The emulsion morphology diagram for the ethanol/benzene/water system is analyzed by a coordinate transformation of emulsion inversion data and by critical-scaling equations in the transformed coordinates. The transformation, which may be called the "Bancroft-Hand" transformation, converts the phase and inversion volume fractions into a new concentration variable. When this new concentration variable is used, the tie lines become parallel with each other. The parallel tie lines extend the validity of the critical-scaling theory to a large fraction of the binodal. The phase and emulsion inversion compositions obtained with these transformed concentration variables are compared to the corresponding phase and inversion compositions in polar coordinates (another coordinate transform for obtaining parallel tie lines). The interpretation of the behavior of this system in the Bancroft-Hand transformation also is compared to the behavior of the binary 2-butoxyethanol/water-temperature system (where the tie lines are rigorously parallel).