150 



Trans. Acad. Sci. of St. Louis. 



cave side (or outside of) the evolute (Fig. 8) only one 

 branch of the hyperbola cuts the ellipse, but this one cuts it 

 in two points, and there are two normals, PN^^, PN'^. 



Figure 8. 



Special Oases. 1. When the point P lies on the ellipse 

 one of the normals through P is the tangent to the auxiliary 

 hyperbola at that point and is readily found by joining with P 

 a point on the asymptote parallel to the axis of ^ at a dis- 



tance -s rr, c from the foot of the ordinate through P. 



a^ — 0^ 



This furnishes a convenient method of drawing a normal to 



the ellipse at a point on the curve because no bisection of 



angles is required and corresponds to the method employed 



for the parabola by using the property that the sub-normal 



is constant. 



