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Observations in 2013 and 2014 of the Centaur 10199 Chariklo and its ring system consistently indicated that the radial width of the inner, more massive ring varies with longitude. That strongly suggests that this ring has a finite eccentricity despite the fast differential precession that Chariklo's large quadrupole moment should induce. If the inferred apse alignment is maintained by the ring's self-gravity, as it is for the Uranian rings, we estimate a ring mass of a few times 10^16 g and a typical particle size of a few meters. These imply a short collisional spreading time of ~10^5 years, somewhat shorter than the typical Centaur dynamical lifetime of a few Myrs and much shorter than the age of the solar system. In light of this time constraint, we evaluate previously suggested ring formation pathways including collisional ejection and satellite disruption. We also investigate in detail a contrasting formation mechanism, the lofting of dust particles off Chariklo's surface into orbit via outflows of sublimating CO and/or N_2 triggered after Chariklo was scattered inward by giant planets. This latter scenario predicts that rings should be common among 100-km class Centaurs but rare among Kuiper belt objects and smaller Centaurs. It also predicts that Centaurs should show seasonal variations in cometary activity with activity maxima occurring shortly after equinox.