310 BROOKS — ORTHIC CURVES. [May 20. 



is tangent to the curve, then 



F{ax) = O, 



the eliminant of x between these two, will have equal roots. But 

 since the equation of a real curve must be self-conjugate, if this has 

 two coincident roots, then 



must also have, and the point a, a, is a focus. It follows that to 

 find the foci of a curve, we have merely to find those values of x 

 which make two values of x coincide. They are the vectors of the 

 foci. Let us apply this method to the orthic cubic. The equation 

 may be taken in the form 



x^ — ^x = 2z = ao -{- la^ 

 where A is a real parameter and the director line is 



^0 + ^(^\ = 2:S, ao -\- Xa^ = 22. 



These relations imply the conjugate expression . 



x^ — ^x =z 2z= ao -\- la^. 

 Two values of x become equal when D'^z = o, /.f.,'when 



x^ — I == o, 

 or 



x^= ± 1, 



These values of x occur when 



^0 -|- M = ± 2, 



or 



— ^n =b 2 



"1 



Either of these values of A when substituted in 



x^ — ^x = a,-]- la^ 

 gives three points which are foci of the cubic. 



