BY MARTIN GARDINER, C.E. 95 



8. The theorem inverse to that made use of in establishing 

 theorem 8 may be enunciated in the following manner : 



THEOREM 11. 



If two conies, S and T, have double contact (real or imaginary), 

 and that from any four points in the conic S there be drawn tangents 

 to the conic T of like or opposite rotatives just according as the points 

 lie on the same side or on opposite sides of the straight line (always 

 real) containing the points of contact of the conies : then will the 

 anhannonic ratio of the four points whence the tangents are drawn 

 be equal to the anharmonic ratio of the four points in which the 

 tangents touch the conic T, and also to the anharmonic ratio of the 

 other four points in which these tangents again cut the conic S. 



This theorem is given in a very imperfect form in Chasles' 

 " Geometrie Superieure," and also in Salmon's " Conies" where 

 its discovery is said to be due to Mr. Townsend, of Trinity 

 College, Dublin. The CORRECT THEOREM is now given for the 

 first time. 



9. It is evident, from this theorem, and from the preceding 

 portions of the paper, we can form theorems in respect to 

 extended data, analogous to those arrived at. I will give the 

 following one as an instance : 



THEOREM 12. 



If the n angular points of a closed n'gon rest on a curve S of the 

 second degree, and that its first n 1 sides pass through n 1 

 points, or that all or any number of them, not passing through fixed 

 points, are tangents of certain prescribable rotatives to fixed conies 

 having double contacts with S ; then if we deform the Tig on subject 

 to these imposed conditions, the envelope of the n^ side will be a conic 

 having double contact with S, or it will be a determinate point, just 

 according to the possibility of inscribing in S an open 2 (n 1) 

 'gon or a closed 2 (n 1)V OW whose two successive series of n 

 sides are distinct and meet in orderly succession with the n 1 

 entities in the manner prescribed for the closed rigons. 



