1886.] 



On Systems of Circles and Spheres. 



243 



Denoting the power of two circles (1, 2) by 7r 1>2 , it is proved that 

 the power of any five circles (1, 2, 3, 4, 5) with respect to any other 

 circles (6, 7, 8, 9, 10) are connected by the relation — 



W U6 > "l,7 > ^l.S J ^ > "LIO 



""2,6 > ""2,7 > ^2,8 > ^2,9 » ""2,10 



""3,6 > ""3,7 ' "3,8 ' "3,9 ' ""3,10 



"4,6 ' "*4,7 5 ""4,8 ' "4.9 ' "4.10 



5.6 » "5.7 



5,8 



5,10 



= 



which may be conveniently written : — 



_n, 2, 3, 4, 5\ n 



"Ae. 7, 8, 9, 10/ — v ' 



This is the fundamental theorem of the paper ; it is shown that if 

 the power of a straight line and a circle be defined as the perpen- 

 dicular from the centre of the circle on the straight line, and the 

 power of two straight lines as the cosine of the angle between them : 

 then the theorem is true if any of the circles of either system be 

 replaced by points, straight lines, or the line at infinity. 



Several special systems of circles are then discussed, the most 

 remarkable perhaps being the case when the circles (1, 2, 3, 4) being 

 given, the circles (5, 6, 7, 8) are orthogonal to the former taken 

 three at a time; then (x, y), denoting any other circles, the 

 equation — 



^(x, 1, 2, 3, 4\ n 



"Ay, 5, 6, 7, Sj — u 



becomes ir x , y 



"?/,2^ 7r *,7 • ">,3 _{_ "*,8 • V 9A 



1,5 



r 3,7 



'4,8 



and as a particular case when the two circles (x, y) are replaced by 

 the line at an infinity, we have 



1.5 



'3,7 



The general theorem is then applied to prove some properties of 

 circles connected with three circles ; a formula is given for the radius 

 of a circle which passes through three of the points of intersection of 

 three given circles ; the eight circles which can be drawn to touch 

 three circles are shown to be each touched by four of eight other 

 circles, called Dr. Hart's circles, these arrange themselves in pairs ; 

 if p, p be the radii of a pair of Dr. Hart's circles, and R, R' the 

 radii of the corresponding pair of the eight circles passing through 

 the points of intersection of the given circles, it is shown that 



p p VB B,/ 



