  Cosmic rays impacting on the atmosphere cause particle-showers. Several
descriptions exist for the evolution of the shower size along the atmospheric
depth. The well known functions for shower profiles, Greisen, Gaisser-Hillas
and `Gaussian in Age', are intimately connected in that they all are
approximate solutions of versions of the Rossi and Greisen diffusion equations.
The mathematical connection will be demonstrated by means of two simple models
for the longitudinal electromagnetic shower profile. Both models can be
regarded either as a generalization of the Heitler model or as a simplification
of the diffusion model of Rossi and Greisen. These models are far closer to
reality than the Heitler model, while they are not as close to reality as the
model of Rossi and Greisen. Therefore, they will be referred to as intermediate
models. For each intermediate model the evolution of the shower is governed by
either a single differential equation or a single integro-differential
equation. The approximate solution of the differential equation is a
Gaisser-Hillas function and can be adjusted such that it almost matches the
Greisen profile. The approximate solution of the integro-differential equation
is a `Gaussian in Age' function. The corresponding profile is, after suitable
adjustment, in excellent agreement with the Greisen profile. The analysis also
leads to an alternative functional form for the age parameter.
