4° 



KANSAS UNIVERSITY OUARTERLV. 



Likewise, if CB be produced to meet the limacon in P" and P'OH' 

 be drawn, malcing ()H'=OC, then H'C trisects the supplementary 

 <OCB. 

 For proof see The. I: Case I, when <NCB is less than 135 ^ ; 



Case II, when <OCB is greater than 135 ° and is less than 180 ° ; 

 Case III, when <OCB is less than 180 ^ . 

 An angle of any magnitude is trisected with equal facility. 

 It is interesting to notice in this connection the following somewhat 

 remarkable property of this particular quartic. 



Since <COP 2 <ocB and <COP' |<OCB', [fig. V.] 



.-. P'OP":- 120°. 

 .\lso <P'^OCB' and <OP"P'r^.l Reflex <OCB\ 



.-. : <P'OP"=6o°. 

 Therefore, any line drawn through the center of the Directric 

 Circle and terminating in the outer branch of the cjuartic, subtends at 

 the double point a constant angle of 1 20 ° , of which the line joining 

 the double point to its intersection with the inner branch is the 

 bisector. 



Also by expanding eipiation (2) and placing the second degree 

 terms equal to zero, we find for the equations of the tangent lines at 

 the double point 



3y2— x2r^o. 

 This equation shows that these tangents also intersect at angles of 

 60 ^ and 120". 



SFXOND METHOD. 



Let BOE (fig. VII) 

 be the given angle. 

 With O as origin and 

 OE as the axis of ordi- 

 nates describe the lim- 

 a9on determined by 

 equation (3). 



On the side of OE 

 take OC equal to the 

 radius of the Directric 

 Circle. 



Produce BO to D, 

 making OD==OC. 



Draw DC and pro- 

 duce it to meet the 

 liraagon in P. 



Then will PO trisect 

 the angle BOC. 



