M.-P.-Vol. I.] DICKSON— THE ORTHOGONAL GROUP. 33 



7. It may be readily verified that O .' . transforms 



Sa given by (3), into S ' a ' whose coefficients are 



a' n = A. a a -f ^ «ji , «'ji = — ^ «u -f *• «ji , 

 a' i2 = X a i2 -}- ft a j2 , a' j2 = -- fi a i2 -|- X a j2 , 



a 'ki= a ki > a 'k2 = a k2 (^ =1 , • • m, k ^ «*,y). 



Since X 2 -f- yu, 2 = 1, we have 



m 2 m 2 m 



(4) 2 « sl =l, 2a 82 =l, Sa al a 82 = o, 



S=l 8=1 3=1 



or precisely the same conditions as those for a sl , a s2 . 



If a 2 u -f- a 2 jX = S 2 , a square not zero, we may make d n = o, 

 d n = 8 by taking X = a n /h, /x, = — a u/<>- 



The condition a' a = a'^ requires 



X (a a — a jl )=— ^ (a a -f a^), X 2 -f- ^ = 1, 



whence X 2 = (a;i + a nY 

 2 (a 2 u 4-a 2 n ; 



It can be satisfied if 2 (a 2 iX -f- a 2 n ) is a square ^ 0. 



If a 2 n -}- a 2 n = 0, with a u and a^ not zero, an arbitrary 

 value t ^ may be assigned to a' u . Thus, X a a -f- ^ a n = T 

 requires ft 2 (a 2 u + a? n ) — 2 a^ /u, t = a 2 a — t 2 . 



8. Similarly the orthogonal substitution: 



f 1 = *"6 -1- /*"!, + »"& 



with X 2 -|- /u, 2 -[- u 2 = 1, etc., transforms «5' a into *$' a ' with 

 the coefficients 



a n = Xa il + ^a J1 + v a kl 



«'ji = *'«u + / a ji + y '«ki 



«ki = x " a ii + ^"«ji + u " a ki 



«si = a si ( s = x » • • w ' s < *»/» *)» 



with analogous formulae for d i2 , a j2 , etc. We may verify 

 that the relations (4) hold here. 



Suppose that a n is not zero. Can we choose X, ft, v, 

 satisfying X 2 -f- /jl 2 -f- v 2 = 1 , so that d n — o ? 



