158 Trans. Acad. Sci. of St. Louis. 



branch, and the other branch of the auxiliary hyperbola cuts 

 the other branch of the given hyperbola each in a single point 

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



Special Cases. 1. When the point P lies on the given 

 hyperbola one of the normals through the point is tangent to 

 the auxiliary hyperbola at that point and is found precisely as 

 in the case of the ellipse, if we observe that the constant base 

 of the triangles on the asymptote parallel to the axis of X 



• ^' ^ 



2. If the given point P is on either of the axes of the given 

 hyperbola, the auxiliary hyperbola degenerates into its asymp- 

 totes, one of which is the axis of the given hyperbola through 

 the point, the other is obtained by the construction. The 

 intersections of these asymptotes with the given hyperbola 

 give the normals. It will be observed that if the given point 

 P is at one of the cusps of the evolute the asymptote parallel 

 to the axis of Y is tangent to the hyperbola and, therefore, 

 three of the normals coincide. 



The constructions for these cases can easily be supplied by 

 the reader. 



THE NORMAL TO THE CONJUGATE HYPERBOLA. 



The equation of the hyperbola conjugate to the one repre- 

 sented by equation (32) is 



— — ^ = —1. (48) 



The auxiliary hyperbola which gives the normals through P 

 for this hyperbola is 



a'% + 6V = (a2 + IjI) xy, (49) 



which is identical with equation (35). From this it appears 

 that the same auxiliary hyperbola gives the normals not only 



