M.-P.— Vol. I.] DICKSON— THE ORTHOGONAL GROUP. 



45 



On On = 



3 4 3 

 -3 5 



— 4 2 5 J 



23 13 0u- 



3 3—4 

 -4 —3 3 

 -3 —4 3 J 



, (6> 12 6> 13 ) ; 



(o 12 o l3 yo 31 = 



-4 5 2 



— 222 



— 5 4 2. 



r-4-251 



2 —5 4 I 

 1-5 42J 



— 2 2 — 2 

 Since ( 6^12 6>i 3 ) 3 = 2 — 5 4 



. -2 4 -5 J 

 #12 O13 is of period six ; also it follows that 



( l2 Onf C 2 C 3 T 13 T n = ( 23 O l3 12 f, 

 so that Zl 3 T i2 belongs to the group Q. 



4. For^ n = 13, the Q y are C ; C] or <9 ±( ?;f. 



/Q6, 2 /O6, 2__ 

 ^1,2 ^1,3 — 



-3 —I 2 



— 2 6 o 

 1 — 4 6 



(6> 12 <9 13 ) 2 --= 



(6> 12 <9 13 ) 4 = 



020 

 — 6 — 1 — 4 

 — 2 3 —1 



, 6'23^'12 6'13 = 



3 3—3 

 2 —3 -~i 



1 3 2 y 



2 3 —1 



1 3 —2 



—3 3 3 



Thus ( <9 12 6> 13 ) 4 T n T 13 = Ci C 2 ( <9 23 6> 12 6> 13 ) C\ C 3 , 



so that ZI2 T\ 3 belongs to the group Q. Note that 0\ 2 0\ 3 

 is of period 7 since (0\ 2 On) s = 0\ 2 X3 . 



±3 



— u i 2 u i 3 — 



.S 4 =: 



T=(%fOttO}* 



9 3 5 



-5 —9 3 



-3 5 -9 



5 . For f = -- 19 , the <2 U are C, C, , 6>f f 4 , 0*; J 



4 3 4l 

 — 4 20 , .S 2 



-8 3 2 J 



-6 6 9^1 

 95-3, & = s-> , S»=i. 

 6-4 5 J 



4-5-6^ 

 —8 -8 —5 



-4 -8 4 



8 -4 4 



, T 2 =T~ 1 . 



o\i o\i 0\i 



— 2 o — 4 

 -3 -2 -8, 

 so that 7^3 7 J2 belongs to the group Q. 



= 7 23 7i 2 C 2 C 3 C/j'o U{ 3 , 



