302 BROOKS— ORTHIC CURVES. [May 20, 



These three lines meet at 



the centre. This point, the origin, and the point which is the 

 centroid of the six points on the circle lie on a line ; and the latter 

 point is midway between the other two. This leads to the interest- 

 ing fact that : 



T/ie centroid of the six points in which any circle meets an orthic 

 cubic bisects the stt'oke from the centre of the curve to the centre of 

 that circle. 



VIII. Triads of the Curve. 



We spoke of the three points a, /5, y, which have the same orienta- 

 tion from every point of the curve, as a triad of the curve. Let us 

 see how many such triads there are, and how they are arranged. 

 The relation 



{x — a){x-~l^){x — r)=z 



may be regarded as establishing a correspondence between points x 

 in one plane and points z in another plane, in such a way that if z 

 describe a line c through the origin, the point x generates an orthic 

 cubic on a^y as a triad. To every position o( z on the director line 

 ^ there correspond three points in the ^-plane. I shall show that each 

 such set of three points is a triad. Write 



^ W = (x—a) (x — /S) (x — y). 



Then, i{ x^, x^, x^, are the three points which correspond to z, 



F (.t) — z = {x — X]) {x — x^) (x — ^3) . 



And also 



F (x) — z^ = (x — ^/) (x — X.2) (x — x/). 



Now this relation is satisfied by x^, or x^, or x^. 



F (x\) —z' = (.Ti — X^') (Xi — x^) {x^ — x^) =z — zf. 



Since z — 2' is a point of the director line, it follows that the three 

 points x^, x^j X3, which correspond to any point 2' of the director 

 line, have the same orientation from every point of the curve. We 

 conclue that : 



To every point of the director line correspofids a triad ; all the 



