Engler — The Normal to the Conic Section. 155 



b 2 x' ¥ ... . 



V = -2~,X 7 ( 41 ) 



a 2 y y 



in which as', y' are the co-ordinates of the point of contact on 

 the given hyperbola. 



The equation of the tangent to the auxiliary hyperbola 

 may be written 



in which as', 2/' are the co-ordinates of the point of contact on 

 the auxiliary hyperbola. 



If equations (41) and (42) are made to represent the same 

 line, which signifies that the tangents to the two curves coin- 

 cide, and if, at the same time, x\ y' in (41) and (42) repre- 

 sent the same point, which means that the curves are tangent 

 to each other at that point, we have as equations of condi- 

 tion, 



^: = _^: 2 , (43) 



a 2 y[ b 2/ nx' 2 ' 



b 2 _ (a 2 + b 2 )y' 2 . 



y' b 2 V ' : ; 



from which for the co-ordinates of the point of contact 



If these values are substituted in equation (32), we have 



W + bV W + b 2 ) * (47) 



This, if £, rj are regarded as variables, is the equation of the 

 evolute of the hyperbola, which is, therefore, the locus of the 



