In a continuation of a study reported previously, rare-earth elements and thorium have been determined in monazite, allanite, cerite, bastnaesite, and a number of miscellaneous cerium-earth minerals. A quantity called sigma (???), which is the sum of the atomic percentages of La, Ce, and Pr, is proposed as an index of composition of all cerium-earth minerals with respect to the rare-earth elements. The value of ??? for all of the minerals analysed falls between 58 and 92 atomic per cent. Monazites, allanites, and cerites cover the entire observed range, whereas bastnaesites are sharply restricted to the range between 80 and 92 atomic per cent. The minimum value of ??? for a cerium-earth mineral corresponds to the smallest possible unit-cell size of the mineral. In monazite, this structurally controlled minimum value of ??? is estimated to be around 30 atomic per cent. Neodymium, because of its abundance, and yttrium, because of its small size, have dominant roles in contraction of the structure. In the other direction, the limit of variation in composition will be reached when lanthanum becomes the sole rare-earth element in a cerium-earth mineral. Cerium-earth minerals from alkalic rocks are all characterized by values of ??? greater than 80 atomic per cent, indicating that the processes that formed these rocks were unusually efficient in fractionating the rare-earth elements-efficient in the sense that a highly selected assemblage is produced without eliminating the bulk of these elements. Analyses of inner and outer parts of two large crystals of monazite from different deposits show no difference in ??? in one crystal and a slightly smaller value of ??? in the outer part of the other crystal compared to the inner part. The ??? of monazites from pegmatites that intrude genetically related granitic rocks in North Carolina is found to be either higher or lower than the ??? of monazites in the intruded host rock. These results indicate that the fractionation of the rare-earth elements is not a simple unidirectional process. When a cerium-earth mineral undergoes replacement, its rare-earth elements may be fractionated into two parts, one forming a new mineral with ??? that is smaller, and the other a second new mineral with ??? that is larger than that of the original mineral. The complete analysis of a cerium-earth mineral to determine its ??? is time consuming. The discovery of a direct relationship between ??? and the Ce/(Nd + Y) atomic ratio in cerium earth minerals allows a rapid determination of ??? from spectrograms obtained in a previously described method for determining thorium in these minerals. ?? 1957.