  Conditions for a maximum or minimum of Poisson's ratio of anisotropic elastic
materials are derived. For a uniaxial stress in the 1-direction and Poisson's
ratio $\nu$ defined by the contraction in the 2-direction, the following three
quantities vanish at a stationary value: $s_{14}$, $[2\nu s_{15} + s_{25}]$ and
$[(2 \nu -1)s_{16} + s_{26}]$, where $s_{IJ}$ are the components of the
compliance tensor. Analogous conditions for stationary values of Young's
modulus and the shear modulus are obtained, along with second derivatives of
the three engineering moduli at the stationary values. The stationary
conditions and the hessian matrices are presented in forms that are independent
of the coordinates, which lead to simple search algorithms for extreme values.
In each case the global extremes can be found by a simple search over the
stretch direction $\bf n$ only. Simplifications for stretch directions in a
plane of orthotropic symmetry are also presented, along with numerical examples
for the extreme values of the three engineering constants in crystals of
monoclinic symmetry.
