Using a cubic equation of state, the phase and critical behavior in type III phase diagrams showing gas-gas immiscibility of the second kind has been studied. The analysis of the pressure-composition diagrams and the corresponding pressure-molar density diagrams has shown new criteria for delimiting the regions of liquid-vapor and liquid-liquid equilibria. Basic to these new criteria was the finding of a molar isopycnic curve on a temperature-pressure projection. This curve follows from a bifurcation of the liquid-vapor critical curve coming from the less volatile component. This bifurcation point provides a criterion for definition of the point of transition from the liquid-vapor critical curve into a liquid-liquid critical curve. In understanding complex phase behavior, the importance of studying molar density phase diagrams alongside the corresponding compositional phase diagrams is emphasized throughout this work.