292 MR TALBOT ON FAGNANl's THEOREM. 



For that normal is tangent to the elHpse, and the asymptote coincides with the 

 conjugate diameter. 



Now, let us draw three vectors to the hyperbola, from the centre, making the 

 angles 0, 6', &', respectively with the axis. The first to be drawn to a point 

 infinitely distant (and, therefore, it will be the asymptote), the second to the point 

 D, and the third to the point P (see fig. 8). Then, if we call tan Q-t, we have the 

 following curious property, — 



tan ^ = ^2 

 tan &'=.t^ 



For tan ^--r-, and tan 6'-- = -^ and tan ^" = -, at the point where the curves inter- 



sect; and .'. =\/^ = -^. 



Since Q is less than 45°, and .-. tan Q less than 1, the angles Q, 6', 6", become suc- 

 cessively smaller. 



Another remarkable property is the following (see fig. 9) : — At the point P, 

 where the curves intersect, and where the elliptic quadrant is algebraically 

 divided according to Fagnani's theorem, the line PN intercepted by the perpen- 

 dicular CN on the tangent, is a maximum. For if we examine in an?/ ellipse 

 what must be the position of the point P, in order that PN may be a maximum, 

 we shall find it to be characterized by those values which we have already shown 

 to belong to P in Fagnani's theorem. This may be shown as follows : — 



In any ellipse, let C be the centre, CP, CQ, conjugate diameters, PN a tangent, 

 and CN or p the perpendicular to it. 



We have CQ=^ .-. Cq^ = ^r- 



Subtract GW^p' 



Let this be a maximum. Therefore since a^ + b' is constant, — ^ + »^ must be a 



pi jr 



71 



minimum, or putting a^b^=n, and ^9^ = ^;, then- + .t% is a minimum. 



Differentiating this, we find cc=\/n; or p^ = ab. From this value we find 

 G(^=ab, and .*. CQ=^, and PN^ =«'' + &'*— «6-a5; .-. PN=a— J. And as these 

 were the values which we found before for the same lines, it is evident that P 

 is the same point which we were considering before. Therefore, at the point P, 

 which may be called Fagnani's point, the line PN is a maximum. 



