298 BROOKS — ORTHIC CURVES. [May 20, 



Now any cubic curve, the asymptotes of which form a regular 

 pencil, can be brought to the form : 



x^ -j- a^x -{- ai -{- a^x -f- x^ = o, 



in which we recognize it as orthic. It follows that : 



T/ie orthic cubic and the equilateral of order three are identical. 



The relation 



{x — a) (^ — /?) {x — y)=pt^ = z 



may be regarded as mapping a line through the origin in the z 

 plane, 



z — r^z = o 



into the orthic cubic. We are thus able to identify the latter with 

 the curves discussed by Holzmiiller^ and by Lucas.^ 



IV. Construction of Points of an Orthic Cubic. 



A figure of the orthic cubic may be obtained without great diffi- 

 culty by constructing points of the curve. In order to show how 

 this may be done, it is necessary to prove the following lemma : 



Elements of the pencil of equilateral {orthic') hyperbolas ^ of which 

 the stroke fiy is a diameter, intersect corresponding elements of the 

 pencil of lines through a on an orthic cubic of which a^y is a triad. 



For the line through a, 



(x — a) = pv', 



and the equilateral hyperbola on t3y as a diameter, 



(x-t3)(x-r) = pr'\ 

 intersect on the orthic cubic 



(x — a) (x — ,3) {x — y)^ pzi 



1 Holzmiiller, Conforvien Abbildungeti, p. 205. 



2 Lucas, Geometric dcs Polynomcs, fournal de V Ecole Poly technique ^ 

 t. XXVIII, p. 23. 



