3-16 PROCEEDINGS OF THE AMERICAN ACADEMY. 



(i = 1, 2, 3, 4, 5, and k I m n is a permutation of the remaining numbers 

 after i is chosen). It is well known, however, that of these five relations 

 fij only three are distinct. 



Now, letting pjj; be the ten homogeneous coordinates of a point in a 

 space of nine dimensions *%, the points which represent lines in /S^ are 

 the points common to the five quadrics 



The intersection of these quadrics is known to be of order five and 

 dimensions six. In our discussions, then, we shall be interested only in 

 the points of this variety which we shall indicate by 4>. The following 

 properties can be easily verified. 

 (a) The hyperplanes 



5 



(■2) ^i^iPiv. = % /-=!, 2, 3, 4, 5, 



1 



cut $ in quadrics 1 7' of order two and dimensions four ; hence <E> contains 

 ex'' such quadrics. 

 ib) The >S; defined by 



Ihk = hPik + hPik' + hPik" 4- hPi 



trrr 



where p\ p'\ ;>'", p'" correspond to lines passing through a common 

 point, lies in 4). 



(c) The /S'a defined by 



Vik — hlhk + kPik' + JoPiu"\ 



where p, p'', p" represent lines which lie in the same plane, also lies 

 in <I>. It is easily seen that the planes (c) and the spaces (IS) do not 

 in general intersect. It can also be easily shown that two quadrics {(t) 

 intersect in a plane (c). 



A pencil of lines in ii^ is represented by a line which is contained in 

 ^. The points of a quadric {a) represent the lines of W^ which lie in a 

 space of three dimensions /SV A space (A) represents the lines in i^^ 

 which pass through a fixed point, and a plane (c) represents the lines 

 which lie in a fixed plane. 



The lines which cut a fixed line />«/ are represented by the intersection 

 of ^ with its tangent /S',; at the point ^7^^'. (Here^^js' is used to denote 

 a line in aS!j or a point on 4>.) 



(o) 2 ,,„^'" = 0- 



