294 MR TALBOT ON FAGNANl's THEOREM. 



And also the following: — Of all the lines, which touch the ellipse, and are 

 terminated by the axes produced, the shortest is OT, which touches at Fagnani's 

 point. 



The simplest proof of this, is by the doctrine of infinitesimals. Let the line 

 of constant length OT (which we called above the generating line) assume another 

 position O'T, it will now be a secant to the ellipse, and P will occupy another 

 point in the curve. But if the position O'T' be taken infinitely near to OT, and 

 P' to P, then OT' must be considered as still being a tangent ; and thus we see 

 that the tangent at P', limited by the axes produced, has the same length which 

 it had before [the part OP having gained an infinitesimal quantity 8, and the part 

 PT having lost exactly the same], which is the character of a minimum. There- 

 fore, OT is the minimum tangent, terminated by the axes produced. It is curious, 

 that while OT=a + b is ih.Q Minimum of its kind, PX=«-& is the Maximum of its 

 kind, as we proved in another part of this Memoir. 



This result may also be obtained by the differential calculus, as thus : — 



And when OT is a minimum, 



a*da; _ b^dy 

 x^ ~ y^ ' 



But by differentiating the equation to the curve, -2 + ^='^^, we find -^ = -^ . 



which is always true. 



Dividing the first of these equations by the second, we get -?= ,-i whence -g = 



•*-■ y y 



T3, which is the property which characterises Fagnani's point. 



I will terminate this paper, by giving some remarkable properties of confocal 

 ellipses and hyperbolas. 



Lemma.— h^i XY be the directrix of an ellipse, and P any point, we have by 

 a property of the ellipse 



HP : PX : : e : 1 



e being the excentricity, 



.-. PX=?? 



e 



