BY MARTIN GARDINER, C.E. 93 



render any point in the curve an answerable position for the first 

 extremity of an inscribable closed 2 n'gon whose two successive series 

 of u sides w ill pass in order through the n points. And no point 

 outside the straight line containing the first extremities of the two 

 closed (n \) 'gons will possess this property. 



5. If we have a curve of the second degree and the first 

 n 2 points of a series of n points, and that we assume the 

 n 1 th point Off^i anywhere in the straight line containing the 

 first extremities p l and q l of the inscribable closed (w 2) 

 'gons whose sides pass in order through the n 2 given points 

 and that we assume the n th point o n co-incident with the pole of 

 the straight line containing the first extremities of the inscribable 

 closed (11 1) 'gons whose sides pass in order through the n 1 

 points Op o 2 , .... o n _ 2 , A _ 1 > it is obvious from theorems 6 and 2 

 that any point in the curve is an answerable position for the 

 first extremity of an inscribable closed w'gon whose sides pass in 

 order through the n points of the series. Moreover, it is evident 

 (by considering p 1 or q l as first points of inscribable w'gons) the 

 w th point o n lies in the line p l q 1 o n _ r Hence 



PORISM 1. 



If any closed n'gon inscribed in a a curve of the second degree 

 have its first n 2 sides passing in order through n 2 given 

 points ; then a straight line xx can be found such that if we look on 

 the points in which it is cut by the n 1 th and n th sides of the rigon 

 as fixed i we can deform the n'gon so that its angular points ivill 

 move along the curve and its n sides continue through the n fixed 

 points composed of the n 2 given ones and the two determined 

 ones. 



HHJ^ In respect to this porism it is well to remember that the 

 straight line xx cuts the given curve in the answerable positions 

 (real or imaginary) for the first extremities of the inscribable 

 closed (n 2) 'gons, the sides of each of which pass in order 

 through the n 2 given points. And when the problem of the 

 inscription of the closed (n 2) 'gons is porismatic, then o 

 and o must be coincident though otherwise unrestricted in the 

 plane. 



G 



