38 CALIFORNIA ACADEMY OF SCIENCES. [Proc. 3D Ser. 



3 



In this case, the transformed of T a C\ by C 2 C 3 gives To C\, 

 where To replaces |i by 



&-1/2 (fc _&_& + &). 

 Thus To C\ is not commutative with every T {j 7se. Hence 

 /contains a substitution .5, given by (1), not commutative 

 with T 12 7^6, for example, but leaving £ 5 , . . f m fixed; hence 

 /contains the substitution, not the identity, 



0~ 7i2 ^56 *5 /l2 ^56 = S~ T\l S ' T\i = Og 7i2, 



where (5*g denotes the substitution (see end of § 12): 



4 



{ 



^g : £'}=& — SiSSjfj (* = i,.. 4). 



14. If S 3 = S.i == #, S§ 7i2 takes the form 



Fl=:-«l«lft+ (I— S 2 2)|2=(a 2 -^ 2 )^1+ 2«^ 2 



ft = (1 - - S 2 i) ft "81 «a & = -- 2aj8fc + (a»-/8«) ft, 



where a = j4 (&i — S 2 ) , /8 = ^ (Si + S 2 ) 



so that a 2 + /3 2 = ^ (8\ + S 2 2 ) = 1 . 



Hence S* T\% becomes Q . Therefore /contains Q 



)rmed of its re< 

 / contains 



the transformed of its reciprocal Q by T i3 T45 and thus 



1 , 2 



with a 1 = 2a/3, a 2 = a 2 — /3 2 , tr 3 = — 1. 



If aft^o, its transformed by T i3 Z45 gives 6* ' 7i2, in 

 which o-' 3 ^ 0. But if a{3 — o, S$ 7\z, not being the 

 identity, reduces to C\ C 2 , when by § 5 the group / contains 



every Q ;j and therefore (for ^ n > 5) a substitution Q ' Q ' 

 in which a/3^o. 



15. We have thus reached in the group /a substitution 

 6g 7*12, not the identity, having S 3 and S 4 not both zero, say 

 &i ^ 0, and leaving | 5 , . . f m fixed. Transforming it by 



Z we can make S' 6 =S' 1 , the conditions being (since 



S 6 = 0) 



a 8 3 4- /3 S 4 == 8, , a 2 + /3 2 4- 7 2 = 1. 



