Abstract: On the principle of the Gibbs energy being minimum in phase equilibria at constant temperature and pressure,the relationships between chemical potential and compositions on phase boundaries of multicomponent systems are derived by using Lagrange's method of indeterminate coefficients.The analytical solutions for two coexisting phase fields of ternary systems are given It is shown that analytical solutions may overcome the difficulties inherent in the older graphical Gibbs-Duhem integration techniques.The utility of this thermodynamic method developed in the present study is demonstrated for thesystems C₆H₆-H₃O-C₂H₅OHand H₂O-C₃H₅OH-CCl₄.The calculated values agree fairly well with the experimental data.The solutions presented have been extended to systems of k components and k-1 phases.