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,^[1] who has shown that it is composed of reflexions, translations, rotations, magnifications, and inversions.^[2] 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

$x'={\frac {kx}{z+is}},\ y'={\frac {ky}{z+is}},\ z'=k{\frac {x^{2}+y^{2}+z^{2}+s^{2}-a^{2}}{2a(z+is)}},$ $s'=k{\frac {x^{2}+y^{2}+z^{2}+s^{2}+a^{2}}{2ia(z+is)}},$

which is quite different from an inversion or simple displacement. This corresponds to the real spherical wave transformation^[3]

$x'={\frac {kx}{z-t}},\ y'={\frac {ky}{z-t}},\ z'=k{\frac {x^{2}+y^{2}+z^{2}-a^{2}-t^{2}}{2a(z-t)}},$ $-t'=k{\frac {x^{2}+y^{2}+z^{2}+a^{2}-t^{2}}{2a(z-t)}}.$

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

$\tanh w=v,$

1. Göttinger Nachrichten (1871), Transformationgruppen, Bd. 3, p. 351.
2. Every transformation belonging to the group is a birational transformation.
3. See a paper by the author "The Conformal Transformations of a Space of Four Dimensions and their Applications to Geometrical Optics," Proc. London Math. Soc., Ser. 2, Vol. 7, p. 70 (1909).