  In this paper, we study the chaotic four-body problem in Newtonian gravity.
Assuming point particles and total encounter energies $\le$ 0, the problem has
three possible outcomes. We describe each outcome as a series of discrete
transformations in energy space, using the diagrams first presented in Leigh \&
Geller (2012; see the Appendix). Furthermore, we develop a formalism for
calculating probabilities for these outcomes to occur, expressed using the
density of escape configurations per unit energy, and based on the Monaghan
description originally developed for the three-body problem. We compare this
analytic formalism to results from a series of binary-binary encounters with
identical point particles, simulated using the \texttt{FEWBODY} code. Each of
our three encounter outcomes produces a unique velocity distribution for the
escaping star(s). Thus, these distributions can potentially be used to
constrain the origins of dynamically-formed populations, via a direct
comparison between the predicted and observed velocity distributions. Finally,
we show that, for encounters that form stable triples, the simulated single
star escape velocity distributions are the same as for the three-body problem.
This is also the case for the other two encounter outcomes, but only at low
virial ratios. This suggests that single and binary stars processed via
single-binary and binary-binary encounters in dense star clusters should have a
unique velocity distribution relative to the underlying Maxwellian distribution
(provided the relaxation time is sufficiently long), which can be calculated
analytically.
