Page:WitteSagnac1.djvu/6

Now, $r\cdot \cos {\tfrac {\vartheta }{2}}$ is equal to the length of the perpendicular $\varrho$ from the center of the circle upon the polygon side, $U\cdot \varrho$ is furthermore the double of area $F$ of the polygon; thus one obtains the end result:

$2\cdot x={\frac {8\pi \cdot N\cdot F}{c\cdot \lambda }}$

5)

Due to reasons of measurement, Sagnac causes the apparatus to turn in one sense first, then in the other sense, thus he measures $4\cdot x$, and his end formula^[1] therefore reads:

$z=4\cdot x={\frac {16\cdot \pi \cdot N\cdot F}{c\cdot \lambda }}$

6)

By this formula 6), Sagnac calculates the fringe displacement to be expected; that the effect is indeed of first order $\left({\tfrac {v}{c}}\right)$ shows variation 4) of the same formula.

As it was said, then the observed value agrees quite satisfactorily with the calculated one.

The circle diameter $2r$ amounts to 50 cm; the rotating number $N$ is of order of magnitude 1 or 2 rotations per second; the interference images are made in a photographic way; the photographic plate, the light source, etc., share the rotation.

5. It is now the question, whether Sagnac's conclusion that the effect is deciding the alternative "aether or relativity principle", namely that it decides against the relativity principle and in favor of the aether, is necessary.^[2]

The following representation seems possible to me:

a) That the effect comes about, doesn't follow from the translatory velocity of observing point $O$, but from the rotational motion of the whole system (and by that of course also point $O$).

1. A. a. O., S. 710 u. 1412.
2. Only this alternative is discussed by me in the following. Regarding another important alternative, namely "emission- or non-emission theory", this effect very much seems to me as capable of being used as an experimentum crucis (against the emission theories), then it is a very valuable terrestrial counterpart of the known astronomical verifications.