1904.] 



BROOKS — ORTHIC CURVES. 308 



points of the curve have the same orientation from any triad, and all 

 the triads of the curve have the same orientation fro?n any point of the 

 curve. 



IX. The System of Confocal Ellipses Connected with the Triads. 



We seek the points of a triad which correspond to a given point 

 z} The map equation can be brought to the form 



x^ — 3.T = 22 



by choosing the centre of the curve as a new origin and making a 

 suitable choice of the unit stroke. We see at once that the sum of 

 the x's for a given z is zero. In other words : The centroid of any 

 triad is the centre of the cubic. 



Making use of the method known as Cardan's solution, put 



mV^ + l^ + Zl^'t'v + ZP-tV' — 3 (/// + Z/) = 2Z 



And we have as two relations between v and [it, 



2Z = ///^ + V% 



and 



(///-|- v) (fitv — i) = 6. 



When z is zero, the values of x are dz i3 and o ; and when 5 is not 

 zero, we must have 



This leads to the expression of .v and z in terms of fxt as follows : 



2z = p.'t'-{--^, 



Now if we assign any value to /x, and let t run around the unit 

 circle, :^ describes an ellipse with foci at x = -{-2 and x = — 2. 

 1 Harkness and Morley, A Treatise on the Theory of Functions, p. 39. 



