On the allometric mass exponent, when it exists.

Allometric power laws of the form q varies; is directly proportional to Mb are commonly used to describe the dependence of a physiological variable q on body mass M in a class of animals that vary widely in size. The theoretical basis for such a law has been ignored or misstated in much of the biological literature. In this paper, an axiomatic basis for establishing the existence of a power law is proposed and a formula for calculating the theoretical mass exponent b is derived. It is shown that the value of b, when it exists, depends not only on the physical dimensions of q but also on biological assumptions about the class of animals under consideration. Two examples are worked through, one concerning the flying speed of birds and one concerning the controversial subject of metabolic scaling in mammals. The wide range of mass exponents reported in the literature, it is argued, express no simple physical principle, but a variety of biological mechanisms including evolution, maturation, and seasonal adaptation.