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However, a sphere will appear to the resting observer as an ellipsoid, with an axis ratio equal to ${\sqrt {1-q^{2}n^{2}}}$. Therefore, ${\sqrt {1-q^{2}/c^{2}}}={\sqrt {1-q^{2}n^{2}}}$. This gives

$n={\frac {1}{c^{2}}}$

(25)

And only from that it follows, that $c$ is constant for all coordinate systems. At the same time we see that the universal space-time constant $n$ is determined by the numerical value of $c$.

Now it is clear that optics lost its special position with respect to the relativity principle by the previous derivation of the transformation equations. By that, the relativity principle itself gains more general importance, because it doesn't depend on a special physical phenomenon any more, but on the universal constant $n$.

Nevertheless we can grant optics or the electrodynamic equations a special position, though not in respect to the relativity principle, but in respect to the other branches of physics, namely in so far as it is possible to determine the constant $n$ from these equations.

On the other hand, when we transform the other physical equations in accordance with the relativity principle, and at that occasion see the occurrence of constant $n$, then we don't have to conclude at all that electrical forces are in play, but we only conclude from the standpoint of the relativity principle, that space and time impress their character upon all physical phenomena by means of constant $n$.

In order to illustrate the meaning of $n$ still further, we use an analogy from optics, namely the relation between image and object. From the pure optical standpoint, object and image are interchangeable. Exactly the same is true when we consider a moving measuring rod that appears to us as being contracted. We can say that space and time project this moving measuring rod to us, so that we can only see the image of it when we assume the resting measuring rod as being an object.

Thus we can completely agree with Minkowski, who says in his lecture "Space and Time"^[1]: "The contraction is not, for example, to be seen as the consequence of resistances in the aether, but as a present from above, as a concomitant of the circumstance of motion", exactly because $n$ is a universal constant.

At the end, I want to mention the velocities possible from the standpoint of the relativity principle.

Consider expression (24) for $p$.

The contraction that we observe at a distance moving with system $K'$ is depending on $p$. Thus it has no meaning to assume that $p$ can become imaginary, i.e., $q$ must always be smaller than $c$. But what is the meaning of $q$? $q$ means the velocity of the coordinate system $K'$, thus it cannot be larger than $c$. In other words: none of the rest coordinate systems can move with superluminal velocity. But we may not see a rest coordinate system only as a mathematical structure, instead we have to think about a material world with its observers and synchronous clocks. Conversely we assume that we can transform any substantial point to rest. By that it follows that a substantial point cannot move with superluminal velocity.

Now the question arises: Is there a velocity, not of substantial points but of phenomena, that are larger than the speed of light, neglecting phase or group velocities? We have to affirm this question.

Without going into details, for which I refer to my last paper in Ann. d. Phys.^[2], I only want to explain this question shortly by an example.

The following can be derived from Lorentz's transformation equations.

$x_{2}-x_{1}$ mean the distance of two fixed points in system $K$ in the direction of ${\mathfrak {c}}_{0}$. This distance is now synchronously measured by the moving observer in $K'$, by which he obtains the distance $l'$. By that, the two synchronous clocks which were placed by the observer in $K'$ at both ends of $l'$ in order to make a synchronous measurement of $x_{2}-x_{1}$, will indicate a time difference $t_{2}-t_{1}$, equal to:

$t_{2}-t_{1}=qn\left(x_{2}-x_{1}\right)$,

(26)

from that if follows

${\frac {x_{2}-x_{1}}{t_{2}-t_{1}}}={\frac {1}{qn}}={\frac {c^{2}}{q}}$

(27)

Now we imagine a rod of length $l'$ in $K'$, and assume that the observers in $K'$ would arrange to elevate the rod at the same time (by aid of their synchronous clocks)

1. l. c. p. 106.
2. Vol. 33, 607, 1910.