1904.] BROOKS — ORTHIC CURVES. 331 



When this is expanded and arranged in powers of x the first and 

 second terms are 



Now the sum of the roots is 



a„a', — a'„a, 



_ ^^ _2a_l ^1 



«2a — «'2n 



•and their centroid is 



I _ I a2„a^ — g Vi _ , 



Now a^n = ^i> and a'jn ^ /ij and we have also the relations 



and 



a\ = a,. + l,a\., 

 from which we obtain 



= n ^1- 



But this is precisely the centre of the centre circle 



We are thus enabled to conclude with the general theorem : 



T/ie centroid of an orthocentric set of points is the centre oj the 

 centre circle of the pencil of orthic curves through tJiose points. 



JoJuis Hopkins University, May 20, igo^.. 



