MR TALBOT ON CONFOCAL CONIC SECTIONS. 57 



any values] ; then the chord of PQ', the middle arc of one series, equals the 

 chord of P'Q, the middle arc of the other series."* 



Demonstration.— In each series of circles the radii have a common difference 

 h, which may be called the interval between them. P and P' are two points in 

 the same ellipse of which S, H, are the foci, because in passing from P to 

 P', SP increases by one interval h, and HP diminishes by the same, therefore 

 SP + HP remains constant. 



By similar reasoning Q and Q' are two points in a second ellipse having same 

 foci. Moreover P and Q are two points in a hyperbola of which S, H, are foci ; 

 because in passing from P to Q, both HP and SP increase by one interval h y and 

 therefore HP — SP remains constant, and equal to HQ — SQ. 



By similar reasoning P' and Q' are two points in a second hyperbola having 

 same foci. Therefore P, P', Q, Q' are the intersections of two ellipses and two 

 hyperbolas, all confocal. Therefore the diagonals PQ', P'Q are equal to each 

 other.— Q.E.D. 



This property of the circle should be readily demonstrable by Euclid's 

 Elements ; a simple geometrical demonstration is, however, at present a desi- 

 deratum. 



* The second or middle circle of one series must be understood to be limited by the first and 

 third circles of the other. 



VOL. XXIV. PART I. Q 



