A thermodynamic model is proposed for calculation of liquidus relations in multicomponent systems of geologic interest. In this formulation of mineral-melt equilibria, reactions are written in terms of the liquid oxide components, and balanced on the stoichiometry of liquidus phases. In order to account for non-ideality in the liquid, a 'Margules solution' is derived in a generalized form which can be extended to systems of any number of components and for polynomials of any degree. Equations are presented for calculation of both the excess Gibbs free energy of a solution and the component activity coefficients. Application to the system CaO-Al 2O 3-SiO 2 at one atmosphere pressure is achieved using linear programming. Thermodynamic properties of liquidus minerals and the melt are determined which are consistent with adopted error brackets for available calorimetric and phase equilibrium data. Constraints are derived from liquidus relations, the CaO-SiO 2 binary liquid immiscibility gap, solid-solid P-T reactions, and measured standard state entropies, enthalpies, and volumes of minerals in this system. Binary and ternary liquidus diagrams are recalculated by computer programs which trace cotectic boundaries and isothermal sections while checking each point on a curve for metastability. The maximum differences between calculated and experimentally determined invariant points involving stoichiometric minerals are 17°C and 1.5 oxide weight per cent. Because no solid solution models have been incorporated, deviations are larger for invariant points which involve non-stoichiometric minerals. Calculated heats of fusion, silica activities in the melt, and heats of mixing of liquids compare favorably with experimental data, and suggest that this model can be used to supplement the limited amount of available data on melt properties.