322 BROOKS — ORTHIC CURVE3. [May 20, 



If we put kI = — I we get a cusp. 



Since multiplication by k' is equivalent to a rotation |-, we see that 

 the locus of cusps is a circle, about the centre of the pencil, and 

 five times as large as the centre circle. A rotation t- sends each 

 cusp into another, and so the cusps are equally spaced along the 

 cusp circle. The intersections of the hypocycloid with the centre 

 circle, 



XX = I , 



are obtained by solving x = x~^ for r. 

 We have 



and 



X = 3r-' — 2T% 



X = :^t' — 2--\ 



The parameters of the points sought are the roots of 



or of 



There are five pair^ of coincident intersections. But since x cannot 

 be less than i, it follows that the curve is tangent to the circle in 

 five places. 



We have obtained this hypocycloid as the locus of one asymptote. 

 But all three asymptotes envelop the same curve, for if we put to for 

 f(T^ we get 



x = 2>^T~'^ — 2r^ 



This has a cusp at «^lv ^5 ; it is, obviously, the same curve. 



V. Perpendicular Tangents of the Hypocycloid. 

 The equation of a tangent to the curve is 



That of a perpendicular tangent, 



— XT~'^ — r- -f- ^ -\- r"^ = o. 



