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

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

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