94 GEOMETRICAL RESEARCHES, 



6 If we have a conic and any even number of points in one 

 straight line, it is evident the points in which this line cuts the 

 conic, are answerable positions for first angular points of inscrib- 

 able closed n'gons, whose sides pass in order through the n points. 



Hence from porism 1, we infer the following theorem : 



THEOREM 8. 



Ifn.be any even number, and that in a conic there be inscribed a 

 closed iigon having n 1 of its sides passing through n 1 fixed 

 points in one straight line; then will the remaining side pass through 

 a point in the same line, such that if we suppose it fixed, we can de- 

 form the closed rigon so that its angular points will move along the 

 curve, and its n sides continue through the n fixed points. 



And from theorem 2 or 7, we at once infer the following 



THEOREM 9. 



If n be any odd number, and that in a conic there be inscribed 

 any closed 2 rigon such that n 1 pairs of its opposite sides cut each 

 other in n 1 points situated in one straight line, then will the re- 

 maining pair of opposite sides cut each other in a point of this same 

 straight line. 



The particular case in which n = 3 is identical with 

 Pascal's famous theorem concerning an inscribed hexagon. 



7. In applying theorem 6 to the finding of the first angular 

 points of the inscribable closed 1'gons, the side of each of which 

 must pass through a given point o^ we immediately perceive that 

 the polar of the point o j cuts the curve in the answerable 

 positions for these angular points. And from this, and porism 1, 

 we have 



THEOREM 10. 



If we have a conic and three points, each point of which is the 

 pole of the straight line containing the other two ; then will any 

 point in the conic be an answerable position for the first angular 

 point of an inscribable closed Qgon whose sides pass through the 

 three points taken in any order whatever. 



