  From 31 Earth-based and three Voyager 2 occultations spanning 1977--2006, we
determine the orbital elements of the nine main Uranian rings with typical RMS
residuals of 0.2 -- 0.4 km and 1-$\sigma$ errors in $a, ae,$ and $a\sin i$ of
order 0.1 km, registered on an absolute radius scale accurate to 0.2 km at the
2-$\sigma$ level. The $\lambda$ ring shows more substantial scatter. In
addition to the free modes $m=0$ in the $\gamma$ ring and $m=2$ in the $\delta$
ring, we find two additional outer Lindblad resonance (OLR) modes ($m=-1$ and
$-2$) and a possible $m=3$ inner Lindblad resonance (ILR) mode in the $\gamma$
ring. No normal modes are detected for rings 6, 5, 4, $\alpha$, or $\beta$.
Five normal modes are forced by small moonlets: the 3:2 inner ILR of Cressida
with the $\eta$ ring, the 6:5 ILR of Ophelia with the $\gamma$ ring, the 23:22
ILR of Cordelia with the $\delta$ ring, the 14:13 ILR of Ophelia with the outer
edge of the $\epsilon$ ring, and the counterpart 25:24 OLR of Cordelia with the
ring's inner edge. We determine the width-radius relations for nearly all of
the detected mode. We find no convincing evidence for librations of any of the
rings. The Uranus pole direction at epoch TDB 1986 Jan 19 12:00 is
$\alpha_P=77.311327\pm 0.000141^\circ$ and
$\delta_P=15.172795\pm0.000618^\circ$. We determine the zonal gravitational
coefficients $J_2=(3509.291\pm0.412)\times 10^{-6},
J_4=(-35.522\pm0.466)\times10^{-6}$, and $J_6$ fixed at $0.5\times 10^{-6}$,
with a correlation coefficient $\rho(J_2,J_4)=0.9861$, for a reference radius
$R=$25559 km. From the amplitudes and resonance radii of normal modes forced by
moonlets, we determine the masses of Cressida, Cordelia, and Ophelia. Their
estimated densities decrease systematically with increasing orbital radius and
generally follow the radial trend of the Roche critical density for a shape
parameter $\gamma=1.6$.
