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§ 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.
An excerpt from the beginning of the Introduction:
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.
 In a letter to Liouville dated October 8th, 1845. Liouville's Journal de Mathématiques 1845).
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.
 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).
 Journal de Mathématiques (1845) ; T. XV. (1850), p. 103.
The group of transformations of this kind is known as the group of conformal transformations of space, it preserves the angles between two surfaces and changes a sphere into either a sphere or a plane.
 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.
 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).