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The actual lengths l', l must thus be different if the light-times are the same. Nevertheless no difference can be detected between the lengths by any test that we can apply, or the principle of relativity will be contravened. Hence we are led to admit a universal contraction of all material systems in the direction of their motion through the ether, according to the law

$l'=l\ \cos \ \beta$, where $\sin \ \beta =v/U$.

The dimensions transverse to the motion are considered unaltered.

If it were objected that the light-times considered above could not be determined with sufficient accuracy to support so astonishing an inference, the objection would apply only to the simple form of argument adopted. In practice the light-times can be compared with extraordinary precision by making the beams interfere on returning to A. In fact we are dealing, as it were, with a schematic representation of the Michelson-Morley experiment, the null result of which is the fundamental fact on the optical side which has to be explained.

3. The uniform contraction in one dimension of the moving material system, the possibility of which we have thus been led to entertain, suffices of itself for the discussion of some interesting optical problems. As an example, the position of the focus of a moving parabolic mirror may be chosen. Let BA₁B' be the mirror (fig. 2) and F₁ its focus when at rest. Let BA₂B' be the figure of the mirror when in motion towards a star situated on its axis and F₂ the simultaneous position of the point of the (contracted) apparatus which corresponds to F₁. The incident wave-front is BCB'. If CF₁ is taken as the axis of x and CB as the axis of y, the equation of BA₁B' will be

$z^{2}+y^{2}=4f(x+b)$