  We study the Lyapunov exponents of a two-dimensional, random Lorentz gas at
low density. The positive Lyapunov exponent may be obtained either by a direct
analysis of the dynamics, or by the use of kinetic theory methods. To leading
orders in the density of scatterers it is of the form
$A_{0}\tilde{n}\ln\tilde{n}+B_{0}\tilde{n}$, where $A_{0}$ and $B_{0}$ are
known constants and $\tilde{n}$ is the number density of scatterers expressed
in dimensionless units. In this paper, we find that through order
$(\tilde{n}^{2})$, the positive Lyapunov exponent is of the form
$A_{0}\tilde{n}\ln\tilde{n}+B_{0}\tilde{n}+A_{1}\tilde{n}^{2}\ln\tilde{n}
+B_{1}\tilde{n}^{2}$. Explicit numerical values of the new constants $A_{1}$
and $B_{1}$ are obtained by means of a systematic analysis. This takes into
account, up to $O(\tilde{n}^{2})$, the effects of {\it all\/} possible
trajectories in two versions of the model; in one version overlapping scatterer
configurations are allowed and in the other they are not.
