78 PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



or, as this is more conveniently written 



/ 1 m 1 n ■ 1 n 



b=c (d-b)(d-c)-d" 2 + c^a (d-c)(d-a)-d" 2 + a~-b (d-a)(d-b)-d" 1 = ' 



viz., considering d, d" as the abscissa and ordinate of a point on a curve, and 

 representing them by x, y respectively, the equation of this curve is 



/ 1 ml n 1 . 



b — c (x—b)(x — c)—y 2 c — a (x—c)(x — a) — y 2 c — a (x — a)(x—b)—y 2 



which is a certain quartic curve ; and we have the original curve 



JTK + y/mB + J^C - 0, 



as the envelope of a variable circle having for its diameter the double ordinate of 

 this quartic curve. 



Write for shortness t— - , — — , -— , = X, M, N respectively, then the equa- 

 tion of the quartic curve may be written 



2 L [{x- a) 2 (x-b) (x-r) - f{x-a) (2x-b-c) + //] = 0, 



viz., this is 



2 L \x {x — a) (x — b) (x — c) 



- y 2 (2x 2 - (a + 6 + c)x + (ab + ac + be)) + y* 



— a (x — a) {x — b) (x— c) + y 2 {ax + be)] = , 



or what is the same thing, the equation is 



(Z + M+N) [x(x-a) (x-b) (x-c)-if(2J-(a + b + ry + ab + ac + bc) + y*~\ 

 — (La + Mb + Ne) (r — a) (x — b) (x — c) 

 + y 2 {La + Mb + Nc)x+ Lbc + Meci + Nab} = 0. 



In the particular case where L + M + X — , that is, where 



1 + m - + A = o, 



b — e c — a. a — b 



the quartic curve becomes a cubic, viz., putting for shortness 



Lbc + Mca + Nab 

 ~ La + Mb + Nc ' 



the equation of the cubic is 



r = 



(x—a) (x — b) (x — c) 



x-b 



viz , this is a cubic curve having three real asymptotes, and a diameter at right 

 angles to one of the asymptotes, and at the inclinations + 45°, — 45° to the other 

 two asymptotes respectively— say that it is a " rectangular" cubic. The relation 



