290 



MR TALBOT ON FAGNAXl'S THEOREM. 



for those co-ordinates verify the equation to the elUpse ^—,+^ = 1. The substitu- 



tion gives 



a 



+ Y =^' which is identically true. 



a + h a + h 



Secondly. By hypothesis, the equation of the hyperbola -ry— ^=1, is satisfied 



by the values x=a, y=h., which gives (1) 



a" 



or 



a + h' 



r = 



a+h' 



which gives (2) 



Subtracting (2) from (1) 



a' 



2 7 2 



-T^ - TvT = 1 , and also by the other values 



= 1. 



{a + h)A? (a + J)B- 



a'h h^a 



{a + h)A? {a + h)W- 



.-. X2 = g2 which gives the remarkable result A.- : 'Q^ : : a : h, showing that the 



elliptic semi-axes are in the duplicate ratio of the hyperbolic ones. Hence if 

 A?=zka, B2=^J {k being an indeterminate.) 



^2 52 



To determine its value, we resume the equation (1), x^ ~ 132 =^ 

 which gives ;^ ~ ^ = ^ ' whence k=a-h. 



Therefore the squares of the semi-axes of the hyperbola, are K'^=a (a-b), 

 W=b (a—b). From whence, by addition, A^ + B- = a^-b'^. 



It remains now to verify the confocahty of the two curves. If C is their com- 

 mon centre, and F the focus of the ellipse, we have CF- = d^-b\ by the property 

 of the ellipse, and if F' is the focus of the hyperbola, we have CF'2=A2 + B2 by the 

 property of the hyperbola. .-. if F and F' are the same point 



CF2 = CF'2 or a^-h'^ = A''+B\ 



But we have shown that this equation exists, and therefore the curves are confocal. 



Thus we have proved that the squares of the 

 co-ordinates of the point P, where the confocal 

 ellipse and hyperbola intersect, are, — 



-3 „ h' 



->.2_^- 



a+ b 



>y 



a + h' 



Therefore (1) CF'=.v^ +y^ = ^'^^=a''- ab + b^ 



Let CQ be the conjugate to CP. 

 .-. (2) CF' + CQ' = a' + b^ by a property of the 

 ellipse. 



Subtracting (1) from (2), we find CQ,'^ = ab. 

 Let CN be perpendicular on the tangent PN, then by another property of the 



