1904.] BROOKS— ORTHIC CURVES. 319 



to a set of four points related in this way. We wish to find out 

 what metrical property distinguishes the ;z"-point, in which two 

 orthic curves of order n intersect. 



II. The Central Pencil and Its Orthoce?itric Set. 



The first generalization which we shall make is to show that any 

 pair of points, a, /3, together with their antipoints, a, /3 and /?, a, 

 form an orthocentric four-point, a and /? determine a central pencil 

 of orthic conies, 



(.r — a) (jc — /?)=r (I-_a) (I — /?), 



and the antipoints are evidently on all the curves of the pencil. 



If we consider - as a parameter in the general equation of an 

 orthic curve, ^ 



{^x — a^) {x — a,). . . (x-—a^) = T {x — a,) (x — a.,) . . . (J— aj, 



we obtain the equation of all the curves of which a^ . . . a^ is an 

 n-ad. The points of the orthocentric «^-point determined by this 

 are the n real points a, and all their antipoints. But as the pencil 

 is determined by the n real points it follows that : 



Any Yi points, with all their antipoints , form a central orthocentric 

 t^-point. 



The centroid of the //^-point determined in this way is the cen- 

 troid of the n real points. The real and imaginary foci of any 

 curve are such a set of orthocentric points. ^ 



III. The Pencil of Orthic Cubics through Five Points of a Circle. 

 / The Locus of Centres. 



We have seen that six points of a circle determine an orthic cubic 

 curve. If the six points are /i, t^, t^, 4, /j, /g* then, as we have seen, 

 the equation of the orthic cubic through them is 



x^ — s^x^ -{- s.x — j-3 -j- s^x — j-jJf" -(- s^x^ = o. 



If we replace /« by a variable parameter /, and put ^'s for the ele- 

 mentary symmetrical combinations of /i . . . t-^, we have 



-^1 = '^l -|- ^> J"4 = ^4 + l<^iJ 



S-l = ^2 -f- ^^), -^5 = ^5 + ^'^4> 



