BY MARTIN GARDINER, C.E. 77 



any answerable straight line L must pass through the point of 



91 1 



intersection of the lines o , o and o _ a . 



n 1 n n 2 n2 



And if we take the point in which the straight line o n _ 1 o n _ 2 

 cuts L _ as an angular point b _ of an answerable ft'gon, we 



71 4 o ir n 4 



can find the corresponding point b l in L r Moreover, it is evident 

 that b n must be co-incident with the known intersection of o n _ 2 

 o n _ 1 and & x o n ; and therefore any answerable straight line L n 

 must pass through the point of intersection of the lines o n _ 2 o n _ 1 

 and b 1 o n . Now let G I c a . . . . C B ^ be any other closed ^'gon, 

 having its n sides passing in the prescribed manner through the 

 n given points, and its first n 2 angular points c^ c g . . . . c n _ 2 

 resting on the n 2 given lines L I} L 2 , .... L n _ 2 . 



It is evident that if we draw the two straight lines \_ 1 ^ n _ l 

 and a n c n , and look on them as fixed, they will be answerable 

 positions for L and L M , because the three closed w'gons a 1 2 



a n a v \~b^ b n b v C 1 C 2 c n i falfil tlie P rescribed 



conditions. 



Hence (in the general state of the data) we may announce 

 the following 



PORISM 5. 



If a closed rfgon have its n sides passing through n given points, 

 and have its first n 2 angular points situated on n 2 given 

 straight lines : two points can be found, such that if we draw a 

 straight line from the first of these determined points through the 

 n 1 th angular point of the n'gon, and that we draw another straight 

 line from the second determined point through the n th angular point 

 of the rfgon, and look on these drawn lines as fixed ; then we can 

 deform the -rfgon so that its n sides will continue through the 

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

 lines composed of the n 2 given ones, and the two determined 

 ones. 



