  We give a self-contained modern linear stability analysis of a system of n
equal mass bodies in circular orbit about a single more massive body. Starting
with the mathematical description of the dynamics of the system, we form the
linear approximation, compute all of the eigenvalues of the linear stability
matrix, and finally derive inequalities that guarantee that none of these
eigenvalues have positive real part. In the end, we rederive the result that
J.C. Maxwell found for large n in his seminal paper on the nature and stability
of Saturn's rings, which was published 150 years ago. In addition, we identify
the exact matrix that defines the linearized system even when n is not large.
This matrix is then investigated numerically (by computer) to find stability
inequalities. Furthermore, using properties of circulant matrices, the
eigenvalues of the large 4nx4n matrix can be computed by solving n quartic
equations, which further facilitates the investigation of stability. Finally,
we have implemented an n-body simulator and we verify that the threshold mass
ratios that we derived mathematically or numerically do indeed identify the
threshold between stability and instability. Throughout the paper we consider
only the planar n-body problem so that the analysis can be carried out purely
in complex notation, which makes the equations and derivations more compact,
more elegant and therefore, we hope, more transparent. The result is a fresh
analysis that shows that these systems are always unstable for 2<=n<=6 and for
n > 6 they are stable provided that the central mass is massive enough. We give
an explicit formula for this mass-ratio threshold.
