M.-P.— VOL. I.] DICKSON— THE ORTHOGONAL GROUP. 43 



We may thus suppose that B u , S 12 , B l3 all differ from 

 zero. Transforming 7?g C\ C 2 by 6^5 we can obtain a 

 substitution commutative with 71 3 or else with 7 23 . Our 

 theorem will then follow by the proof in § 18. The condi- 

 tions that it be commutative with 7i 3 are 



S' 23 EX8 2 3-)-/i 8 2i -\- V S 25 = §21 

 S'i3 = X 6\ 3 = On 



These three conditions combine into a single one : - 



623 



where 60 = 6^1 — -5 — °u • 



013 



Since the coefficient of p* is zero, fi is determined unless 

 &) = o. Similarly the conditions that our substitution be 

 commutative with 7 23 may be satisfied unless 



(O =0 2 2 "j* — 612 = ^. 



Ol3 



It can not happen that both &> and 0' vanish; for 



021 ^22 ^23 



Sll ^12 °\s 



would require r = ± 1 since we have 

 Then would follow 



= 2^ s 2j = ± (S 2 n + 0^0 + o 2 i 3 ) = ± 1 . 



j-1 



APPENDIX. 



I. For p u = 3 , the Q y are simply C s Cj . If m = 3, the 



group of the Q ;j is as follows : 



6?4 : ■{ 1 , C\ C 2 , C\ G% , Ci Cz y . 



