Editing SSC/Stability/InstabilityOnsetOverview (section)

=====Multiple Harmonics=====
In addition to the green circular marker, two small open circular markers have been placed along the dashed segment of the isothermal displacement function that is displayed in the left panel of Figure 4 &#8212; see also the top panel of Figure 2.  As has been [[#Harmonics|explained above]], in connection with the center and bottom panels of Figure 2, these additional markers have been placed at the next two locations along the undulating displacement function where <math>~d\ln x_Y/d\ln\xi = -3</math>.  They also correspond to the next two positions along the isothermal equilibrium sequence where an extremum in the mass occurs.  

In the left panel of Figure 4, the undulating displacement function that provides a solution to the LAWE for pressure-truncated, n = 6 polytropic configurations has been similarly marked at the two additional locations where <math>~d\ln x_P/d\ln\xi = -3</math>.  It is clear that these two additional markers correspond to the next two positions along the companion n = 6 equilibrium sequence (Figure 3) where an extremum in the mass occurs.  By analogy with the isothermal case, then, we can immediately identify the segments of the n = 6 displacement function that represent the eigenfunctions of the marginally unstable, first harmonic and second harmonic modes of radial oscillation.  Presumably, similar undulations occur in, and precisely this same type of structural information about multiple harmonic-mode eigenfunctions can be extracted from, our analytically specified polytropic displacement function, <math>~x_P</math>, for configurations having any value of the polytropic index within the range, <math>~5 < n < \infty</math>.