1904.] BROOKS— ORTHIC CURVES. 301 



in six points, the roots of 



x^ — ^o^v* -|- a^x^ — a^x^ -f a^x"^ — a^x -\- ^5 = 0. 



If we want the cubic to meet the circle in six given points, say r^, 

 Tj, . . . Tg, then this equation must be identical with 



x^ — s^x^ -f sy — s^x^ -|- s^x- — s.^x -{- s^ = o, 



in which the s's stand for the elementary symmetrical combinations 

 of the six r's. This requires 



a^ = i-i, a^ = So, ^2 = -^3, 



^3 = •^4> ^4 = ^5, ^5 = ^6 



The coefficients of the cubic equation are then precisely determined, 

 with the result that : 



Bui one orthic cubic can be constructed through a7iy six points of a 

 circle. 



It remains for us to show that one such curve can always be 

 drawn : that is, that the equation 



x^ — S^X' + s.iX — j-3 -[- s^x — s^x^ -f s^x^ = o 



always represents a real curve. If we so choose the base line that 

 ;"g = I then we have 



and the equation takes the form 



x^ — s^x^ -j- s^x — s^-\- s^x — s^x"' -f- x^ = o, 



which is, obviously, self-conjugate, and is therefore satisfied by the 

 coordinates of real points. As a result : 



An orthic cubic can always be drawn through six points of a circle. 

 It is the?i determined uniquely. 



VII. Tlie Intersections of an Orthic Cubic with a Circle. 



When the orthic cubic is referred to the six points in which it 

 cuts the unit circle, the equations of the asymptotes take the form 



X — :^s^={— s,) "^-"io' {x — i s,s,-'). 



i =^ o, T, 2, 



