hyper-planes in four- dimensional space. 75 



Moreover, since 



(12, 13, 14, 15, 16, 1'2'3', 1'2'40 

 is a plane, it follows immediately that the fifteen points 

 12, 13, 14, 15, 16, 1'2'3', 1'2'4', 1'2'5', 1'2'6', 



1'3'4', 1'3'5', 1'3'6', 1'4'5', 1'4'6', 1'5'6' 

 all lie in a plane. Call this plane 1' and the other five that arise 

 in the same way 2', 3', 4', 5', 6'. 

 Through 16 pass the five planes 



V, 1345, 1245, 1235, 1234. 



On the lines of intersection of V , 1245, 1235, 1234, there 

 lie the points 26, 1'2'5', 1'2'4', 1'2'3', and the plane through 

 these points is the plane 2'. Hence from the configuration of 

 § 1, the planes 1', 2', 3', 4', 5' meet in a point through which 

 6' must clearly pass. Call this point 0'. There is then finally a 

 set of 27 planes, viz. : 



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



1234, 1235, , 8456, 



1', 2', 3', 4', 5', 6'; 

 and 72 points, viz. : 



0, 



123, 124, , 456, 



12, 21, , 56, 65, 



1'2'3', r2'4', , 4'5'6', 



0'. 



Through each point pass six planes and in each plane lie 

 sixteen points, and the configuration is complete. The relations 

 are given by the scheme 



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



0' r, 2', 3', 4', 5', 6'; 



123 1, 2, 3, 1234, 1235, 1236; 



4'5T 4', 5', 6', 1456, 2456, 3456; 



12 1, V, 1456, 1356, 1346, 1345; 



21 2, 2', 2456, 2356, 2346, 2345. 



A very slight modification in the notation, viz. the replacing of 

 the symbol 1234 by (56) (the brackets will prevent confusion 

 between the symbol for a plane and the symbol for a point) gives 

 these relations a well-known form. In fact with Schlafli's nota- 

 tion for the 27 lines on a cubic surface, the table gives the 

 36 double-sixers that can be constituted from them. The tactical 

 analogy between the 27 hyper-planes and the 27 lines on the 

 surface is thus obvious ; and the hyper-planes and points admit a 

 group of 51,940 permutations for which the relations given by the 

 table are invariant. 



