hyper-planes in four -dimensional space. 73 



as any other point (or any other plane). Through each point five 

 planes pass, as given by the table 



1, 2, 8, 4, 5; 



123 1, 2, 3, 1234, 1235; 



16 1, 1345, 1245, 1235; 1234; 



and the points and planes admit simultaneous groups of permuta- 

 tions for which the tactical relations of this table are unaltered. 

 The order of this group is 2* . 5 ! ; and as it affects the planes it is 

 generated by 



(1, 2345) (2, 1345); 

 (1, 2) (2345, 1345); 

 (1, 2, 3, 4, 5) (2345, 1345, 1245, 1235, 1234). 

 It contains an Abelian group of order 16, as a self-conjugate sub- 

 group, and in respect of this is isomorphic with the symmetric 

 group of degree 5. 



2. Take now six planes, 



1, 2, 3, 4, 5, 6, 

 all passing through the base point ; and on the 20 lines of inter- 

 section of each set of three mark arbitrary points 123, etc. With 

 each set of five planes, out of the six, carry out the above construc- 

 tion, so that there arise fifteen fresh planes 1234, etc.; and thirty 

 fresh points 16, etc. For convenience of reference the set of 

 (1 -I- 20 -I- 30 = 51) points thus arrived at may be denoted by 8q. 

 The figure so far is obviously not complete in the sense already 

 explained. That part of the figure which lies in the plane 

 (three-dimensional space) 1, consists of 16 points, 0, 123, ... , 156, 

 12, ... , 16 and 15 three-dimensional planes, viz. the intersections 

 of 2, ... , 6, 1234, ... , 1456 with 1. This is one of the figures 

 referred to in the introduction and is completed by noticing that 



12, 13, 14, 15, 16 

 lie in a three-dimensional plane. 



Take now 123 as a base point. Through it pass just six of the 

 planes connected with 0, viz.: 



1, 2, 3, 1234, 1235, 1236. 

 On each of 19 out of the 20 lines of intersection of these planes 

 one point (besides 123) of the set S^ lie. Thus 



on 1, 2, 3 there lies 0, 



1, 2, 1234 „ 124, 



1, 1234, 1235 „ 16. 



On the line of intersection of the three planes 1234, 1235, 



1236 there is no point of the set S^. On this line mark an 



arbitrary point 1'2'3', distinct from 0. With 123 as base point, 



the six planes through it and the twenty points (including 1'2'3') 



on their lines of intersection, complete the construction of the 



6—2 



