1904.] BROOKS — ORTHIC CURVES. 329 



and any equilateral does not affect the terms of the n^^ and n — i'*" 

 degrees of the equation, we see that the method of proof used above 

 is applicable to equilaterals in general. 



X. A Circle Determined by Atiy Odd Number of Points. 



It is a well-known proposition that the centres of the equilateral 

 hyperbolas circumscribed to a triangle lie on the circle through the 

 mid-points of the sides of the triangle. This circle is usually called 

 the Feuerbach, or nine-point circle of the triangle. Now we have 

 seen that an orthic curve of order n may be made to satisfy 2n lin- 

 ear conditions; it follows that any odd number, 2n — i, of points 

 determine a pencil of orthic curves of the n^^ order. Connected 

 with this pencil is the centre circle, or, as I propose to call it, the 

 Serret circle, which is, in a sense, the generalized nine-point circle. 



Every fi'^ure of an odd number of points has connected with it a 

 unique circle^ the Serret circle, zuhich in the case of three points is 

 identical with the ninepoint circle of Feuerbach. 



Further, every odd number of points, 2n — i, determine the 

 pencil of orthic curves through them, and therefore the remaining 

 (^n — i)- points of the orthocentric ;r-point. In the case of three 

 given points, this set of {n — • i)^ points is a single point, the ortho- 

 centre of the given points. So we are led to the theorem : 



To every figure of 2n — i points belongs a figure of (n — if poi?its. 



In one sense the Serret circle belongs to ;r points, but of these 

 only 2?i — I may be taken at random. 



XI. A Point Determined by An Even Number of Points. 



Now consider an even number, 2n, of points which do not belong 

 to an orthocentric /^^-point. There is a pencil of orthic curves 

 through everv 2n — i points which can be selected from them, or 2« 

 pencils in all. Now these pencils give rise to 2n — i Serret circles, 

 but there is one orthic curve through all 2n points and its centre is 

 on each of the circles. We have, therefore, the result : 



The 2n Serret circles, given by all the sets of 2x1 — i among 2n 

 points, meet in a point. 



XII. The Relation of the Orthocentric vc-point to the Circle of 



Centres. 



In section VIII we obtained the pencil of orthic curves deter- 

 mined by the two given curves, 



