  Narrow planetary rings are eccentric and inclined. Particles within a given
ring must therefore share the same pericenter and node. We solve for the
three-dimensional geometries and mass distributions that enable the Uranian
Alpha and Beta rings, and the Saturnian Maxwell and Colombo (Titan) rings, to
maintain simultaneous apsidal and nodal lock. Ring self-gravity, interparticle
collisions, and the quadrupole field of the host planet balance each other to
achieve this equilibrium. We prove that such an equilibrium is linearly stable.
Predictions for the Saturnian ringlets to be tested by the Cassini spacecraft
include: (1) ringlet masses are of order 1e19 g, (2) surface mass densities
should increase from ring midline to ring edges, and (3) rings are vertically
warped such that the fractional variation of inclination across the ring is of
order 10%. Analogous predictions are made for the Uranian rings. Simultaneous
apsidal and nodal locking forces the narrowest portion of the ring--its
``pinch,'' where self-gravitational and collisional forces are strongest--to
circulate relative to the node, and introduces previously unrecognized
time-varying forces perpendicular to the planet's equator plane. We speculate
that such periodic stressing might drive kilometer-scale bending waves at a
frequency twice that of apsidal precession. Such flexing might be observed over
a few weeks by Cassini.
