500 
ON POLYZOMAL CURVES. 
[414 
whence the equations of 
AJ, A X J are £ — olz = 0, y — /3'z = 0, 
BJ, B X J „ £ — 0z = 0, v — az = 0. 
65. Considering any point P the coordinates of which are rj, z (=1), let 
SI, 35, Sl 1} 33i be its squared distances from the points A, B, A 1} B 1 respectively; then 
by what precedes 
51 = (Ç-uz)(r ) -a , z), 
55 = (Ç-/3z)( v -/3'z), 
2li = (£ - «*) (v ~ P'z), 
35. = (£-£*) (17-«H 
and thence 
21.35 = 2l x . 35j ; 
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 li Bj. This theorem, which was, I believe, first given by me in the 
Educational Times (see reprint, vol. vi. 1866, p. 81), is an important one in the theory 
of foci. It is to be further noticed that we have 
21 + 35 — 2lj — 35 x = (a — /3) {a! — ¡3') z 2 , = Kz 2 = K, 
if K, = (a — a') (/3 — /3'), be the squared distance of the points A, B,= — squared distance 
of points A 1} B x , 
Article No. 66. Antipoints of a Circle. 
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 inter 
section of the two cone spheres having their centres at the two antipoints respectively. 
Article No. 67. Antipoints in relation to a Pair of Orthotomic Circles. 
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 Xt B x ), then these two circles cut at 
right angles. Conversely if a circle pass through the points A, B, then all the ortho 
tomic circles which have their centres on the line A B pass through the antipoints 
A 1} B x . In particular, if on AB as diameter we describe a circle and on A l B l 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 
(I, J)—cut each other at right angles; or say they are concentric orthotomic circles.