situated in a space of four dimensions should lie on a quadric. 211 



on the quadric which belong to different families sometimes have 

 a common line, but geuerally have no common point whatever ; 

 but two 80S that belong to the same family always have one 

 common point. 



Since the lines a, b, c, d, e, lie on Q, through each pass two S 2 's 

 that lie wholly on Q. I confine my attention to five out of these 

 ten S 2 s which are members of the same family, and arrive at the 

 theorem that through the five lines a, b, c, d, e, can be drawn five 

 S 2 's in S 5 of which each two have one point common. 



It is true conversely that, if five S. 2 's situated in S 5 be such 

 that each two of them have one point in common, they lie on 

 a quadric. For a quadric in S 5 can be chosen to fulfil twenty 

 conditions, and may therefore be made to pass through the ten 

 points each common to one pair of the five $ 2 's, and ten others 

 taken at random, two on each of the five $ 2 's. Since this quadric 

 has been made to pass through six points on each of the five S 2 's, 

 each of the five S 2 's must lie wholly upon it, and every 8 4 in the 

 S 5 will cut them in five straight lines which lie on a quadric in 

 the $ 4 . Thus we have the theorem : — 



In order that five lines in S 4 should lie on a quadric, it is 

 necessary and sufficient that they should be sections of five 8 2 s 

 in 8 5 of which each two have one point in common. 



For the sake of investigating the consequences of this pro- 

 perty, let A, B, G, D, E, be five S 2 's situated in S 5 of which each 

 two have a common point. It follows that each two of them lie 

 in an $ 4 ; and I shall denote the point common to A and B by 

 (a/3) and the $ 4 which contains A and B by (AB). Every space 

 of four dimensions in the S 5 intersects A, B, G, D, E, in five lines 

 a, b, c, d, e, which lie on a quadric, and intersects the ten $ 4 's 

 (AB), (AG), ... etc. in ten S 3 's (ab), (ac), ... etc., each of which 

 contains two of the five lines a, b, c, d, e. The points (aft), (ay), . . . 

 etc. are lost when a section is taken, but the line which joins two 

 of them gives a single point, the 8 2 containing three of them a 

 line, ... etc. in the four-dimensional space by which the section is 

 made. For instance, the line joining (a/3) and (ay) lies in the 

 S 2 ,A and in the $ 4 , (BG); it follows that the space of four dimen- 

 sions which meets A, B, G, D, E, in the lines a, b, c, d, e, will 

 meet the line joining (a/3) (ay) in the point where a meets the 

 S 3 (bc). 



Consider now the S 4 which contains the points (ay), (ye), (e/3), 

 (/3S), (Sa), and the lines which join each pair of them : five of 

 these lines join two points such as (ay) and (/38) represented by 

 symbols which have no letter common and are passed over; the 

 other five are of the kind just considered, and, since they lie in 

 an S i} their sections by a space of four dimensions are five points 

 situated in an $,. It is in fact established that 



