BY MARTIN GARDINER, C.E. 87 



points R and S, one in each plane, and through these points draw 

 innumerable pairs of corresponding straight lines (the line through 

 each point being in the plane in which the point lies), such that if we 

 chose any pair of the corresponding lines as the n 1 th and n th straight 

 lines of Hie second system, ive shall render porismatic the problem of 

 the construction of the closed planes ri'gon, whose n planes contain 

 the n lines of the second system, and whose n angular joints rest on 

 the n lines of the first system. 



PORISM 12. 



Given the first n 1 of a system of n straight lines in space, 

 and given also the first n 1 of a second system ofn straight lines 

 in space : 



If a closed planes n'gon have its first n 1 angular joints on 

 the respective n 1 lines of the first system, and its n 1 

 first planes containing the respective n 1 lines of the second 

 system ; two straight lines and a point in each can be found, such 

 that if from either of these two points we draw any staight line L 

 through the n th angular joint of the n'gon, and that through the 

 point where the other found line pierces the -D^ plane of the n'gon, we 

 draw any line K in that plane ; then by taking the lines L and K 

 as fixed n" 1 lines of the first and second systems, we can deform the 

 rigon so that its n planes will continue to contain the n straight lines 

 of the second system, and its angular joints move on the n straight 

 lines of the first system. Moreover, we can give any position to K , 

 and find innumerable corresponding ansiverable positions for L fall 

 in the surface of a determinable hyperboloid of one sheet.} 



The following theorems are also obvious consequences from the 

 theory of homographic figures : 



THEOREM 7. 



If there be three distinct closed straight line w'gons having 

 their first angular points in one straight line xx, each ?i'gon of 

 which has its sides passing through n fixed points, and its angular 

 points in n fixed planes ; then will any point in the straight line 

 xx be an answerable position for the first angular point of another 

 such closed w'gon, and any other point in the plane containing xx 



