  The response of four dimensional quantum field theories to a Weyl rescaling
of the metric in the presence of local couplings and which involve $a$, the
coefficient of the Euler density in the energy momentum tensor trace on curved
space, is reconsidered. Previous consistency conditions for the anomalous
terms, which implicitly define a metric $G$ on the space of couplings and give
rise to gradient flow like equations for $a$, are derived taking into account
the role of lower dimension operators. The results for infinitesimal Weyl
rescaling are integrated to finite rescalings $e^{2\sigma}$ to a form which
involves running couplings $g_\sigma$ and which interpolates between IR and UV
fixed points. The results are also restricted to flat space where they give
rise to broken conformal Ward identities. Expressions for the three loop Yukawa
$\beta$-functions for a general scalar/fermion theory are obtained and the
three loop contribution to the metric $G$ for this theory are also calculated.
These results are used to check the gradient flow equations to higher order
than previously. It is shown that these are only valid when $\beta \to B$, a
modified $\beta$-function, and that the equations provide strong constraints on
the detailed form of the three loop Yukawa $\beta$-function. ${\cal N}=1$
supersymmetric Wess-Zumino theories are also considered as a special case. It
is shown that the metric for the complex couplings in such theories may be
restricted to a hermitian form.
