  Consider the constitutive law for an isotropic elastic solid with the
strain-energy function expanded up to the fourth order in the strain, and the
stress up to the third order in the strain. The stress-strain relation can then
be inverted to give the strain in terms of the stress with a view to
considering the incompressible limit. For this purpose, use of the logarithmic
strain tensor is of particular value. It enables the limiting values of all
nine fourth-order elastic constants in the incompressible limit to be evaluated
precisely and rigorously. In particular, it is explained why the three
constants of fourth-order incompressible elasticity $\mu$, $\bar{A}$, and $\bar
D$ are of the same order of magnitude. Several examples of application of the
results follow, including determination of the acoustoelastic coefficients in
incompressible solids and the limiting values of the coefficients of
nonlinearity for elastic wave propagation.
