GROUP OF THE EQUATION FOR THE 27 STRAIGHT LINES etc. 3Q5 



with the abstract simple group of 273. From its origin [0] is 

 transitive and hence contains a substitution S which replaces E by 

 an arbitrary element A. We proceed to prove that [0] contains a 

 substitution 8 t which leaves JR fixed and replaces E L by an arbitrary 

 one of the ten elements E lf E^, ft z -o, ft*o (i = 1, 2, 3, 4) which lie 

 in sets with E . The substitutions [E 3 ], [E 3 ][E 2 ], [E 9 ][E 9 ][EJ, 

 [^slKlfJEJ 2 replace E t by J? 140 , E 1ZQ , E^ , ^ 110 respectively, without 

 altering E . The transformed of [Jy by [TF] gives the substitution 



(ft n ft 31 E s 21) (ft 22 ft 32 ft 42) (E s 20 ft a 10 E s 30 ) 



which replaces .R 120 by E. 210 , B 110 by E 230 . Then [.EJ and [j 2 ] replace 

 JR 230 by JR 220 and I? 240 respectively. Finally, [E s ] replaces E^^ by E%. 

 It follows that [0] contains a substitution yS^/S which replaces the 

 set E^E^E^ by a set J..Z?(7 in which A is any one of the 27 elements 

 and B any of the 10 elements which lie in sets with A. Hence [0] 

 contains a substitution Z replacing the set E E 1 E 2 by an arbitrary 

 one of the 45 sets. Then Z 1 replaces the given pair ABC, A'B'C' 

 by a pair E Q E l E^ y A 1 B 1 C 1 having no elements in common. The 

 latter sets determine a trieder by the earlier proof. Applying to it 

 the substitution Z, which was derived from [W] and [_EJJ and there- 

 fore replaces sets by sets, we obtain a trieder containing ABC, 

 A'B'C* and determined by them. Hence the above distribution of 

 the 27 elements E into 45 sets is a suitable notation for the con- 

 figuration of the 45 triangles formed by the 27 lines on a general 

 cubic surface. 



284. The next step is to verify that the substitutions [W], 

 [E 2 ~\ and [E 3 ] of 274 permute amongst themselves the 45 triangles. 

 [W] gives rise to the following even substitution: 



where i = 1, 2, 3, 4; j = 2, 3, 4; s = l,2. 



[E 2 ] gives rise to the even substitution on the 45 triangles: 



! , E -R 120 ^220) (-K* 10 E s n Eg 12 , Eg 20 E s n E% a 22) 



EQ E i4Q J2 2 4o) C^ 10 ^* 22 E% g 21 , Eg 20 ^2 * 12 

 12 ; -^2 -^122 ^222) C^ 20 -R* 31 ^ 42 ; Eg 10 ft 81 ft 82) 

 ! 82 3 2 , E 2 E L ^E 242 ) (Eg 20 ft 32 ft 41 , ft 10 ft 42 ft * 4l) 

 (ftllftsoft*32, ftllft40-ft42)(ft2lft32ft40, 

 (ft 2 2ftsoft41, ft s 12 ft 40 ft 4l) (ft 22 ft 81 ft 40, ft 12ft Slftso)- 



Similarly [E^] and [J^ 3 ] give rise to even permutations of the 

 45 triangles. 



DlCKSON, Linear Groups. 20 



