Interchanging $\epsilon_F$ and $\epsilon$ when deriving Density of States (DOS) of Free Electron Fermi Gas (FEFG)

Question

When deriving the Density of States for a Free Electron Fermi Gas, Kittel seems to flamboyantly interchange $\epsilon$ and $\epsilon_F$ when going between equations. I am specifically puzzled as to why he is allowed to do that when going from equations 17 to 19 at page 140 in the eigth edition. Any reason as to why that is allowed?

The equations at hand are:

$ \epsilon_F = \frac{\hbar^2}{2m}\left(\frac{3\pi^2 N}{V} \right)^{2/3}$ (17)

and:

$N = \frac{V}{3\pi^2}\left( \frac{2m \epsilon }{\hbar^2} \right)^{3/2}$ (19)

Answer 1

It is a typo, and I think I can explain how it appeared. I don't have the eighth edition, but in my older copy, close after these expressions is an expression for the density of states, ${\cal D}(\epsilon)$, $${\cal D}(\epsilon)=\frac{V}{2\pi^{2}}\left(\frac{2m}{\hbar}\right)^{3/2}\epsilon^{1/2}.$$ In the expression for ${\cal D}(\epsilon)$, it really is $\epsilon$ (the energy of an arbitrary single-particle state), not the Fermi energy $\epsilon_{F}$. I think the similarity of the two expressions led to the subscript on "$\epsilon$" being dropped in the formula for $N$.