XV. On systems of circles and spheres

Abstract

This Memoir is divided into three Parts: Part I. treats of systems of circles in one plane; Part II. treats of systems of circles on the surface of a sphere; and Part III. of systems of spheres; the method of treatment being that indicated in two papers among Clifford’s ‘Mathematical Papers,ʼ viz., “On Power-Coordinates” (pp. 546—555) and “On the Powers of Spheres” (pp. 332-336). These two papers probably contain the notes of a paper which was read by Clifford to the London Mathematical Society, Feb. 27, 1868, “On Circles and Spheres,” which was not published (‘Lond. Math. Soc. Proc.,ʼ vol. 2, p. 61). The method of treatment indicated in these papers of Clifford’s was successfully applied by the author to prove some theorems given by him in a paper “On the Properties of a Triangle formed by Coplanar Circles” (1885) (‘Quarterly Journal of Mathematics,ʼ vol. 21), and then to the extension of those theorems to the case of spheres. But as Clifford’s papers contained some suggestions as to the application of the same method to the treatment of Bi-circular Quartics, he was induced to develop these ideas and extend the results to the case of the analogous curves on spheres—called by Professor Cayley Spheri-quadrics—and also of cyclides. It is impossible to say whether, if at all, Clifford was indebted to Darboux for any of the ideas contained in the two papers cited above; but it is noticeable that they coincide in a great measure with those expressed by Darboux in several papers published during the years 1869‒1872. In Part I. (§§ 1—124) of this Memoir a general relation is first shown to subsist between the powers of any two groups of five circles; the definition of the power of two circles, as the extension of Steiner’s “power of a point and a circle,” being due to Darboux, but the definition is here slightly modified so as to include the case when the radius of either (or each) circle is infinite. In Chapter II. an extension of the definition so as to apply to a certain system of conics is given; this is practically adapted from Chapter II. in Professor Casey’s Memoir “On Bicircular Quartics” (1867) (‘Irish Acad. Trans.,’ vol. 24). In Chapter III. the general theorem is applied to several interesting cases of circles; some of the results of this chapter are believed to be new. In Chapter IV. the problem of drawing a circle to cut three given circles at given angles is considered, and the circles connected with a triangle formed by three circles, which are analogous to the circumcircle, the inscribed and escribed, and the nine-points circle of an ordinary triangle are discussed. The results are the same, with one or two exceptions which may be new, as arrived at, but in a different manner, in the paper by the author in the ‘Quarterly Journal’ (vol. 21). In Chapter V. the power-coordinates of a point (or circle) are defined, and the equations of circles, &c., discussed; and it is shown that there are two simple coordinate systems of reference; one consisting of four orthogonal circles, mentioned by Clifford (Casey and Darboux consider five orthogonal spheres), the other consisting of two orthogonal circles and their two points of intersection, which seems to have been indicated for the first time by Mr. Homersham Cox in a paper “On Systems of Circles and Bicircular Quartics” (‘Quarterly Journal,’ vol. 19, 1883). In Chapter VI. the general equation of the second degree in power-coordinates is discussed, and in Chapter VII. Bi-circular Quartics are classified according to the number of principal circles which they possess. In Chapter VIII. the connexion between Bi-circular Quartics and their focal conics is briefly indicated, the circle of curvature is found, and an expression for the radius of curvature at any point of a bi-circular quartic is investigated. In these last three chapters the results are probably all old, but as the method employed is different from any previously used to discuss these curves in detail, it may not be without interest.