212 Mr Richmond, On the condition that five straight lines, etc. 



If five lines a, b, c, d, e, situated in a space of four dimensions, 

 lie on a quadric, the five points which are the intersections of 



the line a with (cd), the S 3 containing c and d, 



b ... (de), 



c ... (ea), 



d ... (ab), 



e ... (be), 



lie in a space of three dimensions. 



The point of intersection of a and (cd) is the point of a where 

 it is met by the line (acd), the unique line which intersects a, c 

 and d. There are six points such as this on each of the five lines 

 a, b, c, d, e, that is to say thirty in all, and by taking the different 

 cyclical orders of the five lines we find twelve $ 3 's, each containing 

 five of the thirty points. The twelve $ 3 's fall into two groups of 

 six, corresponding to the following arrangements of order of the 

 five lines a, b, c, d, e ; — 



First group : 



(abode), (abdec), (abecd), (acbed), (aedbe), (adbce) ; 

 Second group : 



(acebd), (adebe), (aedbc), (abdee), (adecb), (abedc); 



one S 3 from each group passing through each of the thirty points. 

 Another form of the condition that the five lines a, b, c, d, e, 

 lie on a quadric may be mentioned, viz. that the three pairs of 

 points of the line e where it meets the $ 3 's 



(ab), (cd), —(ac), (bd), —(ad), (be), 



should be in involution. 



