The study of the formation of minerals from the point of view of the Phase Rule is still in its initial stages, but the results which have already been obtained give promise of a rich harvest in the future.[316] * * * * * {234} CHAPTER XIII SYSTEMS OF THREE COMPONENTS General.--It has already been made evident that an increase in the number of the components from one to two gives rise to a considerable increase in the possible number of systems, and introduces not a few complications into the equilibrium relations of these. No less is this the case when the number of components increases from two to three; and although examples of all the possible types of systems of three components have not been investigated, nor, indeed, any one type fully, nevertheless, among the systems which have been studied experimentally, cases occur which not only possess a high scientific interest, but are also of great industrial importance. On account not only of the number, but more especially of the complexity of the systems constituted of three components, no attempt will be made to give a full account, or, indeed, even a survey of all the cases which have been subjected to a more or less complete experimental investigation; on the contrary, only a few of the more important classes will be selected, and the most important points in connection with the behaviour of these described. On applying the Phase Rule P + F = C + 2 to the systems of three components, we see that in order that the system shall be invariant, no fewer than five phases must be present together, and an invariant system will therefore exist at a _quintuple_ point. Since the number of liquid phases can never exceed the number of the components, and since there can be only one vapour phase, it is evident that in this case, {235} as in others, there must always be at least one solid phase present at the quintuple point. As the number of phases diminishes, the variability of the system can increase from one to four, so that in the last case the condition of the system will not be completely defined until not only the temperature and the total pressure of the system, but also the concentrations of two of the components have been fixed. Or, instead of the concentrations, the partial pressures of the components may also be taken as independent variables.