Page:BatemanElectrodynamical.djvu/18



4. Spherical Wave Transformations and the Group of Conformal Transformations of a Space of Four Dimensions.
The group of spherical wave transformations may be reduced to a known group by putting t = is. The quadratic form which remains invariant is then of the type

$\lambda^{2}\left(dx^{2}+dy^{2}+dz^{2}+ds^{2}\right),$

and so the transformation is a conformal one.

The group of conformal transformations in a space of four dimensions has been studied by Sophus Lie, who has shown that it is composed of reflexions, translations, rotations, magnifications, and inversions. The transformations which are of importance in the present case are the imaginary ones, and it should be noticed that by a combination of two imaginary inversions we can obtain a transformation of the type

which is quite different from an inversion or simple displacement. This corresponds to the real spherical wave transformation

An imaginary rotation in the four-dimensional space may be specified, in a particular case, by the equations

$\begin{array}{lrcc} x'= & x\ \cos iw+s\ \sin iw, & & y'=y,\\ \\s'= & -x\ \sin iw+s\ \cos iw, & & z'=z.\end{array}$

Putting

$\tanh w=v,$