86 GEOMETEICAL RESEARCHES, 



21. The problem solved at the commencement of this paper 

 may be regarded as strictly analogous to the following one : 



" Given a system of n straight lines L I? L* 2 , .... L n in space, 

 and given also a second system of n straight lines K 1? K 2 , .... K^ 

 in space ; through the lines of the second system taken in order 

 to draw n planes forming a closed planes n'gon whose n angular 

 joints will rest on the n lines of the first system taken in order." 



To those who understand the homographic theory (and 

 possess the ability to conjure up figures in the air) the method of 

 solution is obvious ; and the following theorems and porisms are 

 evident consequences. 



THEOREM 5. 



If we can form three closed planes w'gons, such that the n 

 planes of each contain n fixed straight lines in space (each plane 

 containing a certain line), and the n joints rest on n other fixed 

 straight lines in space ; then will any point whatever in any 

 straight line of the second system of n lines be an answerable 

 position for an angular point of a closed planes n'gon fulfilling 

 the like conditions. 



THEOREM 6. 



If we can form one closed planes 2 w'gon, whose first n 

 planes and whose second n planes contain n fixed straight lines 

 (in space) taken in prescribed order, and whose first n angular 

 joints and second n angular joints rest on n other fixed straight 

 lines taken in prescribed order ; then we can deform this planes 

 2 Ti'gon so that its sides will contain the straight lines of the first 

 system, and its angular joints move along the straight lines of 

 the second system. 



PORISM 11. 



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

 n 2 straight lines <>f a second system of n straight lines in space : 



Innumerable straight lines (contained in the surface of a deter- 

 minable hyperboloid of one sheet) can be found, such that if we chose 

 any two of them, and draw a plane through each, we can find two 



