LINEAR GROUP WITH QUADRATIC INVARIANT. 189 



The coefficient of y 2 being not zero, this equation has solutions for 



-f 



in the field and hence solutions /3, y. 



Transforming S a by the orthogonal substitution 



176) rf+fli+yi + ..*!...!, 



we obtain as above a substitution $5 in which 



-f tf 



We proceed to show that solutions of 176) exist in the G-F[p n ~\ 

 which make an= 0. We may suppose that a t -i, c^i, ecu are not zero, 

 since otherwise the result follows by inspection. If either of the sums 



be not zero, the problem is solved as above. If both sums be zero, 

 I = square = 1, ah 4- |i + of i = 0. 



Then the following set of solutions of 176) will make ,-i zero: 



^ = 1. 



192. Transforming 8^ = 8 a C^ by 5 34m, 63 4m, . . ., O m _i 3 4m in 

 succession, we obtain in 6r a substitution $' in which #51, 6i? . . > 

 TO 11? are all zero. Then by 143), 



2222 2 



11 + 21 + 31 + 41 + ^aJni = 1. 

 A1SO 2 2 



ii + aii ^ u + li 4= L 



22 2 



Hence 31 -f ii + fiaJm 4= 0, so that we can transform $' by a 

 suitable Os 4 m into S"=S a "C 1 C 2 in which 



Transforming /S ff by 0/456 (j = 7, 8, . . ., m 1) in succession, 

 we can obtain a substitution S 2 = S^C^C^ which leaves | 7 , | 8 , . . ., ^ TO _i 

 fixed and has /3 41 = 51 = /3 61 == 0. If 42 , /3 62 , /3 62 are all not zero, 

 we transform $ 2 by O^e and obtain a substitution $2 in which we 

 can make 062=0 except in the case 1 ) 



+ +-0, fti+ 0, m i4=0. 



1) If p sl = or j3mi= 0, we transform 2 by 06345 or Oem45 and make 

 0. 



