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70 THE CONFORMAL TRANSFORMATIONS OF A SPACE OF FOUR DIMENSIONS AND THEIR APPLICATIONS TO GEOMETRICAL OPTICS

By.

[Received October 9th, 1908. — Read November 12th, 1908.]

1. The method of inversion which was first applied to problems in electrostatics by Lord Kelvin, and which forms the basis of his theory of electric images, has also been applied with success in other branches of mathematical physics, as, for instance, in hydrodynamics. In geometrical optics, however, the method has been seldom used, probably because the necessary developments are not to be found in books on geometrical optics. The object of this paper is to show that the method can be of real value in both geometrical and physical optics. It is found that the transformation which is really needed is an inversion in a space of four dimensions, the transition to three-dimensional space being made by replacing the fourth coordinate by ict, where t is the time and c the velocity of light.

The first part of the paper is devoted to the general conformal transformation of a space of four dimensions. Shortly after Lord Kelvin's discovery of the method of transforming electrostatical problems by means of inversion, Liouville obtained the most general transformation that can be used for three-dimensional problems in this way.