140 Trans. Acad. Sci. of St. Louis. 



When the point P is so situated that only one branch of 

 the hyperbola cuts the parabola, there will be only one 

 normal. 



When the point P is so situated that both branches of 

 the hyperbola cut the parabola, there will be three normals. 



When the point P is so situated that one branch of the 

 hyperbola cuts the parabola while the other branch of the 

 hyperbola is tangent to it, there will be two normals. 



Both branches of the hyperbola cannot be tangent to the 

 parabola, because the axis of X is one of the asymptotes of 

 the hyperbola; for the same reason there must always be at 

 least one real normal 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 parabola may be written 



y = -,x + ^x' (8) 



y y 



in which x\ y' are the co-ordinates of the point of contact on 

 the parabola. 



The equation of the tangent to the hyperbola may be written 



V = ? , X — ^ ^ ; x' -\- y' ( 9 ) 



^ ^ —p — x' q — p —x' ^ ^ ' 



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

 the hyperbola. 



If equations (8) and (9) 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 (8) and (9) represent 

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

 each other at that point, we have as equations of condition, 



(10) 

 -,x--\-y'; (11) 



y c — p — x 



