorems embody many properties of the actual electron-photon multiplication process.