96 GEOMETRICAL RESEARCHES, 



PROBLEM. 



10. Given a conic and a series of n points o 1 , o^ o n ; to 



inscribe the closed rc'gons, in the conic, the sides of each of which 

 will pass in order through the points. 



Now it is evident that if we can find the answerable positions 

 for the first angular points of the closed rc'gons our object will 

 be attained. 



First method of solution. 



If n be an odd number, it is obvious from theorem 10 that we 

 can replace the point o and the side of any of the closed w'gons 

 which passes through it by two other points and two chords 

 cutting each other in the curve and passing through the deter- 

 mined points and the extremities of the side which passed 

 through o . Hence we may consider n to be an even number. 



/sis. Let a l a^ a^ a l be one of the inscribed closed 



ft'gons whose sides pass in order through the n points o > o ... o 



Looking on the first four sides of this w'gon, let us designate 

 by i l the intersection of the lines o 1 o^ and o^ o^ ; and let b l be 

 the point in which ^ a^ again cuts the conic. Then, a a a b a 

 being an inscribed 4'gon, it is evident from theorem. 8, that the 

 point r^ in which a^ b l cuts the line o l o 2 i^ is known. And, since 

 3 4 5 b 3 is an inscribed 4'gon, and that o o i are in one 

 line, therefore the point ^ in which the line a g ^ cuts the line 

 3 4 *i * s known. Hence we perceive that the inscription of the 

 closed n'gons is reduced to that of the inscription of the closed 

 (n 2)'gons a^ b 5 .... w a l whose sides pass in order through 

 the n 2 known points r^ s^ o^ .... o n . 



And thus, step by step, we can reduce the number of sides 

 repeatedly by two until we arrive at a closed 4'gon a l b^^ a n _ 1 

 a^ o^ whose sides pass in order through four known points r , 2 



