Engler — The Normal to the Conic Section. 145 



a 2 



x = +-f-r 2 Z (19) 



a 1 — b l 



2/ = -n»- < 20 > 



aDcl 



/ "~~a T ^b 2 ''' 



Equation (19) represents a straight line parallel to the axis 



a 2 a 2 



of Y -and at a distance -* r- 9 c from it. As -s To is neces- 



a l — b 2 a 1 — cr 



a? 

 sarily positive, the expression -, j- 2 £ always has the same 



sign as £ ; therefore, the asymptote represented by equation 



(19) always lies on the same side of the axis of Y as the 



a 2 

 point P. And as -j — t- 2 is greater than unity, the point P 



lies between this asymptote and the axis of Y. 



The distance from the point P to this asymptote is 



a 2 b 2 



^-? = -~-rJ. (21) 



a 2 — b 2 ' a 2 — b 2 



Equation (20) represents a straight line parallel to the axis 



b 2 



of Jl and at a distance = — ^ ri from it. 



a 2 — er 



b 2 . b 2 



As -= ^ is necessarily positive, the expression — -= — r^ »? 



<r — o 2 a 2 — 6 2 



always has the sign opposite to that of t] ; therefore, the axis 



of JSTlies between the point P and the asymptote represented 



by equation (20). 



The construction of the auxiliary hyperbola for this case is 

 similar to that already given for the parabola; but it will be 

 observed that neither of the asj'mptotes coincides with one of 

 the co-ordinate axes, and, therefore, a special construction to 

 find each of them is necessary. 



The asymptote parallel to the axis of Y will be found at a 



b 2 

 distance 2 , 2 $ beyond the point P (Fig. 5). To find it, 



join the end of the minor axis of the ellipse, B, with the 



