Manchester Memoirs, Vol. lix. ( 1 9 1 5 ), No. ] & 13 



or in polar co-ordinates, taking as pole and OB as 



initial line 



ar cos 2 + /^sin 2 = ^/jsin (/3 + 6) + da cos (/:> + 0), 

 or 



r cos (2^ - <p) = ^/cos (/3 + d - (p), where tan <p = /*/«. 



The points where the circle, r = d t cuts the hyperbola are 



given by 



cos (26- d>) = cos(/3 -f 6 -(p), or 2ti -<f> = 2Mr + (j3 + -<{>), 

 where n is an integer or zero. 



The plus sign gives 



8=fi+ 2717T, 



which corresponds to the point A. 

 The minus sign gives 



n 2(f) — l3 , 2/lK 



ti = + — . 



3 3 



Three points, P v P v P iy will satisfy this condition and 

 are obtained by taking « = 0, 1 and 2 respectively. The 

 triangle PJ*JP Z is equilateral. 



The method of Chasles is now deduced by taking 

 <p = /3, since then the angles AOP lt AOP n _, A OP., will be 

 one-third of the angles AOB, A0B+2tt, AOB + ^tt 

 respectively, and if <j) = /3 and # = be substituted in the 

 polar equation of the hyperbola, it follows that r cos /3 = d, 

 which is the condition that the hyperbola shall pass 

 through the point where OB cuts the tangent to the 

 circle at A (compare Fig. J). 



The method of Pappus is deduced by simply taking 

 (Jj = 7r/2. 



(2) Use of a hyperbola of eccentricity two. 

 This curve may be used in a very simple manner to 

 trisect a circular arc, and hence to trisect an angle. The 

 following constructions, which only differ slightly, are 

 given by Newton (<?), Pappus (/;) and Clairaut (<;), and are 

 restated in the " Mathematical Recreations " of W. YV. 

 Rouse Ball. 



