for the exponents of systems with more than one pair of exponents. In this paper we present an analytical calculation of the four Lyapunov exponents for a dilute, three-dimensional Lorentz gas with randomly placed hard sphere scatterers, both for an equilibrium system and for a system where the moving particle is placed in a thermostatted applied field. In both cases our results for each of the exponents agree very well with computer simulations of Dellago and Posch [6] and satisfy the conjugate pairing rule to second order in the electric field $ E.