  A solution of the n-body problem in R^d is a relative equilibrium if all of
the mutual distance between the bodies are constant. In other words, the bodies
undergo a rigid motion. Here we investigate the possibility of partially rigid
motions, where some but not all of the distances are constant. In particular, a
{\em hinged} solution is one such that exactly one mutual distance varies. The
goal of this paper is to show that hinged solutions don't exist when n=3 or
n=4. For n=3 this means that if 2 of the 3 distances are constant so is the
third and for n=4, if 5 of the 6 distances are constant, so is the sixth. These
results hold independent of the dimension d of the ambient space.
