814 BROOKS — ORTHIC CURVES. [May 20, 



in a point, C, of the Hessian. Now we know^ that such a line 

 meets the Hessian in the point which corresponds to C, This leads 

 to the theorems that : 



The line joining two points apolar with a cubic curve meets the 

 cubic in three points, the tangents at which meet in a point of the 

 Hessian, and are apolar with the two points apolar ivith the curvel^ 



The line joini7ig two points apolar zvith a cubic curve, and a tangent 

 to the cubic at a point of this line, 7neet the Hessian of the give^i cubic 

 in corresponding points. 



A more novel result is the following. We have seen (XIV, p. 

 28) that the foci of an orthic cubic fall into two sets of three, in 

 such a way that the two sets are triangles of m.aximum area inscribed 

 in two con focal ellipses. Now if we consider tangents from /and 

 /instead of foci, we have the following theorem : 



If 3. andh are a pair of points apolar with a cubic curve, then the 

 tangents from either of these points, say a, fall into two sets of three in 

 such a way that th^ line ab has the same polar pair of lines as to each 

 set of three. 



Part Two — Orthic Curves of any Order. 

 I. Introduction. 



In the preceding pages we have studied the metrical properties 

 of the orthic cubic in some detail. In the following portion of the 

 work I shall indicate an extension of the more important results 

 obtained in the study of the cubic to orthic curves of any order. 



The general equation of the n^^ degree between .v and x contains 

 y^nin — i) product terms. If it is to represent an orthic curve the 

 coefficients of these terms must be made zero. In other words, to 

 make a curve of the z^"* order orthic is equivalent to making it 

 satisfy ^«(a5 — i ) linear conditions. After this has been done there 

 remain 2n degrees of freedom. 



II. The Orthic Curve is Equilateral. 



The kinematical definition which we obtained for the orthic 

 cubic may be extended to curves of any order, that is : 



1 Salmon, Higher J-lane Curves, third edition, articles 70 and 175. 

 2 "On the Algebraic Potential Curves," Dr. Edward Kasner, Bulletins of the 

 American Mathematical Society, June, 1901, p. 393. 



