BY MARTIN GARDINER, C.E. 75 



c c,cJd c .....d , d cL. fulfilling the conditions (because the 



Tl 1 L t n 1 ft A * 



three ?&'gons including these two and the n'gon a^ a . . . 

 & n \ fulfil the conditions). Moreover it is evident that if we 

 draw any straight line XX through q ; then for all points o 

 assumed in XX 3 the corresponding points o will be on a deter- 

 minable straight line passing through q. 

 Hence we have the following porism : 



PORISM 2. 



If all the angular points of a closed n'gon move on n given straight 

 lines meeting in a point q, and that all its sides but the n th pass 

 through determinable fixed points ; then if the point through which 

 the n 1 th side passes be situated on a known straight line passing 

 through q, so will the point through which the n th side passes be on 

 a determinable straight line passing through q. (See Mulcahy's 

 " Modern Geometry," page 77.) 



6. If we have the n 2 given points o v o v .... o n _ 2 , in direc- 

 tum with the intersection of L X and L n ; then (no matter how 

 general otherwise the given lines may be) it is evident the straight 

 line X X will pass through these n 2 given points ; and therefore 

 o n _ 1 will be in directum with the n 2 given points. 



Hence we may announce the following porism : 



PORISM 3. 



If all the angular points of a closed iCgon move on given straight 

 lines, and all its sides, except one, pass through given points which 

 lie in a straight line passing through the intersection of the lines on 

 which the extremities of the free side move, then this side also passes 

 through a fixed determinable point. (See Mulcahy's " Modern 

 Geometry," page 77.) 



7. When the given points o 1 , o a , <? n _ 2 , are in one 



straight line, and that the three lines L J5 L n _ 1 , L n , pass through 

 one point in this line ; then XX and YY are evidently both 

 coincident with the straight line o v o 2 , .... o n _ o and we infer 

 the following theorem : 



