PROFESSOR CAYLEY ON POLYZOMAL CURVES. o7 



(/, J). It would at first sight appear that the only conditions for the circles 

 R, S, T were the conditions of passing through the anti-points of (B, C), of (C,A), 

 and of (A,B) respectively, and that these circles thus became indeterminate; 

 but in fact the definition of the circles is then as follows, viz., R has its centre 

 at A, and passes through the anti-points of (B, C) : (whence squared radius 

 — AB . AC). And similarly, S has its centre at B, and passes through anti-points 

 of (C,A), (squared radius = BA .BC)\ and T has its centre at C, and passes 

 through anti-points of (A , B), (squared radius = CA . CB) ; these three circles 

 cut each other at right angles. As before, A,B,C being in order on the line, the 

 circles R, T are real, but the circle S has its radius a pure imaginary quantity. 



84. That the circles are as just mentioned appears as follows : taking the 

 line as axis of #, and a, b, c, d for the x co-ordinates of the four points respectively, 

 then the co-ordinates of A v D 1 are 



i(a + d), ± -l-i(a — cl); 



whence, m being arbitrary, the general equation of a circle through A v D x is 



x * + V 1 — 2mas3 + \m(a + d) — ad]z 2 = , 



writing herein m = a - - this becomes 



a; 2 + f - 2 (ft - j X* + (a? - P - ~\ 2 = , 



viz., for d = go it is 



(x _ azf +f~ k 2 z 2 = , 



which is a circle having A for its centre, and its radius an arbitrary quantity k. 

 If the circle passes through the anti-points of B, C, the co-ordinates of these are 



i(b + c), ± \i(b - c) , 

 and we find 



W = [i-(b + c) - ay - l(b - cf = (a- b)(a - c) . 



85. Reverting to the general case of four points A, B, C,D on a line, the 

 theorem as to the confocal conies holds good under the form that, drawing 

 any conic whatever through (A v B v C v DJ the points (A 2 , B 2 , C 2 , Z> 2 ), and 

 (A z , B s , C z , D z ) lie in confocal conies, these conies have their centre on the line, and 

 axes in the direction of and perpendicular to tlie line. When D is at infinity, 

 the confocal conies become any three concentric circles through (i? 1} C ), (C n , A ), 

 and (A z , B 3 ) respectively. 



The Involution of Four Circles. — Art. Nos. 86 to 91. 



86. Consider any four points A,B,C, D, the centres of circles denoted by 

 these same letters, and let A°, B°, C°, D° signify as usual, viz., if (in orthogonal 



VOL. XXV. PART I. K 



