92 GEOMETRICAL RESEARCHES, 



4. If o . j be any point in the straight line containing the 

 first extremities a l and b l of the two inscribable closed fi'gons, 

 whose sides pass in order through the n fixed points o^ o 2 , .... o n 

 of a series of ^points, it is evident^ a^ ... a n a l b l b^ ... b n b l a 1 

 is a closed 2 (n -+- 1) 'gon whose two series of n + 1 successive 

 sides pass in order through the n + 1 points o^ o a , .... o n , o l 



Moreover, it is evident (by making a 1 the first point of an 

 inscribed 2 (n -f- 1) 'gen) that if the point o 1 be such as to 

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

 inscribable closed 2 (n + l)'gon whose sides pass in the prescrib- 

 ed manner through the n -f 1 points o^ o 2 o n , o. l9 then will 



, i be situated in the straight line containing a^ and b^ Hence 

 we infer the following theorems : 



THEOREM 6. 



If in a curve of the second degree we inscribe any three distinct 

 tigons^ e 2 ...e^, ^ f a . . . . f i+1 , g 1 g 2 .... g n+1 , the sides of each 

 of ivhich pass in order through n given points ; then will the three 

 pairs of straight lines GI f n+1 , f x e i+1 and G L g n+1 , g L e n+1 and 

 f, g , ,, g, f , ., cw# eacA other in three points in the straight line ivhich 

 contains the first points of the two inscribable closed logons whose 

 sides pass in order through the n 



This theorem is otherwise evident, since homographic 

 pencils having a common vertex in a conic, are such that by 

 taking any two pairs of the corresponding radiants and coupling 

 them transversly they will form an involution with the double 

 radiants of the pencils. 



THEOREM 7. 



Any point in the straight line containing the first points of the 

 two closed (n 1) 'gons inscriptible in a curve of the second degree 

 so that the sides of each pass in order through n 1 fixed points, 

 will be an answerable position for the n th point of the series so as to 



