Page:Elektrische und Optische Erscheinungen (Lorentz) 134.jpg

For the difference $b_{1.2}-b'_{2.1}$ which comes into (123), we may put $b_{1.2}-b'_{1.2}$, which is evidently of order ${\mathfrak {p}}/V$, since the magnitudes $b_{1.2}$ and $b'_{1.2}$ are only different form one another by having different directions of translation.

By (123), also $\sin \left(b_{1.4}-b_{4.1}\right)$ must now be of order ${\mathfrak {p}}/V$. Since we additionally (without changing anything of the matter) can increase or decrease $b_{4.1}$ by a multiple of $\pi$, and also an uneven multiple of $\pi$ as long as the sign of $a_{4.1}$ is reversed, then we may assume, that also the angle $b_{1.4}-b_{4.1}$ itself is of order ${\mathfrak {p}}/V$. The two cosines in (122) thus differ from unity only by magnitudes of second order, so that we may put

$a_{1.2}\ a'_{2.1}+a_{1.4}\ a_{4.1}=1$.

In the same way

$a'_{1.2}\ a_{2.1}+a'_{1.4}\ a'_{4.1}=1$,

and under consideration of (124) and (125) we thus find

$a_{1.4}\ a_{4.1}=a'_{1.4}\ a'_{4.1}$.

Now suppose, similarly to the experiment of Fizeau, a plan-parallel glass plate (at whose two sides the aether is located) will be hit in oblique direction by a light beam, whose light vector has one of the directions previously distinguished, i.e. that it is polarized either in the plane of incidence, or perpendicular to it. The relation, be which the amplitude is diminished during the entrance, can thus be (depending in the direction of translation) represented by $a_{1.4}$ or $a'_{1.4}$, and also, as we can easily see, by the corresponding relation when leaving the plate by $a_{4.1}$ or $a'_{4.1}$. Altogether, the amplitude is thus altered in the ratio of 1 to $a_{1.4}\ a_{4.1}$ or $a'_{1.4}\ a'_{4.1}$. Now, since these products have the same value, the reversal of the translation changes nothing of the intensity of the leaving light, which consequently must be (except magnitudes of second order) the same, as if the plate would stand still: This is true for both main-positions of the polarization plane; consequently, when the incident rays are linearly polarized in an arbitrary way,