Annus MirabilisEinstein in 1905
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special relativity

The equation explained

ζ=aVV2−v2z.\zeta = a \frac{V}{\sqrt{V^2 - v^2}} z .
In wordsThe third coordinate of the moving system is the coefficient times the velocity of light divided by the square root of its square less the square of the speed of the moving system, times the third coordinate of the stationary one.OverviewThe third coordinate transforms exactly as the second does. Across the motion both carry the same factor.Full explanationThe paper says it finds this in an analogous way by considering rays of light moved along the two other axes, and prints the zz line without repeating the working. It is the working of the yy line with zz in place of yy. A ray is sent from the origin of the moving system along its third axis. In the stationary system that ray travels with the velocity VV, but its starting point is carried along the xx axis with the velocity vv, so its progress along zz is only V2−v2\sqrt{V^2 - v^2}. Measured in the moving system the same ray travels with VV, so after the time τ\tau it stands at ζ=Vτ\zeta = V \tau. Putting the two together gives the printed line. Nothing in it depends on the third axis rather than the second, which is the paper's reason for not writing it twice.Every stepzV2−v2=t\frac{z}{\sqrt{V^2 - v^2}} = tThe ray leaves the origin of the moving system along the third axis and stays on it, so the auxiliary coordinate x−vtx - v t is zero throughout. Its progress along zz in the stationary system is V2−v2\sqrt{V^2 - v^2}, because the velocity VV is shared between the motion along zz and the motion along xx that the ray inherits from the system.τ=a(t−vV2−v2(x−vt))=at\tau = a \left( t - \frac{v}{V^2 - v^2} (x - v t) \right) = a tThe expression for τ\tau that stands inside the η\eta display above (p. 900), with the auxiliary coordinate written out. It is zero on this ray, so the second term drops.ζ=Vτ=aVt=aVV2−v2z\zeta = V \tau = a V t = a \frac{V}{\sqrt{V^2 - v^2}} zIn the moving system the ray travels with VV, so its third coordinate after the time τ\tau is VτV \tau. Substituting first τ=at\tau = a t and then t=z/V2−v2t = z / \sqrt{V^2 - v^2} gives the printed line.Historian's marginThe two transverse lines are obtained from rays sent across the motion, not from any assumption that lengths across the motion are unchanged. That is worth keeping in view, because the factor aVV2−v2a \frac{V}{\sqrt{V^2 - v^2}} standing here is not yet 1: it becomes 1 only on p. 902, after the argument from the identity of the transformation from the stationary system to the third system and from the symmetry of a rod moved across its own axis.Also

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