As shown above, also Laguerre's transformation by reciprocal directions or half-lines – later called Laguerre inversion – in the form given by Darboux (1887) leaves the expression x 2 + y 2 + z 2 − − R 2 invariant. Subsequently, the relation to the Lorentz transformation was noted by several authors. For instance, Bateman (1910) argued that this transformation (which he attributed to Ribaucour) is "identical" to the Lorentz transformation. In particular, he argued (1912) that the variant given by Darboux (1887) corresponds to the Lorentz transformation in z direction, if R = c t , R ′ = c t ′ , and the k terms are replaced by velocities. Bateman (1910) also sketched geometric representations of relativistic light spheres using such spherical systems. However, Kubota (1925) responded to Bateman by arguing that the Laguerre inversion is involutory whereas the Lorentz transformation is not. He concluded that in order to make them equivalent, the Laguerre inversion has to be combined with a reversal of direction of the cycles.