1. Bhabha and Heitler (1937) and also Carlson and Oppenheimer (1937) have recently proposed a very ingenious theory of showers which explains nearly all the observed phenomena; both these treatments, however, contain approximations the validity of which cannot always be justified. The calculations of the first two authors make use of the method of successive approximations whose exactness is difficult to estimate, while the second authors replace one of the fundamental equations by a different one, and it is impossible to see how this replacement may distort the final result. Consequently, we have thought it useful to make the calculation in a more rigorous way; the fundamental physical ideas, however, are exactly the same as those of the authors mentioned above. 2. Consider a fast electron penetrating into a layer of matter; as long as its energy is below a certain order of magnitude, the loss of energy of the electron will be determined mainly by ionization; for higher energies, however, loss of energy by radiation (Bremsstrahlung) becomes more important. This critical energy lies in the region of about ϵ = 750/Z̅ million volts, where Z̅ denotes a certain average nuclear charge (there may be several nuclei of different kinds present). Since the collision radiation, which is determined by the interaction between the electron and the nuclear field, increases as Z2, while the ionization is approximately proportional to the number of electrons in the material.