MR TALBOT ON CONFOCAL CONIC SECTIONS. 



55 



For, by a theorem in my former paper (p. 295), if two confocals intersect, the 

 focal distance of their intersection equals the distance between their vertices. 



A A 



V' V H 



B B 



Fig. 1. 



Thus, if AP be the ellipse, and VP the hyperbola (fig. 2), AB the major axis, 



Fig. 2. 



S, H, the foci ; SP will be equal to AV, and HP to BV. 

 Therefore in fig. 1 we have 



HF=VB, HP=VB, SQ-=A'V, SQ=AV 



.-. HP'-HP=VB-VB=VV + BB' 



and SQ'-SQ=A'V-AV=VV + AA' 



.-. (since AA'=BB') HP'-HP=SQ'-SQ. 



This, of itself, is a curious theorem. The other follows immediately from it. 



For, in the particular case, where HPP is a straight line, HP'— HP is the 

 diagonal PP', which is always equal to the diagonal QQ'. 



Therefore, in this case, SQ'— SQ = QQ', and therefore SQQ' is a straight line, 

 which was to be proved. 



Another theorem which I have found concerning these quadrilaterals is the 

 following. 



" If one of the diagonals is a tangent to the inner ellipse, the other diagonal 

 is so likewise." 



I omit, for the present, the demonstration of this, which is not difficult. 



I deduced from Graves's theorem in my former paper the remarkable conse- 

 quence, that if a triangle or other polygon is inscribed to the one, and circum- 



