Manchester Memoirs, Vol lix. (191 5), No. 13. 9 



drb - ^by- = $ay -y'' 



or 



But 



and 





fi/a = tan ABC. 



yl a = =tS = tan HAG = tan Z>## 



Therefore 



tan ^#C=tan $(DBF). 



Thus the three positions for the intersection point F 



are such that the ansrle 



DBF- 



ABC ABC , 2- ABC Air 

 , + — or + -L- 



3 3 3 3 3 



These three points are evidently the corners of an equi- 

 lateral triangle. 



The following method is due to Chasles, 8 and is also 

 given as an example in Taylor's Geometry of Curves, 



Fig. 7. Method of Chasles. 



Ex. 528. If OA and OB {Fig. 7) are the bounding radii 



of a circular arc AB, then a rectangular hyperbola having 



s "Traite des Sections Coniques,''" p. 36. 1S65. 



