174 BOOK OF MATHEMATICAL PROBLEMS. 



Take two circles which have this property, and let R y r be 

 their radii, 8 the distance between their centres ; then 



Reciprocate the system with respect to a point at distances x, y 

 from the centres, and let a be the angle between these distances. 

 Then a will be the angle between the major axes of the conies, 

 and if 2c lt 2c t be the latera recta, e iy e t the eccentricities, 



r ' JK' r' K' 

 whence R* 2i?r = x' + y 1 - 2a?t/ cos a ; 



2 e 



* * 



= a + ~4 -- cos a > 

 , C S* C * C . C . C * 



or c* 2c t c f = e'c* + e'c* - 2e l a c 1 c f cos a ; 



the relation required. 



If a system of confocal conies be reciprocated with respect to 

 one of the foci, the reciprocal system will consist of circles having 

 a common radical axis, the radical axis being the reciprocal of the 

 second focus, and the former focus being one of the limiting points 

 of the system. 



The reciprocal of a conic with respect to any point in its 

 plane is another conic which is an ellipse, parabola, or hyperbola, 

 according as the point lies within, upon, or without the conic. 

 To the points of contact of the tangents from the point correspond 

 the asymptotes, and to the polar of the point the centre of the 

 reciprocal conic. So also to the asymptotes and centre of the 

 original conic correspond the points of contact and polar with 

 respect to the reciprocal conic. 



As an example, we may investigate the elementary property 

 that the tangent at any point of a conic makes equal angles with 

 the focal distances. The reciprocal theorem is, that if we take 

 any point in the plane of a conic, there exist two fixed straight 

 lines (the reciprocals of the foci), such that if a tangent to the 



