MR TALBOT ON FAGNANl's THEOREM. 



289 



difference is equal to the difference of their extremities which touch the ellipse, 

 and are cut off by any line at right angles to them. Call this difference D. The 

 intercepted arc is then a semi-ellipse, because, when tangents are parallel, the 

 line joining the points of contact passes through the centre. 



Now we have PK-QK=PT-QT=D, and BK-AK=a-b; therefore, by sub- 

 traction, PB-QA=D- (a -5). But the arc PB = QZ, .-. QZ-QA=D_(«-J). 

 But ZA is a quadrant of the ellipse, therefore this quadrant is so divided in the 

 point Q, that the difference of QZ and QA is a known straight line = D -(«-&). 



Now, it is by no means obvious, whether or not we have thus obtained a 

 division of the elliptic quadrant different from the one we first obtained. This 

 point can only be settled by a rigorous demonstration ; the result of which gives 

 this curious theorem D=2 (a-5). From which we see that QZ-Q,A=«-^»=BK 

 -K A, so that the elliptic quadrants BA and ZA are divided at corresponding 

 points K and Q, and the arc AK=AQ. And since the tangent at Q or QT is 

 parallel to the asymptote, it follows, by parity of reasoning, that the tangent at K 

 is parallel to the other asymptote. Moreover, if we draw MCN through the 

 centre, at right angles to PT and QT, it is plain that D = PM + QN, or since these 

 lines are equal, 2(«- 5) = D = 2QN .'. Q'N=a-b, hence the point of division Q is 

 such, that the perpendicular let fall from the centre on the tangent at Q, cuts off 

 from it a portion Q'N—a—b. 



It remains, therefore, to demonstrate a prio?'i the theorem we have just indi- 

 cated, viz., that QN=a-b, whence the 

 other properties mentioned will follow. 

 We shall, at the same time, obtain the 

 demonstration of many other theorems. 



Let CA, CB, be the semi-axes of an 

 ellipse, denoted by a, b. Complete the 

 rectangle BCAD. Let A, B, be the 

 semi-axes of a hyperbola, having same 

 centre and focus, and so drawn as to 

 pass through the point D, whose co- 

 ordinates are a, b. Then, according to 

 Salmon (Conic Sections, p. 298), the 

 co-ordinates of the point P, which is the 



intersection of the two curves, are, x^= 



Pig. 8. 



a + h 



T>y' 



a + h' 



We shall reverse this order of reasoning, and suppose a hyperbola drawn with 

 centre C, axis in the line CA, and passing through the points D and P, and then 

 show that such a hyperbola has the same focus with the ellipse. In the first 



place, then, the point whose co-ordinates are cc^ = 



a + b' 



/ = 



a + b 



VOL. XXIIL TART IL 



lies in the ellipse, 

 4k 



