Liouville himself and more extensively Sophus Lie (1871) showed that the related conformal group can be differentiated (Liouville's theorem): For instance, λ λ = 1 includes the Euclidean group of ordinary motions; λ λ ≠ ≠ 1 scale or similarity transformations in which the coordinates of the previous transformations are multiplied by λ λ ; and λ λ = k 4 / ( x 2 + y 2 + z 2 ) 2 gives Thomson's transformation by reciprocal radii (inversions):