154 Trans, Acad. Sci. of St. Louis. 



therefore, (eq. 38) the line through K parallel to the axis of 

 Fis the required asymptote. 



For the other asymptote, draw PR perpendicular to the 

 same asymptote of the given hyperbola and ER parallel to 

 this asymptote. We then have 



ER = 7] sin <p, 

 RQ = ER • sin <p = y sin 2 <p = ^r&v; (*°) 



therefore, (eq. 37) the line through R parallel to the axis of 

 X is the required asymptote. 



To find points on the curve, the same construction as in the 

 previous cases may be employed, if we observe that the con- 

 stant base of the triangles on the asymptote parallel to the 



axis of X is in this case 2 ■ t 2 £ and, therefore, equal to 



the distance ED. 



This hyperbola will intersect the given hyperbola in two, 

 three, or four points according to the position of the point P 

 in the plane. The actual construction of the auxiliary hyper- 

 bola for any given case will give all the real normals possible 

 for the given position of the point. As in the cases of the 

 parabola and the ellipse, the number of points in which the 

 auxiliary hyperbola cuts the given curve will determine the 

 number of normals for any given position of the point P. 

 Since one branch of the auxiliary hyperbola passes through the 

 center of the given hyperbola, as is evident from equation ( 35 ) , 

 both branches of the auxiliary hyperbola cannot be tangent to 

 the given hyperbola ; and since one asymptote of the auxiliary 

 hyperbola is parallel to the axis of X, there must be at least 

 two real normals, one to each branch of the given hyperbola, 

 whatever the position of the point P. 



The following analysis shows in what way the number of 

 normals possible is dependent upon the position of the point 

 in the plane: — 



The equation of the tangent to the given hyperbola may 

 be written 



