PROFESSOR CAYLEY ON POLYZOMAL CURVES. 63 



Hence 6 being determined by the cubic equation as above, we may take = 0, and 

 consequently the equations of the corresponding tangents will be £=6bz, n = Qaz, 

 viz., the foci A,B,C will be given as the intersections of the pairs of lines 

 (£=0^)2, 1 = 6^2), &c. The foci lie therefore in the line a£— 6>? = ; or the curve 

 is symmetrical, the focal axis, or axis of symmetry, passing through the centre. 



On the Property that the Points of Contact of the Tangents from a Pair of Coney die Foci lie in a 



Circle— Art. Nos. 150 to 158. 



150. We have seen that the foci form four concyclic sets {A, B, C, D), (A v B x , 

 Cy, D x ), {A. 2 , B 2 , C 2 , Z> 2 ), (-4 j, i? 3 , C 3 , Z> 3 ), that is, A, B, C, D are in a circle. We 

 may, if we please^say that any one focus is concyclic — viz., it lies in a circle with 

 three other foci; but any two foci taken at random are not concyclic; it is only a pair 

 such as (A, B) taken out of a set of four concyclic foci which are concyclic, viz., 

 there exist two other foci lying with them in a circle. The number of such pairs 

 is, it is clear = 24. Let A, B be any two concyclic foci, I say that the points of 

 contact of the tangents AT, AJ, BI, BJ, lie in a circle. 



151. Consider the case of the bi-circular quartic, and take as before (£ = 0, 

 2 — 0), and (jf = 0, z = 0) for the co-ordinates of the points /, J respectively. Let 

 the two tangents from the focus A be £ — az — 0, n — az = 0, say for shortness 

 p = 0, p' = 0, then the equation of the curve is expressible in the form pp' U = 

 V 2 *, where U — 0, V = are each of them circles, viz., If and Fare each of 

 them quadric functions containing the terms z 2 , z% z£, and £?. Taking an inde- 

 terminate coefficient \ the equation may be written 



pp'(U + 2x V + \ 2 pp') = ( V + -hpp'f, 



and then X may be so determined that U+ 2X V+ \ 2 pp' = 0, shall be a 0-circle, or 

 pair of lines through / and J. It is easy to see that we have thus for X a cubic 

 equation, that is, there are three values of X, for each of which the function 

 U+ 2X V + \ 2 pp' assumes the form (£ — /3z) — /3V), —qq' suppose : taking any 

 one of these, and changing the value of V so as that we may have V in place of 

 V+ Xpp', the equation is pp'qq + V"\ where V= is as before a circle, the equation 

 shows that the points of contact of the tangents p — 0, p* = 0, q = 0, q = lie 

 in this circle V — 0. The circumstance that X is determined by a cubic equation 

 would suggest that the focus q = 0, q' — is one of the three foci B, C, D con- 

 cyclic with A ; but this is the very thing which we wish to prove, and the inves- 

 tigation, though somewhat long, is an interesting one. 



152. Starting from the form pp'qq' = V 2 , then introducing as before an 

 arbitrary coefficient X, the equation may be written 



pp' (qq' + 2XV + y?pp') = (V + ~Kpp') 2 , 



* This investigation is similar to that in Salmon's Higher Plane Curves, p. 196, in regard to 

 the double tangents of a quartic curve. 



