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



where fi n = a' n , 8 15 = a 15 = o , /3 H = \ a u + /a a' 13 + ua 12 . 

 Since a 2 i 4 + « 2 i3 + " 2 i2 = I — a\-^o , we may by § 8 make 

 /3 14 = . 



If kSi be commutative with both C\ C 2 and C\ C3, Si be- 

 comes simply, 



ri=±ti, f 2 =±&, f 3 =±&, 



where o- ^ o since .5 and therefore <5i is not a mere pro- 

 duct of the Cj. But the transformed of Si by T u T 35 gives 

 a substitution 



fi = /»li + a &, fa = ± £2, • • • 

 which is not commutative with C\ C 2 . 



In any case we have in the group 1 a substitution 



S: ri = 2* a "6 (* = i--5) 

 j=i 



with S 14 = S 15 = o and not commutative with C\ C 2 , for ex- 

 ample. Hence /contains the substitution 



S C\ C 2 S~ l C\ Ci = /?g C\ C 2 , 



1=1 j=i 



17. Suppose first that S 2 24-f-S 2 25 is a square not zero. 

 Transforming i?g G C 2 by #45 we may make 8 25 = o. We 

 have therefore in / a substitution affecting only f x , f 2 , &, £4, 

 from which as in § 16 we obtain a substitution 



S: ri=S'*6 (^=i--4) 



with 7 2 u^ 1, 714 = o. It is thus not commutative with C\. 

 Hence /contains the substitution, not the identity, 

 S C\ Cs S~ x C\ Cf> = S C\ S~ C\ = Z^y C\ , 



T„: f', = ft — 271, 2 7 i,ft = 1,2,3). 

 As in § 13, we find that / contains the substitution »Sg T 12 , 

 where 



Sg: &«&_«, 2 *,*, (1=1,2,3), 



with the single condition 8\ -f- S 2 2 -\- 8 2 3 = 2. 



