PROFESSOR CAYLEY ON POLYZOMAL CURVES. 33 



Forms of the Equation of a Circle — Art. Nos. 68 to 71. 



68. In rectangular co-ordinates the equation of a circle, co-ordinates of centre 

 {a, a\ 1) and radius = a, is 



A = (x-azf + (y-a z) 2 -a" 2 z 2 = ; 



and in circular co-ordinates, the co-ordinates of the centre being (a, a, 1), and 

 radius = a" as before, the equation is 



A° = (g- az) {n - a'z) - a" 2 z 2 = 0. 



69. I observe in passing, that the origin being at the centre and the radius 

 being = 1, then writing also z=l, the equation of the circle is & = 1, that is the 

 circular co-ordinates of any point of the circle, expressed by means of a 



variable parameter 0, are ( 0, y., 1 V 



70. Consider a current point P, the co-ordinates of which (rectangular) are 

 ■x, y, z ( = 1), and (circular) are £, *i, z ( = 1), then the foregoing expression 



A = (x - az) 2 + (y- az) 2 - a" 2 z 2 

 = (£-**) (n-*'z) -a" 2 z 2 



denotes, it is clear, the square of the tangential distance of the point P from the 

 circle A° = 0. 



71. But there is another interpretation of this same function A°, viz., writing 

 therein z = 1, and then 



A° = ( x -a)* + (y-a) 2 + (a"i) 2 , 



we see that A° is the squared distance of P from either of the anti-points of the 

 circle (points lying, it will be recollected, out of the plane of the circle), and we 

 have thus the theorem that the square of the tangential distance of any point P 

 from the circle is equal to the square of its distance from either anti-point of 

 the circle. 



On a System of Sixteen Points — Art. Nos. 72 to 77. 



72. Take (A, B, C, D) any four concyclic points, and let the anti-points of 



(B,C) > (A,D)be(B v C 1 ),(A v D 1 ), 

 (C,A),(B,D) „ (C 2 ,A 2 ),(B 2 ,D 2 ), 

 (A,B),{C,D) „ (A Z ,B Z ),(C V D Z ), 



then each of the three new sets (A v B v C v Z> x ), {A 2 , B 2 , C 2 , Z> 2 ), (A z , B s , C 3 , Dj 

 will be a set of four concyclic points. 



73. Let O be the centre of the circle through (A, B, C, D), say of the circle O, 

 and then, the lines BC, AD meeting in R, the lines CA, BD in S, and the lines 

 AB, CD in T, let each of these points be made the centre of a circle orthotomic 

 to O, viz., let these new circles be called the circles R, S, T respectively. 



As regards the circle R, since its centre lies in BC, the circle passes through 

 (/?!, Cj); and since the centre lies in AD, the circle passes through (A^ DJ, that 

 is, the four points [A^ B x , C 15 D x ) lie in the circle R. Similarly {A v B 2 , C 2 , D 2 ) 

 lie in the circle S, and (A ai i? 3 , C 3 , D s ) in the circle T. 



VOL. XXV. PART I. I 



