( 53 ) 



VI.— Note on Confocal Conic Sections. By H. P. Talbot, Esq. 



(Read 17th April 1865.) 



A short paper of mine on Fagnani's theorem, and on Confocal Conic Sections, 

 was inserted in the twenty-third volume of the Transactions of the Royal Society. 

 Some of the conclusions of that paper can, however, be obtained more simply, as 

 I will now proceed to show. 



I will, in the first place, resume the problem— 

 " To find the intersection of a confocal ellipse and hyperbola." 

 Since the curves have the same foci, and therefore the same centre, let the 

 distance between the centre and focus be called unity, since it is the same for 

 both curves. Let a, b, be the axes of the ellipse, A, B, those of the hyperbola 

 Then we have 1 = a 2 — b 2 = A 2 + B 2 , which equation expresses the condition of 

 confocality. 



The equation of the ellipse will be -g- + yg- = 1, and that of the hyperbola 



2 2 



-^2 - ^2 = 1. But at the point of intersection x and y are the same for both 



curves. We have therefore two equations from which to determine two unknown 

 quantities. The result is one of unexpected simplicity. (See Vol. XXIII. p. 295.) 



x = Aa , y = ~Bb . 



The theory of the Conic Sections has been so much studied, that I can scarcely 

 suppose that a result of such extreme simplicity, and so fruitful in remarkable 

 results, should not have occurred to some previous mathematician. I have no 

 had the opportunity of late of consulting many treatises on the Conic Sections, 

 but in those which I have examined I have not found this theorem. 



I will not here repeat the proof which I gave of it in my former paper, since 

 it suffices to show that these values of x and y satisfy both the given equations. 



In fact, if we put x = Aa and y = B5, the equation -^- + ^ = l becomes 



A 2 + B 2 = 1, which is true, and the equation -^ - ^ = * becomes of —b 2 = 1, 



which is also true. 



This fundamental theorem being thus established, I shall proceed to show how 

 easily the theorem of page 296 follows from it, viz., 



VOL. XXIV. PART I. P 



