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Many potential applications of the chemical mass balance involve more sources than species, or sources whose emissions vary with time. This paper demonstrates how standard methodology can be extended to accommodate such situations. Procedures are illustrated with data from a unique study in southern California which characterized fine particle composition in distinct polluted airsheds (sources) and the high desert (receptor). The sample apportionments are consistent with observed meteorological patterns, and identify the San Joaquin Valley as a major contributor to fine particle loadings in the high desert.