102 GEOMETRICAL RESEARCHES, 



the n points, then will the product of the perpendiculars from a l and 

 a on X and Y, respectively , be of constant magnitude. 



The lines are equidistant from the centre of the circle, and 

 are those corresponding to infinity in the homographic figures to 

 which the extremities of the inscribable n'gons belong. 



X. ,14{ ]By the well-known process of reciprocation we can at 

 once^form theorems which are the " duals " or "correlatives" of 

 "tJiQ? & investigated, or we can arrive at them by steps correlative 

 to those already used. It is also obvious we can make the 

 solutions of the problems subservient to the solutions of their 

 " duals," or we can arrive at solutions to the dual problems by 

 steps correlative to those used. As an example of the latter 

 mode of proceeding, I will enunciate the dual problem, and give 

 the method of solution correlative to the first. 



PROBLEM. 



To exscribe a closed ^'gon to a given curve of the second 

 degree, so that its n successive angular points will be situated 

 in n given straight lines L , L 2 , .... L M taken in order. 



Analysis. Let n be considered an even number ; and let 

 a l 2 .... a n j be an exscribed closed M-'gon whose angular points 

 a v a. 2 , .... a n rest on L I? L/ 2 , .... L n respectively. 



Suppose we draw a straight line I through the points of 

 intersection of L X and L 2 , and of L S and L 4 ; and let i be the 

 point in which this line cuts a 2 a y Then if r l be the point in 

 which the other tangent from i cuts a l a^ it follows that the 

 straight line R X through ^ and the intersection of L X and L 2 is 

 known. And if s 1 be the point in which the other tangent from 

 i cuts a^ a^, it follows that the straight line S through 5 and the 

 intersection of L q and L, is known. 



o 4 



Hence we see that the problem is reduced to the exscribing of 



