296 BROOKS — ORTHIC CURVES. '[May 20, 



when applied to a function of x and x^ becomes 



— ^r = O. 



dxdx 

 It follows that : 



T/ie necessary and sufficient condition that a curve be orthic is that 

 its equation i7i conjugate coordinates contain no product- ter^n. 



II. Kineniatical Definition of the Orthic Curve. 



Let us now proceed to the study of the orthic curve of the third 

 order. I shall obtain the equation of an orthic cubic in a way 

 which will suggest immediately a method for the construction of 

 points on the curve. 



The path of a point whicii moves in such a way that it preserves a 

 constant orientation from three fixed points is an orthic cubic curve. 



If X is the moving point, and the three fixed points are a, /5, y^ 

 then the sum of the amplitudes of the strokes which connect x with 

 a, /9, Y, must remain constant. That is, we must have 



(x — a) {x — I3){x — r)= pr,. 



If the curve is to be real, the conjugate relation, 



(^ — a) {x — ^) (x — r)= P^{\ 



must hold simultaneously. 



The equation of the curve is obtained by eliminating the para- 

 meter p between these. It is 



x' — {a-^ ^ ^r) x' ^ {a[^ ^ ^r -\- ya) x — a^y 



This is the most general equation of the third degree which we 

 can have without introducing the product. As a consequence it 

 represents a perfectly general orthic cubic. 



If we transform to 



x = i(aJr^-{-r), 



the centroid of a^y, as a new origin, and so choose the base line 

 that T{^ is real, the equation takes the form 



