The analysis of the solidification curve for the traditional cases of equilibrium and normal non-equilibrium solidification has been extended to include computation for a binary alloy of any phase diagram. An additional case, taking into account diffusion within the solid phase, has been analyzed to give a solidification curve sensitive to the mode of solidification, in particular the size of the solidification element. Expressions have also been derived for each of the three solidification cases in order to compute the rate of solid formation, the interface and average compositions of each phase, the rate of change of the composition with the progress of the solidification, and the relation of the amount of non-equilibrium phase to the size of the solidification element. Numerical analysis was used to calculate the change of the concentration profile on a microscale within a dendrite, due to diffusion within the solid. Apparatus and control equipment were designed and assembled for controlled ingot solidification. In addition to the extended solidification heats described below, the equipment permits a interrupted solidification, b isothermal solidification, c programmed solidification, d unidirectional solidification, e solidification under vibration, and f solidification with stirring.