82 GEOMETRICAL RESEAECHES, 



16. To arrive at porismatic relations of the data let us consider 

 the question when we arrive at that point in the investigation 

 where we have reduced the solution to the forming of a quadri- 

 lateral p l r n _^ s n _ 4 p n p 1 whose sides pass in order through the 

 known points o^ \_^ n _^ n -> and whose angular points rest in 

 order on the known straight lines L p R n _ 4 , &_# ^V It is 

 evident our method of investigation leaves the point o and the 

 lines L^ 5 Lp unimplicated ; and therefore it follows that if we give 

 such positions to these as will render the forming of the quadri- 

 lateral porismatic, then will the problem of the forming of the 

 w'gon p 1 p 2 " P n P 1 be porismatic. Now, if we assume any two 

 positions for L I and L n which pass through q the point of inter- 

 section of R , and S , then (by a well-known porism) we can 

 find a position for o which will render the problem of the form- 

 ing of the quadrilateral porismatic. 



Hence we may announce the following porism : 



PORISM 8. 



If a closed rfgon have all its angular points Taut the 1 st and 

 n 111 resting on given straight lines, and have all its sides 'but the 

 II th passing through given points : a point q can be found, such 

 that if we look on straight lines from it through the 1 st and n^ 

 angular points as fixed, and that we deform the n'gon so that its 

 n angular points will move along the n straight lines composed of 

 the n 2 given ones and the two described ones, and so that its 

 first n 1 sides continue through the n 1 given points, then will 

 the n^or last side of the n'gon continue through a fixed determin- 

 alle 



17. The problem can be investigated in the following manner : 

 Suppose we draw the straight line Q x containing the inter- 

 section of L I and L 2 , and that of L 3 and L 4 . And let q l be the 

 point of intersection ofp 2 p 3 and Q^. 



