  Meaningful laws of nature must be independent of the units employed to
measure the variables. The principle of similitude (Rayleigh 1915) or
dimensional homogeneity, states that only commensurable quantities (ones having
the same dimension) may be compared, therefore, meaningful laws of nature must
be homogeneous equations in their various units of measurement, a result which
was formalized in the $\rm \Pi$ theorem (Vaschy 1892; Buckingham 1914).
However, most relations in allometry do not satisfy this basic requirement,
including the `3/4 Law' (Kleiber 1932) that relates the basal metabolic rate
and body mass, which it is sometimes claimed to be the most fundamental
biological rate (Brown et al. 2004) and the closest to a law in life sciences
(West \& Brown 2004). Using the $\rm \Pi$ theorem, here we show that it is
possible to construct a unique homogeneous equation for the metabolic rates, in
agreement with data in the literature. We find that the variations in the
dependence of the metabolic rates on body mass are secondary, coming from
variations in the allometric dependence of the heart frequencies. This includes
not only different classes of animals (mammals, birds, invertebrates) but also
different exercise conditions (basal and maximal). Our results demonstrate that
most of the differences found in the allometric exponents (White et al. 2007)
are due to compare incommensurable quantities and that our dimensionally
homogenous formula, unify these differences into a single formulation. We
discuss the ecological implications of this new formulation in the context of
the Malthusian's, Fenchel's and the total energy consumed in a lifespan
relations.
