Detailed examination of the Sun's upper atmosphere reveals that elemental abundances vary between different solar regions. Some solar regions exhibit the well-established photospheric abundances, while in other regions the abundances of some elements diverge from photospheric by an order of magnitude, and perhaps even more. Elemental abundances in the solar wind (SW) are also different from elemental abundances in the photosphere. It is postulated that elemental abundances in the SW reflect the elemental abundances of their place of origin near the solar surface. The solar surface is covered, predominantly, by quiet and coronal hole regions. In this paper we measure the intensities of limb-brightening rings over quiet and coronal hole regions in lines of Mg VI and Ne VI (T(e) almost-equal-to 4.3 x 10(5) K) and determine the relative elemental abundances of the features producing the rings. The rings are formed by a plasma component occupying only a small fraction of the immediate volume above the solar surface. Since the brightness of the plasma occupying the rest of the volume is much smaller, it is not discussed in this paper. The findings are correlated with elemental abundances in the SW.