74 GEOMETRICAL RESEARCHES, 



When such is the case, it is evident the line b l 6 n _ 1 

 (or XX) becomes co-incident with L MI , and that the line a 1 a- n _ l 

 (or YY) becomes co-incident with 1^). 



It is also evident o % _ 1 can be taken anywhere in L n _ 1 ; and 

 that o n can be taken anywhere in L r It is also evident the n th 

 angular points of the %'gons are all co-incident in the common 

 point of intersection of the lines L 1? L W _ I and L B . Moreover, 

 we arrive at this under the hypotheses that all the given straight 

 lines do not pass through one point. 



Hence we infer the following theorem : 



THEOREM 2. 



Given n straight lines L,, L 2 , L , of which the first, the 



n 1 th , and the n th pass through one point, the rest not all 

 passing through this point ;. and given likewise n 2 point o v o 2 , 



w _ 2 > f a ser ws of n points : if positions for the n 1 th and 



n th points o % _ 1 and o n of the series be such that any point whatever 

 in the line L is an answerable position for the first angular point of 

 a closed tig on p 1 p 2 .... p M p 1? having its n angular points p p ... 

 P M on the n respective lines L p L 2 , .... L M , and its n sides p p 

 p 2 p 3 , . . . . p n p 1? passing in order through the n respective points 

 o,, o , . . . . o , then will the points o , and o be situated in the 



1 2 n 7 * n 1 n 



given straight lines L W _ I , L /t , each in each respectively. 



5. When all the given straight lines L , L 2 , . . . . L , intersect 

 in one point q, then it is obvious that the infinitely small %'gons 

 a \ a 2 " a n i a n a i> & i ^2 ' ' ' ' ^n ^i are not Distinct, and that 

 * i ^ M _i an( i a i a w _! (XX and YY) are not determined in position. 

 In this case we may evidently assume o^_ 1 anywhere in the plane 

 (because a^ <^ n _ l is not restricted in position) and find corres- 

 ponding answerable positions for o in the intersection of the 

 sides c 1 c n , d l d n , of any other two closed w'gons G I c 2 . . , . o n _ l 



