98 GEOMETRICAL RESEARCHES, 



The result of these operations is obviously an inscribed closed 

 w'gon a l & 2 & 3 . . . . b n a /l whose first n 1 sides are parallels to 

 known straight lines Q I o^ p 2 o^, P$ " P n _^ <> n -, and whose 

 last side b a^ passes through the known point p . 



Now n 1 being an odd number, we know that the chord 

 b n a 1 will be parallel to a fixed determinable direction ; and 

 therefore, as the side is also a transversal through the point 

 p n , we know the point a l of its intersection with the conic. 

 Moreover, it is obvious that the point b n will also be an answer- 

 able position for the first angular point of an inscribable closed 

 n'gon fulfilling the conditions. It is also evident that when the 

 point p n is at infinity and indicated by the infinite production of 

 the chords b n a [7 the problem will be porismatic. 



Third method of solution. 



Theorem 6 intimates to us Poncelet's elegant method of 

 arriving at the first angular points of the closed w'gons fulfilling 

 the imposed conditions. Inscribe any three distinct w'gcns 



a i a 2 ' VK' & 1 5 2 *n+l' C l C 2 ' n+l' the U sideS f 6ach f 



which pass in order through the n given points o lt o 2 , .... o 

 find *', the point of intersection of the chords a. b and b. a 



1 n+l 1 tt-j-i ? 



find k, the point of intersection of the chords a 1 c and 

 c i a n 4-i ' ^ nd ^ ^ e P^ n ^ ^ intersection of the chords b^ c 

 and G I b._ l . Then will the three points *, k, I, be in one straight 

 line which is such that its points of intersection with the conic 

 (real or imaginary as may be) are answerable positions for the 

 first angular points of the inscribable closed ft' 



This method holds whether n is odd or even, and from 

 the present paper it is obvious it holds in the following more 

 extended problem : Given a conic S and n entities G I} o^ .... o^ 

 any number of which represent given points, and the rest given 

 conies having double contacts with S ; to inscribe in S the closed 



