Page:Elektrische und Optische Erscheinungen (Lorentz) 118.jpg

§ 87. If we replace in equation (113), $\mp j$ by $\alpha$, and $\mp k{\mathfrak {p}}_{x}$ by $\beta$, it follows

$4\pi V^{2}{\frac {n'}{m'}}=(\sigma +\alpha m'+\beta n')\left(V^{2}{\frac {m'}{n'}}-{\frac {n'}{m'}}-2{\mathfrak {p}}_{x}\right)$.

Since the terms with $\alpha ,\beta$ and ${\mathfrak {p}}_{x}$ are in any case very small, the value of m following from it, can be represented by a row that progresses with respect to the powers of $\alpha ,\beta$ and ${\mathfrak {p}}_{x}$. The first term independent of these magnitudes, has the value

$m'_{0}=n'{\sqrt {{\frac {4\pi }{\sigma }}+{\frac {1}{V^{2}}}}}$,

and then we also find

$m'=m'_{0}+{\frac {n'}{V^{2}}}{\mathfrak {p}}_{x}-{\frac {2\pi }{\sigma ^{2}}}n'^{2}\alpha -{\frac {2\pi }{\sigma ^{2}}}{\frac {n'^{3}}{m'_{0}}}\beta -{\frac {2\pi }{\sigma ^{2}V^{2}}}{\frac {n'^{3}}{m'_{0}}}\alpha {\mathfrak {p}}_{x}+$ $+A\alpha ^{2}+B\alpha \beta +C\alpha ^{2}{\mathfrak {p}}_{x}$,

where we didn't calculated the three latter terms more closely, and we have neglected all higher powers of $\alpha$ and $\beta$, as well as all terms that include ${\mathfrak {p}}_{x}^{2}$. To these latter ones, also the terms with $\beta ^{2}$ and $\beta {\mathfrak {p}}_{x}$ do belong, since $\beta =\mp k{\mathfrak {p}}_{x}$.

Now, we obtain $m'_{1}$, or $m'_{2}$, depending on whether we put $\alpha =-j,\ \beta =-k{\mathfrak {p}}_{x}$, or $\alpha =+j,\ \beta =+k{\mathfrak {p}}_{x}$. The sought rotation of the polarization consequently becomes

$\omega ={\frac {2\pi }{\sigma ^{2}}}n'^{2}\left(1+{\frac {n'}{m'_{0}}}{\frac {{\mathfrak {p}}_{x}}{V^{2}}}\right)j+{\frac {2\pi }{\sigma ^{2}}}{\frac {n'^{3}}{m'_{0}}}{\mathfrak {p}}_{x}k$,

or, when we denote the propagation velocity ${\frac {n'}{m'_{0}}}$ by W,

$\omega ={\frac {2\pi }{\sigma ^{2}}}n'^{2}\left(1+{\frac {W{\mathfrak {p}}_{x}}{V^{2}}}\right)j+{\frac {2\pi }{\sigma ^{2}}}n'^{2}W{\mathfrak {p}}_{x}k$.

The natural rotation of the polarization plane in stationary bodies would consequently be

${\frac {2\pi }{\sigma ^{2}}}n'^{2}j$;

(116)

if we were allowed to consider as constant $\sigma$ and j, then it would be, as it follows from the meaning of $n'$, proportional to the square of the oscillation time. It's known that all bodies deviate