1904.] BROOKS— ORTHIC CURVES. 295 



asymptotes concurrent and parallel to the sides of a regular polygon. 

 It seems advisable to follow M. Serret's usage, and to denote such 

 a curve by the name equilateral, using another term to express 

 apolarity with the absolute. For this purpose I have adopted the 

 word orthic. 



If we use Cartesian coordinates, a curve 



[/(XV) = o, 



is apolar with the absolute conic, 



c- H- r = o, 



if 



X^ + Jy2 — o- 



In other words, an orthic curve is one which satisfies Laplace's 

 equation in two dimensions. 



Part One — The Orthic Cubic Curve. 



I. T/ie Condition thai a Curve be Orthic. 



In the analysis which may be required, I shall employ conjugate 

 coordinates, x, x, which may be defined as follows : If X and V 

 are rectangular Cartesian coordinates of any point, the conjugate 

 coordinates of that point are 



x = X-^iy, x = X—iV, 



when the origin is retained, and the axis of X is chosen as the axis 

 of reals, or base line. It is sometimes convenient to think of .v as 

 the vector from the origin to the point, and of x as the reflection of 

 this vector in the base line. If ^, :r is a real point of the plane, not 

 on the base line, x — x = o, x and x are conjugate complex 

 numbers. Since if one of its coordinates is known the other is 

 immediately obtainable, we shall, as a rule, name a point by giving 

 only one of its coordinates. It is convenient to reserve the letters 

 / and r for points on the unit circle, 



XX = I . 



Now, Laplace's equation. 



