  We consider a general method for computing the sum of positive Lyapunov
exponents for moderately dense gases. This method is based upon hierarchy
techniques used previously to derive the generalized Boltzmann equation for the
time dependent spatial and velocity distribution functions for such systems. We
extend the variables in the generalized Boltzmann equation to include a new set
of quantities that describe the separation of trajectories in phase space
needed for a calculation of the Lyapunov exponents. The method described here
is especially suitable for calculating the sum of all of the positive Lyapunov
exponents for the system, and may be applied to equilibrium as well as
non-equilibrium situations. For low densities we obtain an extended Boltzmann
equation, from which, under a simplifying approximation, we recover the sum of
positive Lyapunov exponents for hard disk and hard sphere systems, obtained
before by a simpler method. In addition we indicate how to improve these
results by avoiding the simplifying approximation. The restriction to hard
sphere systems in $d$-dimensions is made to keep the somewhat complicated
formalism as clear as possible, but the method can be easily generalized to
apply to gases of particles that interact with strong short range forces.
