Harry Bateman

“ The transition from one solution of Laplace's equation to another is now easily effected.
The effects of combining the different transformations belonging to a group of conformal transformations is most easily studied by interpreting the transformation as a change of axes in a space in which the coordinates are the spherical coordinates . [11] It is important to notice that the angle between two manifolds in this space is equal to the angle between the corresponding manifolds in the space to which the conformal transformations are applied. ”

“ A change in the sign of corresponds to an inversion, a change in the sign of coupled with a change in the sign of corresponds to the other transformation we have mentioned. It is evident that each of these transformations is of period 2. In general, a reflexion in a linear manifold in the a space corresponds to an inversion with regard to the corresponding circle, sphere, or hypersphere, in the space of four dimensions. A displacement of period n in the a space may be obtained by taking successive reflexions in two plane five-folds which cut at an angle π/n. ”

“ Since the transformation enables us to derive the surfaces which are parallel to one surface from the surfaces which are parallel to the inverse surface, it is natural to expect that the above relation between the direction cosines will make the normals to the two surfaces correspond. ”