II. Second memoir on plane stigmatics

Let there be two groups of points upon a plane, termed, for distinction, indices and stigmata respectively, bearing such relations to each other that any one index determines the position of n stigmata, and any one stigma determines the position of m indices. The theory of these relations between indices and stigmata constitutes plane stigmatics . Each related pair of index X and stigma Y constitutes a stigmatic point , henceforth written “the s. point ( xy )." The straight lines joining any index with each of its corresponding stigmata are termed ordinates . If, when the index moves upon a straight line, the ordinate remains parallel to some other straight line, the relation between index and stigma is that expressed by the relation between abscissa and ordinate in the coordinate geometry of Descartes. When only one index corresponds to one stigma and conversely, and both indices and stigmata lie always on one and the same straight line, or the indices upon one and the stigmata upon another, the relations between indices and stigmata are those between homologous points in the homographic geometry of Chasles. The general expression of the stigmatic relation is obtained by a generalization of Chasles’s fundamental lemma in his theory of characteristics ( Comptes Rendus , June 27, 1864, vol. lviii. p. 1175), clinants being substituted for scalars. It results that in certain forms of the law of coordination , which “ coordinates ” the stigmata with the indices, there may be solitary indices which have no corresponding stigmata, and solitary stigmata which have no corresponding indices, and also double points in which the index coincides with its stigma (76). The particular case in which one index corresponds to one stigma and conversely, and no solitary index or stigma occurs, is termed a stigmatic line (henceforth written “s. line”), because the Cartesian case is that of a Cartesian straight line in ordinary coordinate geometry, but in the general s. line the figures described by index and stigma may be any directly similar plane figures (77). The investigation of this particular case occupies almost the whole of the Introductory Memoir . When one index corresponds to one stigma and conversely, but there is one solitary index and one solitary stigma, we have s. homography , provided the solitary index is distinct from the solitary stigma (79), and s. involution when the solitary index coincides with the solitary stigma (78), so called because they generalize the relations treated of under these names by Chasles.