502 
ON POLYZOMAL CURVES. 
[414 
As regards the circle R, since its centre lies in BC, the circle passes through 
(A, A); and since the centre lies in AD, the circle passes through (A> D x ), that is, 
the four points (A> A, A. D x ) lie in the circle R. Similarly (A, B,, C 2 , A) lie in 
the circle S, and (A> B 3 , C 3 , D 3 ) in the circle T. 
74. The points R, S, T are conjugate points in relation to the circle 0 ; that is, 
ST, TR, RS are the polars of R, S, T respectively in regard to this circle; and they 
are, consequently, at right angles to the lines OR, OS, OT respectively; viz., the four 
centres 0, R, S, T are such that the line joining any two of them cuts at right 
angles the line joining the other two of them, and we see that the relation between 
the four sets is in fact a symmetrical one; this is most easily seen by consideration 
of the circular points at infinity I, J, the four sets of points may be arranged thus: 
A, A 3 , A,, A, 
A, B , A, B 2 , 
A, a, c, o s , 
A, A, A, D , 
in such wise that any four of them in the same vertical line pass through I, and 
any four in the same horizontal line pass through J; and this being so, starting for 
instance with (A, B 3 , C 3 , A) we have antipoints 
of (A, AX (A, A) are (A, AX (A, A), 
„ (C 3 , A 3 ), (B 3 , A) „ (A, A), (A, AX 
„ (A, AX (A, A) „ (A, B), (C, D), 
and similarly if we start from (A, B x , C lf A) or (A> B 2 , C 2 , AX 
75. I return for a moment to the construction of (A, A, C lt A) 5 these are 
points on the circle R, and (A, A) are the antipoints of (A 0); that is, they are 
the intersections of the circle R by the line at right angles to BO from its middle 
point, or, what is the same thing, by the perpendicular on BO from 0. Similarly 
(A, A) are the antipoints of (A, D); that is, they are the intersections of the 
circle R by the perpendicular on AD from 0. And the like as to (A, B,, C 2 , A) 
and (A 3 , A, 0 3 , A) respectively. 
7G. Hence, starting with the points A, B, 0, D on the circle 0, and constructing 
as above the circles P, Q, R, and constructing also the perpendiculars from 0 on the 
six chords AB, AC, &c., 
the perpendiculars on BC, AD meet circle R in (A, AX (A, AX 
CA, BD „ „ S „ (A, AX (A, A), 
„ AB, CD „ „ T „ (A, AX (A, AX 
so that the whole system is given by means of the circles P, Q, R, and the six 
perpendiculars.