Analytical conditions for a maximum or minimum

In the case of a set of isothermal equilibrium states we have already seen that if the pressure passes through an extreme value, then a sufficient condition for this is that the two phases shall have identical compositions. (Gibbs-Konovalow theorem). We now require to find the condition that this extreme value is a maximum or minimum, and to do this it is necessary to investigate the second differentials of the coexistence curves. 

We take the reciprocal of equation (18.49) and differentiate with respect to x and obtain 

If the composition of the two phases is the same then the second term vanishes. Furthermore to evaluate dx jdx we may divide (18.50) by (18.49), and put the composition of the two phases the same. This gives simply 

If both phases are stable then the sign of this second derivative is given by the sign of 

We conclude therefore that if the two phases have the same composition, the pressure of the system is a maximum if 

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