816 BROOKS ORTHIC CURVES. [May 20. 



may be regarded as mapping a line through the origin in the s-plane 

 into the orthic curve in the jt-plane. The methods of analysis 

 which were used, in the paragraphs referred to, in the study of the 

 orthic cubic may be extended to any n, and lead to the following 

 general theorems : 



On an orthic curve of order n there is a single infinity of- sets of n 

 points, n-ads of the curve, from which all points of the curve have the 

 same orientation. All the n-ads have the same orientation from any 

 point of the curve (Part One, VIII). 



Any n points may be taken as an n-ad of an orthic curve. If we 

 take n points of the unit circle as an n-ad, and find the remaining 

 intersections of the circle and the curve, we see that they are the 

 vertices of a regular polygon (Part One, XII). 



Every circle through an n-ad of an 07'thic curve of order n meets 

 the curve again in the n vertices of a regular polygon. 



The centre of an orthic curve is the centroid of every n-ad of the 

 curve. 



For when the equation is taken in the form 



.r" + /2jc"-' + , . . + a^_^x = z 



the origin is the centre of the curve, and is also the centroid of the 

 n points which correspond to a point z. This equation will have 

 two coincident roots whenever 



D^z = nx"^'^ -\- n (n — 2) a-^x'^~^ . . . =0. 



In general, this will give n — i branch points in the 2-plane. Each 

 branch point, when reflected in the director line, gives rise to Jt 

 real foci. If the line ^ revolve about a point, each reflection 

 generates a circle (Part One, XIV). All n — i of these circles are 

 concentric; and they map into n — i cassinoids, on which lie the 

 foci of the curves which have the n-ad which corresponds to the 

 centre of the system of circles. These cassinoids are orthogonal 

 trajectories of the central pencil of orthic curves. Since each of 

 the circles must contain a branch point, each cassinoid must have 

 at least one node. 



IV. The Orthic Curve Referred to its Intersections with a Circle. 



We know that we may put 2n linear conditions on an orthic 

 curve. If we make it go through 2n points on the unit circle, its 



