304 CHAPTER XIY. 



2. Any two triangles ABC and A f B J C' having no side in 

 common determine uniquely a third triangle A" B" C n such that the 

 corresponding sides of the three triangles intersect and form three 

 new triangles AA'A", BB'B", CC'C". The former set of three 

 triangles is said to constitute a trieder, which will be designated 

 {ABC, A'B'C',A"B"C"~]. 



These two properties completely define the configuration of the 

 45 triangles formed by the 27 lines on the cubic surface. 



Denoting the lines by E t , E sit (s = 1, 2; i = 1, 2, 3, 4; t = 0, 1, 2), 

 it will be shown that the 45 triangles are given by the notation 1 ) 



[s = 1, 2] 



[t = 0, 1, 2; * = 1, 2, 3, 4] 

 R.*tR.*tR,&t+i [5 = 1,2; *EE 0,1,2 (mod3)] 



[s = 1, 2; j = 2, 3, 4; * EE 0, 1, 2] 



where the subscript 2s is to be replaced by 1 when s = 2. 



Each element E lies in exactly five of these sets. Thus E t lies 

 in the sets R Q R L R i) R t R iit Rsit (i = 1, 2, 3, 4); R 9lt lies in the 5 sets 



E s u)E s iiE s i2, RfRuLfRsitf RtltRajtlRsiijt+l {j = 2, 3, 4); 



finally, E sjt lies in the following 5 sets, in the last two of which % 

 is to be suitably chosen modulo 3: 



Hence each element can be associated with exactly ten other elements 

 to determine a set. Property 1 thus holds for the 45 sets. 



The set E E 1 E 2 lies in exactly the following sixteen trieders: 



1. ^2 > -^110 ^111 -^112 9 -^210 ^211 ^212] 7 



where j = 2, 3 or 4, t = 0, 1 or 2 (mod 3). Property 2 therefore 

 holds for the set E E 1 E 2 in conjunction with any set no one of 

 whose elements is E Q , E L or E 2 . It is next shown that the property 

 holds for an arbitrary pair of sets ABC, A'B'C 1 which have no 

 element in common. By the next section the 45 sets are merely 

 permuted by the substitutions [TF], [.EJ, [J 2 ], [E s ~] given in 274. 

 The latter generate a substitution -group [0] holoedrically isomorphic 



1) The connection with the 27 sets of orthogonal substitutions exhibited 

 in 272 will be shown in the sequel/ 



