1904.] BROOKS— ORTHIC CURVES. 297 



x^ -{- a^x + ^1 -j- a^x -f jc' = o. 



The equation of any orthic cubic can be brought to this form. 

 The three points, a, /5 and y, are on the curve, and form what it is 

 convenient to call a triad of the curve. 



III. The Orthic Curve is an Equilateral Curve. 

 Consider the orthic cubic, 



x^ — ^1^^^ -}- s^x — i-g =ri^ {x^ — s^x^ -f~ •^2'^^' — >^3\ 



where the j's are the elementary symmetrical functions of a, -5 

 The approximation at infinity, 



makes both the square and the cube terms vanish, and therefore 

 represents the asymptotes. The factors of this are : 



x — \sx— fr^ {x — iJi) =. o, 



X — \s^ — ID. f-:{' ( X — -1-J-i) = O, 



X — \Sy — ui^. fr-c {x — \s^) =0. 



where ft»' = i . 



These three lines meet at the point 



which we may call the centre of the curve. We notice that : 

 The centre of the orthic cubic is the centroid of the triad. 

 The clinants of the asymptotes are rjs, wr^l, lo'r^i. They differ 

 only by the constant factor w. Now we know that multiplying the 

 clinant of a line by (o is equivalent to turning the line through an 

 angle -y. A rotation ^ about the centre sends each asymptote 

 into another. It follows that the asymptotes of an orthic cubic are 

 concurrent and parallel to the sides of a regular triangle. M. Serret^ 

 calls such a figure of equally inclined lines which meet in a point a 

 regular pencil, and a curve with asymptotes forming a regular 

 pencil he calls an " equilatere.^' 



1 Comptes Rendus, Sur les hyperboles equilateres d'ordre quelconque. 1895 

 t. 121, p. 340. 



