BY MAKTIN GARDINER, C.E. 83 



Then since L X , L 2 , and Q l pass through one point, it follows 

 (by a well-known porism) that p q l cuts the line o l o in a known 

 point /-j. And since L 3 , L , and Qj pass through one point, it 

 follows that p q cuts the line o., o in a known point . 



Hence evidently the solution of the problem is dependent on 

 that of forming a closed (n l)'gon p n p^ .... p n p 1 whose 

 successive sides pass through the n 1 known points r , ^, o 4 

 . . . . o n , and whose angular points taken in order rest on n 1 

 known straight lines L^, Q 1? L 4 , .... L n . 



And, proceeding with this closed (n l)'gon as we have done 

 with the closed w'gon, we can reduce the solution of the problem 

 to the forming of a closed (n 2)'gon p l q 2 p . .. . P n p^ whose 

 sides pass in order through n 2 known points r g (in 

 line s 1 r 1 ), s 2 (in line s l o^), o_, o g , ---- o^ and whose angular 

 points rest in order on n 2 known straight lines L I} Q 2 , L gj 



And thus, step by step, we can proceed until we make the 

 solution depend on that of forming a triangle p 1 q n _^ p n p v whose 

 sides pass in order through three known points ^ n _3, s n _ 3 > n > an( ^ 

 whose angular points rest in order on three known straight lines 



18. To arrive at porismatic relations, we will (as in last 

 method) consider the question at that point in the investigation 

 where we have reduced the solution of the problem to the forming 

 of the quadrilateral p g n _ 4 p n _ l p n p^ having its angular points, 

 taken in order, on known straight lines L I} Q n4 > ^ J n _ 1 > ^ 

 and its successive sides passing through four known points r^ ^ 

 s ,o , o . For, as the method leaves the points 0,0 , and 



n 4 n 1 n n - 1 



the line L unimplicated, it follows that if we give these such 



