344 Mr. A. Cayley on the Porisin of 



inscribed couic, then the tangents to the circumscribed conic at 

 ABC meet the opposite sides EC, CA, AB in points lying in the 

 chord of contact, the lines Act, B/3, Cy meet in the pole of con- 

 tact, and so on. 



In the case of a quadrangle, if ACEG be the quadrangle, and 

 ^'> (f> f> h the points of contact with the inscribed conic, then 



the tangents to the circumscribed conic at the pair of opposite 

 angles A, E and the corresponding diagonal CG, and in like 

 manner the tangents at the pair of opposite angles C, G and the 

 corresponding diagonal AE, meet in the chord of contact. Again, 

 the pairs of opposite sides AC, EG, and the line dh joining the 

 points of contact of the other two sides with the inscribed conic, 

 and the pairs of opposite sides AG, CE, and the line i/" joining 

 the pairs of contact of the other two sides with the inscribed 

 conic, meet in the chord of contact. The diagonals AE, CG, 

 and the lines bf, dh through the points of contact of pairs of 

 opposite sides with the inscribed conic, meet in the pole of con- 

 tact, &c. 



The beautiful systems of ' focal relations ' for regular polygons 

 (in particular for the pentagon and the hexagon), given in Sir W. 

 11. Haniilton\s Lectures on Quaternions, Nos. 379-393, belong, 

 it is clear, to polygous which are inscribed in and circumscribed 

 about two conies having double contact with each other. In 

 faot, the focus of a conic is a point such that the lines joining 

 such point with the circular points at infinity [i. c. the points in 

 which a circle is intersected by the line infinity) arc tangents to 

 the conic. Iti the case of two concentric circles, these are to be 



