LINEAR GROUP WITH QUADRATIC INVARIANT. 167 



group 0{(3, p n ). Then, by the last section, all the O"'/ generate 

 For p n = 5, i = 2j we have 

 f i -i o 



pi 3i p o 



J ~LO 3J~LO ULO 2-i 



4 L L 



\ 



Hence from JR 123 and Q i = ^M' we rea ch Q i It follows 



as above that -R 12S and 0^l' = C 2 C 5 and C^Q generate 0^(3, 5). 

 The latter is extended to (3, 5) by any 0j *. 



180. Theorem. If ]7 # 2? . ., a m be any set of solutions in the 

 GFWof a , + fll+ ... + a , 1+ i ft 2 =1 



there exists a substitution S derived from the generators 1 ) of 173 which 

 replaces Jj by G^ = cc | t + or 2 ?2 H f" a im 



The proposition having been proven for m = 2 and m = 3, we 

 will give a proof by induction from m 1 to m, supposing m > 3. 



Consider first the case in which every sum of three of the terms 



J, 2> > a mi, <4 is zero. These terms must all be equal and 



therefore 



mal = 1, 3cq = 0, ft = square. 



Hence p = 3, while m is of the form 3& + 2 or 3k -f 1. 



If m = 3k + 2, we have 1 af = af =(= 0, so that the theorem 

 is reduced by 174 to the case of m 1 indices. 



If m = 3^ + l, we must have J = 1. But the product Oi]%S 

 will replace t by a'^ -{ H ttmm, where 



Of the 3 n l sets of values in the #jF[3 n ] satisfying 



at most two give the same value to a[ and hence at most four make 



a[ = 1. Hence, if n > 1, we can avoid the case | = 1. For p n = 3, 

 we may take 



1) For the case p*= 5, m ^ 4, ji = not -square, it would appear that the 

 generator JB 123 were necessary in addition to the Of We can, however, 

 express .R 123 in terms of the generators 



^f,m- 

 leaving invariant gf -f- g + -f gj, _ t -f 3 gj, . Indeed, 



