54 PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



(4', 3', 2', V). And in each case the intersections of the four corresponding tan- 

 gents lie on a conic passing through the two given points on the curve * 



125. Hence taking the points on the curve to be the circular points at infinity, 

 we have the sixteen foci lying in fours upon four different circles — that is, we 

 have four tetrads of concyclic foci. Let any one of these tetrads be A, B, C, D, 

 then if 



Anti-points of (B, C) (A,D) are (B v 0,), (A V I) 1 ) , 

 (C,A){B,D) „ (C 2 ,A 2 ),(B V D 2 ) , 

 (A,B)(C,D) „ (A 3 ,B S ),(C„D 3 ) , 



the four tetrads of concyclic foci are 



A, B, C, D ■ 

 A v B v C v Dj ; 



2' 2' 2' ' ' 



A 3 , B z , C z , D 3 . 



It is to be observed that if A, B, C, D are any four points on a circle, then if, as 

 above, we pair these in any manner, and take the anti-points of each pair, the 

 four anti-points lie on a circle, and thus the original system A, B, C, D, of four 

 points on a circle, leads to the remaining three systems of four points on a circle. 

 The theory is in fact that already discussed ante, No. 72 et seq. 



126. The preceding theory applies without alteration to the bicircular quartic. 

 viz., the quartic curve which has a node at each of the circular points at infinity. 

 The class is here = 8, but among the tangents from a node each of the two 

 tangents at the node is to be reckoned twice, and the number of the remaining 

 tangents is = 4: the number of foci is = 10. And, by the general theorem that 

 in a binodal quartic the pencils of tangents from the two nodes respectively are 

 homologous, the sixteen foci are related to each other precisely in the manner of 

 the foci of the circular cubic. The latter is in fact a particular case of the 

 former, viz., the bicircular quartic may break up into the line infinity, and a 

 circular cubic. 



* It may be remarked that if the equation of the first pencil of lines be 



(x — ay){x — by) (x — cy)(x — dy) = , 

 and that of the second pencil 



',(z — aw)(z — 6w)(e — cio)^z — dw) = , 



then the equations of four conies are 



xw — yz = , 

 (a + d — b — c) xz + (be — ad)(xw + yz) + (ad(b -j- c) — bc(a + d)\w — , 

 (b + d — c — a) xz + {ca — bd)(xw + yz) + (bd(c + a) — ca(b + d)\yw = , 

 (c + d — a — b) xz -\- (ab — cd){xw -)- yz) -\- (cd(a -f b) — ab(c -\- d)\yw — . 



