  We point out that two of Milne's fourth-order integrators are well-suited to
bit-reversible simulations. The fourth-order method improves on the accuracy of
Levesque and Verlet's algorithm and simplifies the definition of the velocity
$v$ and energy $e = (q^2 + v^2)/2$ . ( We use this one-dimensional oscillator
problem as an illustration throughout this paper ). Milne's integrator is
particularly useful for the analysis of Lyapunov ( exponential ) instability in
dynamical systems, including manybody molecular dynamics. We include the
details necessary to the implementation of Milne's Algorithms.
