M.-P.-Vol. I.] DICKSON— THE ORTHOGONAL GROUP. 35 



Taking 1 \= — u,-@±, v = — a- z!t , the first condition 



gives p £72 = M —5- . &»— &2 ) • 



The final condition X 2 -J- /jl 2 -f- v 2 -f a 2 -f- /r = 1 becomes 



which can always be satisfied. Indeed by an earlier trans- 

 formation of S' by 05 6 7 we can avoid the two values of 

 /3 72 which make the coefficient of fi 2 zero. 



12. We have thus reached, in the group /, a substitu- 

 tion S r C\ C 2 affecting only £1 , . . £ 6 , with <y\\ + 7 2 2i < 1 • 

 As in § 10, we may suppose 751 = 761 = 0. Our substitution 

 will thus be commutative with Tw if 752 = 762 , which we may 

 suppose the case from § 7 when 2 and 7 2 52 + 7 2 62 are both 

 not-squares. If 7 2 52 + 7 2 62 be a square not zero, we may by 

 §7 suppose that <y 62 = o, when we are led to the case 

 treated in the next paragraph. 



There remains the case 7 2 52 + 7 2 62 = o, occurring when 

 — 1 is a square. If 731 = o, we transform by Om and have, 

 by § 8, 7^2 = 7a = o, a case treated in the next paragraph. 

 The case y 2 3 i -f- 7 2 4 i = a square is therefore solved ; while 

 7 2 a + 7 2 4i = o is excluded since then <y 2 n -f- 7 2 2i = *• With 

 7 2 52 + 7 2 62 = o, 7 2 3i -f- 7 2 « = a not-square, we may make 



S C\ C 2 commutative with 7s6. Transforming by O 

 1 3 45' 



the conditions are 



7a = X 7a + /"■ 7« = 7gi = o 



7' 32 = \ 732 -f- fl 742 -f- V 752 = 762- 



Writing S = 7 42 — — 73a, these conditions give 



7 a 



7a 752 752 



1 If either /3 31 = or (3.^ = we could at once have made /3 7i = o. 



