II. On systems of circles and spheres

Abstract

This memoir is an attempt to develop the ideas contained in two papers to be found in the volume of Clifford’s Mathematical Papers (Macmillan, 1882), viz., “On Power Coordinates” (pp. 546—555), and “On the Powers of Spheres” (pp. 332—336); the date of the former is stated to be 1866, and of the latter 1868, but the editor explains (see p. xxii, and note, p. 332) that though these papers probably contain the substance of a paper read to the London Mathematical Society, February 27,1868, “On Circles and Spheres” (“Proc. L. M. S.,” vol. ii, p. 61), they were probably not written out before 1876. It is possible, therefore, that Clifford may be indebted to Darboux for the conception of the “ power of two circles,” or spheres, as an extension of Steiner’s use of the “ power of a point with respect to a circle. Darboux was the first to give the definition of the power of two circles, in a paper “Sur les Relations entre les Groupes de Points, de Cercles, et de Sphères” (“Annales de l’École Normale Supérieure,” vol. i, p. 323, 1872), in which some theorems analogous to the fundamental theorem of this memoir are proved.