to have little relation to Gibbs's work on heterogeneous equilibrium. As he wrote ten years later, ''I can remember the time when I thought that people made a good deal of unnecessary fuss over the phase rule. It seemed to me an interesting mathematical relation but nothing more."¹⁹

After returning to the United States, however, Bancroft's research interests and his opinion of the phase rule gradually changed. In a series of papers published while he was at Harvard, Bancroft studied solubility relations in ternary systems, those, for example, composed of two non-miscible liquids and a salt dissolved in both, or of two non-miscible liquids and a liquid miscible in both. Although notable for introducing the word "solute," these papers were, in other respects, dismal failures.²⁰ Convinced that such heterogeneous physical equilibria could be treated by expressions similar to those derived from the law of mass action for homogeneous chemical equilibria, Bancroft had fit his data to equations with up to four arbitrary constants. After several harsh rebukes, Bancroft abandoned his effort to give such systems precise, quantitative treatment and instead began to view them in the context of Gibbs's phase rule.

Shortly after moving to Cornell, this interest found expression in his monograph, The Phase Rule, an extended study not so much of the phase rule itself as of its uses in the classification and analysis of various classes of heterogeneous equilibria. Published in 1897, Bancroft's book was among the earliest works to offer an extended treatment of these applications of the phase rule and served an important role in introducing American chemists to the significance of the work of their countryman, Gibbs. Building upon this work, Bancroft continued to explore the rule and its applications in articles and reviews, making himself the leading American authority on heterogeneous equilibria. Together with his students, notably Charles A. Soch, E. S.