THE HYPERORTHOGONAL AND RELATED LINEAR GROUPS. 139 



S EE S (7j C/2 /Suj (7 2 EE S a Cj (7 2 , 

 where S a ~ S~ C 1 C 2 S is seen to be the substitution 



The coefficient u in $ is therefore 



Hence 



which =(=1 since f 1 s + 1 + f 2 +1 * s ne ^ ner zero nor 1- Applying the 

 above process to 8 in which ^+ 1 =|=l, we reach a substitution in 

 the group I in which all but one of the a^ (j = 2, . . ., m) are zero. 

 Suppose next that p = 2. We have by 128) 



The ratios of lg , a 18 , . . ., a lmi therefore satisfy the equation 



130) T^+^l. 



Hence by transforming S by suitable products of the form 



^,,,-ir,.,, ( = 3,..,), 



where the r z are roots of 130), we reach a substitution $' belonging 

 to I in which 12 = 13 = = i Wl . Transforming S' by the reciprocal 

 of Ogjs, we obtain in J a substitution S" which replaces | t by 



nii + ! (A - M'') it + G + i* 1 ) is + 4 + + sJ- 



If ^ =j= 2, we can choose A and ft [see 146] such that 

 ^+1+^+1= 1, (A-^y+i+l. 



Hence in S" the sum of the (p s + l)* f powers of the coefficients j' 2 

 and j 4 is not zero in the 6rF[2 2 *]. As above we can therefore 

 make " 4 = 0. If p s = 2, we reach at once the same result by 

 transforming S f by (^ | 4 ) W^^), TF being defined at the beginning 

 of 146. 



Repeating the process, we reach finally a substitution in J, not 

 the identity, in which either 



u-0 (j = 3, . ..,w) 



or else 



