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Before we enter into these matters, I will make an important remark. If we have somehow individualized space and time, then to a substantial point at rest corresponds as world-line a straight line parallel to the $t$-axis; to a uniformly moving substantial point, a straight line inclined to the $t$-axis; and to a non-uniformly moving substantial point a worldline curved in some manner. If we take up the world-line passing through an arbitrary world-point $x,y,z,t$, and find it to be there parallel to any radius vector $OA'$ of the hyperboloidal sheet mentioned above; then we may introduce $OA'$ as the new axis of time, and, with the new notions of space and time given thereby, the substance in the world-point in question will appear to be at rest. We will now introduce this fundamental axiom: The substance existent in any arbitrary world-point may be always regarded as at rest, if the space and time are suitably assigned.

This axiom means that in every world-point the expression

$c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}$

turns out to be always positive, or, what comes to the same thing, every velocity $v$ turns out to be always smaller than $c$. Hence $c$ is the upper limit of the velocities of all substances, and herein lies the deep significance of the quantity $c$. In this other form the axiom gives at first the impression of being somewhat unsatisfactory. It should, however, he considered that now we have a modified Mechanics in which the square root appears with the above differential combination of the second degree, so that cases with velocity exceeding that of light will play a part somewhat like that which figures with imaginary co-ordinates play in Geometry.

The impulse and the true motive for the acceptance of the group $G_{c}$ was this, that the differential equation for the propagation of waves of light in empty space possesses that group $G_{c}$.^[1] On the other hand the notion of rigid bodies has a sense only in a Mechanics with the group $G_{\infty }$. Now, let there be an Optics with $G_{c}$ and, further, let there be rigid bodies. Then it is easy to see that with the help of the hyperboloidal sheets corresponding to $G_{c}$ and $G_{\infty }$ a $t$-direction would be marked out, and this would have the following consequence: we should be able to perceive a change of phenomena with the aid of suitable rigid optical instruments in the Laboratory, when these instruments are variously set against the direction of motion of the Earth. However, all exertions having this aim in view, in particular, a famous interference experiment of Michelson, have had a negative result. In order to obtain an explanation of this

1. A true application of this fact has been already given by W. Voigt in the Göttinger Nachrichten, 1887, p. 41.