Page:BatemanConformal.djvu/2

The group of transformations of this kind is known as the group of conformal transformations of space,^[1] it preserves the angles between two surfaces and changes a sphere into either a sphere or a plane.^[2]

The property, however, upon which the applications to electrostatical problems depends is that the transformations enable us to pass from one solution of the equation

${\frac {\partial ^{2}V}{\partial x^{2}}}+{\frac {\partial ^{2}V}{\partial y^{2}}}+{\frac {\partial ^{2}V}{\partial z^{2}}}=0$.

to another.^[3]

Now the group of conformal transformations in a space of four dimensions possesses the analogous property in connection with the two differential equations

${\frac {\partial ^{2}V}{\partial x^{2}}}+{\frac {\partial ^{2}V}{\partial y^{2}}}+{\frac {\partial ^{2}V}{\partial z^{2}}}={\frac {1}{c^{2}}}{\frac {\partial ^{2}V}{\partial t^{2}}},$ $\left({\frac {\partial V}{\partial x}}\right)^{2}+\left({\frac {\partial V}{\partial y}}\right)^{2}+\left({\frac {\partial V}{\partial z}}\right)^{2}={\frac {1}{c^{2}}}\left({\frac {\partial V}{\partial t}}\right)^{2},$

which are fundamental in the wave theory of light.

This has been known for some time,^[4] but the analysis given in § 2 will be useful in indicating the procedure to be adopted to obtain the relation connecting the two solutions for any transformation of the group.

In § 3 a particular solution of the first of the above equations is

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1. A simple method of obtaining the group of conformal transformations is given in Bianchi's Vorlesungen über Differential Geometrie, Leipzig (1899), p. 487. Another investigation is given in Maxwell's Collected Papers, Vol. II., p. 297, where reference is made to a paper by J. N. Haton de Goupillière, Journal de l'Ecole Polytechnique, T. XXV., p. 188 (1867). See also a paper by Bromwich, Proc. London Math. Soc., Vol. XXXIII., p. 185, and three papers by Tait, Collected Papers, Vol. I., pp. 176, 352, Vol. II., p. 329.
2. The effect of combining the elementary transformations of the group is discussed by Darboux, Une Classe remarquable de courbes et de surfaces algébriques, Paris (1896), pp. 236-241. It is shown that any number of successive inversions can be replaced by a single inversion followed by a displacement. It follows from this that any conformal transformation of the group can be replaced by successive inversions with regard to suitably chosen spheres, Cf. Math. Tripos, Part I. (1903).
3. In this connection see a paper by Forsyth, Proceedings of the London Mathematical Society, Vol. XXIX. (1898), p. 165. The transformations which can be applied to the equation

   $\left({\frac {\partial V}{\partial x}}\right)^{2}+\left({\frac {\partial V}{\partial y}}\right)^{2}+\left({\frac {\partial V}{\partial z}}\right)^{2}=0$

   are derived by J. E. Campbell, Messenger of Mathematics, Vol. XXVIII. (1898), p. 97.

4. Liouville's theorem was extended by Lie to a space of n dimensions in 1871. Math. Ann., Vol. V., p. 145; Göttinger Nachrichten, May, 1871. I cannot, however, find any statement with regard to the first of the two equations.