CONIC SECTIONS, ANALYTICAL. ll.i 



If this condition be not satisfied, the three conies intersect in the 

 same four points, real or impossible. 



The relations between the eccentric angles corresponding to 

 normals may be found from the equation 



which if (X, T) be given is a biquadratic whose roots give the 



A 



points, the normals at which meet in (X t Y). If z = tan - , this 

 equation becomes 



z 4 bY + 2z" (aX+ a' - 6 ) + 2z (aX - a + 6) - b Y= 0. 



This equation having four roots, there must be two relations 

 between the roots independent of (X, 7) as is obvious geome- 

 trically. These relations are manifest oil inspection of the 

 equation ; they are 



= 0: 



and the relation between any three is found from these to be 



which is equivalent to 



sin (0, + OJ + sin (0. -I- 0,) + sin (0, + 0,) = 0. 

 Since 1 - (z^ + . . .) + ,*,*A ~ 0> it appears that 



or 0, 4- 0, + 0, 4- 4 iaanodd multiple of ar. 



The following is another method of investigating the same 

 question. If the normal at (x, y) to the ellipse pass through 

 we have 



a t xX-b'yY^(a t -b f )xy ...... (A). 



