BY MARTIN GARDINER, C.E. 91 



THEOREM 3. 



If in a conic S there can be inscribed one open 2 n'gon whose 

 first n sides and whose second n sides pass in order through n fixed 

 points, then will the closing chords of all the inscribable logons whose 

 sides %)ass in order through the n fixed points be tangents to a conic 

 T having double contact with the given conic S, and they will be of 

 Hike or unlike rotatives in respect to T (reckoning from the final 

 extremities of the logons J just according as the final extremities of 

 the logons are on the same side or on opposite sides of the line 

 of contact. 



THEOREM 4. 



If there be a curve of the second degree and a series of n points 

 and that the problem of the inscription of closed n'gons the sides of 

 each of ivhich pass in order through the n points is non-porismatic, 

 there are two and but two answerable positions for the first extremi- 

 ties of such closed rigons. 



THEOREM 5. 



If three closed n'gons be inscribable in a curve of the second 

 degree so that the sides of each pass in order through n fixed points 

 of a series of n points ; then will any point in the curve be an 

 answerable position for the first extremity of an inscribable closed 

 utgon whose sides pass in like manner through the n points. 



This theorem is otherwise evident from the well-known 

 relations of homographic divisions in a conic. It can also be 

 easily deduced from the formula of theorem 1 by supposing a ^ 

 I , and C M , to be co-incident with a^ b^ and c^ respectively. 



For as 



= 1, and that -- : -- = 1 



it would follow that if d l and d be supposed distinct, the chord 

 f/ x d . l should pass through the points of intersection of Aj with 

 B, and 0, 



