BY MARTIN GARDINER, C.E. 121 



20. I will now proceed to indicate methods by which we can 

 graphically find the positions for the first extremities of the closed 

 n'gons inscribable in a surface of the second degree so that the 

 sides of each will pass in order through n given points. And (in 

 doing so) I wo aid have the reader remember that when I speak 

 of a closed w'gon, or of any w'gon, I refer to a n'gon inscribed 

 in the surface whose sides pass in order through the n point of 

 the series ; and wheD I speak of a 2 n'gon, I refer to one inscribed 

 in the surface whose first n sides and whose second n sides pass 

 in order through the n points of the series, and are not coincident 

 in pairs. 



PROBLEM. 



Given a surface S of the second degree, and a series of n 

 points Oj, o 2 , . . . . o n ; to inscribe in the surface the closed w'gons 

 the sides of each of which will pass in order through the n fixed 

 points. 



Analysis of a first method of Solution. 



Suppose ^ 2 . . . . a n a^ to be a closed w'gon such as required, 

 the sides a t a^ a 2 a y .... a n o^ passing through the respective 

 points o v o 2 , .... o n . 



Now let us see whether we could reduce the inscription of this 

 closed w'gon to that of another having a less number of sides. 



If G I , o 2 , 3 are in one straight line we know that in this line 

 we can determine a point g such that a^ a ]L g will be a straight 

 line. And therefore evidently we can reduce the solution of the 

 problem to the inscription of the closed (n 2)'gon ^ & 4 a g .... 

 a n a^ whose sides pass in order through the n 2 known points 



If o v o a , o 3 be such that each one of them has its polar plane 

 passing through the other two, we know that the point h which 

 is the pole of the plane o l o 2 o g is such that a^ a^ h is one straight 

 line. And therefore evidently we can reduce the solution of the 



