1904.] 



BROOKS — ORTHIC CURVES. 307 



maining portion of the curve must then be an ordinary hyperbola, 

 and the inclination of its asymptotes must be either -J or |-'^. The 

 first value refers to the case when the director line is the axis of 

 imaginaries ; and the last, to the case when it is the axis of reals. 



XII. The Intersections of the Circumscribed Circle of a Triad with 



the Cubic. 



Suppose we put a circle through the points of a triad, and ask. 

 Where are the remaining three points in which it cuts the cubic ? 

 For convenience, let three points of the unit circle be taken as a 

 triad. The cubic is then 



{X — A) (^ — A) Gr — 4) =- r^ (x — t-^) (^ ~ t-^) {X — f^). 



On eliminating .v from this and the equation of the circle we 

 obtain 



(.■ - .,) (X - t.) (. - f^ = rJ_ i'.-^)^'.-j)(>.-^ )^ 



or 



as the equation of the three points sought. The roots of this, 



are the coordinates of the vertices of an equilateral triangle. As 

 there is no restriction in taking the triad on the unit circle, we have 

 the following theorem : 



Jf a circle cut an orthic cubic in a triad, then the two curves have 

 three other intersections , which form an equilateral triangle. 



XIII. The Pencil of Orthic Cubics Which Have a Triad in Common. 

 We have seen that the relation 



{x -. a) (x — (S) (x — r) = z 



maps a line through the origin into an orthic cubic of which a^)' is 

 a triad. It must then map all the lines through the origin into a 

 single infinity of orthic curves^ which have the common triad aj3y, 



1 Felix Lucas, Joiir7ial de V Ecole PolytecJmique, t. XVIII, p. 2i. 



