4 i 4 J 
ON POLYZOMAL CURVES. 
501 
Article Nos. 68 to 71. Forms of the Equation of a Circle. 
68. In rectangular coordinates the equation of a circle, coordinates of centre 
{a, a', 1) and radius = a", is 
21° = (x — azf + (y — a'zf — a'^z 1 = 0 ; 
and in circular coordinates, the coordinates of the centre being (a, a, 1), and radius 
= a" as before, the equation is 
21° = (f - az) ( V - oc'z) - a"*z* = 0. 
69. I observe in passing, that the origin being at the centre and the radius 
being =1, then writing also z= 1, the equation of the circle is %r) = 1, that is the 
circular coordinates of any point of the circle, expressed by means of a variable para 
meter 6, are (d, q, 1^ . 
70. Consider a current point P, the coordinates of which (rectangular) are 
x, y, z{— 1), and (circular) are |, rj, z (= 1), then the foregoing expression 
21° = (x - azf + (y- a'zf - a"*z- 
= (£ — az) (?; -- a'z) — a" 2 z 1 
denotes, it is clear, the square of the tangential distance of the point P from the 
circle 21° = 0. 
71. But there is another interpretation of this same function 21°, viz., writing 
therein z = 1, and then 
21° = (x — af + (y — a'f + (a”if, 
we see that 21° is the squared distance of P from either of the antipoints 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 antipoint of the circle. 
Article Nos. 72 to 77. On a System of Sixteen Points. 
72. Take (A, B, C, D) any four coneyclic points, and let the antipoints of 
(.B, C), (A, D) be (A, A), (A lt A), 
(C, A), (B, D) „ (A, A 2 ), (A, A), 
(A, B), (C, D) „ (A 3 , A), (A, A), 
then each of the three new sets (A 1} B 1} A, A), (A 2 , A, A, A), (A 3 , B 3 , C 3 , A) will 
be a set of four concyclic points. 
78. Let 0 be the centre of the circle through (A, B, C, D), say of the circle 0, 
and then, the lines BC, AD meeting in R, the lines CA, BD in S, and the lines 
AD, CD in T, let each of these points be made the centre of a circle orthotomic 
to 0, viz., let these new circles be called the circles R, S, T respectively.