Biological similarity relative to allometric quantities

It is an empirical finding that an allometric quantity with dimensional exponents α, β and γ relative to mass, length, and time, respectively, has a value for its allometric exponentb satisfying the relation

$$tfrac{1}{3}(3alpha + beta + {gamma mathord{left/ {vphantom {gamma 2}} right. kern-nulldelimiterspace} 2}) leqslant b leqslant tfrac{1}{3}(3alpha + beta + gamma ).$$

A theoretical derivation is given of this double inequality using only the fact of constant density and the plausible assumption that metabolic rate is a dominant allometric quantity.