A fourth transform that gives pressure as a function of temperature and the three chemical potentials. Temperature and
are fixed at values giving the equimolar criticalpoint composition.
the base form is
U = f(S,N_{1},N_{2},N_{3},V)
A´´´ = U – TS  µ_{1}N_{1}  µ_{2}N_{2}  µ_{3}N_{3} = PV
scaling with respect to volume gives
As a fourth transform in a ternary system this function has the form of a swallowtail catastrophe. Phaseequilibrium states are given by points along the selfintersection curve where the variables T,P,µ_{1},µ_{2},µ_{3} are common to both the vapor and liquid phases. Again, the red unstable region has been omitted because it would cover the intersection curve and the metastable states.
