Engler — The Normal to the Conic Section. 151 



2. When the given point P is on either axis of the ellipse, 

 the auxiliary hyperbola degenerates into its asymptotes, one 

 of which becomes the axis of the ellipse through the point, 

 and the other is found by the construction. The intersec- 

 tions of these two asymptotes with the ellipse give the 

 normals. It will be observed that when the point P is at 

 one of the cusps of the evolute, one of the asymptotes is 

 tangent to the ellipse, which means that three of the normals 

 coincide. 



3. For the case when the ellipse becomes a circle, a = b, 

 and equation (18) becomes 



fy-VX = 0, (31) 



which is the equation of a straight line through P and the 

 center of the circle; the normals, therefore, coincide with 

 this line. 



The constructions for these cases can easily be supplied by 

 the reader. 



THE NORMAL TO THE HYPERBOLA. 



The equation of the hyperbola referred to its rectangular 

 axes as axes of co-ordinates is, in the usual notation, 



^ 2 _^ 2 = 1. (32) 



a 2 6 2 V ' 



The equation of the normal to the hyperbola is 



a 2 xy 1 4- b 2 yx l = (a- 4- b 2 ) x i y l (33) 



in which x x , y x are the co-ordinates of the point in which the 

 normal crosses the curve. 



If the normal is to be drawn through any point in the plane 

 P whose co-ordinates are c, y, these co-ordinates must satisfy 

 equation (33), so that 



a l qy l 4- b 2 r\x x = (a 2 + b 2 ) x x y v (34) 



The co-ordinates x lt y x also satisfy equation (33). 



