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On the Equations and Properties— (1) of the System oe Circles 

 touching three circles in a plane j (2) oe the system oe spheres 

 touching eour spheres in space ; (3) oe the system oe circles 

 touching three clrcles on a sphere ; (4) on the system oe 

 conics inscribed to a conic, and touching three inscribed 

 Conics in a Plane. 



In the following Paper I shall give — 1°. The method of investigating 

 the equations of the circles in pairs which touch three given circles in 

 a plane ; 2°. the equations of the spheres in pairs which touch four 

 given spheres ; 3°. the equations of the circles in pairs which touch 

 three others on the surface of a sphere ; 4°. the equations in pairs of 

 the conics having double contact with a given conic which touch three 

 other conics having also double contact with the same given conic. 



In the course of the investigation I shall, besides giving the 

 methods by which I discovered the equations, indicate other methods 

 which subsequently occurred to me, and shall show that some of the 

 results are but generalizations of equations with which geometers have 

 been long familiar. 



I. 



EQUATIONS OE THE CIRCLES WHICH TOUCH THREE OTHERS. 



Art. 1. — If A, B, C, D, be four points on a line disposed in any 

 manner, then always, none of the four being at infinity, 



BC'AD+CA-BD + AB-CD = 0, (1) 



regard being had to the signs as well as to the magnitude of the six 

 segments involved — Townsend's " Modern Geometry," vol. i., p. 102. 

 ~Now, let A, B, C, D, be the points of contact of four circles which 

 touch the line, and whose diameters are 8, 8', b" } h ffr , then, dividing 

 equation (1) by ^/l, h", h>", we get 



BC AD OA BB AB CD 



Kow, since, if two circles be inverted from any arbitrary point, the 

 ratio of the square of their common tangent to the rectangle contained 

 by their diameters is constant, that is, remains unaltered by the inver- 

 sion (See Townsend's " Modern Geometry," vol. ii., p. 375), each of 

 the ratios 



BC AD „ 

 • . , — — -, &c, 



x /dd f " 



is unaltered by inversion. Hence we have immediately the following 

 theorem, which is obviously an extension of Ptolemy's theorem con- 

 cerning four points on a circle : — 



