When we use Darboux's representation of a point in S 4 by a spherical wave in S 3 , the group G 15 becomes the group of spherical wave transformations which transform a spherical wave into a spherical wave. This group of transformations has been discussed by S. Lie; it is the group of transformations which transform lines of curvature on a surface enveloped by spherical waves into lines of curvature on the surface enveloped by the corresponding spherical waves.