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



Hence S is simply a product of an even number of the C K , 

 in which certain ones, as C k , are lacking, since S differs 

 from N. Let S=C i C j C t C a C t ' '  . Its transform by 

 TJj T ik gives S'= C i C k C I C s C t •  • . Hence / contains 

 S'S~ 1 = Cj C k , and, by transforming by suitable even sub- 

 stitutions, every product of two C's. 



Lemma. Having- every C- l Cj , the group I contains every 



a 



Q . .■ Thus, 2 being a not-square, // contains one of the 



'■> J 

 two : 



s~) a >P fy * a '——s. a 'iH r p f~* 



1,2' 1,2 1,2 12 I, 



which transform C\ C 3 into Q Ci C 3 and Q ' C 2 C3 



1,2 1)2 



respectively. In either case / contains Q ' . 



1 , 2 



Thus if / n >5, /contains a Q different from the iden- 



1,2 



tity and from C\ C 2 . Taking it in place of S, we are led 

 to the case following. 



6. Suppose on the contrary that S is not commutative 

 with every C x C\, for example 1 not with C\ Ci. Then / 

 contains the substitution, not the identity, 



where S a = 6" -1 Ci C 2 6" , of period 2, is found to be : 



m m 



(3) f i = fc— 2a il 2 «jl lj — 2 a i2 2 «j2 lj • 



j=l 3=1 



The next problem is to simplify S a C\ C 2 by transforming 

 it by substitutions belonging to H. 



•We may assume for later use that a\ x +a>\ x ^ !■ Indeed if 5" be com- 

 mutative with d C 2 then 6" is merely a product of a substitution affecting 

 |i and f 2 by a substitution affecting £ 3 , . . . f m , both of the same deter-" 

 minant dr 1. For w>3 we may make, by a suitable transformation of 6", 

 d\ x + «2, < 1, whence it can not be commutative with C x C 2 . 



