74 Prof. Burnside, On a configuration of twenty -seven 



beginning of this paragraph. Of the thirty additional points that 

 arise it will be found that twenty-one belong to the set Sq, viz.: 



145, 245, 345, 45, 54, 12, 21, 



146, 246, 346, 46, 64, 13, 31, 

 156, 256, 356, 56, 65, 23, 32. 



The remaining nine, adhering to the notation already used, 

 may be represented by 



1'2'4', 1'2'5', 1'2'6', 



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



2'3'4', 2'3'5', 2'3'6' 



(the sequence of the figures is immaterial), 1'2'4' being the point 



common to the five planes 



1234 1245 1246 

 (24, 25, 26, 1'2'3') and '(14, 15, 16, 1'2'3'). 



Since 12, 13, 14, 15, 16 lie in a three-dimensional plane, the 

 two latter planes may be written 



(21, 23, 24, 25, 26, 1'2'30, (12, 13, 14, 15, 16, 1'2'3'). 



Instead of 123 any of the twenty points of which it is one 

 might be taken to make the last construction, and whichever is 

 chosen one point must be arbitrarily chosen on a certain line 

 before the construction can be carried out. The construction 

 however, having led to a point 1'2'4' on the line of intersection of 

 1234, 1245, 1246, it is now possible to take 124 as a base point 

 and, without introducing any further arbitrary element, to carry 

 out the construction from this point. When this is done it is 

 found that of the thirty additional points that arise twenty-one 

 again belong to the set S^, and the remaining nine are 



1^2^3^ V^'b', V2'Q\ 



1%'^', 1%%', 1%'Q', 



2'4'3', 2'4'5', 2'4'6' 



where 1'2'3' is the point common to the five planes 

 1234, 1235, 1236, 

 (12, 13, 14, 15, 16, 1'2'40 and (21, 23, 24, 25, 26, 1'2'4'). 



Comparing this with the specification of 1'2'4', it follows that 

 1'2'3' is the same point as 1'2'3'. 



Similarly 1^2^5^ V2'&, 1%'^' and 2^4^3^ are found to be 



identical with 1'2'5', 1'2'6', 1'4'3' and 2'4'3'. Hence, the point 

 1'2'3' having been once chosen, w^hen the construction of the 

 beginning of this paragraph is carried out with each of the twenty 

 points 123, ... , 456 in turn, besides the set S^ a set of just twenty 

 points (including 1'2'3') and no more will arise 



