1904.] BROOKS — ORTHIC CURVES. 323 



These two lines meet at 



In other words, Perpendicular tangents to the envelope of the asym- 

 ptotes meet on the centre circle. 



We have here a verification of the known property of the hypo- 

 cycloid of this class, that the tangents from a point of the vertex 

 circle are all real and form two regular pencils/ 



VI. The Orthocentric Ninepoint of the Pencil through Five Points 

 of a Circle, and the Extension to 2n — i Points. 



Let us now consider the figure of nine orthocentric points, five of 

 which are on a circle. The equation of the pencil of orthic cubics 

 through five points of a circle is 



a- — (,Ti -f- t)x" + (^, -^ t<T^)x — (^, + /^2) 



+ (^4 + t<y^)x — (^5 -j- tcj^x'' + 6.Jx^ = o. 



We know five of the points of the orthocentric nine-point deter- 

 mined by this pencil, and we seek the remaining four. Rewrite the 

 above equation as 



{X — t) {X] — ff,X -\- (T,) 



-j- (^tX I ) ((73 (T^X -j- (T.X-) = Q. 



Now if both 



and 



X^ (T^X -j-^2 



X (T^ (TiX 



can become zero for conjugate values of .v and x, then those values 

 are the coordinates of a real point which is on every curve of the 

 pencil, and is one of the nine points. If we put (Tg = i, as we may, 

 these two relations become 



.V' — (T^X -f 0-2 = O, 



1 F. Morley, " On the Epicycloid," Avierican Journal or Mathematics Vol 

 XIII, No. 2. 



