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[Nov. 12,
Mr. H. Bateman


By H. Bateman.

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


§ 1. Introduction.

§ 2. The Conformal Transformations of a Space of Four Dimensions.

§ 3. The Relation between Riemann's general Hypergeometric Function
and the Conformal Transformations of a Space of Four Dimensions.

§ 4. Applications to Geometrical Optics.

§ 5. Applications of the preceding results to a Symmetrical Optical Instrument.

1. The method of inversion which was first applied to problems in electrostatics by Lord Kelvin,[1] 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,[2] Liouville[3] obtained the most general transformation that can be used for three-dimensional problems in this way.

  1. In a letter to Liouville dated October 8th, 1845. Liouville's Journal de Mathématiques 1845).
  2. The method of inversion had been used in geometry some time before. It apparently originated with Ptolemy. Quetelet used it in 1827 and Bellavatis gave a general statement of it in 1836. In 1843-4 it was propounded afresh by Ingram and Stubbs (Transactions of the Dublin Philosophical Society, Vol. I., pp. 58, 145, 159 ; Philosophical Magazine, Vol. XXIII., p. 338, Vol. XXV., p. 208).
  3. Journal de Mathématiques (1845) ; T. XV. (1850), p. 103.