288 



MR TALBOT ON FAGNANI S THEOREM. 



over every part of the space intercepted between the ellipses, leaving untouched 

 the whole of the interior ellipse. 



FagnanHs Theorem. 

 Let us now consider MacCullagh's Theorem, given by Salmon, p. 297. 



If any ellipse is cut by a hyperbola, 

 having same centre and focus in the point 

 K, and from any point T of the hyperbola 

 a pair of tangents TP, TQ, is drawn to 

 the ellipse, the difference of the tangents 

 equals the difference of the arcs PK, QK. 

 As an example of this, let tangents 

 be drawn at the extremities of the major 

 and minor axes, meeting in T, and let a 

 hyperbola, having same centre and focus, be so drawn as to pass through T, and 



cut the ellipse in K. Then the elliptic 

 quadrant PQ is so divided in K, that 

 PK-KQ=PT-TQ=«-^», or the dif- 

 ference of the semiaxes of the ellipse, 

 which is Fagnani's theorem (Salmon, 

 p. 298). 



Now, let the point T advance con- 

 Pig. 6. tinually along the hyperbola, and the 



intercepted arc will of course always be divided at the same point K. Ulti- 



Fiff. 5. 



Fig. 7. 



raately, the point T may be supposed to attain the asymptote at an infinite 

 distance. The two tangents TP, TQ. are then parallel to the asymptote. Their 



