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were rigid bodies, it is easy to see that one t-direction is preferred by the two hyperboloidal shells belonging to the groups $G_{\infty }$, and $G_{c}$, which would have got the further consequence, that by means of suitable rigid instruments in the laboratory, we can perceive a change in natural phenomena, in case of different orientations with regard to the direction of progressive motion of the earth. But all efforts directed towards this goal, and even the celebrated interference-experiment of Michelson have given negative results. In order to supply an explanation for this result, H. A. Lorentz formed a hypothesis whose success lies exactly in the invariance of optics for the group $G_{c}$. According to Lorentz every body in motion, shall suffer a contraction in the direction of its motion, namely at velocity v in the ratio

$1:{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}$

This hypothesis sounds rather fantastical. For the contraction is not to be thought of as a consequence of resistances in the ether, but purely as a gift from above, as a condition accompanying the state of motion.

I shall show in our figure, that Lorentz's hypothesis is fully equivalent to the new conceptions about time and space, by which it becomes more intelligible. Let us now, for the sake of simplicity, neglect y and z and imagine a spatially two dimensional world, in which upright strips parallel to the t-axis represent a state of rest and another parallel strip inclined to the t-axis represent a state of uniform motion for a body (see fig. 1). If $OA'$ is parallel to the second strip, we can take $t'$ as time and $x'$ as the space coordinate, then the second body will appear to be at rest, and the first body in uniform motion. We shall now assume that the first body supposed to be at rest, has the length l, i.e., the cross section PP of the first strip upon the x-axis $=l\cdot OC$, where OC is the unit measuring rod upon the x-axis — and that on the other hand, the second body also, when supposed to be at rest, has the same length l, this means that the cross section is $Q'Q'=l\cdot OC'$ when measured parallel to the $x'$-axis. In these two bodies, we have now images of two equal Lorentz-electrons, one of which is at rest and the other moves uniformly. Now if we stick to our original coordinates, then the extension of the second electron is given by the cross section QQ of the corresponding strip is to be given parallel to the x-axis. Now it is clear, since $Q'Q'=l\cdot OC'$, that