MR TALBOT ON FAGNANl'S THEOREM. 



291 



ellipse, CN . CQ=«J ; whence CN' . CQ^ = a"'h\ Divide this equation by CQ^^^ah 

 . •. CN2 = ah, and . •. CN = CQ. But now, since CP' =a' + h''- ab, and CN^ = ah, there- 

 fore PN2=CP2-CN2 = «2 + 52_2^j^ .-. PN=a-5, which is the theorem we under- 

 took to demonstrate.* 



It is curious that CQ when prolonged, becomes the asymptote of the hyper- 

 bola. Perhaps we have already offered sufficient proof of this, but the reader 

 may not object to see it proved in another manner. 



The equation to the ellipse ^+p=l gives ^=-- • ^• 



But the value 



Therefore 



of - at Fagnani's point is j|=^3- 

 dy 



It "P 



= - J-= - y But by the property of the by- 



perbola, if C be the centre and CA, CB (or A, B), 

 the semi axes, the asymptote CD will be found by 

 completing the rectangle ACBD, and joining CD. 

 .-. the asymptote makes, with the axis CA an 



angle, whose tangent =cT~A' ^°^ *^^ other 

 asymptote makes an equal angle below the axis, 

 whose tangent is therefore —-^. But we found —7= — x', therefore, the tangent 



to the ellipse at the point P, or («, p) is parallel to the second asymptote. Con- 

 sequently, the conjugate semi-diameter CQ, is a portion of that asymptote. 



Since the two curves are confocal, they ought to intersect at right angles. 



Let us verify this. We have seen that the equation to the ellipse gives 



Fig. 10. 



dy 



B 



£-= - T- at Fagnani's point. But the equation to the hyperbola T2-g2 = l gives 



. ^dy ^r B2 A^ B2 A ^ _, , ,, B 



at the same pomt ;;j^=- • -^^ 



dx y 



A^ 



W A 



-^ =-g- But these two results, ^ in the ellipse, 



and w in the hyperbola (neglecting the signs) are reciprocals. Therefore at the 



point of intersection the two curves make angles with the axis, whose tangents 

 are reciprocals, and therefore they intersect at right angles. 



Hence this curious theorem " At the point of intersection P, the normal to 

 the hyperbola is parallel to one of its asymptotes." 



* These properties, viz., that CN = CQ = va6, and that PN = a — 6 are proved by Brinkley 

 ia quite a different manner (Trans, of the Royal Irish Academy, vol. ix. p. 146, &c.). He likewise 

 proves, that if CQ produced cuts in O, the circle described on the axis major as diameter, a perpen- 

 dicular let fall from O on the axis, cuts the ellipse in Fagnani's point. But I have shown that CQ 

 produced is the asymptote of the hyperbola, .•. an ordinate to the ellipse at Fagnani's point, passes 

 through the intersection of the asymptote and circle. In other words, tlie common chord of the ellipse 

 and hyperbola, being produced, becomes the common chord of the circle and the two asymptotes. 



