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



for in that case f = 3 so that — 1 is a square. Let p\^p*>. 

 The group / contains the substitution 



(Sq ^12) ~ -*45 7l2 (*->n ^12) ^12 745= ^_ 7*45 ^>p ^45 • 



We may verify that S a = S Z45 S has the form 



5 

 ^o- = fi=li — ^So-^ (# = i--5), 



where 0-4 = 1 + p 4 (/> 5 — /? 4 ) , <r 5 = — 1 + p 5 (p 5 — /04) , 

 °-i = Pi(P5—pt) («'= 1,2,3). 



Thus ov* -f- cr 2 2 -) 1- cr 5 2 — 2. 



If 6*^ 7*45 be the identity, S T n = T& T& ; for we must 

 have p 1 = p 2 = p 3 = o , p i =—p 5 == ±1 . 



If S Z45 be not the identity, we obtain from it a substi- 

 tution affecting four indices only unless 



0- 2 l + G- 2 2 + °"' 2 3 = • 



In the latter case, p\ -f- /a 2 2 + /> 2 3 = o , /a 2 4 -f- p 2 5 = 2. We may 

 thus by § 7 make pi=p 5 = -\-i. Hence p 1 =p 2 =p4=p 5 =: -f-i. 

 Thus iS* is commutative with both 7as and T i5 . The 



theorem then follows by the proof at the end of § 17. 



19. Suppose finally that 8 2 2i + S 2 25 = o , 8 25 ^<?. If 

 S n = 8 12 = o , S 2 13 = 1 . Then S 23 = o follows from the 

 relation 



(5) i 8,^ = 0. 



Thus 7?g (given at end of § 16) becomes 



f' 3 = — fa, fi = & — 2 S 2i 2 S 2j f, (/,y=i,2,4,S). 



i 



Hence / contains a substitution 7?g Ci C 2 commutative 

 with C 3 . Our theorem follows as in § 17. 



With either S u or S 12 not zero, we may make each one 

 <o by transforming by On. 



Similarly the theorem follows if 8u=o. Thus S 2 n -f- h 2 n = 1 , 

 so that we can make S 12 = o. Then S 2i = o by (5). Thus 

 R§ Ci C 2 leaves £1 fixed. 



