### Footnotes:

^{1}They are therefore sometimes called `Moment
Equations.'
^{2}This is often, for historical reasons,
written an the equivalent form (called the Appleton-Hartree
dispersion relation) that can be considered to be
N^{2} = 1 − [(2(A−B+C))/(2A−B±F)].
The benefit of the earlier Appleton Hartree formulation is not this trivial
reorganization, however. It is that in it the coefficients A, B, and C
are effectively multiplied by a factor
r=(Ω_{e}^{2}−ω^{2})(ω^{2}−Ω_{i}^{2}). That has the considerable
benefit of avoiding the necessity to flip the sign of F in order to
get continuous behavior of the solutions through the cyclotron
frequencies. The benefit can also be obtained by writing the solution as
N^{2}=[B±sign(r)F]/2A

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On 14 Nov 2017, 10:06.