32 PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



65. Considering any point P the co-ordinates of which are £, v,z {— 1), let 

 A, B,A 1 ,B l be its squared distances from the points A,B,A l ,B 1 respectively; 

 then by what precedes — 



A = (g - az) (» - u'z), 

 B = ft ~ fr) (u - Pz), 

 \ = it - «-) 0» - fr\ 

 B s = (g - ft (* - «'*), 

 and thence 



A . B = A t . B 1 ; 



that is, the product of the squared distances of a point P from any two points 

 A,B,is equal to the product of the squared distances of the same point P from 

 the two antipoints A V B V This theorem, which was, I believe, first given by me 

 in the Educational Times (see reprint, vol. vi. I860, p. 81), is an important one 

 in the theory of foci. It is to be further noticed that we have 



A + B - Aj - Bj = (« - |8; (a - j8') z 2 



if K, = (a — a') ((3 — /3'), be the squared distance of the points .J , B, = — squared 

 distance of points A V B V 



Antipoints of a Circh — Art. No. 66. 



66. A similar notion to that of two pairs of antipoints is as follows, viz., if 

 from the centre of a circle perpendicular to its plane and in opposite senses, we 

 measure off two distances each = i into the radius, the extremities of these 

 distances are antipoints of the circle. It is clear that the antipoints of the circle 

 and the extremities of any diameter thereof are (in the plane of these four points) 

 pairs of antipoints. It is to be added that each antipoint is the centre of a sphere 

 radius zero, or say of a cone sphere, passing through the circle : the circle is thus 

 the intersection of the two cone spheres having their centres at the two antipoints 

 respectively. 



Antipoints in relation to a Pair of Ortlwtornic Circles — Art. No. 67. 



67. It is a well-known property that if any circle pass through the points 

 {A, B), and any other circle through the antipoints {A x , B x ), then these two circles 

 cut at right angles. Conversely if a circle pass through the points A,B, then all 

 the orthotomic circles which have their centres on the line AB pass through the 

 antipoints A 1 ,B V In particular, if on AB as diameter we describe a circle and 

 on A 1 B 1 as diameter a circle, then these two circles — being, it is clear, concentric 

 circles with their radii in the ratio 1 : i, and as concentric circles touching each 

 other at the points (7, J) — cut each other at right angles ; or say they are con- 

 centric orthotomic circles. 



