1904.] BROOKS — ORTHIC CURVES. 317 



equation, expressed in terms of the elementary symmetrical functions 

 of the points where it meets the circle, becomes 



.r'' _ j-^x"-i -^ s.x''-"^ 1 s.^_,x''-' -f s,Jc'' = o. 



The centre, found by equating to zero the n — i*'' derivative as to 

 X, is 



This is the midpoint of the stroke from the centre of the circle to 

 the centroid of the 2n points. The equation of an asymptote now 

 takes the form 



X -Si |, S2Q yX «f2n— 1 >^'2n )■ 



V. Construction of an Orthic Curve. 



The method which I have proposed (Part One, V) for the con- 

 struction of an orthic cubic might be extended to the construction 

 of any orthic curve. For this purpose the instrument must have 71 

 hands, moved by n weights. The centre of gravity of any number 

 of weights could be held by joining them together in sets of three 

 or less, and then joining again the centres of gravity of these sets. 

 This operation could be repeated until the required number of 

 weights is reached. 



VI. Geometrical Characteristics. 



The geometrical characteristics of an orthic curve of order n are 

 that it is equilateral, and that it ifttersects its asymptotes in points of 

 a second orthic curve of order n — 2. 



For consider the orthic curve referred to its centre, 



.T° -f a^x""-'' — ajx""-^ ... — ^ojc"-' + ^i.T°-' -f j^" = o. 



The asymptotes, which are given by 



are concurrent and equally inclined, so the curve is equilateral. 

 The points common to the curve and its asymptotes lie on the curve 



But this curve is of order n — 2, and is orthic. 



