406 



a > fit 1, &j are all touched by a fourth circle. 



(A) 

 P) 

 (C) 



m 



(BO 

 (C) 

 (D') 



tt; A 7 ', a, 

 «> P> % ^ 



a, /3, 7, B, 



a ', i, 8, 

 * , P, % 

 a', ft, 7', 



a, p, 7', a', 



This proves the theorem for the system of eight circles ; and from 

 the foregoing scheme we see that the relation between the two systems 

 of eight circles, a, /3, 7, c, a, ft ', 7', b\ and A, B, C, D, A', B> C\ D' y 

 is reciprocal, viz., each circle of each system being touched by four 

 circles of the other system — a property which was also noticed by Dr. 

 Hart.— q. e. d. 



EQUATIONS OE THE SIXTEEN" SPHEEES IN PAIES WHICH TOUCH FOTTEt 



15. If A, B, C, D, be four points in a- plane, then denoting their 

 three pairs of connectors by the following notation, 



we have (Salmon's " Geometry of Three Dimensions," Art. 50,) 



l(p - q) (p - r) + m (q - r) q - 1) + n (r -p) (r - q) 

 + Ip(l-m - n) + mq (m - n-l) - nr (n-l-m) + hnn = 0. (24) 



This formula is the expansion of the following determinant : — 



II. 



OTHEES. 



BC, AD, by U, pi; 

 CA, BD, „ *#, qi; 

 AB, CD, „ n\ H; 



0, n, m, p, 1, 

 n, 0, I, q, 1, 

 m, I, 0, r, 1, 

 P, q, r, 0, 1, 



1, 1, 1, 1, 0, 



= 0. 



(25) 



Now, if E be a fifth point in the plane or in space, and if 

 A', B', C', I)', be the points inverse to A, B y C, D, with respect to a 



