The properties of the various factors which arise in the conventional formula for maximum hourly mean ground level concentration (max. g.l.c.) after fixing mean wind speed and source strength are studied with a view to assessing variability and comparing with results from the Tilbury field trial. The analysis indicates some ambiguity in defining vertical spread for dispersion from tall stacks and suggests that the formula might be more profitably rearranged. A natural rearrangement is pointed out and this leads to a simple upper bound for max. g.l.c. in an unbounded atmosphere which is only 20% larger than the familiar result for a uniform atmosphere. This result is obtained using the diffusion equation with simple power laws for wind speed and eddy diffusivity. The general conservation equation is then considered and this leads to a specific definition of mean wind speed below source level and the indication of a general upper bound some 50% larger than the uniform atmosphere value provided certain reasonable conditions are met. The practical implications of these results are discussed and the extra effects introduced by stable layers are pointed out.