44 



KANSAS UNIVERSITY QUARIEHLV, 



The directrix CN inverts into a circle whose radius is e(|iial to the 

 value of (a) in ( lo). Tlie circle inverts into a circle whose center is 

 on CO, passes througli A and V, anil intersects the riglit loop of the 

 quartic in P and P'. 



AC, FC, and PC invert into circles ihrouga (O, A, C), (O, F, C) and 

 (O, P, C), respectively. Then the circle OPC trisects the angle at C, 

 formed bv the circles OAC and ()!"('. 



I 11- III MKMIDII. 



Let Q'JQ' (fi:J- X) bi the given angle. 



Draw the circles O, and O., tangent to QC and Q'C, respectively, 

 at C. so that their oth*r pjlnt of intersection (O^ s':all lie within the 

 given angle; also make their radii satisfy the condition 



\ R, : R„-i -.2, 

 since this is the inverse ratio of the distance of the lines FC and AC 

 (fig. \TII) from (()), the center of inversion. 

 Draw the secant AOF so that 

 OF : OA 1:2. 

 With O as double-point, A as vertex of left loop, and F as focus of 

 right loop, draw the (juartic which is represented by (8). 



With center on CO, describe a circle passing through A and F, 

 and intersecting the right loop of the quartic in P and P'. 



Pass the circle O3 through P, O, and C, and draw Q"C tangent to 



then <QCQ"=^<QCQ'. 



Also, the tangent to a circle through P', O, and C trisects the cor- 

 responding reflex QCQ'. 



