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XY. On Systems of Circles and Spheres. 
By II. Lachlan, B.A., Fellow of Trinity College, Cambridge. 
Communicated by A. Cayley, LL.D., F.R.S., Sadlerian Professor of Mathematics 
in the University of Cambridge. 
Received February 23,—Read March 11, 1886. 
Introduction. 
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/ 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.,’ 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,’ 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 ” 
MDCCCLXXXVI. 3 Q 
