320 BROOKS — ORTKIC CURVES. [May 2(1, 



If we make this substitution we get 



X' ((T^ f /).T- -|- (^j + Ui)x — (^3 + ^^2) 



+ (^i -\- ^""s)^"^ (^r. + ^^i)'>^' + f^r/X^ = O. 



This is the equation of a pencil of orthic cubics through five points 

 of a circle. 



The centre of the curve through the six points is x = y^. If 

 the sixth point move around the unit circle, this becomes 



This is the map equation of a circle. We have thus the theorem : 



T/ie locus of centres of the orthic cubics through five points of a 

 circle is a circle. Its radius is one-third that of the given circle^ and 

 its centre is the point \g^. 



M. Serret^ gives an elegant synthetic proof of the theorem that 

 the locus of centres of the curves of a pencil of equilaterals is a 

 circle. I obtained the same result for orthic curves independently, 

 and, as the analysis is so direct, it seems advisable to let it stand. 



IV. Tlie Hypocycloid Enveloped by the Asymptotes. 



I shall now prove, for the pencil of orthic cubics through five 

 points of a circle, a theorem which M. Serret^ states without proof. 

 The theorem referred to, when stated for orthic cubics of the pencil 

 under discussion, becomes : 



The curve enveloped by the asymptotes of all the orthic cubics 

 through five points of a circle is an hypocycloid of order six and class 

 five. 



It is circumscribed to the centre 'circle of the pencil, and its 

 cusps lie on a concentric circle five times as large. 



We found that the equation of an asymptote, in terms of the six 

 points where the curve cuts the unit circle, is 



{x — 1^1) -|- fs^{x — \s^s-^) = o. 

 If we replace 4 by the parameter /, this becomes 



^ Sur les faisceaux reguhers et les equi'ateres d'ordre n. Paul Serret, Co7nptes 

 Rei7dus, 1895, t. 121, pp. 373-5. 



