CANDY : 'IHF. TIUSF.CTIOX OF AN ANGLF,. 41 



Proof, <COP--| <OCI). (Theorem I, Cor.) 



But <0(:D^ i <BOC. 

 .-. <COP \ <BOC. 

 Similarly if OB be taken eciual to OC, and BC be prolonged to 

 to meet the limacon in P', then 



P'O trisects the supplementary angle COD. 

 Also, if CD be produced to P", and BC intersects the limacon 

 in 7?'", til en 



P"0 trisects the reflex angle COB and 

 P"'0 trisects the reflex angle COD, 

 since <POP"=. <P'OP" ' = 120 ^ . 

 . •. P"()P"' is a straight line. 

 It is now perfectly clear that this method of solution is also appli- 

 cable to an angle of any magnitude. 



INVERSION OF FIG. IV FROM O. 



Interchanging x and y in eciuation (2) so that the curve will be 

 symmetrical witli respect to the axis of x, we get 



(4) x2+y2=..R(2X±| x=^-Ly2)- 



X ' V ' 



(5) Patting x:i^ ,.^ ,, and V: ,:: , , ' , , , , reducing and dropping 

 the [)rimes, (4) becomes 



the equation of a hyperbola with the origin at the left hand focus, 

 and the semi-major axis equal to 



Transferring the origin to the center and then putting A zz , 



equation (6) reduces to 



(7) 3x2^y-=---3a2 



which shows that the excentricity (e) is equal to 2. 

 The equation of the directric circle is 



(8) X— ■-yS^^R'^. 



Substituting formula (5) in (S), reducing, transferring the origin 



as in (6) and putting a^ iT''^^'^ obtain 



(9) ^=-4"' 

 2 



the left hand directrix.'^ 



This fact explains the meaning and abjo the reason for tising the name " Directric 

 Circle."' 



