34 PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



74. The points R, S, T are conjugate points in relation to the circle ; 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 » -^8' ""2> ""l» 



B 3 , B , B v B 3 , 



C 2 , C 1 , C , C 3 , 



A. D v »v If . 



in such wise that any four of them in the same vertical line pass through 7, and 

 any four in the same horizontal line pass through J ; and this being so, starting 

 for instance with (A 3 , B 3 , C 3 , D 3 ) we have anti-points 



of (B v C 3 ), (A 3 ,D 3 ) are (B 2 , C 2 ), (A v D 2 ), 

 „ (C 3 ,A 3 ),(B 3 ,D 3 ) „ (C V A X ),(B V D{), 

 „ (A 3 ,B 3 ),(C 3 ,D 2 ) „ (A, B,),(C ,D), 



and similarly if we start from (A v B v C v D x ) or (A 2 , B,, C. 2 , D.,). 



75. I return for a moment to the construction of (A V B V C v Bj); these are 

 points on the circle R, and (B v C x ) are the anti-points of (B, C); that is, they are 

 the intersections of the circle R by the line at right angles to BC from its middle 

 point, or, what is the same thing, by the perpendicular on BC from 0. Similarly 

 (A v D x ) are the anti-points 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 2 , B 2 , C.,, D 2 ) 

 and (A 3 , B z , C 3 ,i> 3 ) respectively. 



76. Hence, starting with the points A, B, C, D on the circle O, and constructing 

 as above the circles P, Q, R, and constructing also the perpendiculars from O on 

 the six chords AB, AC, &c, 



the perpendiculars on BC, AD meet circle B in (B v C,), (A v DJ , 



CA,BD „ „ S „ (C 2> A 2 ),{B V D 2 ), 

 AB,CD „ „ T „ {A 3 ,B 3 ),{C 3 ,D 3 ), 



so that the whole system is given by means of the circles P, Q, R, and the six 

 perpendiculars. 



77. If to fix the ideas (A, B, C, D) are real points taken in order on the real 

 circle 0, then the points R, S, T are each of them real ; but R and T lie outside, 

 S inside the circle 0. The circles R and T are consequently real, but the circle 

 S imaginary, viz., its radius is = i into a real quantity; the imaginary points 

 (A v B v C x , BJ are thus given as the intersections of a real circle by a pair of 

 real lines, and the like as to the imaginary points (A 3 , B 3 , C 3 , D 3 ) ; but the 



