BY MARTIN GARDINER, C.E. 73 



line L .. Now if we take any point in ^ b n _ l as a position for 

 O M _ I , and that we draw o c n __ l to cnt L^ in c^ ; and then draw 

 c c, to cut a a in o : it is evident o . and o so determined 



n 1 1 n 1 n * i *> 



are answerable positions : for we have the three closed w'gons, 



c n c i Dialling the 



conditions. 



Hence in the general states of the data, we have the following 

 porism : 



PORISM I. 



If a closed n'gon have its n angular points on n given straight 

 lines, and have its first n 2 sides passing through n 2 given 

 points; then its n 1 th and upsides will cut two determinate 

 straight lines XX, TY in points o n _ 1 and o n , such that if we look 

 on these points a$ fixed, we can " deform" the n'gon so that its 

 angular points will move along the n given straight lines, and its 

 sides continue through the n fixed points. 



In respect to this porism it may be proper to observe that 

 when the two lines L , and L are parallels (and therefore a 



n 1 n n1 



at infinity) and that a^ is at infinity on 1^, then the line a^ a^_ 1 

 is at infinity ; and the point o n where G I c n cuts a l (^ n _ 1 is at in- 

 finity ; and therefore the last sides of the w'gons (the sides 

 through o ) must be all parallels to each other. And if L , L 



72 A ft fl 1* 



and L be parallels, and that a { is at infinity, then b l b n _ 1 and 

 a a n _ l are at infinity ; and it is evident we can assume c^ any- 

 where on L^, and that c n _ l c- n and c n c l will continue through o n _^ l 

 and O M at infinity. 



4. We arrived at porism 1 under the hypotheses that L X , '^ n _ v 

 and L do not intersect each other in one point ; and therefore it 

 is necessary to inquire into the nature of the relations when these 

 three lines pass through one point. 



