414] 
ON POLYZOMAL CURVES. 
499 
61. We have from either of the equations in (x, y, z) 
(x — az) 2 4- (y — a'z) 2 = 0, 
that is, the distance from each other of any two points (x, y, 1), and (a, a', 1) in a 
line through I or J is =0. And in particular, if z = 0, then x 2 + y 2 = 0 ; that is, the 
distance of the point (a, a', 1) from / or / is in each case = 0. 
62. Consider for a moment any three points P, Q, A ; the perpendicular distance 
of P from QA is = 2 triangle PQA -r- distance QA ; if Q be any point on the line 
through A to either of the points I, J, and in particular if Q be either of the points 
I, J, then the triangle PQA is finite, but the distance QA is = 0 : that is, the 
perpendicular distance of P from the line through A to either of the points I, J, 
that is, from any line through either of these points, is = oo. But, as just stated, the 
triangle PQA is finite, or say the triangles PI A, PJA are each finite; viz., the 
coordinates (rectangular) of P, A being (x, y, z= 1), (a, a', 1) or (circular) (£, rj, z = 1), 
(a, a', 1), the expressions for the doubles of these triangles respectively are 
X, 
y> 
z 
J 
X, 
y> 
z 
- i, 
1, 
0 
i, 
1, 
0 
a, 
a', 
1 
a, 
a', 
1 
that is, they are (rectangular coordinates) x — az + i(y — a'z), x — az — i (y — a'z), or 
(circular coordinates) £ — az, rj — a'z. 
Representing the double areas by PI A, PJA, respectively, and the squared distance 
of the points A, P, by SI, we have— 
SI = (x — az) 2 + (y — a'z) 2 
= (f-az) ( v - a'z), = PI A . PJA. 
Article No. 63. Antipoints; Definition and Fundamental Properties. 
63. Two pairs of points (A, B) and (Aj, BQ which are such that the lines 
AB, A^B X bisect each other at right angles in a point 0 in such wise that 
0A = OB=iOA 1 = iOB l , are said to be antipoints, each of the other. In rectangular 
coordinates, taking the coordinates of (AB,) to be (a, 0, 1) and (—a, 0, 1), those of 
(A,, Pj) will be (0, ai, 1) and (0, —ai, 1) respectively, whence joining the points (A, P) 
with the points (I, J), the points A 1} B x are given as the intersections of the lines 
AI and BJ, and of the lines AJ and BI respectively. Or, what is the same thing, 
in any quadrilateral wherein I, J are opposite angles, the remaining pairs (A, P) and 
(A 1} Pj) are antipoints each of the other. 
64. In circular coordinates, if the coordinates of A are (a, a', 1), and those of P 
are (/3, /S', 1), then the equations of 
AI, AJ are % — az = 0, r) — dz — 0, 
BI, BJ „ £-/3z = 0, v- ffz = 0, 
63—2