S18 BROOKS — ORTHIC CURVES. [May 20, 



To require a curve to be equilateral is to impose 2n — 3 con- 

 ditions, and to require the curve of order n — 2, along which it cuts 

 its asymptotes, to be orthic is to impose \{n — 2) {n — 3) further 

 conditions, in all \ii {11 — i). But \n {n — i) is the number of 

 conditions required to make a curve of order 71 orthic. 



Part Three — Pencils Determined by Two Orthic Curves and 

 Orthocentric Sets of Points. 



I. Introduction. 



We shall now take up the study of the pencils of curves deter- 

 mined by two orthic curves. The main purpose of this investiga- 

 tion shall be to learn what we can about the figure of n!- points in 



Figure 2. The hypocycloid of class five and order six, which is enveloped by the 

 asymptotes of curves in a pencil of orthic cubics. 



which two orthic curves intersect. Such a figure of n^ points we 

 shall call an Orthocentric Set, or an Orthocentric 7i--point. 



There is a well-known proposition that all the equilateral hyper- 

 bolas (orthic conies) which can be circumscribed to a given triangle 

 pass through the orthocentre of the triangle. The four points, the 

 Vertices and the orthocentre of a triangle, or, what is the same thing, 

 the intersections of two orthic curves of the second order, have the 

 property that the line joining any two of them is perpendicular to 

 the line joining the other two. The term orthocentric is applied 



