Transforming straight lines to straight lines

One important feature concerning connections betweens the points in the two planes ($\rho/\lambda$, $\mu/\lambda$) and ($\rho/\mu$, $\lambda/\mu$) is the fact that (with only a few exceptions that will be noted) straight lines in one plane transform into straight lines in the other. For example, points satisfying

= A + B   in the ($\rho/\mu$, $\lambda/\mu$)-plane (where A and B are constant intercept and slope, respectively), then satisfy

= A^-1 - A^-1B   in the ($\rho/\lambda$, $\mu/\lambda$)-plane. So long as $A \ne 0$in (AB1), the straight line in (AB1) transforms into the straight line in (AB2). This observation is very important because the straight line in (patchy) corresponds to a straight line in the saturation-proxy plot in the ($\rho/\mu$, $\lambda/\mu$)-plane. But this line transforms into a straight line in data-sorting plot in the ($\rho/\lambda$, $\mu/\lambda$)-plane. In fact the apparent straight line along which the data align themselves in these plots is just this transformed patchy saturation line.

When A = 0 in (AB1) [which seems to happen rarely if ever in the real data examples, but needs to be considered in general], the resulting transformed line will just be one of constant $\rho/\lambda = B^{-1}$,which is a vertical line on the ($\rho/\lambda$, $\mu/\lambda$)-plane. The more interesting special case is when B=0, in which situation $\lambda/\mu = A$ or $\mu/\lambda = A^{-1}$. But this case includes that of Gassmann-Domenico for homogeneous mixing of the fluids at low to moderate saturation values. For B = 0, on both planes we have horizontal straight lines, but their lengths can differ significantly on the two displays.