312 BROOKS — ORTHIC CURVES. [May 20, 



two ^'s coincide depends on X alone, and that either of three values 

 of Jt: give A a particular value. It is clear that the reflection must be 

 in every sheet of the surface. 



In general, the orthic cubic is of class six. Since it cuts the line 

 at infinity in three points apolar with the circular points, it cannot 

 contain one of the circular points unless it is as a point of inflection. 

 There should, therefore, be six tangents from each of the circular 

 points and, consequently, thirty-six foci. The thirty foci still to 

 be accounted for are the antipoints^ of the six real foci, paired in 

 all possible ways. When the cubic has a node it is of class four, 

 and has but four real foci. The node takes the place of the two 

 foci which coincide-there. 



The circular rays 



X — ay = o 

 and 



meet at a^, a^. So the thirty-six foci of an orthic cubic may be 

 represented by the scheme of coordinates : 



where / and y run from one to six. It follows that the centroid of 

 the whole thirty-six is the centroid of the six real points ; that is, 

 the centre of the cubic. 



Consider any selection of three foci. All their antipoints are 

 foci, and the nine points together make up a central orthocentric 

 set. 



XVI. T/ie Foci of the Orthic Cubics which Have a Triad in 

 Cojnmon Lie on Two Cassinoids. 



The foci of all the orthic cubics which have a common triad a^y ^^^ 

 on two cassinoids which have their foci at a, /?, a7td y, and are ortho- 

 gonal to the orthic curves. 



We know, that these cubics correspond to all the lines through a 

 point, and that their foci correspond to the reflections of the branch 

 points in those lines. Now the reflections of a fixed point in all the 

 lines through a second point lie on a circle which goes through the 

 first point, and which has its centre at the second point. Accord- 



1 Salmon, Higher Plane Curves, third edition, p. 122. 



