BY MARTIN GARDINER, C.E. 103 



a closed (n 2)'gon r s^a^ . . . . a^ T I having its successive angular 

 points on the n 2 known straight lines B , S 1? L & , .... L^. 



And thus, step by step, we can reduce the problem until we 

 make the solution dependent on that of exscribing a closed 4'gon 

 r 2 s a n \ a r * having its angular points on the four 

 known straight lines R , S T , L , L . 



f Z fn 2 n 1 n 



Now our object is to find out how to form this closed 4'gon. 



Suppose we draw the straight line which contains the inter- 

 section of the lines R and L^, and the intersection of S* and 

 L _ ; and suppose # to be the point in which this line cuts the 



Then as the point in which the other tangent from x cuts the 

 side a a n must be on each one of two known straight lines 

 (one through the intersection of B A 9 and L , and the other 

 through that of S A 9 and L ) it is known. And therefore the 

 tangent a a n through it is known, &c. Moreover it is evident 

 the point x is such that if L and L^ pass through it, the 

 problem of the construction of the closed 4'gon will be porismatic, 

 and therefore also the construction of the exscribed closed ?i'gon. 

 Hence we may announce the following porism, which is the dual 

 of porism 1 already given : 



PORISM 2. 



If a closed n'gon be exscribed to a fixed conic, and have its first 

 n 2 angular points on n 2 faed straight lines ; a point x can be 

 found, such that if we draw straight lines from it through the 

 n 1 th and n th angular points of the n'gon, and regard these two 

 lines as fixed, we can deform the n'gon so that its angular points will 

 move along the n fixed straight lines, and its n sides continue tan- 

 gents to the fixed conic. 



