328 BROOKS — ORTHIC CURVES. [May 20, 



a' 

 circle, i.e., adopt | — ^ | as the unit length, the equation takes the 



form 



This is the equation ol" an hypocycloid of the kind found as the 

 locus of asymptotes of a special pencil of orthic cubics. Its vertex 

 circle is the centre circle of the pencil. It has cusps when 



D-x = o, 

 and 



in =^> 



simultaneously, or when 



r-°-^ -j- I =z o. 



The parameters of the cusps are the 2/1 — i" roots of — i. 

 If we let /^-""^ = — I, a cusp is 



X = ««^~° -f- (i — n) k'' 

 or 



xk"" ^ = n — (i — n) 



The absokite value of a cusp is, therefore, 2/1 — i. 

 Since the equation of a tangent, 



X — ia'^T"" + rx — ir^-° a\ = o 



is of the 2n — i'' degree in the parameter r, the hypocycloid is of 

 class 2n — I. If we eliminate x between the equation of the curve 

 and the equation of any line, 



I — r 



we get an equation of the 2n^^ degree to determine the parameters 

 of the points of intersection. The curve meets any line in 2;/ 

 points, and is therefore of order 2n. We have now established 

 analytically the theorem stated by M. Serret, as far as orthic curves 

 are concerned. It is : 



T/ie curve enveloped by the asymptotes of a pencil of orthic curves of 

 order n is an hypocycloid of order 2n, and of class 2n — i. Its vertex 

 circle is ike centre circle of the pencil, and its cusp circle is concentric 

 with that circle, and 2n — i times as large. 



If we bear in mind that any difference between an orthic curve 



