  The conjugate pairing of Lyapunov exponents for a field-driven system with
smooth inter-particle interaction at constant total kinetic energy was first
proved by Dettmann and Morriss [Phys. Rev. E {\bf 53}, R5545 (1996)] using
simple methods of geometry. Their proof was extended to systems interacting via
hard-core inter-particle potentials by Wojtkowski and Liverani [Comm. Math.
Phys. {\bf 194}, 47 (1998)], using more sophisticated methods. Another, and
somewhat more direct version of the proof for hard-sphere systems has been
provided by Ruelle [J. Stat. Phys. {\bf 95}, 393 (1999)]. However, these
approaches for hard-sphere systems are somewhat difficult to follow. In this
paper, a proof of the pairing of Lyapunov exponents for hard-sphere systems at
constant kinetic energy is presented, based on a very simple explicit geometric
construction, similar to that of Ruelle. Generalizations of this construction
to higher dimensions and arbitrary shapes of scatterers or particles are
trivial. This construction also works for hard-sphere systems in an external
field with a Nos\'e-Hoover thermostat. However, there are situations of
physical interest, where these proofs of conjugate pairing rule for systems
interacting via hard-core inter-particle potentials break down.
